# On regular ∗-algebras of bounded linear operators: A new approach towards a
theory of noncommutative Boolean algebras

Michiya Mori  Interdisciplinary Theoretical and Mathematical Sciences Program
(iTHEMS), RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198 Japan.
<EMAIL_ADDRESS>
###### Abstract.

We study (von Neumann) regular ∗-subalgebras of $B(H)$, which we call
R∗-algebras. The class of R∗-algebras coincides with that of “E∗-algebras that
are pre-C∗-algebras” in the sense of Z. Szűcs and B. Takács. We give examples,
properties and questions of R∗-algebras. We observe that the class of unital
commutative R∗-algebras has a canonical one-to-one correspondence with the
class of Boolean algebras. This motivates the study of R∗-algebras as that of
noncommutative Boolean algebras. We explain that seemingly unrelated topics of
functional analysis, like AF C∗-algebras and incomplete inner product spaces,
naturally arise in the investigation of R∗-algebras. We obtain a number of
results on R∗-algebras by applying various famous theorems in the literature.

###### Key words and phrases:

nonclosed self-adjoint operator algebra; von Neumann regular ring; Boolean
algebra; AF C∗-algebra; projection lattice; complemented modular lattice;
inner product space

###### 2020 Mathematics Subject Classification:

Primary 06E75, Secondary 03G12, 06C20, 16E50, 47C15, 47L40, 81P10.

The author is supported by RIKEN Special Postdoctoral Researcher Program.

##  1\. Introduction

Throughout the paper, $H$ is a complex Hilbert space, and $B(H)$ denotes the
von Neumann algebra of bounded linear operators on $H$. A motivation for our
research comes from a problem raised by Kadison. In the celebrated conference
in Baton Rouge in 1967, Kadison proposed a list of 20 problems on operator
algebras. The list was shared among operator algebraists worldwide, which
brought a great influence to the development of the theory of operator
algebras over the last half-century. In 2003, Ge published a survey article
[23] on Kadison’s list of problems. Among the 20 problems, 19 are about C∗\-
or von Neumann algebras. The only exception is listed as the 15th problem in
Ge’s article, which asks the following.

######  Problem 1.1 (Kadison).

If a self-adjoint operator algebra is finitely generated (algebraically) and
each self-adjoint operator in it has finite spectrum, is it finite-
dimensional?

In the recent paper [48], Szűcs and Takács discovered that for a ∗-subalgebra
$A$ of $B(H)$ several conditions are equivalent to the condition every self-
adjoint element of $A$ has finite spectrum. In this paper, we first observe in
Theorem 2.1 that these conditions are also equivalent to the condition $A$ is
regular in the sense of von Neumann. We call a ∗-subalgebra of $B(H)$ that
satisfies these equivalent conditions an R∗-algebra (“R” meaning “regular”).
The abundance of the conditions for a ∗-subalgebra of $B(H)$ to be an
R∗-algebra implies that R∗-algebras form more or less an important class of
∗-algebras.

Even though we assume no completeness, it turns out that R∗-algebras share
some important properties with C∗-/von Neumann algebras. For example, we may
freely use the technique of functional calculus. Recall that there exists a
correspondence between commutative C∗-algebras and locally compact Hausdorff
spaces (Gelfand–Naimark theorem), and similarly, between commutative von
Neumann algebras and certain measure spaces. Is there such a correspondence
for R∗-algebras? By von Neumann’s study of ∗-regular rings in [43, Part II],
projections of a general R∗-algebra form a lattice. It follows that the
lattice of projections of a commutative R∗-algebra is a generalized Boolean
algebra. In fact, there exists a complete correspondence between the class of
commutative R∗-algebras and that of generalized Boolean algebras. By
restricting ourselves to the unital case, we see that unital commutative
R∗-algebras correspond to Boolean algebras. In that sense (the projection
lattice of) an R∗-algebra can be considered as a noncommutative (generalized)
Boolean algebra. This correspondence is closely tied with Stone’s theorem on
the duality between Boolean algebras and Boolean spaces.

Are there plenty of noncommutative R∗-algebras? The answer is yes. The class
of ultramatricial R∗-algebras, that is, directed unions of finite-dimensional
C∗-algebras, provides us with many examples of noncommutative R∗-algebras. It
turns out that Problem 1.1, which remains to be open, is equivalent to asking
whether every R∗-algebra is ultramatricial. Ultramatricial R∗-algebras with
countable Hamel dimension are highly relevant to the theory of separable AF
C∗-algebras, whose classification by Elliott [19] is one of the best-known
results of the theory of C∗-algebras. Another subclass of ultramatricial
R∗-algebras, which we call purely atomic R∗-algebras, is constructed from
inner product spaces. We give a complete classification of purely atomic
R∗-algebras in terms of linear isometries of inner product spaces. By applying
Banach space theory and results on operator ranges, we prove that there are
many mutually nonisomorphic purely atomic R∗-algebras.

The literature of regular rings, Boolean algebras, and inner product spaces in
connection with Banach spaces, as well as C∗-/von Neumann algebras provides us
with many deep results and further questions. The paper intends to collect
some of them in order to give supporting evidence for the author’s belief that
the study of R∗-algebras is interesting enough to be considered in more depth.
This paper should give much motivation to the research on Problem 1.1, which
seems to have gotten less attention among operator algebraists compared to
other problems in the list [23]. The choice of materials in this paper is more
or less biased by the author’s preference.

This paper is formulated in the following manner. In the next section, we give
the definition and basic properties of R∗-algebras. In particular, we
characterize R∗-algebras by closedness of the ranges of operators, the
uniqueness of C∗-norm of ∗-subalgebras as well as von Neumann regularity
(Theorems 2.1, 2.12). Section 3 provides examples of R∗-algebras and discusses
Problem 1.1. The correspondence between commutative R∗-algebras and Boolean
algebras is given in Section 4. Section 5 is devoted to giving the relation
between purely atomic R∗-algebras and inner product spaces. Some results that
arise by comparing the projection lattice of an R∗-algebra with that of a von
Neumann algebra or a Baer ∗-ring are given in Section 6. The consequence of
Elliott’s classification theorem of AF C∗-algebras is summarized in Section 7.
We also revisit Elliott’s theorem from a lattice theoretic viewpoint (Theorem
7.5). In Section 8 we collect results that derive from von Neumann, Dye and
Gleason’s theorems. We close this paper by proposing questions on R∗-algebras
for further research (Section 9).

In the paper, we frequently use basic results on C∗-/von Neumann algebras as
in [31, 32, 49], and von Neumann’s theory on complemented modular lattices and
regular rings as in (the first four chapters of) [43, Part II]. None of the
proofs is dependent on any results by Szűcs and Takács in [48], although the
underlying idea may have some overlap implicitly.

##  2\. Regular ∗-subalgebras of $B(H)$

If there is no confusion, for each $z\in{\mathbb{C}}$, we often write $z$
instead of the scalar multiplication operator $z\cdot\operatorname{id}_{H}\in
B(H)$. In particular, $1$ means the unit of $B(H)$. An _algebra_ means an
algebra over the field of complex numbers, possibly without unit. A subspace
or a subalgebra in this paper need not be closed. For a subset $S\subset
B(H)$, the symbol $\operatorname{span}S$ means the linear space spanned by
$S$, that is, the smallest complex-linear subspace of $B(H)$ that contains
$S$. Similarly, ${\mathcal{A}}(S)$ and ${\mathcal{A}}^{*}(S)$ denote the
algebra and the ∗-algebra generated by $S$, respectively. Namely,
${\mathcal{A}}(S)$ (resp. ${\mathcal{A}}^{*}(S)$) is the smallest subalgebra
(resp. ∗-subalgebra) of $B(H)$ that contains $S$. For a finite subset
$\\{x_{1},\ldots,x_{n}\\}\subset B(H)$, we define
${\mathcal{A}}(x_{1},\ldots,x_{n}):={\mathcal{A}}(\\{x_{1},\ldots,x_{n}\\})$
and
${\mathcal{A}}^{*}(x_{1},\ldots,x_{n}):={\mathcal{A}}^{*}(\\{x_{1},\ldots,x_{n}\\})$.
A _projection_ of $B(H)$ is an operator $p\in B(H)$ with $p=p^{2}=p^{*}$.

Let $x\in B(H)$. The range of $x$ is written as $\operatorname{ran}x$ or $xH$,
and the kernel as $\ker x$. The symbol $\sigma(x)$ stands for the spectrum of
$x$, i.e., $\sigma(x):=\\{z\in{\mathbb{C}}\mid z-x\text{ is not invertible in
}B(H)\\}$. Let $l(x)$ and $r(x)$ denote the left and right support projections
of $x$. That is, $l(x)$ is the projection onto the closure of
$\operatorname{ran}x$, and $r(x)$ is the projection onto $(\ker x)^{\perp}$.
If in addition $x$ is self-adjoint, then $s(x):=l(x)=r(x)$ is called the
_support_ of $x$. Recall that every $x\in B(H)$ has the unique polar
decomposition $x=v\lvert x\rvert$ such that $\lvert x\rvert=(x^{*}x)^{1/2}$
and $vv^{*}=l(x)$, $v^{*}v=r(x)$.

Although the theorem below can be obtained as a corollary of [48, Theorems 3.8
and 4.1] by Szűcs and Takács, our discovery of (or at least the emphasis on)
its relation with von Neumann regularity is new. This viewpoint motivates a
deeper study on this class of algebras in connection with Boolean algebras in
the subsequent sections. For the convenience of the reader, we give a proof
that is independent of [48]. Our proof needs no background beyond the standard
first course of functional analysis.

######  Theorem 2.1.

Let $A$ be a ∗-subalgebra of $B(H)$. The following are equivalent.

  1. (1)

$A$ is regular (in the sense of von Neumann), i.e., for each $x\in A$ there
exists $y\in A$ with $xyx=x$.

  2. (2)

$A$ is algebraic, i.e., for each $x\in A$ the algebra ${\mathcal{A}}(x)$
generated by $x$ has finite dimension.

  3. (3)

Each operator in $A$ has closed range.

  4. (4)

Each operator in $A$ has finite spectrum.

  5. (5)

Each self-adjoint operator in $A$ has closed range.

  6. (6)

Each self-adjoint operator in $A$ has finite spectrum.

Before proving it, let us first give a preliminary result. Let $x,y\in B(H)$.
Then $y$ is called a _Moore–Penrose inverse_ of $x$ if $y$ satisfies
$l(y)=r(x)$, $r(y)=l(x)$, $yx=r(x)$ and $xy=l(x)$. It is easy to see that such
an operator (if there exists) is unique.

######  Lemma 2.2.

For an operator $x\in B(H)$ the following are equivalent.

  * •

The range of $x$ is closed.

  * •

There exists $y\in B(H)$ with $xyx=x$.

  * •

$x$ has a Moore–Penrose inverse.

###### Proof.

Let $x\in B(H)$ have closed range. Then $x$ restricts to a bounded bijection
$x_{0}$ from $(\ker x)^{\perp}$ onto $\operatorname{ran}x$, so $x_{0}$ has a
bounded inverse by the inverse mapping theorem. We define $y$ as the unique
operator in $B(H)$ with $y=x_{0}^{-1}$ on $\operatorname{ran}x$ and $y=0$ on
$(\operatorname{ran}x)^{\perp}$. Then we have $xyx=x$ by a direct calculation.
If $xyx=x$, then we have $x=xyx=xr(x)yl(x)x$, and it is easy to show that the
operator $r(x)yl(x)$ is the Moore–Penrose inverse of $x$. If $x$ has a
Moore–Penrose inverse, the restriction of $x$ as a mapping from $(\ker
x)^{\perp}$ into the closure of $\operatorname{ran}x$ has a bounded inverse,
which implies the closedness of $\operatorname{ran}x$. ∎

###### Proof of Theorem 2.1.

The implications $(2)\Rightarrow(4)\Rightarrow(6)$ and $(3)\Rightarrow(5)$ are
obvious. The above lemma implies $(1)\Rightarrow(3)$.
$(5)\Rightarrow(6)$ Recall that a self-adjoint operator $a\in B(H)$ has closed
range if and only if $0$ is not an accumulation point of the spectrum
$\sigma(a)$ of $a$. Assume that a self-adjoint operator $a\in A$ has infinite
spectrum. Take an accumulation point $s\in{\mathbb{R}}$ in the spectrum
$\sigma(a)$. By the spectral mapping theorem, the self-adjoint operator
$a^{2}-sa\in A$ has $0$ as an accumulation point of its spectrum, so the range
of $a^{2}-sa$ is not closed.
$(6)\Rightarrow(1)$ Let $x\in A$ and $x=v\lvert x\rvert$ be the polar
decomposition of $x$ in $B(H)$. Since $x^{*}x$ is a self-adjoint operator with
finite spectrum, we may take a polynomial $P_{1}$ with real coefficients that
coincides with the square root map on $\sigma(x^{*}x)\cup\\{0\\}$. We thus
have $\lvert x\rvert=(x^{*}x)^{1/2}=P_{1}(x^{*}x)\in A$. We may also take a
polynomial $P_{2}$ with real coefficients such that $P_{2}(t)=t^{-1}$ for
every $t\in\sigma(\lvert x\rvert)\setminus\\{0\\}$ and $P_{2}(0)=0$. We thus
obtain $v=v\lvert x\rvert P_{2}(\lvert x\rvert)=xP_{2}(\lvert x\rvert)\in A$.
Set $y:=P_{2}(\lvert x\rvert)v^{*}\in A$. Then we obtain $xyx=v\lvert x\rvert
P_{2}(\lvert x\rvert)v^{*}v\lvert x\rvert=v\lvert x\rvert=x$.
$(6)\Rightarrow(2)$ Let $x\in A$. Take self-adjoint operators $a,b\in A$ with
$x=a+ib$, and put $A_{0}:={\mathcal{A}}(a,b,1)={\mathcal{A}}^{*}(a,b,1)\subset
A+{\mathbb{C}}\operatorname{id}_{H}\subset B(H)$. Observe that $A_{0}$ has
countable Hamel dimension. Fix $s\in{\mathbb{R}}$. For $z\in{\mathbb{C}}$, set
$f(z):=\exp(sa+zb)\in B(H)$. Then $f\colon{\mathbb{C}}\to B(H)$ is a
holomorphic map. If $t\in{\mathbb{R}}$, then $sa+tb$ has finite spectrum, and
we obtain $f(t)=\exp(sa+tb)\in{\mathcal{A}}^{*}(sa+tb,1)\subset A_{0}$. Take a
Hamel basis $\\{e_{n}\\}_{n\geq 1}$ of $A_{0}$. For each $n\geq 1$, the subset
$F_{n}:=\\{t\in{\mathbb{R}}\mid
f(t)\in\mathrm{span}\\{e_{1},\ldots,e_{n}\\}\\}$ is closed in ${\mathbb{R}}$.
Moreover, we have $\cup_{n\geq 1}F_{n}={\mathbb{R}}$ because
$f({\mathbb{R}})\subset A_{0}$. By the Baire category theorem, we see that
some $F_{n}$ has nonempty interior $U$. Take a bounded linear surjective
operator $\pi\colon B(H)\to\mathrm{span}\\{e_{1},\ldots,e_{n}\\}$ such that
$\pi^{2}=\pi$. We see that $\pi\circ f=f$ on $U$. On the other hand, both $f$
and $\pi\circ f$ are holomorphic maps from ${\mathbb{C}}$ into $B(H)$. By the
identity theorem, we obtain $\pi\circ f=f$ on ${\mathbb{C}}$. Thus
$f({\mathbb{C}})\subset\mathrm{span}\\{e_{1},\ldots,e_{n}\\}$. It follows that
$\exp(sx)=\exp(s(a+ib))\in A_{0}$ for every $s\in{\mathbb{R}}$. Define the
holomorphic map $g\colon{\mathbb{C}}\to B(H)$ by $g(z)=\exp(zx)$. By the same
reasoning as above, we see that the image of $g$ is contained in a finite-
dimensional subspace of $A_{0}$. This plainly forces $x$ to be algebraic. ∎

We call $A$ that satisfies the equivalent conditions of Theorem 2.1 a _regular
∗-subalgebra_ of $B(H)$, or an _R ∗-algebra_, for short. The same object is
called an “E∗-algebra that is a pre-C∗-algebra” in [48].

######  Corollary 2.3.

Every ∗-subalgebra of an R∗-algebra is an R∗-algebra.

###### Proof.

Clear from any item of Theorem 2.1 (2)–(6). ∎

The equivalence of (1)–(4) of the above theorem does not hold in general if we
consider an algebra instead of a ∗-algebra. Indeed, it is known that there
exists a _closed_ subalgebra $A$ of $B(H)$ such that (3) and (4) hold but (2)
does not hold [51, Theorems 5 and 6]. Since $\\{x\in B(H)\mid
x^{*}x=0\\}=\\{0\\}$, we obtain

######  Lemma 2.4.

An R∗-algebra $A$ is a ∗-regular ring, i.e., $A$ is a ∗-ring that is regular
and satisfies $\\{x\in A\mid x^{*}x=0\\}=\\{0\\}$.

The following lemma is a consequence of an elementary fact about algebraic
elements.

######  Lemma 2.5.

Let $x\in B(H)$ be an algebraic operator (i.e., ${\mathcal{A}}(x)$ is finite-
dimensional). If $x$ is invertible in $B(H)$, then $x$ is invertible in
${\mathcal{A}}(x,1)$.

###### Proof.

Assume that $x$ is an algebraic operator that is invertible in $B(H)$. Then
the minimal polynomial $P_{1}(t)=\sum_{j=0}^{k}c_{j}t^{j}$ of $x$ must satisfy
$c_{0}\neq 0$. Indeed, if $c_{0}=0$, then the polynomial $P_{2}(t)=P_{1}(t)/t$
satisfies $P_{2}(x)=x^{-1}P_{1}(x)=0$, contradicting the minimality of
$P_{1}$. Thus $1=(c_{0}-P_{1}(x))/c_{0}=-\sum_{j=1}^{k}c_{j}x^{j}/c_{0}$,
which clearly implies that $x$ is invertible in ${\mathcal{A}}(x,1)$. ∎

Although we cannot make use of tools that involve completeness (like taking
limits) in the setting of R∗-algebras in general, we may take advantage of a
number of techniques from operator theory.

######  Proposition 2.6.

Let $A\subset B(H)$ be an R∗-algebra, and $x\in A$.

  1. (1)

Assume that $A$ contains the unit of $B(H)$. Let $U\subset{\mathbb{C}}$ be an
open set with $\sigma(x)\subset U$. Let $f\colon U\to{\mathbb{C}}$ be a
holomorphic map. Then the operator $f(x)$ (which is obtained by holomorphic
functional calculus in $B(H)$) belongs to $A$.

  2. (2)

Assume that $x$ is normal ($xx^{*}=x^{*}x$). Let
$f\colon\sigma(x)\to{\mathbb{C}}$ be a function with $f(0)=0$. Then the
operator $f(x)$ (which is obtained by Borel functional calculus in $B(H)$)
belongs to $A$. In particular, if $x$ is self-adjoint, then the positive part
$x_{+}:=f_{+}(x)$ and the negative part $x_{-}:=f_{-}(x)$ belong to $A$, where
$f_{+}(t):=\max\\{0,t\\}$, $f_{-}(t):=\max\\{-t,0\\}$, $t\in{\mathbb{R}}$.

  3. (3)

Let $x=v\lvert x\rvert$ be the polar decomposition of $x$ in $B(H)$. Then
$v,\lvert x\rvert\in A$, hence $l(x),r(x)\in A$.

  4. (4)

The Moore–Penrose inverse $x^{\dagger}$ of $x$ belongs to $A$.

###### Proof.

(1) Let $x\in A$ and $z\in{\mathbb{C}}$. If $x-z$ is invertible in $B(H)$, the
preceding lemma implies that $(x-z)^{-1}\in{\mathcal{A}}(x,1)$. Since
${\mathcal{A}}(x,1)$ is finite-dimensional, it is a closed subalgebra of
$B(H)$. Thus it is clear by the definition of holomorphic functional calculus
that $f(x)\in{\mathcal{A}}(x,1)\subset A$.
(2) Since $\sigma(x)$ is a finite set, $f(x)$ coincides with some polynomial
in $x$ and $x^{*}$. Thus we obtain $f(x)\in{\mathcal{A}}^{*}(x)\subset A$.
(3), (4) See Proof of Theorem 2.1 $(6)\Rightarrow(1)$. The Moore–Penrose
inverse of $x$ is the $y$ constructed there. ∎

We call a linear mapping (resp. a linear bijection) $\phi\colon A\to B$
between two ∗-algebras a _∗ -homomorphism_ (resp. a _∗ -isomorphism_) provided
that it is multiplicative and ∗-preserving, i.e., $\phi(xy)=\phi(x)\phi(y)$
and $\phi(x^{*})=\phi(x)^{*}$ for any $x,y\in A$.

In view of the theory of C∗-/von Neumann algebras, we also give an abstract
axiom for a ∗-algebra to be ∗-isomorphic to an R∗-algebra. Let $A$ be a
∗-algebra (over ${\mathbb{C}}$). Recall that a _C ∗-norm_ of $A$ is a norm
$\lVert\cdot\rVert\colon A\to[0,\infty)$ satisfying $\lVert xy\rVert\leq\lVert
x\rVert\lVert y\rVert$ (submultiplicativity) and $\lVert x^{*}x\rVert=\lVert
x\rVert^{2}$ (C∗-identity) for all $x,y\in A$. A _pre-C ∗-algebra_ is a
∗-algebra endowed with a C∗-norm. It is clear that every ∗-subalgebra of
$B(H)$ is a pre-C∗-algebra.

The (noncommutative) Gelfand–Naimark theorem [49, Theorem I.9.18] states that
a complete pre-C∗-algebra is ∗-isomorphic to a closed ∗-subalgebra of $B(H)$.
On the other hand, Sakai’s theorem [49, Theorem III.3.5] states that a
C∗-algebra $A$ is ∗-isomorphic to a von Neumann algebra if and only if $A$ is
a dual Banach space.

######  Theorem 2.7.

Let $A$ be a pre-C∗-algebra. Then the following conditions are mutually
equivalent.

  * •

$A$ is ∗-isomorphic to an R∗-algebra.

  * •

$A$ is regular.

  * •

$A$ is algebraic.

###### Proof.

The completion of $A$ is a C∗-algebra, which is ∗-isomorphic to a closed
∗-subalgebra of some $B(H)$ by the Gelfand–Naimark theorem. Now the
equivalence easily follows from Theorem 2.1. ∎

We call a pre-C∗-algebra with the above equivalent conditions an _abstract R
∗-algebra_. (This can also be referred to as an _algebraic pre-C ∗-algebra_,
or a _regular pre-C ∗-algebra_.) As is the case for C∗-algebras, the above
theorem ensures that in many situations there is no need to distinguish
between abstract R∗-algebras and R∗-algebras in $B(H)$ (see also Proposition
2.11 below, which implies the uniqueness of C∗-norm of an R∗-algebra).

######  Remark 2.8.

In this paper, an element of a pre-C∗-algebra is said to be _positive_ if it
is positive in the completion (as an operator of a C∗-algebra).

In what follows we collect some basic properties of ideals, quotients and
∗-homomorphisms of R∗-algebras, which were first proved in [48]. We give a
proof for the reader’s convenience. Our proof is based on C∗-algebra theory.

######  Proposition 2.9 ([48]).

Let $I$ be a (two-sided) ideal of an R∗-algebra $A$. Then $I$ is closed under
∗-operation and in the norm topology in $A$, and the quotient by the ideal
forms an R∗-algebra.

###### Proof.

Let $I\subset A$ be an ideal. Let $x\in I$ and let $x=v\lvert x\rvert$ be its
polar decomposition. Then $v,\lvert x\rvert\in A$ by Proposition 2.6 (3). Thus
we have $I\ni v^{*}v\lvert x\rvert=\lvert x\rvert$ and $x^{*}=\lvert x\rvert
v^{*}\in I$.

Let $x_{n}\in I$, $n\geq 1$, converge to $x\in A$. Then $x_{n}^{*}x_{n}\to
x^{*}x$, hence

$y_{n}:=\lvert x\rvert^{\dagger}x_{n}^{*}x_{n}\lvert
x\rvert^{\dagger}\to\lvert x\rvert^{\dagger}x^{*}x\lvert
x\rvert^{\dagger}=s(\lvert x\rvert)$

as $n\to\infty$. Since $y_{n}$ is positive and $s(y_{n})\leq s(\lvert
x\rvert^{\dagger})=s(\lvert x\rvert)$, the above convergence implies that
$s(y_{n})=s(\lvert x\rvert)$ for a sufficiently large $n$. For each $n$, we
have $y_{n}=\lvert x\rvert^{\dagger}x_{n}^{*}x_{n}\lvert x\rvert^{\dagger}\in
I$ and hence $s(y_{n})=y_{n}y_{n}^{\dagger}\in I$. It follows that $s(\lvert
x\rvert)=s(y_{n})\in I$ for a sufficiently large $n$, thus we have
$x=xs(\lvert x\rvert)\in I$. Therefore, $I$ is closed in $A$.

Consider the completion $\widehat{A}$ of $A$, which is a C∗-algebra. It is
easy to see that the closure $J$ of $I$ in $\widehat{A}$ is a closed ideal of
$\widehat{A}$ and $J\cap A=I$. Let $\pi$ denote the quotient map
$\widehat{A}\to\widehat{A}/J$. Then the restriction of $\pi$ to $A$ has kernel
$I$, and the image $\pi(A)$ is endowed with the C∗-norm of $\widehat{A}/J$.
This implies that $A/I$ forms a pre-C∗-algebra, which is clearly algebraic. ∎

Recall that there are C∗-algebras with no projections (see for example [16,
Sections IV.8 and VII.8]). In contrast, every R∗-algebra has plenty of
projections because of Theorem 2.1 (6). This is somehow akin to von Neumann
algebras, so we introduce the following concept borrowed from the theory of
von Neumann algebras.

######  Proposition 2.10 (Reduced/induced R∗-algebras (cf. [49, Definition
II.3.11])).

Let $A\subset B(H)$ be an R∗-algebra.

  1. (1)

If $p\in A$ is a projection, then $pAp=\\{x\in A\mid pxp=x\\}$ forms an
R∗-algebra in $B(H)$ with unit $p$.

  2. (2)

If $p^{\prime}\in A^{\prime}=\\{y\in B(H)\mid xy=yx\text{ for all }x\in A\\}$
is a projection in the commutant of $A$, then $p^{\prime}A=\\{p^{\prime}x\mid
x\in A\\}\subset B(H)$ forms an R∗-algebra.

###### Proof.

Easy. Just check that $pAp$ and $p^{\prime}A$ form ∗-subalgebras of $B(H)$ and
that each operator in them has closed range. ∎

If $A\subset B(H)$ is an R∗-algebra and $A$ has a unit $e$, then $e$ need not
coincide with the unit of $B(H)$. However, it is easy to check that $e$ is a
projection and $A=eAe$ can be considered as a ∗-subalgebra of $B(eH)$, and the
embedding $A=eAe\to B(eH)$ preserves the unit. With this in mind, when we
consider a unital R∗-algebra in $B(H)$, we may always assume that
$1=\operatorname{id}_{H}\in A$. This enables us to consider functional
calculus as in Proposition 2.6 (2) for functions $f$ without assuming $f(0)=0$
if $A$ is unital.

Let $A\subset B(H)$ be an R∗-algebra without unit. Let $\widetilde{A}$ denote
the algebra $A+{\mathbb{C}}\cdot\operatorname{id}_{H}\subset B(H)$, which is
called the _unitization_ of $A$. By Theorem 2.1 (4), $\widetilde{A}$ is a
unital R∗-algebra.

######  Proposition 2.11 ([48]).

Let $A$ be an R∗-algebra and $K$ be a Hilbert space.

  1. (1)

Every injective ∗-homomorphism $\phi\colon A\to B(K)$ is an isometry.

  2. (2)

For every ∗-homomorphism $\phi\colon A\to B(K)$ the kernel of $\phi$ is an
ideal of $A$.

  3. (3)

Let $\pi$ be the canonical projection of $A$ onto $A/\ker\phi$. There exists a
unique injective ∗-homomorphism $\phi_{0}\colon A/\ker\phi\to B(K)$ with
$\phi=\phi_{0}\circ\pi$.

###### Proof.

$(1)$ Let $\phi\colon A\to B(K)$ be an injective ∗-homomorphism. Clearly the
image $\phi(A)$ is an R∗-algebra. Let $A$ act on the Hilbert space $H$. If $A$
is unital, then we may assume that $\operatorname{id}_{H}\in A$ and
$\phi(\operatorname{id}_{H})=\operatorname{id}_{K}$ by the comment preceding
this proposition. If $A$ is not unital, then $\phi$ uniquely extends to an
injective ∗-homomorphism $\widetilde{\phi}\colon\widetilde{A}\to B(K)$ by the
formula
$\widetilde{\phi}(x+z\operatorname{id}_{H})=\phi(x)+z\operatorname{id}_{K}$,
$x\in A$, $z\in{\mathbb{C}}$. Therefore, it suffices to consider the case $A$
is unital and $\phi(\operatorname{id}_{H})=\operatorname{id}_{K}$. It follows
from Lemma 2.5 that for each $x\in A$, $x$ is invertible (in
${\mathcal{A}}(x,\operatorname{id}_{H})$, or equivalently, in $B(H)$) if and
only if $\phi(x)$ is invertible (in
${\mathcal{A}}(\phi(x),\operatorname{id}_{K})$, or equivalently, in $B(K)$).
Therefore, for each positive operator $a\in A$, we obtain $\lVert
a\rVert=\sup\sigma(a)=\sup\sigma(\phi(a))=\lVert\phi(a)\rVert$. This implies
that for every $x\in A$,

$\lVert x\rVert^{2}=\lVert
x^{*}x\rVert=\lVert\phi(x^{*}x)\rVert=\lVert\phi(x)^{*}\phi(x)\rVert=\lVert\phi(x)\rVert^{2},$

thus $\phi$ is an isometry.
The rest of the statement is a consequence of a routine straightforward
argument combined with Proposition 2.9. ∎

A (pre-C)∗-algebra $A$ is said to _have a unique C ∗-norm_ if there exists a
unique norm of $A$ that makes $A$ a pre-C∗-algebra. Recall that every
C∗-algebra has a unique C∗-norm [49, Corollary I.5.4]. The preceding
proposition shows that every R∗-algebra has a unique C∗-norm. More generally,
it is easy to see that a ∗-subalgebra $A\subset B(H)$ has a unique C∗-norm if
it satisfies the following property:

  * (LC∗)

The closure of the (∗-)subalgebra ${\mathcal{A}}(a)$ in $B(H)$ is contained in
$A$ for every positive element $a\in A$.

(Proof: Let $\lVert\cdot\rVert$ denote the operator norm of $B(H)$, and let
$\lVert\cdot\rVert_{1}$ be any C∗-norm of $A$. If $a\in A$ is positive, then
the closure of ${\mathcal{A}}(a)$ in $B(H)$ is a C∗-algebra that is a
subalgebra of $A$. Since a C∗-algebra has a unique C∗-norm, we have $\lVert
a\rVert=\lVert a\rVert_{1}$. For a general $x\in A$, by the C∗-identity and
the positivity of $x^{*}x$ we obtain $\lVert x\rVert^{2}=\lVert
x^{*}x\rVert=\lVert x^{*}x\rVert_{1}=\lVert x\rVert_{1}^{2}$.) In particular,
the union of any increasing net of C∗-algebras has a unique C∗-norm.

In [48] Szűcs and Takács introduced the following property (E∗) for a
∗-algebra $A$, and proved that it is equivalent to some conditions in Theorem
2.1 in a general setting.

  * (E∗)

For any ∗-subalgebra $B\subset A$, any Hilbert space $K_{1}$, any
∗-homomorphism $\pi\colon B\to B(K_{1})$ and any vector $v\in K_{1}$ such that
$\pi(B)v$ is dense in $K_{1}$, there are a Hilbert space $K_{2}$, a
∗-homomorphism $\pi_{0}\colon A\to B(K_{2})$ and a vector $w\in K_{2}$ such
that $\pi_{0}(A)w$ is dense in $K_{2}$ and
$\langle\pi(x)v,v\rangle=\langle\pi_{0}(x)w,w\rangle$ for every $x\in B$.

Let us deduce the equivalence of the conditions in Theorem 2.1 and (E∗) for
pre-C∗-algebras without resorting to results of [48]. We also clarify that
these conditions are related to the uniqueness of C∗-norm.

######  Theorem 2.12.

Let $A\subset B(H)$ be a ∗-subalgebra. The following are equivalent.

  1. (1)

$A$ is an R∗-algebra.

  2. (2)

Every ∗-subalgebra of $A$ has a unique C∗-norm.

  3. (3)

For every positive element $a\in A$ the ∗-subalgebra
${\mathcal{A}}(a)={\mathcal{A}}^{*}(a)\subset A$ has a unique C∗-norm.

  4. (4)

$A$ satisfies $(E^{*})$.

###### Proof.

Proposition 2.11 combined with Corollary 2.3 shows $(1)\Rightarrow(2)$, and
$(2)\Rightarrow(3)$ is clear.
$(3)\Rightarrow(1)$ Assume that there is a self-adjoint operator $a\in A$ with
infinite spectrum. By replacing $a$ with $a^{2}$ we may assume that $a$ is
positive. We prove that ${\mathcal{A}}^{*}(a)$ has more than one C∗-norm. Let
$A_{0}$ denote the algebra of polynomials $P(t)=\sum_{k=1}^{n}z_{k}t^{k}$ with
$n\geq 1$, $z_{k}\in{\mathbb{C}}$. It is clear that $A_{0}\ni P(t)\mapsto
P(a)\in{\mathcal{A}}^{*}(a)$ is a ∗-isomorphism. The algebra of polynomials
$A_{0}$ has more than two C∗-norms, thus ${\mathcal{A}}^{*}(a)$ has the same
property. Indeed, define e.g. $\lVert
P(t)\rVert_{j}:=\sup_{\lambda\in[j-1,j]}\lvert P(\lambda)\rvert\in[0,\infty)$,
$j=1,2$. Then $\lVert\cdot\rVert_{1}$ and $\lVert\cdot\rVert_{2}$ are distinct
C∗-norms of $A_{0}$.
$(1)\Rightarrow(4)$ Let $B\subset A$ be a ∗-subalgebra, $K_{1}$ a Hilbert
space, $\pi\colon B\to B(K_{1})$ a ∗-homomorphism, $v\in K_{1}$ a vector and
assume that $\pi(B)v$ is dense in $K_{1}$. Corollary 2.3 combined with
Proposition 2.11 implies that the mapping $f\colon B\to{\mathbb{C}}$ defined
by $f(x):=\langle\pi(x)v,v\rangle$ is a bounded positive linear functional. It
follows by the Hahn–Banach theorem (see [49, Lemma III.3.2]) that $f$ extends
to a bounded positive linear functional $g$ on $A$. By the
Gelfand–Naimark–Segal (GNS) construction [49, Theorem I.9.14] there are a
Hilbert space $K_{2}$, a ∗-homomorphism $\pi_{0}\colon A\to B(K_{2})$ and a
vector $w\in K_{2}$ such that $\pi_{0}(A)w$ is dense in $K_{2}$ and
$g(x)=\langle\pi_{0}(x)w,w\rangle$ for every $x\in A$. It follows that
$\langle\pi(x)v,v\rangle=f(x)=g(x)=\langle\pi_{0}(x)w,w\rangle$ for every
$x\in B$.
$(4)\Rightarrow(1)$ Assume that there is a self-adjoint operator $a\in A$ with
infinite spectrum. We prove that ${\mathcal{A}}^{*}(a)$ does not satisfy (E∗),
which clearly implies that $A$ does not satisfy (E∗), either. Let $B$ denote
the ∗-subalgebra of ${\mathcal{A}}^{*}(a)$ generated by $a^{2}$. Take a
∗-homomorphism $\pi\colon B\to{\mathbb{C}}$ that is uniquely determined by
$\pi(a^{2})=-1$. (Note that $a$ has infinite spectrum and so does $a^{2}$,
thus $a^{2}$ is not an algebraic operator.) Assume that there are a Hilbert
space $K$, a ∗-homomorphism $\pi_{0}\colon{\mathcal{A}}^{*}(a)\to B(K)$ and a
vector $w\in K$ such that $\pi_{0}({\mathcal{A}}^{*}(a))w$ is dense in $K$ and
$\pi(x)=\langle\pi_{0}(x)w,w\rangle$ for every $x\in B$. Then $b:=\pi_{0}(a)$
is a self-adjoint operator in $B(K)$ such that

$-1=\pi(a^{2})=\langle\pi_{0}(a^{2})w,w\rangle=\langle b^{2}w,w\rangle=\lVert
bw\rVert^{2}\geq 0,$

which is a contradiction. ∎

##  3\. Examples

Although the definition of R∗-algebra might seem somehow restrictive, we can
construct various examples of R∗-algebras. In this section, we give a number
of examples of R∗-algebras, as well as means of obtaining new examples from
known ones. More examples can be found in Sections 4 and 5.

First of all, for each $n\geq 1$, the $n\times n$ _full matrix algebra_
${\mathbb{M}}_{n}={\mathbb{M}}_{n}({\mathbb{C}})$ is the collection of all
$n\times n$ complex matrices, which is an example of a finite-dimensional
C∗-algebra and also an R∗-algebra.

######  Example 3.1 (Finite-dimensional R∗-algebras).

Every finite-dimensional pre-C∗-algebra is a complete R∗-algebra (thus a
C∗-algebra). In fact, a finite-dimensional (pre-)C∗-algebra is the direct sum
of finitely many full matrix algebras.

######  Non-example 3.2.

An infinite-dimensional C∗-algebra is never an R∗-algebra. Indeed, it is well-
known that every infinite-dimensional C∗-algebra has an operator with infinite
spectrum. See for example [31, Exercise 4.6.14].

It follows that the intersection of C∗\- and R∗-algebras consists of finite-
dimensional C∗-algebras.

######  Example 3.3 (Sequence space and direct sums).

Let $c_{00}$ denote the algebra of complex sequences with finitely many
nonzero terms. This acts on the sequential $\ell_{2}$ space by termwise
multiplication, and it is clear that $c_{00}$ is an R∗-algebra in
$B(\ell_{2})$.

More generally, let $I$ be a set, and let $A_{i}\subset B(H_{i})$ be an
R∗-algebra for each $i\in I$. Let $H$ be the direct sum of $H_{i}$, and let
$p_{i}$ be the projection of $H$ onto $H_{i}$. Each operator $x\in B(H_{i})$
determines an operator $p_{i}xp_{i}\in B(H)$. Then

$\bigoplus_{i\in I}A_{i}:={\mathcal{A}}^{*}(\\{p_{i}xp_{i}\mid x\in
A_{i},\,\,i\in I\\})\subset B(H)$

forms an R∗-algebra.

######  Example 3.4 (Algebras of operators with finite rank).

Let $F(H)$ denote the algebra of bounded linear operators with finite rank on
a Hilbert space $H$. Then $F(H)$ is an R∗-algebra in $B(H)$. On the other
hand, the collection ${\mathbb{M}}_{\infty}$ of infinite matrices
$(a_{ij})_{i,j\geq 1}$ with finitely many nonzero entries forms an R∗-algebra,
which acts on the sequential $\ell_{2}$ space in the usual manner. It is clear
that ${\mathbb{M}}_{\infty}\subset F(\ell_{2})$. Let us observe that these are
nonisomorphic. Indeed, it is clear that ${\mathbb{M}}_{\infty}$ has countable
Hamel dimension. However, $F(\ell_{2})$ does not. (To prove it, consider for
example the subspace of $F(\ell_{2})$ that consists of the collection of
operators $x$ with $x^{*}(\ell_{2})\subset{\mathbb{C}}h$ for a fixed unit
vector $h\in\ell_{2}$. This subspace is linearly isometric to $\ell_{2}$ by
the map $x\mapsto xh$. Recall that $\ell_{2}$ has uncountable Hamel
dimension.) It is worth mentioning that both of the completions of
${\mathbb{M}}_{\infty}$ and $F(\ell_{2})$ are the C∗-algebra $K(\ell_{2})$ of
compact operators. In Section 5 we will take a closer look at examples of this
type.

######  Example 3.5 (Directed unions).

Let $I$ be a directed set and $(A_{i})_{i\in I}$ be an increasing net of
abstract R∗-algebras. Since every injective ∗-homomorphism between R∗-algebras
is isometric, the union $A=\bigcup_{i\in I}A_{i}$ forms an R∗-algebra. We say
such an $A$ is _ultramatricial_ provided that each $A_{i}$ is finite-
dimensional. If in addition $(A_{i})_{i\in I}$ is a sequence (rather than a
net), then we say $A$ is _sequentially ultramatricial_.

In this paper, a C∗-algebra that arises as the completion of an ultramatricial
R∗-algebra is called an _AF_ (approximately finite-dimensional) C∗-algebra. A
C∗-algebra that arises as the completion of a sequentially ultramatricial
R∗-algebra is called a _sequentially AF_ C∗-algebra. It is well-known that a
separable AF C∗-algebra is sequentially AF (see [16, Theorem III.3.4]).
CAUTION: In C∗-algebra theory it is common to consider only separable
C∗-algebras, so an AF C∗-algebra often means a separable one in the
literature, which is different from our choice of terminology.

######  Proposition 3.6 (Matrix algebras).

Let $A$ be an R∗-algebra. For $n\geq 1$, the algebra
${\mathbb{M}}_{n}(A)=A\otimes{\mathbb{M}}_{n}$ of the collection of $n\times
n$ matrices with entries in $A$ forms an R∗-algebra.

###### Proof.

It is well-known that the algebraic tensor product $A\otimes{\mathbb{M}}_{n}$
is a regular ring for every unital regular ring $A$. See for example [43,
Theorem II.2.13]. If $A\subset B(H)$ is a unital R∗-algebra, then
$A\otimes{\mathbb{M}}_{n}$ can be considered as a ∗-subalgebra of $B(H^{\oplus
n})$. Thus $A\otimes{\mathbb{M}}_{n}$ forms an R∗-algebra. If $A$ is
nonunital, consider $A\otimes{\mathbb{M}}_{n}$ as a subalgebra of
$\widetilde{A}\otimes{\mathbb{M}}_{n}$. ∎

In [48, Proof of Proposition 4.7], the above proposition was shown by means of
the fact that the matrix algebra over an algebraic algebra is algebraic [1,
Theorem 9].

######  Example 3.7 (Tensor products with ultramatricial algebras).

Let $A$, $B$ be R∗-algebras. Suppose that $B$ is ultramatricial. Take a
directed set $I$ and an increasing net $(B_{i})_{i\in I}$ of finite-
dimensional R∗-algebras with $B=\bigcup_{i\in I}B_{i}$. Since a finite-
dimensional R∗-algebra decomposes into a direct sum of matrix algebras, the
preceding proposition implies that for each $i$ the algebraic tensor product
$A\otimes B_{i}$ forms an R∗-algebra. Moreover, for any $i,j\in I$ with $i\leq
j$ we may take a canonical inclusion from $A\otimes B_{i}$ into $A\otimes
B_{j}$ (as algebraic tensor products). We define $A\otimes B$ to be the union
of the increasing net $(A\otimes B_{i})_{i\in I}$.

In particular, for an R∗-algebra $A$,
${\mathbb{M}}_{\infty}(A)=A\otimes{\mathbb{M}}_{\infty}$ is an R∗-algebra,
which can be considered as the collection of ${\mathbb{N}}\times{\mathbb{N}}$
matrices with finitely many nonzero entries in $A$. This R∗-algebra is used to
consider the $K_{0}$-group in Section 7.

######  Example 3.8 (Infinite tensor products).

Let $I$ be a set and $(A_{i})_{i\in I}$ be a family of unital ultramatricial
R∗-algebras. Then the above argument implies that for each finite subset $F$
of $I$ we may construct the tensor product R∗-algebra $\bigotimes_{i\in
F}A_{i}$. Moreover, if $F_{1}\subset F_{2}\subset I$ are finite subsets, there
exists a natural embedding $\iota\colon\bigotimes_{i\in
F_{1}}A_{i}\to\bigotimes_{i\in F_{2}}A_{i}$ determined by the condition
$\iota(\bigotimes_{i\in F_{1}}x_{i})=(\bigotimes_{i\in
F_{1}}x_{i})\otimes(\bigotimes_{j\in F_{2}\setminus F_{1}}1_{A_{j}})$. Thus
the family $(\bigotimes_{i\in F}A_{i})_{F}$ forms an increasing net of
R∗-algebras, whose union over all finite subsets $F\subset I$ will be written
as $\bigotimes_{i\in I}A_{i}$.

######  Example 3.9 (Group algebras).

Let $G$ be a discrete group and consider the group algebra ${\mathbb{C}}[G]$.
As usual, we endow ${\mathbb{C}}[G]$ with the ∗-operation that is determined
by $(zg)^{*}=\overline{z}g^{-1}$, $z\in{\mathbb{C}}$, $g\in G$. Since the
study of C∗-/von Neumann algebras arising from ${\mathbb{C}}[G]$ plays a
dominant role in the theory of operator algebras, it is natural to ask when
${\mathbb{C}}[G]$ forms an R∗-algebra. Recall that $G$ is said to be _locally
finite_ if every finite subset generates a finite subgroup of $G$. It is easy
to see that the group algebra ${\mathbb{C}}[G]$ forms an R∗-algebra if and
only if $G$ is locally finite. Indeed, if ${\mathbb{C}}[G]$ forms an
R∗-algebra, then it is algebraic, which plainly leads to local finiteness of
$G$. (For a finite subset $F\subset G$, consider the algebra generated by
$\sum_{g\in F}g$.) The converse implication is even easier: The left regular
representation gives a C∗-norm of ${\mathbb{C}}[G]$, and it is clear that
every element in ${\mathbb{C}}[G]$ is algebraic if $G$ is locally finite. More
generally, in [48, Theorem 6.5] it was proved that the convolution ∗-algebra
of complex-valued compactly supported continuous functions on a locally
compact group $G$ is an R∗-algebra if and only if $G$ is discrete and locally
finite. Von Neumann regularity of more general group algebras (not necessarily
over ${\mathbb{C}}$) is discussed in [36, Problem 8].

###  3.1. Problems of Kurosh and Kadison

In [48], Szűcs and Takács found that Kadison’s Problem 1.1 is related to
Kurosh’s problem. Let us revisit it. Recall that an algebra $A$ is defined to
be _locally finite-dimensional_ if every finite subset of $A$ generates a
finite-dimensional subalgebra of $A$. The classical Kurosh’s problem asks
whether every algebraic algebra $A$ (i.e., such that every single element of
$A$ generates a finite-dimensional subalgebra of $A$) is locally finite-
dimensional. This problem has a negative answer in the general setting.
However, it is known that for certain classes of algebras (e.g., the class of
PI-algebras), Kurosh’s problem has a positive answer. See Kaplansky’s list of
problems [36] for references and several related topics. We would like to
consider Kurosh’s problem for the class of R∗-algebras. Since an R∗-algebra is
algebraic by Theorem 2.1, this is equivalent to asking whether every
R∗-algebra is locally finite-dimensional. This problem clearly coincides with
Kadison’s Problem 1.1.

######  Lemma 3.10.

An R∗-algebra $A$ is locally finite-dimensional if and only if $A$ is
ultramatricial.

###### Proof.

Let $A$ be locally finite-dimensional. Consider the directed set $\mathcal{F}$
made of all finite subsets of $A$, ordered by inclusion. Then $A$ is the union
of the increasing net $({\mathcal{A}}(F\cup F^{*}))_{F\in\mathcal{F}}$ of
finite-dimensional R∗-algebras. Conversely, let $A$ be equal to the union of
an increasing net $(A_{i})_{i\in I}$ of finite-dimensional R∗-subalgebras. For
each finite subset $F\subset A$, we may find some $i\in I$ such that $F\subset
A_{i}$, and thus ${\mathcal{A}}(F)\subset A_{i}$ is finite-dimensional. ∎

Therefore, Problem 1.1, as well as Kurosh’s problem for R∗-algebras, is
equivalent to asking whether every R∗-algebra is ultramatricial.

##  4\. Lattice of projections as noncommutative Boolean algebras

Let us recall some definitions of lattice theory for beginners. A _lattice_ is
a partially ordered set ${\mathcal{P}}$ such that every pair
$p,q\in{\mathcal{P}}$ has the least upper bound $p\vee q\in{\mathcal{P}}$ and
the greatest lower bound $p\wedge q\in{\mathcal{P}}$. A mapping (resp.
bijection) $\psi$ between two lattices is called a _lattice homomorphism_
(resp. a _lattice isomorphism_) if it satisfies $\psi(p\vee
q)=\psi(p)\vee\psi(q)$ and $\psi(p\wedge q)=\psi(p)\wedge\psi(q)$ for every
pair of elements $p,q$. A lattice ${\mathcal{P}}$ with greatest element $1$
and least element $0$ is _complemented_ if for every $p\in{\mathcal{P}}$ there
exists $q\in{\mathcal{P}}$ with $p\vee q=1$ and $p\wedge q=0$. Such a $q$ is
called a _complement_ of $p$. A lattice ${\mathcal{P}}$ is _modular_ if
$p\vee(q\wedge r)=(p\vee q)\wedge r$ for all $p,q,r\in{\mathcal{P}}$ with
$p\leq r$.

An _orthocomplemented lattice_ is a complemented lattice ${\mathcal{P}}$ with
a bijection ${\mathcal{P}}\ni p\mapsto p^{\perp}\in{\mathcal{P}}$ (called the
orthocomplementation) with the following properties.

  * •

$(p^{\perp})^{\perp}=p$ for all $p\in{\mathcal{P}}$.

  * •

$p^{\perp}$ is a complement of $p$ for all $p\in{\mathcal{P}}$.

  * •

$p\leq q\iff p^{\perp}\geq q^{\perp}$ for all $p,q\in{\mathcal{P}}$.

An _orthoisomorphism_ $\psi$ between two orthocomplemented lattices is a
lattice isomorphism such that $\psi(p^{\perp})=\psi(p)^{\perp}$ for all $p$.

A family $\emptyset\neq{\mathcal{S}}\subset 2^{X}$ of subsets of a set $X$ is
called a _field of sets_ if $A\cap B$, $A\cup B$, $A^{c}\in{\mathcal{S}}$ for
any $A,B\in{\mathcal{S}}$. A field of sets forms an orthocomplemented lattice
relative to the order by inclusion and the orthocomplementation by set-
complement. A _Boolean algebra_ is an orthocomplemented lattice that is
orthoisomorphic to a field of sets. It is clear that each element of a Boolean
algebra has a unique complement. It follows that a lattice isomorphism between
two Boolean algebras is an orthoisomorphism. The book [25] is a gentle
introduction to the theory of Boolean algebras. A _generalized field of sets_
is a family $\emptyset\neq{\mathcal{S}}\subset 2^{X}$ of subsets of a set $X$
such that $A\cap B$, $A\cup B$, $A\setminus B\in{\mathcal{S}}$ for any
$A,B\in{\mathcal{S}}$. A lattice that is lattice isomorphic to a generalized
field of sets is called a _generalized Boolean algebra_.

Recall that for a von Neumann algebra $M$ the collection ${\mathcal{P}}(M)$ of
projections forms an orthocomplemented lattice. It is well-known that the
projection lattice of a von Neumann algebra is modular if and only if the
algebra is of finite type. Indeed, if a von Neumann algebra $M$ is not finite,
then $M$ has $B(\ell_{2})$ as a $\sigma$-weakly closed ∗-subalgebra, but
${\mathcal{P}}(B(\ell_{2}))$ is not modular by [30, Solution 14]. The proof of
modularity of ${\mathcal{P}}(M)$ when $M$ is of finite type can be found in
e.g. [49, Theorem V.1.37].

Let $A$ be an R∗-algebra with unit. By von Neumann’s theory in [43, Part II],
the collection of principal right ideals of $A$, ordered by inclusion, forms a
complemented modular lattice. Moreover, by [43, Theorem II.4.5] together with
Lemma 2.4, this lattice can be identified with the lattice of projections
${\mathcal{P}}(A)=\\{p\in A\mid p=p^{2}=p^{*}\\}$. Here ${\mathcal{P}}(A)$ is
endowed with the usual order, namely, for two projections $p,q\in B(H)$,
$p\leq q$ means $pH\subset qH$, or equivalently, $pq=p$. The correspondence
between principal right ideals and projections can be described in the
following manner: Each principal right ideal of $A$ coincides with $pA$ for a
unique $p\in{\mathcal{P}}(A)$. Observe that $p^{\perp}:=1-p$, $p\in A$,
determines an orthocomplementation of ${\mathcal{P}}(A)$.

Let $A$ be a unital R∗-algebra in $B(H)$. Then each element in
${\mathcal{P}}(A)$ can be identified with its range $pH$, which is a closed
subspace of the Hilbert space $H$. The lattice structure of ${\mathcal{P}}(A)$
can be written more explicitly:

######  Proposition 4.1.

Let $A$ be a unital R∗-algebra in $B(H)$ with $\operatorname{id}_{H}\in A$.
For $p,q\in{\mathcal{P}}(A)$,

  * •

$p\vee q\in{\mathcal{P}}(A)$ is the projection onto $pH+qH$;

  * •

$p\wedge q\in{\mathcal{P}}(A)$ is the projection onto $pH\cap qH$; and

  * •

$p^{\perp}\in{\mathcal{P}}(A)$ is the projection onto the orthogonal
complement $(pH)^{\perp}=\\{h\in H\mid\langle h,k\rangle=0\text{ for all }k\in
pH\\}$.

###### Proof.

Let us prove the first item. The projection $p\vee q$ is determined as the
smallest projection in $A$ with $p\leq p\vee q$, $q\leq p\vee q$. Therefore,
it is clear that the range of $p\vee q$ contains $pH+qH$. It suffices to show
that $pH+qH$ is closed and the projection onto $pH+qH$ belongs to $A$. Since
$A$ is an R∗-algebra, the positive operator $p+q$ has closed range by Theorem
2.1 (3). Since $0\leq p\leq p+q$, we have $p=s(p)\leq s(p+q)$, which implies
$pH\subset(p+q)H$. Similarly, we have $qH\subset(p+q)H$. It is clear that
$(p+q)H\subset pH+qH$. Thus we obtain $pH+qH=(p+q)H$. Recall that $(p+q)H$ is
closed and the projection onto $(p+q)H$ is equal to $s(p+q)\in A$. This
completes the proof of the first item.

To prove the second item, check that the projection onto $pH\cap qH$ is equal
to $\chi_{\\{1\\}}(pqp)$, which belongs to $A$ by Proposition 2.6 (2). The
third item is easy. ∎

This property is very similar to a basic property of the lattice of
projections of a von Neumann algebra. However, there is a big difference: For
projections $p,q$ of a von Neumann algebra in $B(H)$, $p\vee q$ is the
projection onto the _closure_ of $pH+qH$, and taking closure is inevitable.

In that respect, the lattice ${\mathcal{P}}(A)$ resembles that of operator
ranges, namely, the lattice formed of ranges of bounded linear operators on a
fixed Hilbert space. It is a curious fact that the sum and intersection of any
pair of operator ranges are operator ranges. However, the lattice of operator
ranges is never complemented unless $H$ is finite-dimensional. In fact, an
operator range has a complement if and only if it is closed. See [22] for the
proofs of these facts and a more detailed study of operator ranges.

######  Remark 4.2.

In the above argument, we discussed only the case of unital R∗-algebras, just
for simplicity. All of these results carry over to the nonunital case after a
suitable change. For example, for a nonunital R∗-algebra $A$, the collection
of projections forms a modular lattice that is not complemented. (To show
this, consider the unitization of $A$.)

Although the following theorem is easy to prove, it gives an enormous
motivation to the research on R∗-algebras and their lattices of projections.
For a locally compact Hausdorff space $K$, let $C_{0}(K)$ denote the
commutative C∗-algebra of continuous complex-valued functions on $K$ that
vanish at infinity.

######  Theorem 4.3.

If $A$ is a commutative R∗-algebra then the lattice ${\mathcal{P}}(A)$ forms a
generalized Boolean algebra. Every generalized Boolean algebra arises in this
way. Two commutative R∗-algebras are ∗-isomorphic if and only if their
projection lattices are isomorphic. The commutative R∗-algebra $A$ is unital
if and only if ${\mathcal{P}}(A)$ is a Boolean algebra.

###### Proof.

It is a basic fact that the lattice of principal right ideals of a commutative
regular ring forms a generalized Boolean algebra. See for example [43, Theorem
II.2.10], in which unital regular rings are considered. However, let us give a
proof of this fact for the particular case of R∗-algebras. The proof below
visualizes the relation among commutative R∗-algebras, Boolean algebras and
commutative C∗-algebras.

Let $A$ be a commutative R∗-algebra. Take the completion $\widehat{A}$ of $A$,
which is a commutative C∗-algebra. By the Gelfand–Naimark theorem there exists
a locally compact Hausdorff space $K$ such that $\widehat{A}\cong C_{0}(K)$.
Then each projection $p\in{\mathcal{P}}(A)$ can be identified with a
projection $f_{p}$ in $C_{0}(K)$, which is a continuous function with value in
$\\{0,1\\}$, and $f_{p}$ is in turn identified with a compact open subset
$f_{p}^{-1}(1)\subset K$. Set ${\mathcal{S}}:=\\{f_{p}^{-1}(1)\mid
p\in{\mathcal{P}}(A)\\}\subset 2^{K}$. Then it follows from a quite
straightforward argument that the map $p\mapsto f_{p}^{-1}(1)$ is a lattice
isomorphism from ${\mathcal{P}}(A)$ onto ${\mathcal{S}}$, which forms a
generalized field of sets.

Let ${\mathcal{S}}\subset 2^{X}$ be a generalized field of sets. Consider the
Hilbert space $\ell_{2}(X)$, which is the space of square-summable complex-
valued functions on $X$. Then the space of functions
$R({\mathcal{S}}):=\\{f\colon
X\to{\mathbb{C}}\mid\\#f(X)<\infty,\,\,f^{-1}(z)\in{\mathcal{S}}\text{ for
every }z\in{\mathbb{C}}\setminus\\{0\\}\\}$ acts on $\ell_{2}(X)$ by pointwise
multiplication. It is clear that $R({\mathcal{S}})$ is a commutative
R∗-algebra. Moreover, for each $f\in{\mathcal{P}}(R({\mathcal{S}}))$, we have
$f^{-1}(1)\in{\mathcal{S}}$, and this gives an identification
${\mathcal{P}}(R({\mathcal{S}}))={\mathcal{S}}$ as generalized Boolean
algebras.

Let $A,B$ be commutative R∗-algebras. It is obvious that if $A$ is
∗-isomorphic to $B$ then ${\mathcal{P}}(A)$ is lattice isomorphic to
${\mathcal{P}}(B)$. Assume that there is a lattice isomorphism
$\psi\colon{\mathcal{P}}(A)\to{\mathcal{P}}(B)$. We concretely construct a
∗-isomorphism from $A$ onto $B$. Let $x\in A$. By Theorem 2.1, $x$ has finite
spectrum. Since $A$ is commutative, $x$ is a normal operator. Thus there exist
$n\geq 1$, complex numbers $z_{1},z_{2},\ldots z_{n}\in{\mathbb{C}}$, and
projections $p_{1},p_{2},\ldots,p_{n}\in{\mathcal{P}}(A)$ such that
$x=\sum_{k=1}^{n}z_{k}p_{k}$. We define
$\phi(x):=\sum_{k=1}^{n}z_{k}\psi(p_{k})$. It is a routine exercise to show
that $\phi$ is well-defined (independent of the choice of $z_{k}$, $p_{k}$)
and is a ∗-isomorphism from $A$ onto $B$.

It is clear that $A$ is unital if and only if ${\mathcal{P}}(A)$ is a Boolean
algebra. ∎

For a generalized Boolean algebra ${\mathcal{S}}$, let $R({\mathcal{S}})$
denote the commutative R∗-algebra whose projection lattice is isomorphic to
${\mathcal{S}}$. By the preceding theorem, such an R∗-algebra exists uniquely
up to ∗-isomorphism.

The above theorem is highly reminiscent of a similar characterization of
commutative C∗-/von Neumann algebras. The Gelfand–Naimark theorem tells us
that the research of C∗-algebras is considered as “noncommutative locally
compact Hausdorff spaces”. Similarly, von Neumann algebras can be considered
as “noncommutative measure spaces”. The author believes that the theory of
R∗-algebras can be considered as a theory of “noncommutative (generalized)
Boolean algebras”.

It is known as _Stone’s duality theorem_ that the category of Boolean algebras
is dually equivalent to that of Boolean spaces. Recall that a _Boolean space_
(also known as a _Stone space_) is a zero-dimensional compact Hausdorff space.
The lattice of clopen subsets of a Boolean space forms a Boolean algebra,
which gives one direction of the correspondence between Boolean spaces and
Boolean algebras.

More generally, the category of generalized Boolean algebras is dually
equivalent to that of zero-dimensional locally compact Hausdorff spaces. A
zero-dimensional locally compact Hausdorff space is also called a locally
compact Boolean space, and can be characterized as a locally compact Hausdorff
space such that every connected component is a singleton. Let us revisit this
correspondence from the viewpoint of R∗\- and C∗-algebras. Let ${\mathcal{S}}$
be a generalized Boolean algebra. Then the lattice of projections of the
commutative R∗-algebra $R({\mathcal{S}})$ is ${\mathcal{S}}$. We consider the
completion $\widehat{R({\mathcal{S}})}$ of $R({\mathcal{S}})$, which is a
commutative C∗-algebra.

######  Proposition 4.4.

Let ${\mathcal{S}}$ be a generalized Boolean algebra. Then every projection in
$\widehat{R({\mathcal{S}})}$ is an element of
${\mathcal{P}}(R({\mathcal{S}}))$, which can be identified with
${\mathcal{S}}$. In other words, if one takes a locally compact Hausdorff
space $K$ with $\widehat{R({\mathcal{S}})}\cong C_{0}(K)$ (whose existence is
guaranteed by the Gelfand–Naimark theorem), then

  * •

$K$ is zero-dimensional, and

  * •

the generalized Boolean algebra formed of compact open subsets of $K$
coincides with ${\mathcal{S}}$.

###### Proof.

Let $p\in{\mathcal{P}}(\widehat{R({\mathcal{S}})})$. Then there exists a
sequence $x_{n}\in R({\mathcal{S}})$ such that $x_{n}\to p$ (in norm). Regard
$p$ and $x_{n}$ as functions in $C_{0}(K)$. Then $p$ is a function with
$p(K)\subset\\{0,1\\}$, and $x_{n}\to p$ means that $x_{n}$ converges to $p$
uniformly. Take an open disc $D$ with center $1$ and radius $1/2$ in
${\mathbb{C}}$. It follows that $\chi_{D}(x_{n})\to p$. Remark that
$R({\mathcal{S}})$ is a commutative R∗-algebra, so Proposition 2.6 (2) implies
that $\chi_{D}(x_{n})\in{\mathcal{P}}(R({\mathcal{S}}))$. Moreover,
$\chi_{D}(x_{n})$ can be regarded as a function in $C_{0}(K)$ with
$\chi_{D}(x_{n})(K)\subset\\{0,1\\}$. It follows by $\chi_{D}(x_{n})\to p$
that $\chi_{D}(x_{n})=p$ for a sufficiently large $n$, which implies
$p\in{\mathcal{P}}(R({\mathcal{S}}))$. Therefore, the generalized Boolean
algebra formed of compact open subsets of $K$ coincides with
${\mathcal{P}}(R({\mathcal{S}}))={\mathcal{S}}$. Since every function in
$C_{0}(K)$ can be uniformly approximated by elements in $R({\mathcal{S}})$, it
is easy to see that each connected component of the locally compact Hausdorff
space $K$ is a singleton, which means that $K$ is zero-dimensional. ∎

Using this, we may interpret Stone’s theorem as the equivalence of the
following three categories:

  * •

the category of generalized Boolean algebras with lattice homomorphisms that
preserve $0$,

  * •

the category of commutative R∗-algebras with ∗-homomorphisms,

  * •

the category of all $C_{0}(K)$ for zero-dimensional locally compact Hausdorff
spaces $K$, with ∗-homomorphisms.

We may give a proof of this equivalence straightforwardly. The details are
omitted, but let us just observe how the morphisms correspond. Let $A$, $B$ be
commutative R∗-algebras. Let $\psi\colon{\mathcal{P}}(A)\to{\mathcal{P}}(B)$
be a lattice homomorphism such that $\psi(0)=0$. The corresponding
∗-homomorphism $\phi\colon A\to B$ can be described just as in Proof of
Theorem 4.3: Each element $x$ of $A$ can be written as
$x=\sum_{k=1}^{n}z_{k}p_{k}$ for some $n\geq 1$, complex numbers
$z_{1},z_{2},\ldots,z_{n}\in{\mathbb{C}}$, and
$p_{1},p_{2},\ldots,p_{n}\in{\mathcal{P}}(A)$. Define $\phi\colon A\to B$ by
$\phi(x):=\sum_{k=1}^{n}z_{k}\psi(p_{k})$, and check that it is a well-defined
∗-homomorphism. Secondly, given a ∗-homomorphism $\phi\colon A\to B$ (which is
contractive), we may take its continuous extension
$\varphi\colon\widehat{A}\to\widehat{B}$ to completions, which is clearly a
∗-homomorphism. Finally, a ∗-homomorphism
$\varphi\colon\widehat{A}\to\widehat{B}$ maps
${\mathcal{P}}(\widehat{A})={\mathcal{P}}(A)$ into
${\mathcal{P}}(\widehat{B})={\mathcal{P}}(B)$, thus restricts to a lattice
homomorphism that sends $0$ to $0$.

Is it possible to generalize this equivalence to the setting of noncommutative
R∗-algebras? Apparently, there is no that beautiful correspondence applicable
to general R∗-algebras. Indeed, there exist distinct R∗-algebras $A,B$ with
$\widehat{A}\cong\widehat{B}$ (see Example 3.4 and Section 5), so we cannot
recover full information about $A$ from $\widehat{A}$. Moreover, in
Proposition 8.8 we prove that there exists a pair of noncommutative
R∗-algebras which are not ∗-isomorphic but the lattices of projections are
orthoisomorphic. Thus the structure of an R∗-algebra cannot be fully recovered
from the lattice of projections. Nonetheless, there do exist several nice
correspondence results that apply to certain classes of R∗-algebras and their
lattices of projections. See Sections 7 and 8.

##  5\. Purely atomic R∗-algebras and inner product spaces

Recall that a von Neumann algebra can be classified into several types by
means of projections. In particular, a von Neumann algebra is said to be
atomic if the least upper bound of minimal projections is the unit. It is a
well-known fact, which follows from e.g. the type decomposition theorem of von
Neumann algebras, that an atomic von Neumann algebra is an
$\ell_{\infty}$-direct sum of type I factors.

Let us define a class of R∗-algebras that is similar to the class of atomic
von Neumann algebras. Since it is hard to consider the greatest lower bound of
an infinite set of projections in an R∗-algebra, we want a definition that
involves only finitely many projections. An _atom_ of an R∗-algebra $A$ is a
minimal projection in ${\mathcal{P}}(A)\setminus\\{0\\}$. Note that
$p\in{\mathcal{P}}(A)$ is an atom if and only if the R∗-algebra $pAp$ is one-
dimensional. We say an R∗-algebra $A$ is _purely atomic_ if every projection
of $A$ is a sum of finitely many atoms. We give a complete description of
purely atomic R∗-algebras.

######  Lemma 5.1.

Let $A$ be an R∗-algebra and $p,q\in{\mathcal{P}}(A)$ atoms. Then the linear
space $pAq$ is at most one-dimensional.

###### Proof.

Let $x,y\in pAq$ be elements with norm one. Then $xx^{*}\in pAp$ is positive
and $\lVert xx^{*}\rVert=\lVert x\rVert^{2}=1$. Since $pAp$ is one-
dimensional, we have $xx^{*}=p$. Similarly, we obtain $yy^{*}=p$ and
$x^{*}x=y^{*}y=q$. It follows that $x,y$ are partial isometries that share
initial and final spaces. Since $xy^{*}\in pAp$, there is a complex number
$\lambda\in{\mathbb{C}}$ with $\lvert\lambda\rvert=1$ such that
$xy^{*}=\lambda p$. It follows that $x=\lambda y$, proving that $x$ and $y$
are linearly dependent. ∎

######  Lemma 5.2.

Let $A$ be an R∗-algebra. Then $A$ is purely atomic if and only if $pAp$ is
finite-dimensional for every $p\in{\mathcal{P}}(A)$.

###### Proof.

If $pAp$ is finite-dimensional for every $p\in{\mathcal{P}}(A)$, it is clear
that $A$ is purely atomic. Suppose that $A$ is purely atomic and let
$p\in{\mathcal{P}}(A)$. Take a collection of atoms $p_{1},\ldots,p_{n}$ with
$p=p_{1}+\cdots+p_{n}$. For each pair $k,l\in\\{1,\ldots,n\\}$, the preceding
lemma implies that $p_{k}Ap_{l}$ is at most one-dimensional. Therefore,
$pAp=\operatorname{span}\\{p_{k}Ap_{l}\mid k,l\in\\{1,\ldots,n\\}\\}$ is
finite-dimensional. ∎

It follows that a purely atomic R∗-algebra is ultramatricial. Indeed, for a
purely atomic R∗-algebra $A$ and a collection $x_{1},\ldots,x_{n}\in A$,
define $p:=\bigvee_{1\leq i\leq n}{l(x_{i})}\vee\bigvee_{1\leq i\leq
n}{r(x_{i})}$, then $pAp$ is finite-dimensional and $x_{1},\ldots,x_{n}\in
pAp$.

Let $V$ be an inner product space. Let $H$ be the completion of $V$. Consider
the collection $F(V)$ of operators $x$ in $B(H)$ with finite rank such that
$r(x)H,l(x)H\subset V$ (namely, such that the ranges of $x$ and $x^{*}$ are
included in $V$). It is easy to see that $F(V)$ is a purely atomic R∗-algebra.

######  Theorem 5.3.

Let $(V_{i})_{i\in I}$ be a family of complex inner product spaces. Then the
R∗-algebra $\bigoplus_{i\in I}F(V_{i})$ is purely atomic. Every purely atomic
R∗-algebra is ∗-isomorphic to some R∗-algebra that can be obtained in this
manner.

###### Proof.

It is clear that the space $\bigoplus_{i\in I}F(V_{i})$ as in the statement of
this theorem forms a purely atomic R∗-algebra. We prove that every purely
atomic R∗-algebra is of this form. The following proof is based on the theory
of von Neumann algebras.

Let $A$ be a purely atomic R∗-algebra. Realize $A$ as a ∗-subalgebra of some
$B(H)$. Take the $\sigma$-weak closure (or equivalently, the $\sigma$-strong
closure) $M$ of $A$ in $B(H)$. Consider ${\mathcal{P}}(A)$ as an increasing
net of projections in $M$. The fact that for every $x\in A$ the projection
$p=l(x)\vee r(x)\in{\mathcal{P}}(A)$ satisfies $pxp=x$ implies that
$\bigvee_{p\in{\mathcal{P}}(A)}p\in{\mathcal{P}}(M)$ is equal to the unit of
$M$. Moreover, for each $p\in{\mathcal{P}}(A)$, the finite-dimensionality of
$pAp$ implies that $pMp=pAp$. It follows that $M$ is an atomic von Neumann
algebra. Thus we may find a collection $(H_{i})_{i\in I}$ of Hilbert spaces
such that $M$ is ∗-isomorphic to the $\ell_{\infty}$-direct sum of
$(B(H_{i}))_{i\in I}$. Since $pAp=pMp$ is finite-dimensional for each
$p\in{\mathcal{P}}(A)$, each operator in $A$ can be thought of as an operator
of finite rank in the $\ell_{\infty}$-direct sum of $(B(H_{i}))_{i\in I}$.
Therefore, we may and do identify $A$ with a ∗-subalgebra of the R∗-algebra
$\bigoplus_{i\in I}F(H_{i})$ ($\subset B(K)$, where $K:=\bigoplus_{i\in
I}H_{i}$).

Let us check that there exists a dense subspace $V_{i}\subset H_{i}$ for each
$i\in I$ such that $A$ is identified with $\bigoplus_{i\in
I}F(V_{i})\,\,(\subset\bigoplus_{i\in I}F(H_{i}))$. Fix $i\in I$. Define
$V_{i}:=\\{h\in H_{i}\mid h\in\operatorname{ran}p\text{ for some
}p\in{\mathcal{P}}(A)\\}$. Let $p_{i}$ be the projection from $K$ onto
$H_{i}$, which lies in the center of $M$. Observe that for each
$p\in{\mathcal{P}}(A)$ we have $pp_{i}\in pMp=pAp$. Thus we see that $V_{i}$
is equal to $\\{h\in H_{i}\mid h\in\operatorname{ran}p\text{ for some
}p\in{\mathcal{P}}(A)\text{ with }p\leq p_{i}\\}$. By the fact that
${\mathcal{P}}(A)$ forms a lattice, we see that $V_{i}$ is actually a linear
subspace of $H_{i}$. Since $A$ is $\sigma$-strongly dense in $M$, we also see
that $V_{i}$ is dense in $H_{i}$. For each $x\in A$, there exist
$i_{1},\ldots,i_{n}\in I$ such that $x=\sum_{k=1}^{n}p_{i_{k}}x$. Set
$q_{k}=p_{i_{k}}(l(x)\vee r(x))$, $1\leq k\leq n$. Then we have
$q_{k}\in{\mathcal{P}}(A)$ and $q_{k}\leq p_{i_{k}}$, thus
$p_{i_{k}}x=q_{k}xq_{k}\in F(V_{i_{k}})$. This shows that
$A\subset\bigoplus_{i\in I}F(V_{i})$. Let $i\in I$ and $p\in
F(V_{i})\,\,(\subset F(H_{i}))$ be a projection of rank one. The definition of
$V_{i}$ implies that there is a projection $q\in{\mathcal{P}}(A)$ with $p\leq
q$. Thus we have $p\in qMq$, but $q\in{\mathcal{P}}(A)$ implies that $p\in
qMq=qAq\subset A$. Since the collection of rank-one projections (atoms) in
$F(V_{i})$ linearly spans $F(V_{i})$, we obtain $F(V_{i})\subset A$. This
implies $\bigoplus_{i\in I}F(V_{i})\subset A$, and the proof is complete. ∎

######  Corollary 5.4.

A purely atomic simple R∗-algebra is ∗-isomorphic to $F(V)$ for some inner
product space $V$.

Therefore, the classification of purely atomic R∗-algebras reduces to that of
R∗-algebras of the form $\bigoplus_{i\in I}F(V_{i})$.

######  Theorem 5.5.

Let $(V_{i})_{i\in I}$, $(W_{j})_{j\in J}$ be two families of complex inner
product spaces. Let $H_{i}$, $K_{j}$ be the completions of $V_{i}$, $W_{j}$,
respectively, for $i\in I$, $j\in J$. Let $\phi\colon\bigoplus_{i\in
I}F(V_{i})\to\bigoplus_{j\in J}F(W_{j})$ be a ∗-isomorphism. Then there exist
a bijection $\tau\colon I\to J$ and a linear surjective isometry $u_{i}\colon
H_{i}\to K_{\tau(i)}$ for each $i\in I$ such that $u_{i}(V_{i})=W_{\tau(i)}$
and $\phi(x)=u_{i}xu_{i}^{-1}\in F(W_{\tau(i)})\,\,(\subset\bigoplus_{j\in
J}F(W_{j}))$ for every $x\in F(V_{i})$, $i\in I$.

###### Proof.

Since $\phi$ is a ∗-isomorphism between two R∗-algebras, it is isometric and
preserves the collection of atoms. Thus $\phi$ maps each connected component
of the set of atoms in $\bigoplus_{i\in I}F(V_{i})$ onto some connected
component of the set of atoms in $\bigoplus_{j\in J}F(W_{j})$. Fix $i_{0}\in
I$. Then the set $S$ of atoms in $F(V_{i_{0}})$ is a connected component of
the set of atoms in $\bigoplus_{i\in I}F(V_{i})$. There exists a unique
$\tau(i_{0})=j_{0}\in J$ such that $\phi(S)$ coincides with the collection of
atoms in $F(W_{j_{0}})$. Set $V:=V_{i_{0}}$, $H:=H_{i_{0}}$, $W:=W_{j_{0}}$
and $K:=K_{j_{0}}$. Assume that $V$ is at least two-dimensional. Observe that
$S$ can be identified with a dense subset of the collection of atoms in
$F(H)$. Since a ∗-isomorphism is isometric, we may apply Wigner’s theorem (see
for example [24]) to show that there exists a linear or conjugate-linear
surjective isometry $u\colon H\to K$ such that $u(V)=W$ and $\phi(p)=upu^{-1}$
for every $p\in S$. Take a pair of self-adjoint operators $a,b\in F(V)$ with
$ab\neq ba$. Since $\phi$ is linear and every self-adjoint operator in $F(V)$
can be written as a finite real-linear combination of $S$, we have

$\phi(ab)=\phi\left(\frac{ab+ba}{2}+i\frac{ab-
ba}{2i}\right)=u\frac{ab+ba}{2}u^{-1}+iu\frac{ab-ba}{2i}u^{-1}.$

This is equal to $uabu^{-1}$ if $u$ is linear, and to $ubau^{-1}$ if $u$ is
conjugate-linear. On the other hand, since $\phi$ is multiplicative, we have

$\phi(ab)=\phi(a)\phi(b)=uau^{-1}ubu^{-1}=uabu^{-1}.$

Since $ab\neq ba$, we conclude that $u$ is actually linear. It follows that
$\phi(x)=uxu^{-1}$ for every $x\in F(V)$. If $V=H$ is one-dimensional then so
is $W=K$. In this case it is clear that for any linear isometry $u\colon H\to
K$ we have $\phi(x)=uxu^{-1}$ for every $x\in F(V)$. It is easy to see that
$\tau\colon I\to J$ constructed in the above manner is a bijection. ∎

In particular, the classification of purely atomic simple R∗-algebras is
equivalent to that of inner product spaces. How rich is the class of inner
product spaces? If we restrict ourselves to inner product spaces with
countable Hamel dimension, then there is nothing interesting: Every inner
product space with countably infinite Hamel dimension is linearly isometric to
$c_{00}$ as a subspace of $\ell_{2}$. (Proof: Consider the Gram–Schmidt
procedure.) On the other hand, we know that complete inner product spaces ($=$
Hilbert spaces) are completely classified by the cardinality of orthonormal
basis.

In contrast, the behavior of incomplete inner product spaces with uncountable
Hamel dimension is mysterious (even in the separable case). A class of inner
product spaces with uncountable Hamel dimension is given by ranges of
operators: If $X$ is a Banach space and $T\colon X\to H$ is a bounded linear
operator, then the range $TX$ of $T$ is a linear subspace of $H$, which forms
an inner product space.

######  Proposition 5.6.

  1. (1)

For every separable Banach space $X$, there exists an injective bounded linear
operator $T\colon X\to\ell_{2}$.

  2. (2)

Let $X_{i}$ be a Banach space, $H_{i}$ be a Hilbert space, and $T_{i}\colon
X_{i}\to H_{i}$ be an injective bounded linear operator with image
$T_{i}X_{i}=V_{i}\subset H_{i}$, $i=1,2$. If the R∗-algebras $F(V_{1})$ and
$F(V_{2})$ are ∗-isomorphic, then $X_{1}$ and $X_{2}$ are isomorphic as Banach
spaces, i.e., there exists a bounded linear bijection between $X_{1}$ and
$X_{2}$.

###### Proof.

This proposition is a consequence of well-known facts (see for example [15,
Proposition 3.1 and Remark 3.14]). For convenience, we give a proof.
(1) Take a sequence $x_{n}$, $n\geq 1$, of elements that is dense in the unit
sphere of $X$. By the Hahn–Banach theorem for each $n\geq 1$ we may take a
linear functional $f_{n}\colon X\to{\mathbb{C}}$ with $\lVert f_{n}\rVert=1$,
$f_{n}(x_{n})=1$. Define the map $T\colon X\to\ell_{2}$ by
$T(x)=(f_{n}(x)/2^{n})_{n\geq 1}$. It is a routine exercise to show that $T$
is well-defined, bounded, linear, and injective.
(2) Take a ∗-isomorphism $\phi\colon F(V_{1})\to F(V_{2})$. By Theorem 5.5 we
may take a linear surjective isometry $u\colon K_{1}\to K_{2}$ such that
$u(V_{1})=V_{2}$ and $\phi(x)=uxu^{-1}$, $x\in F(V_{1})$, where $K_{i}$ is the
completion of $V_{i}$, $i=1,2$. It is easy to see that the mapping
$T_{2}^{-1}\circ u\circ T_{1}\colon X_{1}\to X_{2}$ is a linear bijection with
closed graph. It follows by the closed graph theorem that this mapping is a
continuous bijection (with continuous inverse). ∎

Since there are many isomorphism classes of separable Banach spaces, we see
that there are many simple purely infinite R∗-algebras arising from ranges of
operators. What if we restrict ourselves to ranges of operators in $B(H)$?

######  Proposition 5.7.

Let $x,y\in B(H)$ be injective operators with dense range. Then the
R∗-algebras $F(\operatorname{ran}x)$ and $F(\operatorname{ran}y)$ are
∗-isomorphic if and only if there exist bounded linear bijections $a,b\in
B(H)$ such that $y=axb$.

###### Proof.

This is a corollary to [22, Theorem 3.4]. For convenience we repeat a proof.
Suppose that $F(\operatorname{ran}x)$ and $F(\operatorname{ran}y)$ are
∗-isomorphic. By Theorem 5.5 there exists a unitary operator $u\colon H\to H$
with $u(\operatorname{ran}x)=\operatorname{ran}y$. It follows by the closed
graph theorem that the map $a=y^{-1}\circ u\circ x$ is a bounded linear
bijection on $H$, and $y=uxa^{-1}$. Let $a,b\in B(H)$ be bounded linear
bijections and $y=axb$. Then
$\operatorname{ran}y=\operatorname{ran}(ax)=\operatorname{ran}(a(xx^{*})^{1/2})$.
Let $a(xx^{*})^{1/2}=v\lvert a(xx^{*})^{1/2}\rvert$ be the polar decomposition
of $a(xx^{*})^{1/2}$. Then $v$ is unitary and

$\operatorname{ran}y=\operatorname{ran}(v\lvert
a(xx^{*})^{1/2}\rvert)=v(\operatorname{ran}(\lvert
a(xx^{*})^{1/2}\rvert))=v(\operatorname{ran}((xx^{*})^{1/2}a^{*})=v(\operatorname{ran}x),$

which implies that $F(\operatorname{ran}x)$ is ∗-isomorphic to
$F(\operatorname{ran}y)$. ∎

In the above proposition, we see for example that if in addition $x$ belongs
to the Schatten–von Neumann $p$-ideal for some $1\leq p<\infty$, then so does
$y$. Therefore, there are many examples of mutually nonisometric inner product
spaces that arise as ranges of compact operators in $B(H)$. See [22] and [15]
for more on operator ranges.

Let us show that purely atomic R∗-algebras can also be algebraically
characterized by an Artinian-type condition.

######  Proposition 5.8.

An R∗-algebra $A$ is purely atomic if and only if there exists no strictly
decreasing infinite sequence of projections $p_{1}\geq p_{2}\geq\cdots$ in
$A$, or in other words, there exists no infinite chain of descending principal
right ideals of $A$.

###### Proof.

Every purely atomic R∗-algebra $A$ satisfies the latter condition because for
every $p\in{\mathcal{P}}(A)$ the algebra $pAp$ is finite-dimensional. Suppose
that $A$ is not purely atomic. Consider the collection $S$ of all projections
$p$ with the following property, which is nonempty by the assumption: For any
pairwise orthogonal subprojections $q_{1},q_{2},\ldots,q_{n}$ of $p$ with
$p=q_{1}+q_{2}+\cdots+q_{n}$, at least one of $q_{1},q_{2},\ldots,q_{n}$ is
not an atom. If $p\in S$, then $p$ is not an atom, so there exists a
subprojection $q$ of $p$ such that $q\neq 0\neq p-q$. It is easy to see that
at least one of the two projections $q,p-q$ is in $S$. Therefore, we may
choose a strictly decreasing infinite sequence of projections $p_{1}\geq
p_{2}\geq\cdots$ in $S$. ∎

What if one considers ascending chain condition instead of descending one?

######  Proposition 5.9.

For an R∗-algebra $A$ the following three conditions are equivalent.

  * •

There exists no strictly increasing infinite sequence of projections
$p_{1}\leq p_{2}\leq\cdots$ in $A$, or in other words, there exists no
infinite chain of ascending principal right ideals of $A$.

  * •

There is no infinite family of mutually orthogonal nonzero projections in $A$.

  * •

$A$ is finite-dimensional.

###### Proof.

It is clear that the first and the second conditions are equivalent and led by
the third condition. Let $A$ satisfy the first condition. Then clearly
${\mathcal{P}}(A)$ has a maximal element. Since ${\mathcal{P}}(A)$ is a
lattice it is the unique maximal element of ${\mathcal{P}}(A)$. Moreover, it
is easy to see that the unique maximal element of ${\mathcal{P}}(A)$ must be a
unit of $A$. It follows that there exists no strictly decreasing infinite
sequence of projections $p_{1}\geq p_{2}\geq\cdots$ in $A$, because if there
were then the sequence $1-p_{1},1-p_{2},\ldots$ would be strictly increasing.
Therefore, $A$ is purely atomic and unital, which together with Theorem 5.3
implies the finite-dimensionality of $A$. ∎

##  6\. R∗-algebras in comparison with von Neumann algebras, Baer ∗-rings and
continuous geometries

Again let us recall the theory of von Neumann algebras. In the type
classification of von Neumann algebras, two important concepts appear: abelian
projections and finite projections. Recall that a projection $p$ of a von
Neumann algebra $M$ is finite if every partial isometry $v\in M$ with
$vv^{*}\leq v^{*}v=p$ satisfies $vv^{*}=p$. Finiteness of projections in
R∗-algebras is automatic.

######  Proposition 6.1.

Let $A$ be an R∗-algebra and $v\in A$ be a partial isometry. If $vv^{*}\leq
v^{*}v$, then $vv^{*}=v^{*}v$.

###### Proof.

Assume that $v\in A$ is a partial isometry with $vv^{*}\leq v^{*}v$ and
$vv^{*}\neq v^{*}v$. Then the operator $v$ restricted to $(\ker v)^{\perp}$ is
a proper isometry into $(\ker v)^{\perp}$. It is well-known that a proper
isometry has infinite spectrum. (To prove it, use e.g. the Wold–von Neumann
decomposition theorem and the fact that the unilateral shift has infinite
spectrum.) This contradicts the assumption that $A$ is an R∗-algebra. ∎

In what follows we consider abelian projections. For a von Neumann algebra
$M$, recall that $M$ is of type I if and only if there is a family of abelian
projections whose least upper bound is the unit.

We say a projection $p$ of an R∗-algebra $A$ is _abelian_ if the R∗-algebra
$pAp$ is commutative. Clearly, an atom is an abelian projection. The author is
not aware whether there is a natural class of R∗-algebras that plays a role
similar to that played by the class of type I von Neumann algebras in the
theory of von Neumann algebras.

Let $A$ be an R∗-algebra. Two projections $p,q\in{\mathcal{P}}(A)$ are said to
be _equivalent_ in $A$ (in the sense of Murray–von Neumann, and written $p\sim
q$) if there exists a partial isometry $v\in A$ with $v^{*}v=p$ and
$vv^{*}=q$. This is an equivalence relation in ${\mathcal{P}}(A)$.

######  Lemma 6.2.

Let $A$ be an R∗-algebra. For $x\in A$, $l(x)\sim r(x)$.

###### Proof.

Let $x=v\lvert x\rvert$ be the polar decomposition. By Proposition 2.6 (3),
$v$ is a partial isometry in $A$. Since $vv^{*}=l(x)$ and $v^{*}v=r(x)$, we
have $l(x)\sim r(x)$. ∎

######  Lemma 6.3.

Let $A$ be an R∗-algebra and $p\in{\mathcal{P}}(A)$. Then $p$ is nonabelian if
and only if there exists a pair of nonzero subprojections $p_{1},p_{2}$ of $p$
in $A$ with $p_{1}p_{2}=0$ and $p_{1}\sim p_{2}$.

###### Proof.

If the latter condition is satisfied then the algebra $pAp$ has a partial
isometry $v$ with $v^{*}v=p_{1}\neq p_{2}=vv^{*}$, thus $p$ is nonabelian.
Suppose that $p$ is nonabelian, which means that $pAp$ is noncommutative.
Since $pAp$ is an R∗-algebra, it is spanned by projections in $pAp$. It
follows that there exists a pair of projections
$p_{1},p_{2}\in{\mathcal{P}}(pAp)$ such that $p_{1}p_{2}\neq p_{2}p_{1}$. Thus
$p_{1}p_{2}-p_{2}p_{1}p_{2}$ is not self-adjoint, and we obtain
$x:=(p-p_{2})p_{1}p_{2}=p_{1}p_{2}-p_{2}p_{1}p_{2}\neq 0$. Then $l(x)\leq
p-p_{2}$ and $r(x)\leq p_{2}$, thus $l(x)$ and $r(x)$ are mutually orthogonal
and equivalent. ∎

The _center_ of an R∗-algebra $A$ is the collection $\\{x\in A\mid xy=yx\text{
for all }y\in A\\}$. An element of $A$ is said to be central if it belongs to
the center of $A$.

######  Lemma 6.4.

Let $A$ be an R∗-algebra and $p\in{\mathcal{P}}(A)$. Then $p$ is noncentral if
and only if there exists a pair of nonzero projections
$p_{1},p_{2}\in{\mathcal{P}}(A)$ such that $p_{1}\leq p$, $p_{2}p=0$ and
$p_{1}\sim p_{2}$.

###### Proof.

If $p$ is central then for every partial isometry $v$ with $v^{*}v\leq p$ we
have $vv^{*}=vpv^{*}=pvv^{*}p\leq p$, so there is no pair $p_{1},p_{2}$ with
the property. Assume that $p$ is not central. Since $A$ is spanned by
projections there is a projection $q\in A$ such that $pq\neq qp$. It follows
that $x:=pq(p\vee q-p)=pq-pqp$ is not self-adjoint and thus nonzero. Then
$p_{1}:=l(x)\leq p$ and $p_{2}:=r(x)\leq p\vee q-p$ satisfy the desired
property. ∎

Recall that a ring $R$ is _Baer_ if the right annihilator of every subset of
$R$ is generated by an idempotent as a principal right ideal, that is, if for
every $X\subset R$ there exists an idempotent $e\in R$ (i.e., $e=e^{2}$) such
that $\\{r\in R\mid Xr=0\\}=eR$. A von Neumann algebra is a Baer ∗-ring. The
structure of Baer ∗-rings is studied by Kaplansky [35] (see also [4]) as a
purely algebraic generalization of von Neumann algebras. Which R∗-algebras are
Baer? Below we give the answer (Proposition 6.8). Recall that a lattice
${\mathcal{P}}$ is said to be _complete_ if any family $(p_{i})_{i\in I}$ of
elements in ${\mathcal{P}}$ has the least upper bound $\bigvee_{i\in I}p_{i}$
(upward complete) and the greatest lower bound $\bigwedge_{i\in I}p_{i}$
(downward complete). First let us recall a fact, which is actually true for
general ∗-regular rings [4, Proposition 4.1], and sketch the proof.

######  Lemma 6.5.

For an R∗-algebra $A$ the following conditions are equivalent.

  1. (1)

$A$ is Baer.

  2. (2)

${\mathcal{P}}(A)$ is complete.

  3. (3)

${\mathcal{P}}(A)$ is upward complete.

###### Proof.

Recall that every principal right ideal of a ∗-regular ring is generated by a
projection.
$(1)\Rightarrow(3)$ Assume that $A$ is Baer. It clearly follows that $A$ is
unital. Let $(p_{i})_{i\in I}$ be a family of elements in ${\mathcal{P}}(A)$.
Let the right annihilator of $(p_{i})_{i\in I}$ be equal to $qA$ with
$q\in{\mathcal{P}}(A)$. Then we may check that $1-q$ is the least upper bound
of $(p_{i})_{i\in I}$, hence ${\mathcal{P}}(A)$ is upward complete.
$(3)\Rightarrow(2)$ If ${\mathcal{P}}(A)$ is upward complete then the least
upper bound of ${\mathcal{P}}(A)$ needs to be a unit of $A$, hence $A$ is
unital. Then ${\mathcal{P}}(A)$ is downward complete, too. Indeed, for each
family $(p_{i})_{i\in I}$ of projections the greatest lower bound is
determined by $1-\bigvee_{i\in I}(1-p_{i})$.
$(2)\Rightarrow(1)$ For a subset $X\subset R$, the right annihilator of $X$ is
the principal right ideal generated by $\bigwedge_{x\in X}(1-r(x))$. ∎

It is also common to consider completeness with respect to countably infinite
collections. A lattice ${\mathcal{P}}$ is said to be _$\sigma$ -complete_ if
any countable family $(p_{i})_{i\in I}$ of ${\mathcal{P}}$ has the least upper
bound $\bigvee_{i\in I}p_{i}$ (upward $\sigma$-complete) and the greatest
lower bound $\bigwedge_{i\in I}p_{i}$ (downward $\sigma$-complete). Recall
that these concepts are important in measure theory. A $\sigma$-field is a
$\sigma$-complete Boolean algebra, and for a measure space
$(X,\mathcal{F},\mu)$ the collection $\\{A\in\mathcal{F}\mid\mu(A)<\infty\\}$
forms a downward $\sigma$-complete generalized Boolean algebra. Notice that a
$\sigma$-complete generalized Boolean algebra need not be Boolean. For
example, for an uncountable set $X$, consider the generalized field of sets
formed of all (at most) countable subsets of $X$. It should be also mentioned
that a $\sigma$-complete Boolean algebra need not be orthoisomorphic to a
$\sigma$-field (see [25, Chapter 40]). Let us characterize R∗-algebras whose
lattice of projections are upward/downward $\sigma$-complete.

######  Proposition 6.6.

For an R∗-algebra $A$ the following conditions are equivalent.

  1. (1)

${\mathcal{P}}(A)$ is $\sigma$-complete.

  2. (2)

${\mathcal{P}}(A)$ is upward $\sigma$-complete.

  3. (3)

There exist a $\sigma$-complete generalized Boolean algebra ${\mathcal{S}}$
and a finite-dimensional R∗-algebra $B$ such that $A$ is ∗-isomorphic to
$R({\mathcal{S}})\oplus B$.

###### Proof.

$(3)\Rightarrow(1)$ Easy. $(1)\Rightarrow(2)$ This is trivial.
$(2)\Rightarrow(3)$ Assume ${\mathcal{P}}(A)$ is upward $\sigma$-complete.
First we claim that there exists no sequence $(p_{n})_{n\geq 1}$ of pairwise
orthogonal nonzero nonabelian projections in $A$. Indeed, if there exists such
a sequence, then Lemma 6.3 implies that for each $n\geq 1$ there exist
mutually equivalent nonzero subprojections $q_{n},r_{n}\leq p_{n}$ with
$q_{n}r_{n}=0$. Take a partial isometry $v_{n}\in A$ with
$v_{n}^{*}v_{n}=q_{n},v_{n}v_{n}^{*}=r_{n}$. Note that
$\\{q_{n},v_{n},v_{n}^{*},r_{n}\\}$ forms a system of $2\times 2$ matrix
units. Set $q=\bigvee_{n\geq 1}q_{n}$ and

$r=\bigvee_{n\geq
1}\left((1-n^{-1})q_{n}+\sqrt{(1-n^{-1})n^{-1}}(v_{n}+v_{n}^{*})+n^{-1}r_{n}\right)$

Then we have $p_{n}qp_{n}=q_{n}$ and

$p_{n}rp_{n}=(1-n^{-1})q_{n}+\sqrt{(1-n^{-1})n^{-1}}(v_{n}+v_{n}^{*})+n^{-1}r_{n}$

for each $n\geq 1$. From this, it is not difficult to see that the sum $q+r$
has infinite spectrum, which contradicts Theorem 2.1. Thus we get the desired
claim.

Consider the collection $S$ of all projections in ${\mathcal{P}}(A)$ that
cannot be written as a sum of finitely many mutually orthogonal abelian
projections. We prove that for each $p\in S$ there exists a subprojection $e$
of $p$ such that $e\in S$ and $p-e$ is nonabelian. Let $p\in S$. Since $p$ is
clearly nonabelian, there is a pair of subprojections
$p^{\prime},p^{\prime\prime}\in{\mathcal{P}}(A)$ of $p$ such that
$p^{\prime}p^{\prime\prime}=0$ and $p^{\prime}\sim p^{\prime\prime}$. If
$p^{\prime}$ is not abelian, then it is clear that both $p^{\prime}$ and
$p-p^{\prime}$ are nonabelian and at least one of $p^{\prime},p-p^{\prime}$ is
in $S$, which leads to the desired conclusion. If $p^{\prime}$ is abelian,
then it is clear that $p^{\prime}+p^{\prime\prime}$ is nonabelian and does not
belong to $S$, and hence $p-(p^{\prime}+p^{\prime\prime})$ is in $S$, which
also leads to the desired conclusion. Therefore, if $S\neq\emptyset$, then we
may inductively take a sequence of mutually orthogonal nonzero nonabelian
projections in $A$, which contradicts the above claim. It follows that
$S=\emptyset$.

Let $p\in{\mathcal{P}}(A)$ be an abelian projection. We prove that there
exists a subprojection $f(p)\leq p$ such that

  1. (a)

there exists a collection $e_{1},e_{2},\ldots,e_{n}$ of mutually orthogonal
atoms in $A$, none of which is central in $A$, such that
$f(p)=e_{1}+\cdots+e_{n}$, and

  2. (b)

$p-f(p)$ is a central abelian projection in $A$.

To prove it, take a maximal family $(p_{i})_{i\in I}$ of pairwise orthogonal
nonzero subprojections of $p$ with the following property: For each $i\in I$
there exists a projection $q_{i}\in{\mathcal{P}}(A)$ such that $p_{i}q_{i}=0$
and $q_{i}\sim p_{i}$ (the existence of a maximal family with this property is
ensured by Zorn’s lemma). Then $(q_{i})_{i\in I}$ is pairwise orthogonal.
Indeed, if $q_{i}q_{j}\neq 0$ for some $i\neq j\in I$ then
$l(q_{i}q_{j})\,\,(\leq q_{i})$ is equivalent to $r(q_{i}q_{j})\,\,(\leq
q_{j})$, which must imply that some nonzero subprojection of $p_{i}$ is
equivalent to a subprojection of $p_{j}$, and this contradicts the
commutativity of $pAp$. Similarly, we have $p_{i}q_{j}=0$ if $i\neq j\in I$.
It follows that $(p_{i}+q_{i})_{i\in I}$ is a family of pairwise orthogonal
nonzero nonabelian projections in $A$, which forces $I$ to be a finite set.
Set $f(p):=\sum_{i\in I}p_{i}$. The above reasoning also shows that there is
$q\in{\mathcal{P}}(A)$ with $f(p)\sim q$ and $f(p)q=0$. Take a partial
isometry $v$ such that $v^{*}v=f(p)$ and $vv^{*}=q$. Assume that $f(p)Af(p)$
is infinite-dimensional. Proposition 5.9 implies that there is a sequence of
pairwise orthogonal nonzero subprojections $(p_{n})_{n\geq 1}$ of $f(p)$. It
follows that the sequence $(p_{n}+vp_{n}v^{*})_{n\geq 1}$ is a pairwise
orthogonal sequence of nonzero nonabelian projections, which contradicts the
above claim. Therefore, we see that $f(p)Af(p)$ is finite-dimensional. It is
now clear that $f(p)$ is a sum of finitely many pairwise orthogonal atoms none
of which is central in $A$. By Lemma 6.4 combined with the maximality of the
family $(p_{i})_{i\in I}$, $p-f(p)$ is central in $A$.

Let $p\in{\mathcal{P}}(A)$. Then $p$ is the sum of some finitely many pairwise
orthogonal abelian projections. It follows from the preceding paragraph that
there is a subprojection $f(p)$ of $p$ with (a) and (b). A moment’s reflection
shows that such an $f(p)$ is unique and that $A$ can be decomposed into the
direct sum of two unions of increasing nets of R∗-algebras
$(f(p)Af(p))_{p\in{\mathcal{P}}(A)}$ and
$((p-f(p))A(p-f(p)))_{p\in{\mathcal{P}}(A)}$. Clearly, the directed union of
the family $(f(p)Af(p))_{p\in{\mathcal{P}}(A)}$ is purely atomic and that of
$((p-f(p))A(p-f(p)))_{p\in{\mathcal{P}}(A)}$ is abelian. It follows that there
exist a $\sigma$-complete generalized Boolean algebra ${\mathcal{S}}$ and a
purely atomic R∗-algebra $B$, none of whose atoms is central, such that $A$ is
∗-isomorphic to $R({\mathcal{S}})\oplus B$.

Lastly, let us prove that $B$ is actually finite-dimensional. By Theorem 5.3
$B$ is ∗-isomorphic to $\bigoplus_{j\in J}F(V_{j})$ for some family of inner
product spaces $(V_{j})_{j\in J}$. Since none of the atoms of $B$ is abelian,
none of $(V_{j})_{j\in J}$ is one-dimensional. Assume that $B$ is infinite-
dimensional. Then it is an easy exercise to find an infinite family of
pairwise orthogonal nonzero nonabelian projections in $B$, which contradicts
the above claim. This means that $B$ is finite-dimensional, which completes
the proof. ∎

######  Proposition 6.7.

For an R∗-algebra $A$, ${\mathcal{P}}(A)$ is downward $\sigma$-complete if and
only if there exist a downward $\sigma$-complete generalized Boolean algebra
${\mathcal{S}}$ and a purely atomic R∗-algebra $B$ such that $A$ is
∗-isomorphic to $R({\mathcal{S}})\oplus B$.

###### Proof.

Assume that ${\mathcal{P}}(A)$ is downward $\sigma$-complete. Let
$p\in{\mathcal{P}}(A)$. Then $pAp$ is upward $\sigma$-complete. Indeed, for a
countable family of projections $(p_{i})_{i\in I}$ in $pAp$, its least upper
bound is equal to $p-\bigwedge_{i\in I}(p-p_{i})$. Thus we may apply the proof
of the preceding theorem verbatim to show that $A$ can be decomposed into the
direct sum of two unions of increasing nets of R∗-algebras
$(f(p)Af(p))_{p\in{\mathcal{P}}(A)}$ and
$((p-f(p))A(p-f(p)))_{p\in{\mathcal{P}}(A)}$, where $f(p)\leq p$ satisfies (a)
and (b). It follows that there exist a downward $\sigma$-complete generalized
Boolean algebra ${\mathcal{S}}$ and a purely atomic R∗-algebra $B$ (none of
whose atom is central) such that $A$ is ∗-isomorphic to
$R({\mathcal{S}})\oplus B$. It is easy to prove the converse implication. ∎

The characterization of Baer R∗-algebras is a special case of Proposition 6.6.

######  Proposition 6.8.

Let $A$ be an R∗-algebra. Then $A$ is Baer if and only if there exist a
complete Boolean algebra ${\mathcal{S}}$ and a finite-dimensional R∗-algebra
$B$ such that $A$ is ∗-isomorphic to $R({\mathcal{S}})\oplus B$.

###### Proof.

Use Lemma 6.5 and apply Proposition 6.6. (Beware of the fact that a Baer ring
is unital.) ∎

######  Proposition 6.9.

For an R∗-algebra $A$, ${\mathcal{P}}(A)$ is downward complete if and only if
there exist a downward complete generalized Boolean algebra ${\mathcal{S}}$
and a purely atomic R∗-algebra $B$ such that $A$ is ∗-isomorphic to
$R({\mathcal{S}})\oplus B$.

Recall that a complete complemented modular lattice ${\mathcal{P}}$ is called
a (possibly reducible) _continuous geometry_ if it satisfies

$\left(\bigvee_{i\in I}p_{i}\right)\wedge q=\bigvee_{i\in I}(p_{i}\wedge q)$

and

$\left(\bigwedge_{i\in I}p_{i}\right)\vee q=\bigwedge_{i\in I}(p_{i}\vee q)$

for any linearly ordered family $(p_{i})_{i\in I}$ of elements in
${\mathcal{P}}$ and $q\in{\mathcal{P}}$. Von Neumann studied continuous
geometries in [43, Parts I and III]. Kaplansky [37] proved that every
orthocomplemented complete modular lattice is actually a continuous geometry.
Thus we obtain:

######  Corollary 6.10.

Let $A$ be an R∗-algebra. The projection lattice ${\mathcal{P}}(A)$ is a
continuous geometry if and only if the equivalent conditions of Proposition
6.8 hold.

We may prove it without resorting to Kaplansky’s theorem.

###### Proof.

If ${\mathcal{P}}(A)$ is a continuous geometry, then ${\mathcal{P}}(A)$ is
complete, so $A$ is of the form of Proposition 6.8. It is easy to check that
the projection lattice of an R∗-algebra of this form is a continuous geometry.
∎

Therefore, R∗-algebras themselves do not provide interesting examples of
continuous geometries. However, it should be worth mentioning that two
important examples of continuous geometries can be constructed from
sequentially ultramatricial R∗-algebras.

For a von Neumann algebra $M$ of finite type, the projection lattice of $M$
forms a continuous geometry. (It is a remarkable fact that ${\mathcal{P}}(M)$
is identified with the projection lattice of the ∗-regular algebra
$\mathcal{U}(M)$ formed of all affiliated operators relative to $M$. Observe
that for the algebra $A=\mathcal{U}(M)$, which can be considered as an algebra
of unbounded operators on a Hilbert space, the equivalence of Theorem 2.1 is
far from true.) In particular, the projection lattice of the AFD II1 factor
${\mathcal{R}}$ is a continuous geometry. Recall that ${\mathcal{R}}$ arises
as the $\sigma$-weak closure of the R∗-algebra $\bigotimes_{k\geq
1}{\mathbb{M}}_{n_{k}}$ (represented on a Hilbert space through the GNS-
construction with respect to the trace), and it does not depend on the choice
of integers $n_{k}\geq 2$.

On the other hand, von Neumann also considered the completion of
$\bigotimes_{k\geq 1}{\mathbb{M}}_{n_{k}}$ with respect to a metric called the
rank metric, and proved that the lattice of projections forms a continuous
geometry that does not depend on the choice of $(n_{k})_{k\geq 1}$. He also
proved that the lattice obtained in this way is different from any projection
lattice of a finite von Neumann algebra. See [44] for the details.

##  7\. Classification of sequentially ultramatricial R∗-algebras

In this section, we give a classification result on locally finite-dimensional
R∗-algebras with countable Hamel dimension. Clearly, an R∗-algebra has a
countable Hamel basis if and only if it is countably generated as a ∗-algebra.
It follows that the class of locally finite-dimensional R∗-algebras with
countable Hamel dimension coincides with that of sequentially ultramatricial
R∗-algebras.

It was shown by Bratteli [11] (after Glimm’s classification of UHF algebras
[27]) that the classification problem of separable AF C∗-algebras is
equivalent to that of sequentially ultramatricial R∗-algebras. The
classification of sequentially ultramatricial R∗-algebras was completed by
Elliott in the seminal paper [19].

There are at least two formulations of this result: One is by dimension local
semigroups, and the other is by $K_{0}$-groups. Although the former
formulation appears in Elliott’s original paper [19], it is seemingly less
commonly used among specialists nowadays (indeed, it does not appear
explicitly in standard textbooks like [16, 7]). Let us briefly recall both of
these two formulations because the former formulation will be essentially used
in the proof of Theorem 7.5.

Let $A$ be an R∗-algebra. Let ${\mathcal{E}}(A)$ denote the collection of
idempotents, i.e., ${\mathcal{E}}(A):=\\{e\in A\mid e^{2}=e\\}$. For
$e,f\in{\mathcal{E}}(A)$, we write $e\sim_{0}f$ if there exist $x,y\in A$ with
$xy=e,yx=f$. Then $\sim_{0}$ is an equivalence relation. Set
$d_{0}(A):={\mathcal{E}}(A)/{\sim_{0}}$. For $e\in{\mathcal{E}}(A)$, let
$[e]_{0}\in d_{0}(A)$ denote the equivalence class with $e\in[e]_{0}$. We
define the partial binary operation $+$ in $d_{0}(A)$ in the following manner
(_partial_ in the sense that the operation is defined only for a certain
collection of pairs of elements). For $[e]_{0},[f]_{0}\in d_{0}(A)$,
$[e]_{0}+[f]_{0}$ is defined if and only if there are
$e^{\prime}\in[e]_{0},f^{\prime}\in[f]_{0}$ with
$e^{\prime}f^{\prime}=f^{\prime}e^{\prime}=0$, and in that case, we define
$[e]_{0}+[f]_{0}=[e^{\prime}]_{0}+[f^{\prime}]_{0}:=[e^{\prime}+f^{\prime}]_{0}$.
This is actually a well-defined partially defined binary operation.

The pair $(d_{0}(A),+)$ can also be obtained by means of projections. Consider
the set $d(A):={\mathcal{P}}(A)/{\sim}$, where the equivalence relation $\sim$
means the Murray–von Neumann equivalence. For $p\in{\mathcal{P}}(A)$, let
$[p]\in d(A)$ denote the equivalence class with $p\in[p]$. We define the
partial operation $+$ in $d(A)$ in the following manner. For $[p],[q]\in
d(A)$, $[p]+[q]$ is defined if and only if there are
$p^{\prime}\in[p],q^{\prime}\in[q]$ with $p^{\prime}q^{\prime}=0$, and in that
case, we define $[p]+[q]=[p^{\prime}]+[q^{\prime}]:=[p^{\prime}+q^{\prime}]$.
Then this partial operation is well-defined, and moreover, by a routine
argument as in [35] or [7, Section 4], $(d(A),+)$ is isomorphic to
$(d_{0}(A),+)$ as sets endowed with partially defined operations. The
_dimension local semigroup_ of $A$ is the pair $(d(A),+)$.

Below we give minimal information to obtain the $K_{0}$-group of an
R∗-algebra. See for example [7, Chapter III] for more details about this.
(Note that an R∗-algebra $A$ is a local C∗-algebra in the sense of [7],
namely, $A$ is a pre-C∗-algebra that is closed under holomorphic functional
calculus.) For an R∗-algebra $A$, we consider the R∗-algebra
${\mathbb{M}}_{\infty}(A)$, which can be considered as the collection of
${\mathbb{N}}\times{\mathbb{N}}$ matrices with finitely many nonzero entries
in $A$. For $n\geq 1$, we identify ${\mathbb{M}}_{n}(A)$ with the collection
of all operators in ${\mathbb{M}}_{\infty}(A)$ whose nonzero entries are in
the $n\times n$ upper-left corner. In that way we identify
${\mathbb{M}}_{\infty}(A)$ with the directed union of
$({\mathbb{M}}_{n}(A))_{n\geq 1}$.

Observe that for every pair of projections $p,q\in{\mathbb{M}}_{\infty}(A)$
there exists a pair of mutually orthogonal projections
$p^{\prime},q^{\prime}\in{\mathbb{M}}_{\infty}(A)$ with $p\sim p^{\prime}$,
$q\sim q^{\prime}$. Thus the operation $+$ on the set
$d({\mathbb{M}}_{\infty}(A))$ is defined for every pair of projections. We
define $V(A):=d({\mathbb{M}}_{\infty}(A))$, which is an abelian semigroup
endowed with operation $+$. The $K_{0}$-group $K_{0}(A)$ of $A$ is defined as
the Grothendieck group of $(V(A),+)$.

In the setting of nonunital C∗-algebras, it is mathematically more natural to
define the $K_{0}$-group using the $K_{0}$-group of the unitization (see [7,
Section 5]). However, the above naive definition works in the setting of
R∗-algebras: Recall that the projection lattice of an R∗-algebra (in
particular ${\mathbb{M}}_{\infty}(A)$) forms a net, which is an approximate
identity. Therefore, by [7, Proposition 5.5.5], our definition of the
$K_{0}$-group coincides with the usual definition.

Recall that a unital (C∗-, or more general normed) algebra is said to have
_stable rank one_ if the collection of invertible elements is dense. Clearly,
a unital R∗-algebra and its completion have stable rank one. It follows by [7,
Proposition 6.5.1] that $V(A)$ is a cancellation semigroup, which means that
there is a canonical injection from $V(A)$ into $K_{0}(A)$. It is easy to see
that the same is true for nonunital R∗-algebras. The image of this injection
is written as $K_{0}(A)_{+}$. Then $K_{0}(A)_{+}$ forms a cone of $K_{0}(A)$
which makes $K_{0}(A)$ an ordered group. The _scale_ of $K_{0}(A)$ is defined
as the subset $\Sigma(A):=\\{[p]\mid
p\in{\mathcal{P}}(A)\,\,(={\mathcal{P}}({\mathbb{M}}_{1}(A))\subset{\mathcal{P}}({\mathbb{M}}_{\infty}(A)))\\}$
of $K_{0}(A)_{+}$. We are now in the position of stating Elliott’s theorem
(see [19], [7, Section 7] and [16, Chapter IV]).

######  Theorem 7.1.

Let $A,B$ be two sequentially ultramatricial R∗-algebras. The following are
equivalent.

  1. (1)

$A$ is ∗-isomorphic to $B$.

  2. (2)

$d(A)$ is isomorphic to $d(B)$ as sets endowed with partially defined binary
operations.

  3. (3)

There exists an ordered group isomorphism from $K_{0}(A)$ onto $K_{0}(B)$ that
sends $\Sigma(A)$ onto $\Sigma(B)$.

  4. (4)

$\widehat{A}$ is ∗-isomorphic to $\widehat{B}$.

The equivalence $A\cong B\iff\widehat{A}\cong\widehat{B}$ means that distinct
sequentially ultramatricial R∗-algebras have distinct completions (separable
AF C∗-algebras), and moreover, for each separable AF C∗-algebra any pair of
dense R∗-subalgebras with countable Hamel dimension is mutually ∗-isomorphic.
Recall that the same is true for the class of commutative R∗-algebras by
Stone’s theorem. Theorem 7.1 is invalid for ultramatricial R∗-algebras with
uncountable Hamel dimension. For example, it is easy to see that both

$(\widehat{F(\ell_{2})},d(F(\ell_{2})),K_{0}(F(\ell_{2})),K_{0}(F(\ell_{2}))_{+},\Sigma(F(\ell_{2})))$

and

$(\widehat{{\mathbb{M}}_{\infty}},d({\mathbb{M}}_{\infty}),K_{0}({\mathbb{M}}_{\infty}),K_{0}({\mathbb{M}}_{\infty})_{+},\Sigma({\mathbb{M}}_{\infty}))$

are isomorphic to

$(K(\ell_{2}),{\mathbb{Z}}_{\geq 0},{\mathbb{Z}},{\mathbb{Z}}_{\geq
0},{\mathbb{Z}}_{\geq 0}).$

In what follows we give a lattice theoretic condition that is equivalent to
the conditions of Theorem 7.1. We say two projections $p,q$ of an R∗-algebra
$A$ are _perspective_ if there exists a projection $r\in{\mathcal{P}}(A)$ such
that $p\wedge r=0=q\wedge r$ and $p\vee r=q\vee r$. We write $p\approx q$ if
there is a collection $p_{0}:=p,p_{1},\ldots,p_{n}:=q$ of finitely many
projections in $A$ such that $p_{j-1}$ and $p_{j}$ are perspective,
$j=1,\ldots,n$.

######  Lemma 7.2.

Let $A$ be an R∗-algebra. Let $p,q\in{\mathcal{P}}(A)$. The following are
equivalent.

  * •

$p\sim q$.

  * •

$p\vee q-p\sim p\vee q-q$.

  * •

$p\approx q$.

###### Proof.

By restricting our attention to the unital R∗-subalgebra $(p\vee q)A(p\vee
q)$, and applying [7, Propositions 6.4.1 and 6.5.1], we see the equivalence of
the first two items. Suppose that the first two items hold. There is a partial
isometry $v\in A$ such that $vv^{*}=p\vee q=v^{*}v$ and $vpv^{*}=q$. We may
consider $v$ as a unitary operator in $(p\vee q)A(p\vee q)$. Since $v$ has
finite spectrum, there is a (unique) positive operator $a\in(p\vee q)A(p\vee
q)$ such that $\lVert a\rVert<2\pi$ and $v=e^{ia}$. Fix a sufficiently large
$n$. It is an elementary exercise to show that
$p_{j-1}:=e^{ia(j-1)/n}pe^{-ia(j-1)/n}$ and $p_{j}:=e^{ia{j}/n}pe^{-iaj/n}$
are perspective (set e.g. $r:=p_{j-1}\vee p_{j}-p_{j-1}$), $j=1,\ldots,n$,
showing that $p=p_{0}\approx p_{n}=e^{ia}pe^{-ia}=q$.

Let $r\in{\mathcal{P}}(A)$ be a projection such that $p\wedge r=0=q\wedge r$
and $p\vee r=q\vee r$. Then it is easy to see that the operator $x=q(p\vee
r-r)p$ satisfies $r(x)=p$ and $l(x)=q$. By Lemma 6.2, we have $p\sim q$. Since
$\sim$ is transitive, we see that $p\approx q$ implies $p\sim q$. ∎

As a corollary, we see that a unital R∗-algebra has some special properties as
a regular ring.

######  Corollary 7.3.

A unital R∗-algebra $A$ is directly finite (i.e., if $x,y\in A$ and $xy=1$,
then $yx=1$) and unit regular (i.e., for every $x\in A$ there exists an
invertible $y\in A$ with $xyx=x$).

###### Proof.

Let $x\in A$. By the preceding lemma combined with Lemma 6.2, we see that
$1-l(x)\sim 1-r(x)$. Take a partial isometry $v\in A$ with $vv^{*}=1-l(x)$ and
$v^{*}v=1-r(x)$. A direct calculation shows that $x^{\dagger}+v^{*}$ has
inverse $x+v$ and $x(x^{\dagger}+v^{*})x=xx^{\dagger}x=x$. Thus $A$ is unit
regular. Recall that unit regularity implies direct finiteness. Indeed, if $A$
is unit regular and $xy=1$, take invertible $a\in A$ with $xax=x$, then
$xa=xaxy=xy=1$, thus $x=a^{-1}$ and $y=a$, which implies $yx=1$. ∎

######  Lemma 7.4.

Let $A$ be an R∗-algebra. Let $p,q\in{\mathcal{P}}(A)$. Then $[p]+[q]$ is
defined in $d(A)$ if and only if there exist projections
$p^{\prime},q^{\prime}\in{\mathcal{P}}(A)$ with $p\sim p^{\prime}$, $q\sim
q^{\prime}$ and $p^{\prime}\wedge q^{\prime}=0$, and moreover, if there exist
such $p^{\prime},q^{\prime}$, then $[p]+[q]=[p^{\prime}\vee q^{\prime}]$.

###### Proof.

Assume that $p^{\prime},q^{\prime}\in{\mathcal{P}}(A)$ satisfy $p\sim
p^{\prime}$, $q\sim q^{\prime}$ and $p^{\prime}\wedge q^{\prime}=0$. Then
$x:=q^{\prime}(p^{\prime}\vee q^{\prime}-p^{\prime})$ satisfies
$l(x)=q^{\prime}$ and $r(x)=p^{\prime}\vee q^{\prime}-p^{\prime}$, which
implies that $q^{\prime}\sim p^{\prime}\vee q^{\prime}-p^{\prime}$. Thus we
obtain $q\sim p^{\prime}\vee q^{\prime}-p^{\prime}$, which together with
$p\sim p^{\prime}$ implies that $[p]+[q]$ is defined and equal to
$[p^{\prime}+(p^{\prime}\vee q^{\prime}-p^{\prime})]=[p^{\prime}\vee
q^{\prime}]$. The converse implication is trivial. ∎

######  Theorem 7.5.

Let $A,B$ be sequentially ultramatricial R∗-algebras. If ${\mathcal{P}}(A)$ is
lattice isomorphic to ${\mathcal{P}}(B)$, then $A$ is ∗-isomorphic to $B$.

###### Proof.

By Theorem 7.1 $(2)\Rightarrow(1)$ it suffices to show that we may construct
$(d(A),+)$ only from the lattice structure of ${\mathcal{P}}(A)$. Observe that
the relation $\approx$ is defined by means of lattice operations. It follows
by Lemma 7.2 that the Murray–von Neumann equivalence relation is recovered
from the lattice structure. Moreover, the preceding lemma implies that for a
pair $p,q$ of projections, $[p]+[q]$ is defined if and only if there exist
$p^{\prime},q^{\prime}\in{\mathcal{P}}(A)$ such that $p^{\prime}\sim p$,
$q^{\prime}\sim q$, and $p^{\prime}\wedge q^{\prime}=0$, and in that case, we
also have $[p]+[q]=[p^{\prime}\vee q^{\prime}]$. This implies that the local
semigroup structure of $d(A)$ can be fully recovered only from the lattice
structure of ${\mathcal{P}}(A)$. Thus the proof is complete. ∎

In Section 8, we will see that lattice isomorphisms of projection lattices of
certain R∗-algebras can be described more explicitly by means of ring
isomorphisms, as a consequence of von Neumann’s theorem (Theorem 8.1).

More results on separable AF C∗-algebras are summarized in [16, Chapters III,
IV]. Mundici and Panti [42] studied separable AF C∗-algebras $\widehat{A}$
such that $d(A)$ forms a lattice.

##  8\. Von Neumann, Dye, Gleason

In this section, we give several results concerning mappings on the lattice of
projections of an R∗-algebra.

First, we give a consequence of von Neumann’s result. A _ring isomorphism_ is
an additive and multiplicative bijection. It is not necessarily linear.
Indeed, there exist many nonlinear ring automorphisms of ${\mathbb{C}}$.

Let $A,B$ be R∗-algebras. Recall that each principal right ideal of $A$ can be
written as $pA$ for a unique projection $p\in{\mathcal{P}}(A)$. Therefore, if
$\phi\colon A\to B$ is a ring isomorphism, then there exists a unique lattice
isomorphism $\psi\colon{\mathcal{P}}(A)\to{\mathcal{P}}(B)$ such that

$\phi(pA)=\psi(p)B$

for all $p\in{\mathcal{P}}(A)$. (Note that this equation implies
$l(\phi(p))=\psi(p)$.) Von Neumann’s theorem states that the converse also
holds under a certain condition.

######  Theorem 8.1 (von Neumann’s theorem for R∗-algebras).

Let $A,B$ be unital R∗-algebras, and $n\geq 3$ an integer. If
$\psi\colon{\mathcal{P}}({\mathbb{M}}_{n}(A))\to{\mathcal{P}}(B)$ is a lattice
isomorphism, then there exists a ring isomorphism
$\phi\colon{\mathbb{M}}_{n}(A)\to B$ such that

$\phi(p{\mathbb{M}}_{n}(A))=\psi(p)B$

for all $p\in{\mathcal{P}}({\mathbb{M}}_{n}(A))$.

###### Proof.

This is a special case of [43, Theorem II.4.2]. ∎

The uniqueness of $\phi$ in the above theorem can be obtained from the
proposition below.

######  Lemma 8.2.

Let $A\subset B(H)$ be an R∗-algebra and $e\in A$ an idempotent, i.e.,
$e^{2}=e$. If $r(e)\leq l(e)$, then $e$ is a projection.

###### Proof.

By $e^{2}=e$, we have $l(e)=ee^{\dagger}=e^{2}e^{\dagger}=el(e)$. Since
$r(e)\leq l(e)$, we have $el(e)=e$. Thus $e=l(e)$ is a projection. ∎

######  Proposition 8.3.

Let $A$ be an R∗-algebra, and let $n\geq 2$ be an integer. If
$\phi\colon{\mathbb{M}}_{n}(A)\to{\mathbb{M}}_{n}(A)$ is a ring automorphism
with

$\phi(p{\mathbb{M}}_{n}(A))=p{\mathbb{M}}_{n}(A)$

for all $p\in{\mathcal{P}}({\mathbb{M}}_{n}(A))$, then $\phi$ is the identity
mapping on ${\mathbb{M}}_{n}(A)$.

###### Proof.

If $A$ is unital, then the desired conclusion is a special case of [43,
Theorem II.4.1]. Let $p\in{\mathcal{P}}({\mathbb{M}}_{n}(A))$. It is clear
that $\phi(p)$ is an idempotent. The equation
$\phi(p{\mathbb{M}}_{n}(A))=p{\mathbb{M}}_{n}(A)$ implies $l(\phi(p))=p$.
Moreover, for every $q\in{\mathcal{P}}({\mathbb{M}}_{n}(A))$ with $pq=0$, we
have
$\phi(p)q{\mathbb{M}}_{n}(A)=\phi(p)\phi(q{\mathbb{M}}_{n}(A))=\phi(pq{\mathbb{M}}_{n}(A))=\\{0\\}$.
Therefore, we have $\phi(p)q=0$ for every
$q\in{\mathcal{P}}({\mathbb{M}}_{n}(A))$ with $pq=0$. This means
$r(\phi(p))\leq p=l(\phi(p))$. The preceding lemma implies that $\phi(p)$ is a
projection and hence $\phi(p)=l(\phi(p))=p$. It follows that $\phi$ restricts
to a ring automorphism of ${\mathbb{M}}_{n}(qAq)$ for every
$q\in{\mathcal{P}}(A)$. Using the proposition for the unital case, we see that
$\phi(x)=x$ for every $x\in{\mathbb{M}}_{n}(qAq)$. Since ${\mathbb{M}}_{n}(A)$
is the directed union of $({\mathbb{M}}_{n}(qAq))_{q\in{\mathcal{P}}(A)}$, we
have $\phi(x)=x$ for every $x\in{\mathbb{M}}_{n}(A)$. ∎

If $\phi$ and $\phi^{\prime}$ are ring isomorphisms from ${\mathbb{M}}_{n}(A)$
onto $B$ that satisfy the condition of Theorem 8.1, then
$\phi^{-1}\circ\phi^{\prime}$ maps each principal right ideal of
${\mathbb{M}}_{n}(A)$ onto itself. Thus the preceding proposition implies
$\phi^{-1}\circ\phi^{\prime}=\operatorname{id}_{{\mathbb{M}}_{n}(A)}$, and
$\phi^{\prime}=\phi$. A somewhat surprising fact is that we cannot drop the
assumption of the existence of unit in Theorem 8.1.

######  Example 8.4.

For an inner product space $V$, the lattice ${\mathcal{P}}(F(V))$ can be
identified with the lattice of finite-dimensional subspaces of $V$ (ordered by
inclusion). Therefore, if two inner product spaces $V,W$ have the same Hamel
dimension, then ${\mathcal{P}}(F(V))$ is lattice isomorphic to
${\mathcal{P}}(F(W))$. Let $H$ be an infinite-dimensional Hilbert space, and
let $f\colon H\to H$ be any linear bijection that is not bounded. Remark that
$F(H)$ is ∗-isomorphic to ${\mathbb{M}}_{n}(F(H))$ for every $n\geq 1$. Since
$f$ maps each finite-dimensional subspace of $H$ onto another finite-
dimensional subspace of $H$, $f$ determines a lattice automorphism $\psi$ of
the lattice of finite-dimensional subspaces of $H$, which is identified with
${\mathcal{P}}(F(H))$. Let us prove that there exists no ring automorphism
$\phi\colon F(H)\to F(H)$ that satisfies $\phi(pF(H))=\psi(p)F(H)$ for all
$p\in{\mathcal{P}}(F(H))$. Assume for a contradiction that there exists such a
ring automorphism $\phi\colon F(H)\to F(H)$. By unboundedness of $f$, we may
find a nonzero vector $h\in H$ such that $f(\\{h\\}^{\perp})$ is dense in $H$.
Take the rank-one projection $p\in{\mathcal{P}}(F(H))$ onto ${\mathbb{C}}h$.
For each projection $q\in{\mathcal{P}}(F(H))$ of rank one with $pq=0$, we see
that $\phi(q)\in F(H)$ is an idempotent with $\phi(p)\phi(q)=\phi(pq)=0$.
However, the equation $\phi(qF(H))=\psi(q)F(H)$ implies $\phi(q)H=f(qH)$. It
follows that $\phi(p)|_{f(\\{h\\}^{\perp})}=0$. Since $f(\\{h\\}^{\perp})$ is
dense in $H$, the boundedness of $\phi(p)$ implies $\phi(p)=0$, a
contradiction.

Recall that a mapping $\phi\colon A\to B$ between ∗-algebras is called a
_Jordan ∗-homomorphism_ (resp. a _Jordan ∗-isomorphism_) if it is a linear
mapping (resp. a linear bijection) such that $\phi(x)^{*}=\phi(x^{*})$ and
$\phi(x^{2})=\phi(x)^{2}$ for every $x\in A$. Clearly, if $A$ and $B$ are
R∗-algebras, $\phi\colon A\to B$ is a Jordan ∗-homomorphism and $x\in A$ is
self-adjoint (resp. positive, a projection), then so is $\phi(x)\in B$.

Jordan ∗-isomorphisms appear quite often in studying mappings between operator
algebras that preserve certain structures. See Molnár’s recent article [40]
for more about Jordan ∗-isomorphisms between C∗-algebras.

######  Example 8.5.

Let $A$ be the unitization of $\bigoplus_{n\in{\mathbb{Z}}}{\mathbb{M}}_{2}$,
which is formed of all bi-infinite sequences $(x_{n})_{n\in{\mathbb{Z}}}$ of
$2\times 2$ matrices $x_{n}$ with the following property: There exists $N\geq
1$ such that $x_{n}=x_{N}\in{\mathbb{C}}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}$ for all $n$ with $\lvert n\rvert\geq N$. Define $\phi\colon
A\to A$ by $\phi((x_{n})_{n\in{\mathbb{Z}}})=(y_{n})_{n\in{\mathbb{Z}}}$,
$y_{n}=x_{n}$ if $n\geq 0$, and $y_{n}$ is the transpose of $x_{n}$ otherwise.
Then $\phi$ is a Jordan ∗-automorphism of $A$.

######  Proposition 8.6.

Let $A,B$ be R∗-algebras. Let $\phi\colon A\to B$ be a linear map such that
$\phi({\mathcal{P}}(A))\subset{\mathcal{P}}(B)$. Then $\phi$ is a Jordan
∗-homomorphism.

###### Proof.

It is easy to check that $\phi$ maps mutually orthogonal projections to
mutually orthogonal projections. Since every self-adjoint operator $a$ in $A$
can be written as a real-linear combination of mutually orthogonal
projections, the linearity of $\phi$ implies that $\phi(a)^{*}=\phi(a)$ and
$\phi(a^{2})=\phi(a)^{2}$. It follows that $\phi$ preserves involution. If
$a,b$ are self-adjoint, then so is $a+b$, hence we have
$\phi((a+b)^{2})=\phi(a+b)^{2}$, which together with $\phi(a^{2})=\phi(a)^{2}$
and $\phi(b^{2})=\phi(b)^{2}$ implies that
$\phi(ab+ba)=\phi(a)\phi(b)+\phi(b)\phi(a)$. Thus we obtain

$\begin{split}\phi((a+ib)^{2})&=\phi(a^{2})+i\phi(ab+ba)-\phi(b^{2})\\\
&=\phi(a)^{2}+i(\phi(a)\phi(b)+\phi(b)\phi(a))-\phi(b)^{2}\\\
&=\phi(a+ib)^{2}.\end{split}$

∎

By strengthening the assumption of Theorem 8.1, we obtain a Feldman–Dye type
result. Let $A,B$ be R∗-algebras. A bijection
$\psi\colon{\mathcal{P}}(A)\to{\mathcal{P}}(B)$ is called an orthoisomorphism
if $pq=0\iff\psi(p)\psi(q)=0$ for every pair $p,q\in{\mathcal{P}}(A)$. It is
easy to verify that an orthoisomorphism
$\psi\colon{\mathcal{P}}(A)\to{\mathcal{P}}(B)$ is a lattice isomorphism and
$\psi(p^{\perp})=\psi(p)^{\perp}$ if $A$ and $B$ are unital, so this
definition is consistent with the definition of orthoisomorphism in Section 4.
Dye [18] (see also Feldman’s [21]) proved that every orthoisomorphism between
the projection lattices of two von Neumann algebras extends to a Jordan
∗-isomorphism if one of the von Neumann algebras is without type I2 direct
summands. To obtain this theorem, Dye needed to appropriately modify the proof
of von Neumann’s [43, Theorem II.4.2]. In the setting of R∗-algebras, we may
directly apply Theorem 8.1 to consider orthoisomorphisms of the projection
lattices. The proof below is partly based on [21, Proof of Theorem 3]. It
should be worth mentioning that we do not need the assumption of the existence
of unit in this theorem.

######  Theorem 8.7 (Dye’s theorem for R∗-algebras).

Let $A,B$ be R∗-algebras. If $\phi\colon A\to B$ is a Jordan ∗-isomorphism,
then it restricts to an orthoisomorphism from ${\mathcal{P}}(A)$ onto
${\mathcal{P}}(B)$. Let $n\geq 3$. If
$\psi\colon{\mathcal{P}}({\mathbb{M}}_{n}(A))\to{\mathcal{P}}(B)$ is an
orthoisomorphism, then there exists a unique Jordan ∗-isomorphism
$\phi\colon{\mathbb{M}}_{n}(A)\to B$ such that $\phi(p)=\psi(p)$ for all
$p\in{\mathcal{P}}({\mathbb{M}}_{n}(A))$.

###### Proof.

It is clear that a Jordan ∗-isomorphism restricts to an orthoisomorphism
between the projection lattices. In what follows, let $n\geq 3$ and suppose
that $\psi\colon{\mathcal{P}}({\mathbb{M}}_{n}(A))\to{\mathcal{P}}(B)$ is an
orthoisomorphism.

Assume that $A$ is unital. Then $B$ is also unital because $\psi$ preserves
the maximal projection. Since $\psi$ is a lattice isomorphism, there exists a
ring isomorphism $\phi\colon{\mathbb{M}}_{n}(A)\to B$ such that
$\phi(p{\mathbb{M}}_{n}(A))=\psi(p)B$ for all
$p\in{\mathcal{P}}({\mathbb{M}}_{n}(A))$. Let
$p\in{\mathcal{P}}({\mathbb{M}}_{n}(A))$. It is clear that $\phi(p)$ is an
idempotent. The equation
$\phi(p^{\perp}{\mathbb{M}}_{n}(A))=\psi(p^{\perp})B=\psi(p)^{\perp}B$ implies
that $l(\phi(p^{\perp}))=\psi(p)^{\perp}$. Since $pp^{\perp}=0$, we have
$\phi(p)\phi(p^{\perp})=0$. Thus we have
$0=r(\phi(p))l(\phi(p^{\perp}))=r(\phi(p))\psi(p)^{\perp}$ and hence
$r(\phi(p))\leq\psi(p)=l(\phi(p))$. By Lemma 8.2, we see that $\phi(p)$ is a
projection, and we have $\phi(p)=l(\phi(p))=\psi(p)\in{\mathcal{P}}(B)$.

Consider the ring automorphism $x\mapsto\phi^{-1}(\phi(x^{*})^{*})$ of
${\mathbb{M}}_{n}(A)$. This fixes every projection, hence Proposition 8.3
implies that $x=\phi^{-1}(\phi(x^{*})^{*})$, or equivalently,
$\phi(x)^{*}=\phi(x^{*})$ for each $x\in{\mathbb{M}}_{n}(A)$. It follows that
$\phi$ maps the self-adjoint part of ${\mathbb{M}}_{n}(A)$ onto that of $B$.
Since $\phi$ preserves squares, $\phi$ restricted to self-adjoint parts is an
order isomorphism. It follows that $\phi(t)=t$ for all $t\in{\mathbb{R}}$.
Define the complex-linear map $\Phi\colon A\to B$ by
$\Phi(a+ib)=\phi(a)+i\phi(b)$ for self-adjoint $a,b\in A$. By Proposition 8.6,
$\Phi$ is a Jordan ∗-homomorphism that extends $\psi$. Since projections of
$B$ linearly span $B$, $\Phi$ is surjective.

If $A$ is not unital, then for each $p\in{\mathcal{P}}(A)$, $\psi$ restricts
to an orthoisomorphism from ${\mathcal{P}}({\mathbb{M}}_{n}(pAp))$ onto
${\mathcal{P}}(\psi(p\otimes 1)B\psi(p\otimes 1))$. Since $pAp$ is a unital
R∗-algebra, it follows that $\psi$ restricted to
${\mathcal{P}}({\mathbb{M}}_{n}(pAp))$ extends to a Jordan ∗-isomorphism from
${\mathbb{M}}_{n}(pAp)$ onto $\psi(p\otimes 1)B\psi(p\otimes 1)$. Moreover,
since ${\mathcal{P}}({\mathbb{M}}_{n}(pAp))$ linearly spans
${\mathbb{M}}_{n}(pAp)$, this is the unique linear extension of
$\psi|_{{\mathcal{P}}({\mathbb{M}}_{n}(pAp))}$. Since ${\mathbb{M}}_{n}(A)$ is
the directed union of $({\mathbb{M}}_{n}(pAp))_{p\in{\mathcal{P}}(A)}$, it
follows that $\psi$ extends to a linear mapping, and this extension is a
Jordan ∗-isomorphism from ${\mathbb{M}}_{n}(A)$ onto $B$. The uniqueness of
the extension is again the consequence of the fact that an R∗-algebra is
linearly spanned by projections. ∎

The assumption $n\geq 3$ is essential in Theorem 8.7 (and Theorem 8.1).
Indeed, it is easy to find wild examples of orthoisomorphisms from
${\mathcal{P}}({\mathbb{M}}_{2})$ onto itself.

A _∗ -antiisomorphism_ $\phi\colon A\to B$ between ∗-algebras is a complex-
linear bijection with $\phi(x^{*})=\phi(x)^{*}$, $\phi(xy)=\phi(y)\phi(x)$ for
every pair $x,y\in A$. Clearly, a ∗-antiisomorphism is a Jordan ∗-isomorphism.
Theorem 8.7 motivates the following question: How far are Jordan
∗-isomorphisms from ∗-isomorphisms? In particular, how different are
∗-antiisomorphisms from ∗-isomorphisms? A more specific question is: Is there
a pair of mutually ∗-antiisomorphic R∗-algebras that are not ∗-isomorphic, or
equivalently, is there an R∗-algebra that is not ∗-isomorphic to its opposite
R∗-algebra? Here, for an R∗-algebra $A$, its _opposite R ∗-algebra_
$A^{\mathrm{op}}$ is defined in the following manner. As a complex normed
space $A^{\mathrm{op}}$ is equal to $A$. For each $x\in A$ we write the
corresponding element in $A^{\mathrm{op}}$ as $x^{\mathrm{op}}$ in order to
distinguish $A$ from $A^{\mathrm{op}}$. We define the ∗-algebra structure of
$A^{\mathrm{op}}$ by $(x^{\mathrm{op}})^{*}:=(x^{*})^{\mathrm{op}}$ and
$x^{\mathrm{op}}y^{\mathrm{op}}:=(yx)^{\mathrm{op}}$ for $x,y\in A$. Then it
is easy to show that $A^{\mathrm{op}}$ forms an R∗-algebra that is
∗-antiisomorphic to $A$.

It is a trivial fact that every commutative R∗-algebra is ∗-isomorphic to its
opposite. By the Elliott classification theorem, it is plain that every
sequentially ultramatricial R∗-algebra is ∗-isomorphic to its opposite. How
about general R∗-algebras? Banach space theory gives an answer. For a complex
normed space $X$, its _complex conjugate normed space_ $\overline{X}$ is
defined in the following manner. As a real normed space $\overline{X}$ is
equal to $X$. For each $v\in X$ we write the corresponding element in
$\overline{X}$ as $\overline{v}$ in order to distinguish $X$ from
$\overline{X}$. We define the complex-linear structure of $\overline{X}$ by
$\lambda\overline{v}:=\overline{\overline{\lambda}v}$ $v\in X$,
$\lambda\in{\mathbb{C}}$.

######  Proposition 8.8.

There is a purely atomic simple separable R∗-algebra $F(V)$ that is not
∗-isomorphic to its opposite.

###### Proof.

It is well-known (see for example [10, 33]) that there exists a separable
Banach space $X$ that is not isomorphic (as complex Banach space) to its
complex conjugate $\overline{X}$. For such an $X$, take an injective bounded
linear operator $T\colon X\to\ell_{2}$ with dense range (see Proposition 5.6),
and set $V:=TX$. Consider the complex conjugate
$\overline{T}\colon\overline{X}\to\overline{\ell_{2}}$ of $T$ (defined by
$\overline{T}(\overline{v}):=\overline{T(v)},\,\,v\in X$), which is clearly a
bounded linear operator with dense range, and observe that the range is
canonically identified with $\overline{V}$. It is easy to see that the
opposite R∗-algebra of $F(V)$ is ∗-isomorphic to $F(\overline{V})$ by the
mapping $x^{\mathrm{op}}\mapsto\overline{x}^{*}$, $x\in F(V)$. If $F(V)$ is
∗-isomorphic to its opposite, then it follows by Proposition 5.6 that $X$ and
$\overline{X}$ are isomorphic as complex Banach spaces, which contradicts our
choice of $X$. ∎

The difference between an algebra and its opposite for operator algebras has
been considered by many hands. In the setting of von Neumann algebras, Connes
[14] discovered an example of a II1 factor which is not ∗-isomorphic to its
opposite, giving a negative solution to Problem 4 of Kadison’s list [23]. By
now many examples of C∗-algebras that are not ∗-isomorphic to their opposites
are known. However, it remains as an open problem whether there exists a
simple nuclear separable C∗-algebra that is not ∗-isomorphic to its opposite.
The case of AF C∗-algebras is still under investigation in the literature.
Indeed, in [20] Farah and Hirshberg proved the existence of a (nonseparable)
AF C∗-algebra that is not ∗-isomorphic to its opposite under some set-
theoretic axiom that is consistent with ZFC. It is an open problem whether
such an example can be constructed inside ZFC. See [20] for more details about
the state-of-the-art on this topic.

Let $A$ be an R∗-algebra or a von Neumann algebra. A function
$\mu\colon{\mathcal{P}}(A)\to{\mathbb{R}}$ is called a _finitely additive_
(signed) _measure_ if it satisfies

$\mu(p+q)=\mu(p)+\mu(q)$

for every pair of mutually orthogonal projections $p,q\in{\mathcal{P}}(A)$.

Mackey’s problem asks whether a finitely additive measure extends to a linear
mapping under a suitable assumption. This question is motivated by the theory
of quantum logic. Gleason proved that every positive countably-additive
measure on the projection lattice of $B(H)$, except for the case $H$ is
$2$-dimensional, extends to a bounded normal functional [26]. More generally,
it was proved by Bunce and Wright that every finitely additive measure with
bounded range extends uniquely to a bounded linear functional in the setting
of general von Neumann algebras without type I2 direct summands [12, 13]. We
give a Gleason-type result for a certain class of R∗-algebras. Let $A$ be an
ultramatricial R∗-algebra. We say $A$ satisfies the property (N2) if $A$ can
be written as the union of an increasing net $(A_{i})_{i\in I}$ of finite-
dimensional R∗-algebras such that no $A_{i}$ has ${\mathbb{M}}_{2}$ as its
direct summand.

Let $A$ be an R∗-algebra and $\mu$ a finitely additive measure on
${\mathcal{P}}(A)$. Since every operator $x\in A$ has cartesian decomposition
$x=a+bi$ with $a,b\in A$ self-adjoint, $x$ is written as a finite linear
combination of projections. Thus the uniqueness of linear extension of a
finitely additive measure is clear. If in addition $\lVert x\rVert\leq 1$,
then $0\leq a_{\pm},b_{\pm}\leq 1$ (see Proposition 2.6 (2)) and they are in
the convex hull of ${\mathcal{P}}(A)$. If $\mu$ extends to a linear functional
$f$ on $A$, then we have $f(x)=f(a_{+})-f(a_{-})+if(b_{+})-if(b_{-})$, and
$f(a_{+}),f(a_{-}),f(b_{+}),f(b_{-})$ belong to the convex hull of
$\mu({\mathcal{P}}(A))$. Therefore, if $\mu({\mathcal{P}}(A))$ is a bounded
subset of ${\mathbb{R}}$ then the linear extension, if exists, is bounded.

######  Theorem 8.9 (Gleason’s theorem for ultramatricial R∗-algebras).

Let $A$ be an ultramatricial R∗-algebra with (N2). If
$\mu\colon{\mathcal{P}}(A)\to{\mathbb{R}}$ is a finitely additive measure such
that $\mu({\mathcal{P}}(A))$ is a bounded subset of ${\mathbb{R}}$, then $\mu$
extends uniquely to a bounded linear functional on $A$.

###### Proof.

Assume that $A$ is the union of an increasing net $(A_{i})_{i\in I}$ of
finite-dimensional R∗-algebras such that no $A_{i}$ has ${\mathbb{M}}_{2}$ as
its direct summand. Then Gleason’s theorem for a finite-dimensional von
Neumann algebra (see [12, Section 2], which is based on Gleason’s result [26])
implies that the map $\mu$ restricted to each ${\mathcal{P}}(A_{i})$ extends
uniquely to a linear mapping on $A_{i}$. It follows that $\mu$ extends
uniquely to a linear mapping on $A$, and the extension is bounded by the
observation preceding this theorem. ∎

######  Corollary 8.10.

Let $A$ and $B$ be R∗-algebras. Assume that $A$ is ultramatricial and
satisfies (N2). If $\psi\colon{\mathcal{P}}(A)\to{\mathcal{P}}(B)$ is a map
such that $\psi(p)+\psi(q)=\psi(p+q)$ for every pair of mutually orthogonal
projections $p,q\in{\mathcal{P}}(A)$, then $\psi$ uniquely extends to a Jordan
∗-homomorphism $\phi\colon A\to B$.

###### Proof.

For each bounded linear functional $f$ on $B$, the preceding theorem implies
that the map $f\circ\psi\colon{\mathcal{P}}(A)\to{\mathbb{C}}$ extends
uniquely to a bounded linear functional of $A$. As a consequence, we may
easily prove the following: If complex numbers
$z_{1},z_{2},\ldots,z_{n}\in{\mathbb{C}}$ and projections
$p_{1},\ldots,p_{n}\in{\mathcal{P}}(A)$ satisfy $\sum_{k=1}^{n}z_{k}p_{k}=0$,
then $\sum_{k=1}^{n}z_{k}\psi(p_{k})=0$. It follows that $\psi$ extends
(uniquely) to a linear mapping $\phi\colon A\to B$. By Proposition 8.6, we
conclude that $\phi$ is a Jordan ∗-homomorphism. ∎

##  9\. Questions

In this section, we propose naive questions on R∗-algebras. These questions
are not complete: some of them might be trivially easy to solve, and some
might be totally meaningless to consider, but some should have essential
importance.

First of all, the author suspects that the theory of R∗-algebras falls into
one of the following two possibilities:

  * •

If every R∗-algebra is locally finite-dimensional or ultramatricial, then
chances are that we can capture a lot about the structure of general
R∗-algebras. Indeed, we already have a complete classification result of
locally finite-dimensional R∗-algebras with countable Hamel dimension (Section
7).

  * •

If not, then the theory of R∗-algebras should be so deep that the full
understanding might be far out of the reach. That is because once we get one
wild example of an R∗-algebra we may easily construct many other wild examples
by Section 3. In that case, R∗-algebras would be all the more interesting as a
new area of “noncommutative” mathematics.

######  Question 9.1.

Which of the above two is true? If the first option is true, then is it
possible to give a classification result that applies to a class broader than
that in Section 7? (For example, to the class of separable R∗-algebras under
some appropriate condition?) If the second option is true, is it still
possible to give a satisfactory classification theorem that is valid for all
R∗-algebras with countable Hamel dimension?

Szűcs and Takács proved that there exists a finitely generated infinite-
dimensional R∗-algebra if and only if there exists a singly generated
infinite-dimensional R∗-algebra $A$ (which means that there exists $x\in A$
with ${\mathcal{A}}^{*}(x)=A$) [48, Proposition 4.7]. The latter remains to be
a hard problem, it seems. The difficulty of considering singly generated
∗-algebras can also be observed in the following relevant famous open problem,
which is listed in Kadison’s problems [23] right before Problem 1.1. Is every
von Neumann algebra acting on a separable Hilbert space $H$ singly generated
(as a $\sigma$-weakly closed ∗-subalgebra of $B(H)$)? For more details on this
famous problem, see [47, Chapter 16].

We have seen the importance of the uniqueness of C∗-norm for R∗-algebras in
Theorem 2.12. Recall that uniqueness of C∗-norm is important in the theory of
C∗-algebras, too. Indeed, nuclearity of a C∗-algebra, which lies in the core
of the study of C∗-algebras, is defined in terms of the uniqueness of C∗-norm:
A C∗-algebra $A$ is nuclear if for every C∗-algebra $B$ the algebraic tensor
product $A\otimes B$ has a unique C∗-norm. It is natural to ask the following
question.

######  Question 9.2.

Does the study of R∗-algebras give a better understanding of the uniqueness of
C∗-norm? In particular, can we obtain a nice necessary/sufficient condition
for a ∗-algebra to have a unique C∗-norm?

Property (LC∗) in Section 2 gives one sufficient condition. In the case of the
group algebra ${\mathbb{C}}[G]$ of a discrete group $G$, Grigorchuk, Musat and
Rørdam asked in [29, Question 6.8] whether ${\mathbb{C}}[G]$ has a unique
C∗-norm if and only if $G$ is locally finite. It was pointed out by N. Ozawa
that this question has a negative solution, see [3]. Let us also give a
related question, which makes sense only in the case there exists a non-
ultramatricial R∗-algebra.

######  Question 9.3.

Given a pair of non-ultramatricial R∗-algebras $A,B$, does $A\otimes B$ form
an R∗-algebra?

Note that if the answer is affirmative then $A\otimes B$ has a unique C∗-norm,
which further implies that $\widehat{A}\otimes\widehat{B}$ has a unique
C∗-norm, or in other words, $(\widehat{A},\widehat{B})$ is a nuclear pair [45,
Definition 9.1]. Another question that makes sense only in the case there
exists a non-ultramatricial R∗-algebra is the following.

######  Question 9.4.

Does Theorem 8.9 (Gleason-type theorem) generalize to a wider class of
R∗-algebras? In particular, is the following statement true? For every
R∗-algebra $A$ and an integer $n\geq 3$, a finitely additive measure
$\mu\colon{\mathcal{P}}({\mathbb{M}}_{n}(A))\to{\mathbb{R}}$ with bounded
range extends linearly to ${\mathbb{M}}_{n}(A)$.

Let us propose a question about the property of R∗-algebras that is shared
with C∗-/von Neumann algebras.

######  Question 9.5.

Let $H$ be a Hilbert space. Study the class of ∗-subalgebras $A\subset B(H)$
that satisfy the property of Proposition 2.6 (1). How about (2) and (3)? (In
(2) we assume Borel measurability and boundedness of $f$.)

We know that every C∗-algebra and every R∗-algebra satisfy (1). So does the
union of an increasing net of C∗-algebras. In Blackadar’s book [7] a
∗-subalgebra $A\subset B(H)$ that satisfies (1) is called a local C∗-algebra.
Note that in the article [9] the same term local C∗-algebra is used in another
sense: It means a ∗-subalgebra $A\subset B(H)$ with the property (LC∗) in
Section 2, which is a condition far weaker than (2). Every von Neumann algebra
and every R∗-algebra satisfy (2) and (3). If $A\subset B(H)$ is a ∗-subalgebra
and the $\sigma$-weak closure of ${\mathcal{A}}^{*}(x)$ is contained in $A$
for every normal element $x\in A$, then $A$ satisfies (2). We do not know
whether (2) implies (1).