# Symmetry and Spectral Invariance for Topologically
Graded $C^{*}$-Algebras and Partial Action Systems

D. Jauré and M. Măntoiu  D. Jauré and M. Măntoiu 111 2020 Mathematics Subject
Classification: Primary 43A20, Secondary 47L65,47L30.
Key Words: partial group action, symmetric Banach algebra, inverse closed,
Wiener-Hopf operator, ordered group, graph algebra.
D. J. acknowledges financial support from ”Beca de Doctorado Nacional
Conicyt”. M. M. has been supported by the Fondecyt Project 1200884.

###### Abstract

A discrete group ${\sf G}$ is called rigidly symmetric if the projective
tensor product between the convolution algebra $\ell^{1}({\sf G})$ and any
$C^{*}$-algebra $\mathcal{A}$ is symmetric. We show that in each topologically
graded $C^{*}$-algebra over a rigidly symmetric group there is a
$\ell^{1}$-type symmetric Banach ∗-algebra, which is inverse closed in the
$C^{*}$-algebra. This includes new general classes, as algebras admitting dual
actions and partial crossed products. Results including convolution dominated
kernels, inverse closedness with respect with ideals or weighted versions of
the $\ell^{1}$-decay are included. Various concrete examples are presented.

###### Contents

  1. 1 Introduction
  2. 2 The abstract theory
    1. 2.1 The main result
    2. 2.2 Topologically graded algebras from dual actions and partial crossed products
    3. 2.3 Inverse closedness modulo ideals.
    4. 2.4 Other types of decay
  3. 3 Examples
    1. 3.1 Inverse closed Banach ∗-algebras from topological partial actions
    2. 3.2 Toeplitz-Bunce-Deddens operators
    3. 3.3 UHF algebras
    4. 3.4 CAR algebras
    5. 3.5 Wiener-Hopf algebras associated to quasi-lattice ordered groups
    6. 3.6 Higher rank graph algebras

##  1 Introduction

Spectral invariance and symmetry of normed algebras have been important
concepts, starting with the classical article [33]. Some general references
which are related with the point of view of this article include [12, 14, 16,
17, 20, 24, 25]. We also refer to [15] for a very readable general
presentation, making connections with related mathematical topics.

The basic definition is the following (we restrict to unital algebras; this
will be enough in examples):

######  Definition 1.1.

  1. (i)

The unital ∗-subalgebra $\mathfrak{B}$ of the $C^{*}$-algebra $\mathfrak{C}$
is called inverse closed (or spectrally invariant) if for each
$\Phi\in\mathfrak{B}$ which is invertible in $\mathfrak{C}$ , one actually has
$\Phi^{-1}\\!\in\mathfrak{B}$ .

  2. (ii)

A unital Banach ∗-algebra $\mathfrak{B}$ is called symmetric if the spectrum
of $b^{*}b$ is positive for every $b\in\mathfrak{B}$ (this happens if and only
if the spectrum of any self-adjoint element is real.)

The two notions are related, but in this Introduction we only discuss
symmetry. A great interest lies in algebras connected somehow to a discrete
group. The following notions will be relevant in the article; the first two
are classical, while the third constitutes our main concern:

######  Definition 1.2.

  1. (i)

The discrete group ${\sf G}$ is called symmetric if the convolution Banach
∗-algebra $\ell^{1}({\sf G})$ is symmetric.

  2. (ii)

The discrete group ${\sf G}$ is called rigidly symmetric if given any
$C^{*}$-algebra $\mathcal{A}$ , the projective tensor product $\ell^{1}({\sf
G})\otimes\mathcal{A}$ is symmetric.

  3. (iii)

The discrete group ${\sf G}$ is called hypersymmetric if given any
topologically graded $C^{*}$-algebra $\mathfrak{C}$ , the Banach ∗-algebra
$\ell^{1}(\mathfrak{C})$ (to be defined below) is symmetric.

The class of topologically graded $C^{*}$-algebras, described for example in
[9] and reviewed briefly in Subsection 2.1, is much larger than the one
appearing in (ii). Simplifying, the main result of the article is

######  Theorem 1.3.

If a discrete group is rigidly symmetric, then it is hypersymmetric.

######  Remark 1.4.

Clearly, a rigidly symmetric group is symmetric. It is still not known if the
two notions are really different. Anyhow, as a consequence of [19, Thm. 1], if
${\sf G}$ is supposed only symmetric, but the $C^{*}$-algebra $\mathcal{A}$ is
type $I$, the projective tensor product $\ell^{1}({\sf G})\otimes\mathcal{A}$
is symmetric. Classes of rigidly symmetric discrete groups are (cf. [20]): (a)
Abelian, (b) finite, (c) finite extensions of discrete nilpotent. This last
class includes all the finitely generated groups with polynomial growth. A
central extension of a rigidly symmetric group is rigidly symmetric, by [20,
Thm. 7]. In [21, Cor. 2.16] it is shown that the quotient of a discrete
rigidly symmetric group by a normal subgroup is rigidly symmetric.

There are intermediary notions between rigid symmetry and hypersymmetry. It
has already been proved [14, 2, 21] that if a discrete group is rigidly
symmetric, it is also symmetric in the sense of global crossed products: For
an action $\beta$ of ${\sf G}$ in a $C^{*}$-algebra $\mathcal{A}$ , one
canonically constructs a crossed product $C^{*}$-algebra
$\mathcal{A}\\!\rtimes_{\beta}\\!{\sf G}$ , which is the enveloping algebra of
a Banach ∗-algebra $\ell^{1}_{\beta}({\sf G};\mathcal{A})$ . If for every such
global action $({\sf G},\beta,\mathcal{A})$ the algebra $\ell^{1}_{\beta}({\sf
G};\mathcal{A})$ is symmetric, ${\sf G}$ may be called symmetric in the sense
of global crossed products. For a trivial action $\beta$ one gets the
projective tensor product $\ell^{1}({\sf
G})\otimes\mathcal{A}\cong\ell^{1}({\sf G};\mathcal{A})$ and plenty of
interesting $C^{*}$-algebras proved to be global crossed products for some
non-trivial action, so [14] supplied a lot of new examples of symmetric
algebras.

Our initial aim was to extend this further to partial actions $\theta$ of
discrete groups ${\sf G}$ on $C^{*}$-algebras $\mathcal{A}$ . The class of
$C^{*}$-algebras that may be realized as partial crossed products
$\mathcal{A}\\!\rightthreetimes_{\theta}\\!{\sf G}$ , but which are not
accessible to global actions, is quite impressive. We soon understood that
topologically graded algebras over the discrete group ${\sf G}$ provide an
even larger setting, which is also simpler and more natural in a certain
sense. So Theorem 2.4 and a consequence on covariant convolution dominated
kernels are stated in this framework.

However, since the subcase of partial crossed products is very important, and
many $C^{*}$-algebras are proved to be of such a type (a non-trivial task,
very often), in Subsection 2.2 we made our results explicit for this
situation. In the same Subsection 2.2, we indicated the connection between
topologically ${\sf G}$-graded $C^{*}$-algebras $\mathfrak{C}$ and ”dual
actions” of ${\sf G}$. The right non-commutative concept is that of a coaction
of ${\sf G}$ on $\mathfrak{C}$ , but for simplicity and having in mind the
examples we want to treat by this tool, we only discussed the case of Abelian
groups. In this case one works with a (global) action of the compact Abelian
Pontryagin dual $\widehat{\sf G}$ and the spaces of the grading are defined to
be the spectral subspaces of this dual action.

######  Remark 1.5.

In [30, Cor. 4.8] it is shown that a symmetric group is amenable. Since our
results below require ${\sf G}$ to be rigidly symmetric, we will assume ${\sf
G}$ to be (at least) amenable. This will simplify the grading theory. In
particular, the (universal) partial crossed product
$\mathcal{A}\rightthreetimes_{\theta}\\!{\sf G}$ defined in Subsection 2.2
coincides with the reduced one [22, Sect. 3].

In Subsection 2.3 we treat inverse closednes with respect to a suitable ideal.
The most important application is to Fredholm properties of operators, the
ideal being (isomorphic to) the ideal of all the compact operators in a
Hilbert space.

In Subsection 2.4 we explore others types of decay, using weighted
$\ell^{1}$-algebras. It is possible to implement the result in all the
examples, but we will not do it explicitly.

Very often one encounters $C^{*}$-algebras that are not obviously involving a
grading or a partial action of a group. Showing that such a structure can be
put into evidence is a valuable tool to study various properties, in
particular the existence of inverse closed subalgebras. In our last Section,
using various references to the literature, we present such examples and find
out what the $\ell^{1}$-algebra is composed of in each case. Our treatment is
not meant to be exhaustive, but it will be clear that many new interesting
cases are covered. In addition, some examples belong to different classes or
can be treated from several points of view; we stick to the one that seems to
be simpler. When the group is Abelian, we mostly use dual actions. In other
cases partial actions or a direct presentation of the topological grading is
the tool.

Some conventions and notations: $C^{*}$-algebraic morphisms and
representations are assumed to be involutive, the ideals are self-adjoint, bi-
sided and closed. $\mathbb{B}(\mathcal{H})$ will denote the $C^{*}$-algebra of
linear bounded operators in a Hilbert space; $\mathbb{K}(\mathcal{H})$ is the
ideal of compact operators.

##  2 The abstract theory

###  2.1 The main result

All over the article, ${\sf G}$ will be a discrete group with unit ${\sf e}$ .
As explained in Remark 1.5, there is no loss if we assume it amenable. For
constructions involving graded $C^{*}$-algebras we followed [8, 11] ; we
decided not to mention Fell bundles explicitly (with one exception), but they
can be perceived in the background.

######  Definition 2.1.

We say that the unital $C^{*}$-algebra $\mathfrak{C}$ is ${\sf G}$-graded if
that there is a family $\big{\\{}\mathfrak{C}_{g}\,|\,g\in{\sf G}\big{\\}}$ of
closed subspaces of $\mathfrak{C}$ such that the algebraic direct sum
$\,\bigoplus_{g\in{\sf G}}\mathfrak{C}_{g}$ is dense in $\mathfrak{C}$ and
such that

  1. (i)

$\mathfrak{C}_{g}\mathfrak{C}_{h}\subset\mathfrak{C}_{gh}\,$ for every
$g,h\in{\sf G}$ ,

  2. (ii)

$\mathfrak{C}_{g}^{*}\subset\mathfrak{C}_{g^{-1}}\,$ for every $g\in{\sf G}$ ,

For every $h\in{\sf G}$ we denote by $P_{h}:\bigoplus_{g\in{\sf
G}}\\!\mathfrak{C}_{g}\to\mathfrak{C}_{h}$ the canonical projection (in
general it might not be continuous). If $P_{\sf e}$ is continuous,
$\mathfrak{C}$ is a topologically ${\sf G}$-graded $C^{*}$-algebra. We are
going to write

$\mathfrak{C}=\widetilde{\bigoplus}_{g\in{\sf G}}\mathfrak{C}_{g}\,.$ (2.1)

For $\Phi,\Psi\in\bigoplus_{g}\mathfrak{C}_{g}$ and $g\in{\sf G}$ one verifies
easily that

$P_{g}(\Phi\Psi)=\sum_{hk=g}P_{h}(\Phi)P_{k}(\Psi)\,,$ (2.2)
$P_{g}(\Phi)^{*}=P_{g^{-1}}(\Phi^{*})\,.$ (2.3)

When the grading is topological, there are several facilities which will be
crucial for our approach. Let us denote by $\widetilde{P}_{\sf
e}:\mathfrak{C}\to\mathfrak{C}_{\sf e}$ the extension of $P_{\sf e}$ to a
linear bounded map. By Tomiyama’s Theorem ([3, Thm. 1.5.10], see also [8, Thm.
3.3]), it is a positive contractive conditional expectation. Then all the
projections $P_{g}$ do extend to contractions
$\widetilde{P}_{g}:\mathfrak{C}\to\mathfrak{C}_{g}$ . We refer to the proof of
Theorem 2.4,(ii) for another use of topological grading, in conjunction with
amenability.

######  Definition 2.2.

Let $\mathfrak{C}$ be a topologically graded $C^{*}$-algebra. On
$\,\bigoplus_{g}\\!\mathfrak{C}_{g}$ we can introduce the new norm

$\parallel\\!\Phi\\!\parallel_{\ell^{1}(\mathfrak{C})}\,:=\sum_{g\in{\sf
G}}\parallel\\!P_{g}(\Phi)\\!\parallel.$

The completion $\ell^{1}(\mathfrak{C})$ of $\bigoplus_{g}\\!\mathfrak{C}_{g}$
in this norm is called the $\ell^{1}$-algebra of the graded $C^{*}$-algebra
$\mathfrak{C}$ .

Since
$\parallel\\!\cdot\\!\parallel\,\leq\,\parallel\\!\cdot\\!\parallel_{\ell^{1}(\mathfrak{C})}$
on $\bigoplus_{g}\\!\mathfrak{C}_{g}$ , one may interpret
$\ell^{1}(\mathfrak{C})$ as a subspace of $\mathfrak{C}$ :

$\ell^{1}(\mathfrak{C})=\Big{\\{}\Phi\in\mathfrak{C}\;\Big{|}\,\sum_{g\in{\sf
G}}\parallel\\!\widetilde{P}_{g}(\Phi)\\!\parallel\,<\infty\Big{\\}}\,.$

In fact it is a Banach ∗-subalgebra with the algebraic structure borrowed from
$\mathfrak{C}$ and its new norm. Its elements may be written as
unconditionally convergent series $\Phi=\sum_{g\in{\sf
G}}\widetilde{P}_{g}(\Phi)$ in the norm topology.

The next result, besides being interesting in itself, is basic for our
approach.

######  Proposition 2.3.

Let $\mathfrak{C}=\widetilde{\bigoplus}_{g\in{\sf G}}\mathfrak{C}_{g}$ be a
topologically graded $C^{*}$-algebra over a discrete group ${\sf G}$ . There
exists a $C^{*}$-algebra $\mathcal{B}$ and an isometric ∗-morphism

$T:\ell^{1}(\mathfrak{C})\hookrightarrow\ell^{1}({\sf
G};\mathcal{B})\equiv\ell^{1}({\sf G})\otimes\mathcal{B}\,.$

###### Proof.

Let $\pi:\mathfrak{C}\to\mathbb{B}(\mathcal{H})$ be a faithful representation.
For any $g\in{\sf G}$ we denote by $\pi_{g}$ the restriction of $\pi$ to the
closed subspace $\mathfrak{C}_{g}$ ; it is a linear isometry. One has

$\pi_{g}(\Phi)\pi_{h}(\Psi)=\pi_{gh}(\Phi\Psi)\,,\quad\forall\,g,h\in{\sf
G}\,,\ \Phi\in\mathfrak{C}_{g}\,,\ \Psi\in\mathfrak{C}_{h}\,,$ (2.4)
$\pi_{g}(\Phi^{*})=\pi_{g^{-1}}(\Phi)^{*},\quad\forall\,g\in{\sf G}\,,\
\Phi\in\mathfrak{C}_{g}\,.$ (2.5)

Then we set

$\big{[}T(\Phi)\big{]}(g):=\pi_{g}\big{[}\widetilde{P}_{g}(\Phi)\big{]}\,,\quad\Phi\in\ell^{1}(\mathfrak{C})\,,\
g\in{\sf G}\,.$ (2.6)

Clearly $T$ is a well-defined linear isometry from $\ell^{1}(\mathfrak{C})$ to
$\ell^{1}\big{(}{\sf G},\mathbb{B}(\mathcal{H})\big{)}$ :

$\parallel\\!T(\Phi)\\!\parallel_{\ell^{1}({\sf
G},\mathbb{B}(\mathcal{H}))}\,=\sum_{g\in{\sf
G}}\parallel\\!\pi_{g}\big{[}\widetilde{P}_{g}(\Phi)\big{]}\\!\parallel_{\mathbb{B}(\mathcal{H})}\,=\sum_{g\in{\sf
G}}\parallel\\!\widetilde{P}_{g}(\Phi)\\!\parallel\,=\,\parallel\\!\Phi\\!\parallel_{\ell^{1}(\mathfrak{C})}.$

To check that $T$ respects the algebraic operations, it is enough to work with
elements of the algebraic direct sum. By taking
$\Phi,\Psi\in\bigoplus_{g}\\!\mathfrak{C}_{g}$ , one can write

$\displaystyle\big{[}(T\Phi)\star(T\Psi)\big{]}(g)$
$\displaystyle=\sum_{h\in{\sf G}}(T\Phi)(h)(T\Psi)(h^{-1}g)$
$\displaystyle=\sum_{h\in{\sf
G}}\pi_{h}\left[P_{h}(\Phi)\right]\,\pi_{h^{-1}g}\big{[}P_{h^{-1}g}(\Psi)\big{]}$
$\displaystyle\overset{\eqref{primika}}{=}\sum_{h\in{\sf
G}}\pi_{g}\big{[}P_{h}(\Phi)P_{h^{-1}g}(\Psi)\big{]}$
$\displaystyle=\pi_{g}\Big{(}\sum_{h\in{\sf
G}}P_{h}(\Phi)P_{h^{-1}g}(\Psi)\Big{)}$
$\displaystyle\overset{\eqref{multiplix}}{=}\pi_{g}\big{[}P_{g}(\Phi\Psi)\big{]}=\big{[}T(\Phi\Psi)\big{]}(g)\,,$

showing that $T$ is an algebraic morphism. Finally we treat the involution:

$\displaystyle\left[T(\Phi)\right]^{\star}\\!(g)$
$\displaystyle=\pi_{g^{-1}}\\!\left[P_{g^{-1}}(\Phi)\right]^{*}\overset{\eqref{secundika}}{=}\pi_{g}\big{[}P_{g^{-1}}(\Phi)^{*}\big{]}$
$\displaystyle\overset{\eqref{multiplix}}{=}\pi_{g}[P_{g}(\Phi^{*})]=\big{[}T(\Phi^{*})\big{]}(g)\,.$

∎

With Proposition 2.3 in hand, we prove now our main abstract result. Recall
that a Banach ∗-algebra is called reduced if its universal $C^{*}$-seminorm is
in fact a norm. In this case, its enveloping $C^{*}$-algebra is simply its
completion in this $C^{*}$-norm (no quotient is needed).

######  Theorem 2.4.

Let $\mathfrak{C}=\widetilde{\bigoplus}_{g\in{\sf G}}\mathfrak{C}_{g}$ be
topologically graded over a rigidly symmetric discrete group ${\sf G}$ .

  1. (i)

$\ell^{1}(\mathfrak{C})$ is a symmetric Banach ∗-algebra.

  2. (ii)

$\ell^{1}(\mathfrak{C})$ is an inverse closed subalgebra of $\,\mathfrak{C}$ .

  3. (iii)

Let $\Pi:\mathfrak{C}\to\mathbb{B}(\mathcal{H})$ be a faithful representation.
Then $\Pi\left[\ell^{1}(\mathfrak{C})\right]$ is inverse closed in
$\mathbb{B}(\mathcal{H})$ .

###### Proof.

(i) By [24, Th.11.4.2], a closed ∗-algebra of a symmetric Banach ∗-algebra is
also symmetric. Proposition 2.3 and the fact that $\ell^{1}({\sf
G};\mathcal{B})$ was assumed symmetric proves the result.

(ii) It is known [24, 11.4] that a reduced Banach ∗-algebra is symmetric if
and only if it is inverse closed in its enveloping $C^{*}$-algebra. If we show
that we are in such a case, we then apply (i). Our $\ell^{1}(\mathfrak{C})$ is
clearly reduced, being given as a ∗-subalgebra of $\mathfrak{C}$ . Actually,
we claim that $\mathfrak{C}$ can be seen as (a copy of) the enveloping
$C^{*}$-algebra of $\ell^{1}(\mathfrak{C})$ . In [30, Corol. 4.8] it is shown
that a symmetric group (as our rigidly symmetric ${\sf G}$) is surely
amenable. For such groups, one may identify $\mathfrak{C}$ with both the full
and the reduced $C^{*}$-algebras of the Fell bundle associated to the
topological grading [8, Prop. 4.2 ,Thm. 4.7] and then the assertion about
enveloping follows [11, VIII.17].

(iii) Is a consequence of (ii), since any $C^{*}$-algebra is inverse closed in
a larger $C^{*}$-algebra. ∎

In the context of group actions on $C^{*}$-algebras one deals with convolution
dominated kernels and operators [12, 14, 15, 17, 21]. We briefly indicate an
extension for the topologically graded case, leading to another form of a
symmetric Banach ∗-algebra at the level of kernels. Representations by
integral operators and their inverse closedness properties are left to the
reader.

######  Definition 2.5.

A convolution dominated kernel (or matrix) is a function $K:{\sf G}\times{\sf
G}\to\mathfrak{C}$ such that $K(g,h)\in\mathfrak{C}_{gh^{-1}}$ for every
$g,h\in{\sf G}$ and such that the norm

$\parallel\\!K\\!\parallel_{\mathscr{K}}\,:=\inf\big{\\{}\\!\parallel\\!\kappa\\!\parallel_{\ell^{1}({\sf
G})}\,\big{|}\,\parallel\\!K(g,h)\\!\parallel_{\mathfrak{C}}\,\leq|\kappa(gh^{-1})|\,,\,\forall\,g,h\in{\sf
G}\big{\\}}$

is finite. We denote by $\mathscr{K}_{\sf G}(\mathfrak{C})$ the vector space
of all these convolution dominated kernels. An element $K$ of
$\mathscr{K}_{\sf G}(\mathfrak{C})$ is called covariant, and we write
$K\in\mathscr{K}^{\rm co}_{\sf G}(\mathfrak{C})$ , if

$K(gk,hk)=K(g,h)\,,\quad\forall\,g,h,k\in{\sf G}\,.$

It is straightforward to show that endowed with the norm
$\parallel\\!\cdot\\!\parallel_{\mathscr{K}}$ , the multiplication

$(K\bullet L)(g,h):=\sum_{k\in{\sf G}}K(g,k)L(k,h)$

and the involution $K^{\bullet}(g,h):=K(h,g)^{*}$, the space
$\mathscr{K}_{G}(\mathfrak{C})$ is a Banch ∗-algebra.

######  Corollary 2.6.

The subspace $\mathscr{K}^{\rm co}_{\sf G}(\mathfrak{C})$ is a symmetric
Banach ∗-algebra.

###### Proof.

It is easy to check that $\mathscr{K}^{\rm co}_{\sf G}(\mathfrak{C})$ is a
closed ∗-subalgebra of $\mathscr{K}_{\sf G}(\mathfrak{C})$ . Let us define

$\big{[}\Upsilon(\Phi)\big{]}(g,h):=\widetilde{P}_{gh^{-1}}(\Phi)\,,\quad\forall\,g,h\in{\sf
G}\,.$

Then $\Upsilon:\ell^{1}(\mathfrak{C})\to\mathscr{K}_{\sf G}(\mathfrak{C})$ is
an isometric ∗-morphism with range $\mathscr{K}^{\rm co}_{\sf
G}(\mathfrak{C})$ . For the algebraic relations use (2.2) and (2.3)
(extended). On this range, the inverse reads

$\big{[}\Upsilon^{-1}(K)\big{]}_{g}:=K(g,{\sf e})\,,\quad\forall\,g\in{\sf
G}\,.$

Since $\mathscr{K}^{\rm co}_{\sf G}(\mathfrak{C})$ and
$\ell^{1}(\mathfrak{C})$ are isomorphic, the result follows from Theorem 2.4
(i). ∎

######  Remark 2.7.

In [28] it is shown how to deform Fell bundles by a $2$-cocycle $\omega:{\sf
G}\\!\times\\!{\sf G}\to\mathbb{T}$ , bound to satisfy for every $g,h,k\in{\sf
G}$ the axioms

$\omega(g,h)\omega(gh,k)=\omega(h,k)\omega(g,hk)\,,\quad\omega(g,{\sf
e})=1=\omega({\sf e},g)\,.$

Starting with a topologically graded $C^{*}$-algebra
$\mathfrak{C}=\widetilde{\bigoplus}_{g\in{\sf G}}\mathfrak{C}_{g}$ , it is
possible to construct a new one
$\mathfrak{C}\\{\omega\\}=\widetilde{\bigoplus}^{\omega}_{g\in{\sf
G}}\mathfrak{C}_{g}$ . The Banach ∗-algebra
$\ell^{1}\big{(}\mathfrak{C}\\{\omega\\}\big{)}$ is different from
$\ell^{1}(\mathfrak{C})$ and it is symmetric and inverse closed in
$\mathfrak{C}\\{\omega\\}$ .

###  2.2 Topologically graded algebras from dual actions and partial crossed
products

Given a $C^{*}$-algebra, it is often not obvious whether it has an interesting
grading or not. In this subsection we are going to review some ways to solve
this problem, following especially results of Exel, Quigg and Raeburn. This
will lead to two reformulations of Theorem 2.17 that are useful when studying
examples and which will also provide the reader with a more extended picture.
We will take advantage of the fact that our group ${\sf G}$ is discrete and
amenable; this will allow for some simplifications.

The most general statement is that in this setting, topologically graded
$C^{*}$-algebras over ${\sf G}$ are equivalent to coactions of $\,{\sf G}$ ,
cf. [26] and [28, App. A]. In this direction we only indicate the case of
Abelian groups, that will be enough for the examples we intend to treat via
this approach.

So we fix for a while a discrete Abelian group ${\sf G}$ . Its Pontryagin dual
$\widehat{\sf G}$ is a compact Abelian group with normalized Haar measure
$d\chi$ . Let $\big{(}\mathfrak{C},\alpha,\widehat{\sf G}\,\big{)}$ be a
(usual, full) action of $\widehat{\sf G}$ on a $C^{*}$-algebra. For every
$g\in{\sf G}$ let

$\mathfrak{C}_{g}:=\big{\\{}\Phi\in\mathfrak{C}\,\big{|}\,\alpha_{\chi}(\Phi)=\chi(g)\Phi\,,\,\forall\,\chi\in\widehat{\sf
G}\big{\\}}$

be the $g$’th spectral subspace of the action. It is easy to see that one gets
a grading, as in (2.1). We have now the concrete projections

$\widetilde{P}_{g}:\mathfrak{C}\to\mathfrak{C}_{g}\,,\quad\widetilde{P}_{g}(\Phi):=\int_{\widehat{\sf
G}}\,\overline{\chi(g)}\,\alpha_{\chi}(\Phi)\,d\chi\,,$

which are obviously contractive; the grading is topological. We can
reformulate Theorem 2.4 in this Abelian setting; the $\ell^{1}$-algebra will
deserve a special notation, to recall the action $\alpha$ and the type of
decay.

######  Theorem 2.8.

Let $\big{(}\mathfrak{C},\alpha,\widehat{\sf G}\,\big{)}$ be a continuous
action of a compact Abelian group $\widehat{\sf G}$ on a unital
$C^{*}$-algebra. Denoting by ${\sf G}$ the dual of $\,\widehat{\sf G}$ ,

$\ell^{1}(\mathfrak{C})\equiv\bigoplus_{g\in{\sf
G}}^{1,\alpha}\mathfrak{C}_{g}=\Big{\\{}\Phi\in\mathfrak{C}\;\Big{|}\,\sum_{g\in{\sf
G}}\parallel\\!\widetilde{P}_{g}(\Phi)\\!\parallel\,<\infty\Big{\\}}$

is a reduced and symmetric Banach ∗-algebra, which is inverse closed in
$\mathfrak{C}$ .

######  Remark 2.9.

Using normal subgroups of ${\sf G}$ one can increase the family of symmetric
subalgebras. If, as above, $\big{(}\mathfrak{C},\alpha,\widehat{\sf
G}\,\big{)}$ is a continuous action of the dual of the Abelian discrete group
${\sf G}$ and ${\sf N}$ is a normal subgroup, the dual $\widehat{{\sf G}/{\sf
N}}$ of the quotient is isomorphic to the closed subgroup

${\rm Ann\big{(}{\sf N}\\!\mid\\!\widehat{\sf
G}\big{)}}:=\big{\\{}\chi\in\widehat{\sf G}\,\big{|}\,\chi|_{\sf
N}=1\big{\\}}\,.$ (2.7)

Thus from $\alpha$ we deduce a continuous action $\alpha^{\sf N}$ of
$\widehat{{\sf G}/{\sf N}}$ on $\mathfrak{C}$ , first by restricting $\alpha$
to ${\rm Ann\big{(}{\sf N}\\!\mid\\!\widehat{\sf G}\big{)}}$ and then
composing with the isomorphism. In this way we obtain the new symmetric Banach
∗-algebra

$\ell^{1}\big{(}\mathfrak{C}^{{\sf G}/{\sf
N}}\big{)}\equiv\bigoplus_{\gamma\in{\sf G}/{\sf N}}^{1,\alpha^{\sf
N}}\mathfrak{C}_{\gamma}\,.$

Extreme cases are $\ell^{1}\big{(}\mathfrak{C}^{{\sf G}/\\{{\sf
e}\\}}\big{)}=\ell^{1}(\mathfrak{C})$ and $\ell^{1}\big{(}\mathfrak{C}^{{\sf
G}/{\sf G}\\}}\big{)}=\mathfrak{C}$ .

Crossed products assigned to partial actions of (discrete) groups ${\sf G}$ on
(unital) $C^{*}$-algebras are among the most important examples of graded
$C^{*}$-algebras, and they were our initial motivation. In [9, Thm. 27.11]
they are characterized among the larger class, while [27, Thm. 4.1] gives a
Landstad-type description of them in terms of the associated coaction. In many
cases what we are actually given is the partial action; the grading comes
afterwards. Now the discrete group ${\sf G}$ is no longer Abelian.

######  Definition 2.10.

A partial action of ${\sf G}$ on the $C^{*}$-algebra $\mathcal{A}$ is a pair
$\big{(}\\{\mathcal{A}_{g}\\}_{g\in{\sf G}},\\{\theta_{g}\\}_{g\in{\sf
G}}\\}\big{)}$ , where for each $g\in{\sf G}$ , $\mathcal{A}_{g}$ is a closed
two-sided ideal in $\mathcal{A}$ , $\theta_{g}$ is a ∗-isomorphism from
$\mathcal{A}_{g^{-1}}$ onto $\mathcal{A}_{g}$ , satisfying the following
postulates, for all $g,h\in{\sf G}$ :

  1. (i)

$\mathcal{A}_{\sf e}=\mathcal{A}$ and $\theta_{\sf e}$ is the identity
automorphism of $\mathcal{A}$ ,

  2. (ii)

$\theta_{g}\\!\left(\mathcal{A}_{g^{-1}}\\!\cap\mathcal{A}_{h}\right)\subset\mathcal{A}_{gh}$
,

  3. (iii)

$\theta_{g}\big{(}\theta_{h}(a)\big{)}=\theta_{gh}(a)$ ,
$\forall\,a\in\mathcal{A}_{h^{-1}}\\!\cap\mathcal{A}_{(gh)^{-1}}$ .

We also denote by $(\mathcal{A},\theta,{\sf G})$ the partial dynamical system.
It follows easily that we have $\theta_{g^{-1}}\\!=\theta_{g}^{-1}$. Our main
object is the following vector space:

$\ell^{1}_{\theta}({\sf G};\mathcal{A}):=\left\\{\Phi\in\ell^{1}({\sf
G};\mathcal{A})\mid\Phi(g)\in\mathcal{A}_{g}\,,\,\forall\,g\in{\sf
G}\right\\},$

which is obviously closed. For $\Phi,\Psi\in\ell^{1}_{\theta}({\sf
G};\mathcal{A})$ we define the product

$\left(\Phi\star_{\theta}\Psi\right)(g)=\sum_{h\in{\sf
G}}\theta_{h}\\!\left[\theta_{h}^{-1}\big{(}\Phi(h)\big{)}\Psi\big{(}h^{-1}g\big{)}\right]$

and the involution
$\Phi^{\star_{\theta}}(g)=\theta_{g}\\!\left[\Phi\big{(}g^{-1}\big{)}\right]^{*}.$
Then $\big{(}\ell^{1}_{\theta}({\sf
G};\mathcal{A}),\star_{\theta},^{\star_{\theta}}\\!,\|\cdot\|_{\ell^{1}_{\theta}({\sf
G};\mathcal{A})}\big{)}$ is a Banach ∗-algebra. We will refer to it as the
$\ell^{1}$-partial crossed product. Its enveloping $C^{*}$-algebra is denoted
by $\mathcal{A}\\!\rightthreetimes_{\theta}\\!{\sf G}$ and called the partial
crossed product associated to $(\mathcal{A},\theta,{\sf G})$ . We used the
fancy symbol $\rightthreetimes$ instead of the more usual one $\rtimes$ to
stress that our crossed product is assigned to a partial action.

We indicate the particularization of Theorem 2.4 to partial actions. Since
${\sf G}$ must be amenable, by [8] the full and the reduced partial crossed
products are identical and possess a (faithful) positive contractive
conditional expectation.

######  Theorem 2.11.

Let $(\mathcal{A},\theta,{\sf G})$ be a partial $C^{*}$-dynamical system with
rigidly symmetric discrete group ${\sf G}$ and unital $C^{*}$-algebra
$\mathcal{A}$ . Then $\ell^{1}_{\theta}({\sf G};\mathcal{A})$ is symmetric and
inverse closed in $\mathcal{A}\rightthreetimes_{\theta}\\!{\sf G}$ . If
$\,\Pi:\mathcal{A}\rightthreetimes_{\theta}\\!{\sf
G}\to\mathbb{B}(\mathcal{H})$ is a faithful representation,
$\Pi\left[\ell^{1}_{\theta}({\sf G};\mathcal{A})\right]$ is inverse closed in
$\mathbb{B}(\mathcal{H})$ .

###### Proof.

The crossed product
$\mathfrak{C}:=\mathcal{A}\\!\rightthreetimes_{\theta}\\!{\sf G}$ is a
topologically graded $C^{*}$-algebra [9], where

$\mathfrak{C}_{g}=\big{\\{}\Phi\in\ell^{1}_{\theta}({\sf
G};\mathcal{A})\,\big{|}\,{\rm
supp}(\Phi)\subset\\{g\\}\big{\\}}\,,\quad\forall\,g\in{\sf G}\,.$

Its $\ell^{1}$-algebra (with a slight reinterpretation) is

$\ell^{1}(\mathfrak{C})\equiv\ell^{1}\big{(}\mathcal{A}\\!\rightthreetimes_{\theta}\\!{\sf
G}\big{)}=\ell^{1}_{\theta}({\sf G};\mathcal{A})\,.$

Thus we can apply Theorem 2.4. ∎

######  Remark 2.12.

The result can also be obtained directly, by means specific to partial action.
To prove the isometric linear embedding $\ell^{1}_{\theta}({\sf
G};\mathcal{A})\hookrightarrow\ell^{1}\big{(}{\sf G};\mathbb{B}(\mathcal{H}))$
one uses the map $\big{[}T(\Phi)\big{]}:=\rho\big{[}\Phi(g)\big{]}u_{g}$ ,
where $(\rho,u)$ is a covariant representation of $(\mathcal{A},\theta,{\sf
G})$ in the Hilbert space $\mathcal{H}$ , as in [9, Sect. 13]. Here $\rho$
should be a faithful representation of $\mathcal{A}$ and $u$ a partial
representation of ${\sf G}$ , connected by an equivariance condition.

###  2.3 Inverse closedness modulo ideals.

######  Definition 2.13.

Let $\mathfrak{K}$ be an ideal of the unital C∗-algebra $\mathfrak{C}$ . The
∗-subalgebra $\mathfrak{B}$ of $\mathfrak{C}$ is called $\mathfrak{K}$-inverse
closed if $\mathfrak{B}/(\mathfrak{B}\cap\mathfrak{K})$ is inverse closed in
$\mathfrak{C}/\mathfrak{K}$ .

######  Example 2.14.

If $\,\mathfrak{K}=\\{0\\}$ , the notion coincides with that introduced in
Definition 1.1. But the main motivating case is as follows: Let $\mathfrak{C}$
be a $C^{*}$-algebra of bounded operators in the Hilbert space $\mathcal{H}$ ,
containing the ideal $\mathbb{K}(\mathcal{H})$ of all the compact operators.
We recall that $T\in\mathbb{B}(\mathcal{H})$ is called Fredholm if its range
is closed and its kernel and its cokernel are both finite-dimensional. By
Atkinson’s Theorem, this happens exactly if the image of $T$ in the Calkin
algebra $\mathbb{B}(\mathcal{H})/\mathbb{K}(\mathcal{H})$ is invertible. A
∗-subalgebra $\mathfrak{B}$ of $\mathbb{B}(\mathcal{H})$ will be called
Fredholm inverse closed if the situation in Definition 2.13 holds with
$\mathfrak{K}=\mathbb{K}(\mathcal{H})$ . For $\mathfrak{C}$ we can take any
$C^{*}$-algebra of $\mathbb{B}(\mathcal{H})$ containing $\mathfrak{B}$ .

######  Definition 2.15.

Let $\mathfrak{C}=\widetilde{\bigoplus}_{g}\mathfrak{C}_{g}$ be a graded
$C^{*}$-algebra over the discrete group ${\sf G}$ . The ideal $\mathfrak{K}$
of $\mathfrak{C}$ is a graded ideal (also called induced ideal) if the ideal
generated by $\mathfrak{K}\cap\mathfrak{C}_{\sf e}$ coincides with
$\mathfrak{K}$ (or, equivalently, that
$\bigoplus_{g}\mathfrak{K}\cap\mathfrak{C}_{g}$ is dense in $\mathfrak{K}$).

######  Theorem 2.16.

Let $\mathfrak{C}=\widetilde{\bigoplus}_{g}\mathfrak{C}_{g}$ be a
topologically graded $C^{*}$-algebra over the discrete rigidly symmetric group
${\sf G}$ and $\mathfrak{K}$ a graded ideal. Then $\ell^{1}(\mathfrak{C})$ is
$\mathfrak{K}$-inverse closed in $\mathfrak{C}$ .

###### Proof.

One needs to show that
$\ell^{1}(\mathfrak{C})/\ell^{1}(\mathfrak{C})\cap\mathfrak{K}$ is inverse
closed in $\mathfrak{C}/\mathfrak{K}$ . Let us denote by
$\kappa:\mathfrak{C}\to\mathfrak{C}/\mathfrak{K}$ the quotient map. By [9,
Prop. 23.10] (or by [8, Prop. 3.11]), under the stated conditions upon the
graded algebra and the ideal, $\mathfrak{C}/\mathfrak{K}$ is topologically
graded by the subspaces

$\big{\\{}(\mathfrak{C}/\mathfrak{K})_{g}:=\kappa(\mathfrak{C}_{g})\,\big{|}\,g\in{\sf
G}\big{\\}}\,.$

The corresponding linear contraction
$\widetilde{Q}_{g}:\mathfrak{C}/\mathfrak{K}\to(\mathfrak{C}/\mathfrak{K})_{g}$
satisfies

$\widetilde{Q}_{g}\circ\kappa=\kappa\circ\widetilde{P}_{g}\,,\quad\forall\,g\in{\sf
G}\,.$ (2.8)

Using this and the form of the $\ell^{1}$-norms, one shows immediately that
$\kappa$ sends $\ell^{1}(\mathfrak{C})$ into
$\ell^{1}(\mathfrak{C}/\mathfrak{K})$ contractively. Its kernel is clearly
$\ell^{1}(\mathfrak{C})\cap\mathfrak{K}$ . Let us show its surjectivity,
starting from

$\varphi:=\sum_{g\in{\sf G}}\varphi_{g}\equiv\sum_{g\in{\sf
G}}\widetilde{Q}_{g}(\varphi)\in\ell^{1}(\mathfrak{C}/\mathfrak{K})$

(unconditional convergence in $\mathfrak{C}/\mathfrak{K}$) . Let
$\\{\beta_{g}\\!\mid\\!g\in{\sf G}\\}$ be a summable family of positive
numbers. For each $g$, there is an element $\Phi_{g}\in\mathfrak{C}_{g}$ such
that $\kappa(\Phi_{g})=\varphi_{g}$ and
$\parallel\\!\Phi_{g}\\!\parallel_{\mathfrak{C}}\,\leq\,\parallel\\!\varphi_{g}\\!\parallel_{\mathfrak{C}/\mathfrak{K}}+\beta_{g}$
. Then $\Phi:=\sum_{g}\Phi_{g}\in\ell^{1}(\mathfrak{C})$ and
$\kappa(\Phi)=\varphi$ . So
$\ell^{1}(\mathfrak{C})/\ell^{1}(\mathfrak{C})\cap\mathfrak{K}$ may be
identified with $\ell^{1}(\mathfrak{C}/\mathfrak{K})$ . We apply Theorem 2.4
(ii) to the topologically graded quotient algebra $\mathfrak{C}/\mathfrak{K}$
and finish the proof. ∎

The particular case of partial actions is worth mentioning. Suppose that
$(\mathcal{A},\theta,{\sf G})$ is a partial dynamical system with discrete
group ${\sf G}$ and that $\mathcal{K}$ is an ideal of $\mathcal{A}$ that is
$\theta$-invariant:

$\theta_{g}\big{(}\mathcal{A}_{g^{-1}}\\!\cap\mathcal{K}\big{)}\subset\mathcal{K}\,,\quad\forall\,g\in{\sf
G}\,.$

We denote by the same letter $\theta$ the action of ${\sf G}$ by partial
automorphisms of $\mathcal{K}$ defined by restrictions. One gets the
C∗-partial dynamical system $(\mathcal{K},\theta,{\sf G})$ . The partial
crossed product $\mathcal{K}\rightthreetimes_{\theta}{\sf G}$ may be
identified with an ideal of $\mathcal{A}\rightthreetimes_{\theta}{\sf G}$ .
Under this identification, $\ell^{1}_{\theta}({\sf G};\mathcal{K})$ becomes an
ideal of $\ell^{1}_{\theta}({\sf G};\mathcal{A})$ in the natural way: the
$\ell^{1}$-function $\Phi:{\sf G}\rightarrow\mathcal{K}$ is taken to be
$\mathcal{A}$-valued. One can use the exactness of the partial crossed product
construction [10, Sect. 3] to prove

######  Theorem 2.17.

Assume that the discrete group ${\sf G}$ is rigidly symmetric. The Banach
∗-algebra $\ell^{1}_{\theta}({\sf G};\mathcal{A})$ is
$\mathcal{K}\\!\rightthreetimes_{\theta}\\!{\sf G}$-inverse closed in the
partial crossed product $\mathcal{A}\\!\rightthreetimes_{\theta}\\!{\sf G}$
for any $\theta$-invariant ideal $\mathcal{K}$ of $\mathcal{A}$ .

Theorem 2.17 can also be obtained as a consequence of Theorem 2.16, since
partial crossed products are topologically graded, and the ideal
$\mathcal{K}\\!\rightthreetimes_{\theta}\\!{\sf G}$ is indeed graded under the
stated assumptions.

The situation in Exemple 2.14 is the most interesting:

######  Corollary 2.18.

Let $(\mathcal{A},\theta,{\sf G})$ be a partial C${}^{*}\\!$-dynamical system
with discrete rigidly symmetric group ${\sf G}$ and $\mathcal{K}$ a
$\theta$-invariant ideal in $\mathcal{A}$ . Let
$\Pi\\!:\\!\mathcal{A}\rightthreetimes_{\theta}\\!{\sf
G}\to\mathbb{B}(\mathcal{H})$ be a faithful representation such that
$\Pi\big{(}\mathcal{K}\rightthreetimes_{\theta}\\!{\sf
G}\big{)}=\mathbb{K}(\mathcal{H})$ . Then the Banach ∗-algebra
$\Pi[\ell^{1}_{\theta}({\sf G};\mathcal{A})]$ is Fredholm inverse closed.

###  2.4 Other types of decay

######  Definition 2.19.

A weight on the discrete group ${\sf G}$ is a function $\nu:{\sf
G}\to[1,\infty)$ satisfying everywhere

$\nu(gh)\leq\nu(g)\nu(h)\,,\quad\nu(g^{-1})=\nu(g)\,.$ (2.9)

Let $\mathfrak{C}$ be a topologically graded $C^{*}$-algebra. On
$\,\bigoplus_{g}\\!\mathfrak{C}_{g}$ we can introduce the norm

$\parallel\\!\Phi\\!\parallel_{\ell^{1,\nu}(\mathfrak{C})}\,:=\sum_{g\in{\sf
G}}\nu(g)\\!\parallel\\!P_{g}(\Phi)\\!\parallel.$ (2.10)

The completion in this norm, denoted by $\ell^{1,\nu}(\mathfrak{C})$ , can be
seen as a Banach ∗-algebra with the algebraic structure inherited from
$\ell^{1}(\mathfrak{C})$ (or from $\mathfrak{C}$) .

######  Theorem 2.20.

Let ${\sf G}$ be a rigidly symmetric discrete group and $\nu$ a weight. Assume
that there exists a generating subset $V$ of $\,{\sf G}$ containing the unit
${\sf e}$ such that

  1. (a)

the following uGRS (uniform Gelfand-Raikov-Shilov) condition holds:

$\lim_{n\rightarrow\infty}\sup_{g_{1},\dots,g_{n}\in V}\nu(g_{1}\cdots
g_{n})^{\frac{1}{n}}=1\,,$ (2.11)

  2. (b)

for some finite constant $C$ one has for any $n\in\mathbb{N}$

$\sup_{g\in V^{n}\setminus V^{n-1}}\nu(g)\leq C\\!\inf_{g\in V^{n}\setminus
V^{n-1}}\nu(g)\,.$ (2.12)

Then $\ell^{1,\nu}(\mathfrak{C})$ is a symmetric Banach ∗-algebra for every
topologically ${\sf G}$-graded $C^{*}$-algebra $\mathfrak{C}$ .

###### Proof.

The problem of the symmetry of a weighted $\ell^{1}$-algebra as a ∗-subalgebra
of an unweighted one has been solved in [14, Th. 3] in a more particular
context. We are going to check that the arguments can be adapted to the more
general case. The main idea is proving that, given an element
$\Phi\in\ell^{1,\nu}(\mathfrak{C})$ , its spectral radius is the same as in
$\ell^{1}(\mathfrak{C})$ . Then one applies Hulanicki’s Lemma.

As $\nu(\cdot)\geq 1$ , we have
$\parallel\\!\cdot\\!\parallel_{\ell^{1}(\mathfrak{C})}\,\leq\,\parallel\cdot\,\parallel_{\ell^{1,\nu}(\mathfrak{C})}$
; the spectral radius formula implies that

$r_{\ell^{1}(\mathfrak{C})}(\Phi)\leq
r_{\ell^{1,\nu}(\mathfrak{C})}(\Phi)\,,\quad\forall\,\Phi\in{\ell^{1,\nu}(\mathfrak{C})}\,.$

To prove the opposite inequality, pick $V$ a generating set for ${\sf G}$ and
define on $\mathbb{Z}$ the function

$v(n):=\sup_{g\in V^{|n|}}\nu(g)\,.$

Due to the uGRS condition, it is a weight and one has the obvious associated
weighted space $\ell^{1,\nu}(\mathbb{Z})$ . By induction, from the equation
(2.10) we get

$\parallel\\!\Phi^{n}\\!\parallel_{\ell^{1,\nu}(\mathfrak{C})}\,\leq\sum_{g_{1}\in{\sf
G}}\cdots\sum_{g_{n}\in{\sf G}}\nu(g_{1}\dots
g_{n})\parallel\\!P_{g_{1}}(\Phi)\\!\parallel\cdots\\!\parallel\\!P_{g_{n}}(\Phi)\\!\parallel.$

Since ${\sf G}=\bigsqcup_{m\in\mathbb{N}}(V^{m}\setminus V^{m-1})$ , where
$V^{0}=\emptyset$ , we may split the sum accordingly. This yields

$\parallel\\!\Phi^{n}\\!\parallel_{\ell^{1,\nu}(\mathfrak{C})}\,\leq\\!\sum_{m_{1},\dots,m_{n}=1}^{\infty}\sum_{V^{m_{1}}\setminus
V^{m_{1}-1}}\cdots\sum_{V^{m_{n}}\setminus V^{m_{n}-1}}\nu(g_{1}\dots
g_{n})\parallel P_{g_{1}}(\Phi)\\!\parallel\cdots\\!\parallel
P_{g_{n}}(\Phi)\\!\parallel.$ (2.13)

If $g_{j}\in V^{m_{j}}\setminus V^{m_{j}-1}$, then $g_{1}\cdots g_{n}\in
V^{m_{1}+\cdots+m_{n}}$ and so the weight is majorized by

$\nu(g_{1}\cdots g_{n})\leq\sup_{h\in
V^{m_{1}+\cdots+m_{n}}}\nu(h)=v(m_{1}+\cdots m_{n})\,.$

Set $b_{m}:=\sum_{g\in V^{m}\setminus V^{m-1}}\\!\parallel
P_{g}(\Phi)\\!\parallel$ and $b=(b_{m})_{m\in\mathbb{N}}$ . Then we have
$\\!\parallel\Phi\\!\parallel_{\ell^{1}(\mathfrak{C})}\,=\,\parallel
b\\!\parallel_{\ell^{1}(\mathbb{N})}$ . Also the condition (2.12) implies
immediately that

$C^{-1}\\!\parallel
b\\!\parallel_{\ell^{1,v}(\mathbb{Z})}\,\leq\,\parallel\Phi\\!\parallel_{\ell^{1,\nu}(\mathfrak{C})}\,\leq\,\parallel\,b\\!\parallel_{\ell^{1,v}(\mathbb{Z})}.$

Returning to (2.13) we obtain

$\\!\parallel\Phi^{n}\\!\parallel_{\ell^{1,\nu}(\mathfrak{C})}\,\leq\\!\sum_{m_{1},\dots,m_{n}=1}^{\infty}\\!v(m_{1}+\cdots+m_{n})\,b_{m_{1}}\cdots
b_{m_{n}}\\!=\;\parallel
b^{n}\\!\parallel_{\ell^{1,v}(\mathbb{Z})}\,<\infty\,.$

By its definition the weight $v$ on $\mathbb{Z}$ satisfies the GRS-condition,
and $\ell^{1,v}(\mathbb{Z})$ is symmetric by [13]. Hence

$\displaystyle r_{\ell^{1,\nu}(\mathfrak{C})}(\Phi)$
$\displaystyle=\lim_{n\rightarrow\infty}\\!\parallel\Phi^{n}\\!\parallel_{\ell^{1,\nu}(\mathfrak{C})}^{1/n}\leq\lim_{n\rightarrow\infty}\\!\parallel
b^{n}\\!\parallel_{\ell^{1,\nu}(\mathbb{Z})}^{1/n}$
$\displaystyle=r_{\ell^{1,\nu}(\mathbb{Z})}(b)=r_{\ell^{1}(\mathbb{Z})}(b)$
$\displaystyle\leq\,\parallel
b\\!\parallel_{\ell^{1}(\mathbb{Z})}=\,\parallel\Phi\\!\parallel_{\ell^{1}(\mathfrak{C})}.$

So for all $k\in\mathbb{N}$ we have

$r_{\ell^{1,\nu}(\mathfrak{C})}(\Phi)=r_{\ell^{1,\nu}(\mathfrak{C})}(\Phi^{n})^{1/n}\leq\,\parallel\Phi^{n}\\!\parallel_{\ell^{1}(\mathfrak{C})}^{1/n},$

and by letting $n\rightarrow\infty$ we obtain the required inequality
$\,r_{\ell^{1,\nu}(\mathfrak{C})}(\Phi)\leq r_{\ell^{1}(\mathfrak{C})}(\Phi)$
. ∎

######  Remark 2.21.

There is a different way to prove Theorem 2.20. We define the Beurling algebra
associated to the weight $\nu$ and to the $C^{*}$-algebra $\mathcal{B}$ as

$\ell^{1,\nu}({\sf G};\mathcal{B}):=\big{\\{}\Psi\,|\,\nu\Psi\in\ell^{1}({\sf
G};\mathcal{B})\big{\\}}\,,$

with norm $\parallel\cdot\\!\parallel_{\ell^{1,\nu}({\sf
G},\mathcal{B})}\,:=\,\parallel\nu\,\cdot\,\\!\parallel_{\ell^{1}({\sf
G};\mathcal{B})}$. By [14], the assumptions on the weight imply that
$\ell^{1,\nu}({\sf G};\mathcal{B})$ is a symmetric Banach ∗-algebra. One then
shows as in Proposition 2.3 an isometric embedding
$\ell^{1,\nu}(\mathfrak{C})\hookrightarrow\ell^{1,\nu}({\sf G};\mathcal{B})$ ,
for some $C^{*}$-algebra $\mathcal{B}$ , and use this as in Theorem 2.4 to
deduce the symmetry of $\ell^{1,\nu}(\mathfrak{C})$ .

##  3 Examples

###  3.1 Inverse closed Banach ∗-algebras from topological partial actions

We introduce now certain partial crossed product $C^{*}$-algebras associated
to partial topological actions of the discrete group. We do not consider the
most general case; in particular, we restrict generality in such a way to have
certain natural faithful Hilbert space representations. In this way,
transparent results on inverse closedness will be available.

So let $(X,\Theta,{\sf G})$ be a partial action of the group ${\sf G}$
(assumed here to be countable) on the compact Hausdorff space $X$. In more
detail, it could be denoted by $\big{\\{}\Theta_{g}:X_{g^{-1}}\\!\to
X_{g}\,\big{|}\,g\in{\sf G}\big{\\}}$ , where each $X_{g}$ is an open subset
of $X$ and $\Theta_{g}$ is a homeomorphism with domain $X_{g^{-1}}$ and image
$X_{g}$ . One requires

  * •

$X_{\sf e}=X$ and $\Theta_{\sf e}={\rm id}_{X}$ ,

  * •

for every $g,h\in{\sf G}$ , the homeomorphism $\Theta_{gh}$ extends
$\Theta_{g}\circ\Theta_{h}$ (this one with maximal domain).

In fact each locally compact partial dynamical system may be deduced by
restricting a global one in an essentially canonical way, but sometimes the
total space $\tilde{X}$ might fail to be Hausdorff [1, 9].

We deduce from $(X,\Theta,{\sf G})$ a partial $C^{*}$-dynamical system
$\big{(}C(X),\theta,{\sf G}\big{)}$ , where the ideals are

$\mathcal{A}_{g}\\!:=C\big{(}X_{g}\big{)}\equiv\big{\\{}a\in
C(X)\,\big{|}\,a(x)=0\,,\,\forall\,x\notin X_{g}\big{\\}}$

and the isomorphisms are

$\theta_{g}:C\big{(}X_{g^{-1}}\big{)}\to
C\big{(}X_{g}\big{)}\,,\quad\theta_{g}(a):=a\circ\Theta_{g^{-1}}\,.$

Therefore, one can construct the Banach ∗-algebra
$\ell^{1}_{\theta}\big{(}{\sf G};C(X)\big{)}$ and its enveloping partial
crossed product $C(X)\\!\rightthreetimes_{\theta}\\!{\sf G}$ . We deduce from
Theorem 2.4

######  Proposition 3.1.

If $\,{\sf G}$ is rigidly symmetric, $\ell^{1}_{\theta}\big{(}{\sf
G},C(X)\big{)}$ is reduced, symmetric and inverse closed in the partial
crossed product.

Assume now that under the partial action there is an open dense orbit $Y$ with
trivial isotropy. Thus, for some (and then any) $y\in Y$ one has

${\sf G}(y):=\big{\\{}g\in{\sf G}\,\big{|}\,y\in
X_{g^{-1}}\,,\Theta_{g}(y)=y\big{\\}}=\\{{\sf e}\\}\,.$

Then $X$ will be a regular compactification of the discrete orbit $Y$ and for
any $y\in Y$ the orbit map $g\to\Theta_{g}(y)$ will be a homeomorphism on its
image.

In the Hilbert space $\ell^{2}\big{(}Y\big{)}$ we introduce the representation
by multiplication operators

$\pi:C(X)\to\mathbb{B}\big{[}\ell^{2}(Y)\big{]}\,,\quad\big{[}\pi(a)\xi\big{]}(y):=a(y)\xi(y)\,.$

For any $h\in{\sf G}\,,\,y\in Y$ and $\xi\in\ell^{2}(Y)$ one defines

$\big{[}u_{h}(\xi)\big{]}(y):=\begin{cases}\xi\big{(}\Theta_{h^{-1}}(y)\big{)}&{\rm
if}\ \;y\in X_{h}\,,\\\ 0\,&{\rm if}\ \;y\notin X_{h}\,.\end{cases}$

Then $\big{(}\pi,u,\ell^{2}(Y)\big{)}$ is a covariant representation of
$\big{(}C(X),\theta,{\sf G}\big{)}$ and the integrated form representation

$\Pi\equiv\pi\\!\rightthreetimes u\\!:C(X)\\!\rightthreetimes_{\theta}\\!{\sf
G}\to\mathbb{B}\big{[}\ell^{2}(Y)\big{]}$

acts on $\ell^{1}_{\theta}\big{(}{\sf G};C(X)\big{)}$ as
$\Pi(\Phi)=\sum_{h\in{\sf G}}\pi\big{[}\Phi(h)\big{]}u_{h}$ and satisfies

$\Pi\big{(}a\otimes\delta_{h}\big{)}=\pi(a)u_{h}\,,\quad\forall\,h\in{\sf
G}\,,\,a\in C(X_{h})\,.$ (3.1)

To express it explicitly, we need some notations. For $y,z\in Y$ we set
$g_{yz}$ for the unique element of the group such that $\Theta_{g_{yz}}(y)=z$
; we use the fact that $Y$ is an orbit with trivial isotropy. Note the
relations

$g_{yz}^{-1}=g_{zy}\,,\quad g_{zy}=g_{xy}g_{zx}\,,\quad\forall\,x,y,z\in Y.$
(3.2)

A direct computation, relying on (3.2), shows that on
$\ell^{1}_{\theta}\big{(}{\sf G};C(X)\big{)}$ the representation $\Pi$ reads

$\big{[}\Pi(\Phi)\xi\big{]}(y)=\sum_{z\in
Y}\Phi\big{(}g_{zy},y\big{)}\xi(z)\,,$

where we set $\big{[}\Phi(k)\big{]}(y)=:\Phi(k,y)$ .

######  Theorem 3.2.

Suppose that ${\sf G}$ is countable and rigidly symmetric and that $Y$ is a
dense open orbit of the space $X$, with trivial isotropy. Let
$\Phi\in\ell^{1}_{\theta}\big{(}{\sf G};C(X)\big{)}$ .

  1. (i)

If the operator $\,\Pi(\Phi)$ is invertible, there exists
$\Psi\in\ell^{1}_{\theta}\big{(}{\sf G};C(X)\big{)}$ such that
$\Pi(\Phi)^{-1}\\!=\Pi(\Psi)$ .

  2. (ii)

If $\,\Pi(\Phi)$ is Fredholm, at least one of its inverses $T$ modulo
$\mathbb{K}\big{[}\ell^{2}(Y)\big{]}$ belongs to
$\Pi[\ell^{1}_{\theta}\big{(}{\sf G};C(X)\big{)}]$ .

###### Proof.

(i) Having in view Proposition 3.1, we only need to check that the
representation $\Pi$ is faithful; see also Theorem 2.4(iv). The universal and
the reduced partial crossed products here are the same, cf. Remark 1.5. Note
that the restriction of $\Pi$ to $C(X)$ coincides with $\pi$ (set $h={\sf e}$
in (3.1)). By [10, Th. 2.6], injectivity of $\Pi$ would follow if we prove
that $\pi$ is injective and the action is topologically free. The injectivity
of $\pi$ is obvious: $\pi(a)=0$ means that the restriction of $a$ to $Y$ is
null, which implies that $a$ is null, since the orbit $Y$ was supposed dense.
Topological freeness (cf. [10, Def. 2.1.]) is clear, since $Y$ is dense:
having trivial isotropy, it is disjoint from any set of the form

$F_{g}:=\big{\\{}x\in X_{g^{-1}}\,\big{|}\,\Theta_{g}(x)=x\big{\\}}\,,\quad
g\neq{\sf e}\,,$

so this one must have empty interior.

(ii) Since $Y$ is discrete, the elements of $C_{0}(Y)$ are transformed by the
representation $\pi$ into compact multiplication operators in $\ell^{2}(Y)$ .
The $\theta$-invariant ideal $C_{0}(Y)$ gives rise to the ideal
$C_{0}(Y)\\!\rightthreetimes_{\theta}\\!{\sf G}$ of the partial crossed
product $C(X)\\!\rightthreetimes_{\theta}\\!{\sf G}$ and one has
$\Pi\big{[}C_{0}(Y)\\!\rightthreetimes_{\theta}\\!{\sf
G}\big{]}=\mathbb{K}\big{[}\ell^{2}(Y)\big{]}$ . Therefore one can apply
Corollary 2.18 and conclude that $\Pi[\ell^{1}_{\theta}\big{(}{\sf
G};C(X)\big{)}]\subset\mathbb{B}\big{[}\ell^{2}(Y)\big{]}$ is Fredholm inverse
closed. Then the assertion follows easily from Atkinson’s Theorem; see also
Example 2.14. ∎

###  3.2 Toeplitz-Bunce-Deddens operators

In the Hilbert space $\ell^{2}(\mathbb{N})$ , with canonical orthonormal base
$\\{\delta_{n}\\}_{n\in\mathbb{N}}$ , each function
$a\in\ell^{\infty}(\mathbb{N})$ defines a bounded multiplication operator
${\sf M}_{a}$ . Let us denote by ${\sf S}$ the right shift uniquely determined
by ${\sf S}\,\delta_{n}\\!:=\delta_{n+1}$ for every $n\in\mathbb{N}$ . For
$q\in\mathbb{N}$ , we say that $a\in\ell^{\infty}(\mathbb{N})$ is $q$-periodic
if $a(n+q)=a(n)\,,\,\forall\,n\in\mathbb{N}$ . We denote by
$\ell^{\infty}(\mathbb{N};q)$ the $C^{*}$-subalgebra of all $q$-periodic
functions. The starting point in constructing a Toeplitz-Bunce-Deddens algebra
is an infinite family $\mathbf{q}:=\\{q_{i}\\!\mid\\!i\in\mathbb{N}\\}$ of
strictly increasing positive integers such that each $q_{i}$ divides $q_{i+1}$
, i.e. $q_{i+1}=r_{i}q_{i}$ with $r_{i}\geq 2$ .

######  Definition 3.3.

The Toeplitz-Bunce-Deddens algebra $\mathfrak{D}(\mathbf{q})$ associated to
the multi-integer $\mathbf{q}$ is the $C^{*}$-algebras of operators in
$\ell^{2}(\mathbb{N})$ generated by the family

$\Big{\\{}{\sf S}_{a}:={\sf S}{\sf
M}_{a}\,\Big{|}\,a\in\bigcup_{i\in\mathbb{N}}\ell^{\infty}(\mathbb{N};q_{i})\Big{\\}}\,.$

######  Proposition 3.4.

The Toeplitz-Bunce-Deddens algebra is topologically graded over $\mathbb{Z}$ .
Its $\ell^{1}$-Banach ∗-algebra
$\ell^{1}\big{[}\mathfrak{D}(\mathbf{q})\big{]}$ is reduced, symmetric and
inverse closed in $\mathfrak{D}(\mathbf{q})$ and in
$\mathbb{B}\big{[}\ell^{2}(\mathbb{N})\big{]}$ (an explicit description is
included in the proof).

###### Proof.

By [5], a canonical action $\alpha$ of the torus
$\mathbb{T}=\widehat{\mathbb{Z}}$ on $\mathfrak{D}(\mathbf{q})$ is defined as
follows: For every $z\in\mathbb{T}$ the unitary multiplication operator
determined by $U_{z}\delta_{n}:=z^{n}\delta_{n}$ ($n\in\mathbb{N}$) defines on
$\mathbb{B}\big{[}\ell^{2}(\mathbb{N})\big{]}$ the inner automorphism $T\to
U_{z}TU_{z}^{*}$. Because of

$\alpha_{z}(S_{a})=U_{z}S_{a}U_{z}^{*}=zS_{a}\,,\quad\forall\,a\in\ell^{\infty}(\mathbb{N})\,,$

$\big{(}\mathfrak{D}(\mathbf{q}),\alpha,\mathbb{T}\big{)}$ is indeed a (full)
continuous action. We also write $\ell^{\infty}(\mathbb{N};\mathbf{q})$ for
the (unital, Abelian) $C^{*}$-subalgebra of $\ell^{\infty}(\mathbb{N})$
generated by $\bigcup_{i\in\mathbb{N}}\ell^{\infty}(\mathbb{N};q_{i})$ . The
spectral subspace $\mathfrak{D}(\mathbf{q})_{k}$ is generated by operators of
the ordered form

$S_{a_{1}}\dots S_{a_{n}}S^{*}_{b_{1}}\dots S^{*}_{b_{m}}\,,\quad
a_{1},\dots,a_{n},b_{1},\dots,b_{m}\in\ell^{\infty}(\mathbb{N};\mathbf{q})\,,\
n-m=k\,.$

This follows easily from the relations $S^{*}_{a}S_{b}=M_{\overline{a}b}$ and
$M_{a}S_{b}=S_{\tilde{a}b}$ , where $\tilde{a}_{n}:=a_{n-1}$ if $n\geq 1$ .
Since $\tilde{a}_{0}$ does not appear in the definition of the weighted shift
operator, it can be fixed such that $\tilde{a}$ is periodic. Other forms of
the elements in $\mathfrak{D}(\mathbf{q})_{k}$ could involve the final
projection $Q:=SS^{*}$. These having been settled, the assertions concerning
the $\ell^{1}$-Banach algebra follow from Theorem 2.8. ∎

######  Remark 3.5.

In [5] one gets the isomorphism

$\mathfrak{D}(\mathbf{q})\overset{\sim}{\longrightarrow}\big{[}c_{0}(\mathbb{N})\oplus\ell^{\infty}(\mathbb{N};\mathbf{q})\big{]}\\!\rightthreetimes_{\theta}\mathbb{Z}\cong
C\big{[}\mathbb{N}(\mathbf{q})\big{]}\\!\rightthreetimes_{\theta}\mathbb{Z}\,.$
(3.3)

We refer to [5] for the Gelfand spectrum
$\mathbb{N}(\mathbf{q})=\mathbb{N}\sqcup\Sigma(\mathbf{q})$ of the Abelian
$\alpha$-fixed point $C^{*}$-algebra

$\mathfrak{D}(\mathbf{q})_{0}\\!:=c_{0}(\mathbb{N})\oplus\ell^{\infty}(\mathbb{N};\mathbf{q})$

(a compactification of the discrete set $\mathbb{N}$ by a Cantor set
$\Sigma(\mathbf{q})$) and for the interesting explicit form of the partial
action $\theta$ in the Gelfand realization (induced by a partial homeomorphism
of $\mathbb{N}(\mathbf{q})$).

We may also state that
$\ell^{1}\big{[}\mathfrak{D}(\mathbf{q})\big{]}=\bigoplus_{k\in\mathbb{Z}}^{1,\alpha}\mathfrak{D}(\mathbf{q})_{k}$
is Fredholm inverse closed in
$\mathfrak{D}(\mathbf{q})\subset\mathbb{B}\big{[}\ell^{2}(\mathbb{N})\big{]}$
. This follows from Corollary 2.18, where $\mathcal{K}=c_{0}(\mathbb{N})$ and
$\Pi$ is the inverse of the isomorphism appearing in (3.3), since
$\Pi\big{[}c_{0}(\mathbb{N})\\!\rightthreetimes_{\theta}\\!\mathbb{Z}\big{]}=\mathbb{K}\big{[}\ell^{2}(\mathbb{N})\big{]}$
. One calls the quotient

$\mathfrak{D}(\mathbf{q})/\mathbb{K}\big{[}\ell^{2}(\mathbb{N})\big{]}\cong\ell^{\infty}(\mathbb{N};\mathbf{q})\\!\rtimes_{\theta}\mathbb{Z}\cong
C\big{[}\Sigma(\mathbf{q})\big{]}\\!\rtimes_{\theta}\mathbb{Z}$

the Bunce-Deddens algebra. The action ”at infinity” of $\mathbb{Z}$ on the
Cantor set $\Sigma(\mathbf{q})$ is induced by an odometer map. It is a global
action, so this quotient can be dealt with by usual crossed products.

###  3.3 UHF algebras

An UHF-algebra is the inductive limit

$\mathfrak{C}:=\underset{m\in\mathbb{N}}{\rm lim\,ind}\,\mathbf{M}^{p_{m}}$
(3.4)

of a sequence of full matricial $C^{*}$-algebras, with unital and injective
connecting morphisms, where each $p_{m}$ divides $p_{m+1}$. In [6] Exel
treated general approximately finite algebras. He defined on $\mathfrak{C}$ a
regular and semi-saturated $\mathbb{T}$-action and then he applied his theory
from [4] to get an isomorphism between $\mathfrak{C}$ and a partial crossed
product $\mathcal{A}\rightthreetimes_{\theta}\mathbb{Z}$ , where $\mathcal{A}$
is an Abelian almost finite $AF$-algebra. To simplify, we decided to treat
only UHF algebras; the general case is similar, but it would involve more
complicated notations. To state a result on symmetric subalgebras, we make use
of Theorem 2.8, only invoking the circle action and its spectral subspaces;
the partial crossed product will not be mentioned.

On every full matrix algebra $\mathbf{M}^{p}$ one defines

$\alpha^{p}:\mathbb{T}\to{\rm
Aut}\big{(}\mathbf{M}^{p}\big{)}\,,\quad\alpha^{p}_{z}\big{[}(c_{ij})_{i,j}\big{]}=\big{(}z^{i-j}c_{ij}\big{)}_{i,j}\,.$

The connecting morphisms
$\big{\\{}\mu^{m}\\!:\\!\mathbf{M}^{p_{m}}\to\mathbf{M}^{p_{m+1}}\big{\\}}$
are covariant with respect to the actions $\big{\\{}\alpha^{p_{m}}\big{\\}}$.
The inductive limit comes with the canonical monomorphisms
$\big{\\{}\nu^{m}\\!:\mathbf{M}^{p_{m}}\\!\to\mathfrak{C}\big{\\}}$ and
$\bigcup_{m}\nu^{(m)}\\!\big{(}\mathbf{M}^{p_{m}}\big{)}$ is dense in
$\mathfrak{C}$ . For every $z\in\mathbb{T}\,,m\in\mathbb{N}$ and
$\Phi\in\mathbf{M}^{p_{m}}$ we set

$\alpha_{z}\big{[}\nu^{m}(\Phi)\big{]}:=\nu^{m}\big{[}\alpha^{p_{m}}_{z}(\Phi)\big{]}\,.$

It is shown in [6, Sect. 2] that $\alpha$ extends to an action on
$\mathfrak{C}$ (which is semi-saturated and regular).

To make Theorem 2.8 concrete, one needs the spectral subspaces. For each
$|k|\leq p$ , let us denote by $\mathbf{M}^{p}_{k}$ the vector subspace of
$p\\!\times\\!p$-matrices with non-empty entries only on the $k^{\prime}$th
diagonal ($c_{ij}=0$ if $i-j\neq k$). If $|k|>p$ , we set simply
$\mathbf{M}^{p}_{k}\\!:=\\{0\\}$ . Then $\mathbf{M}^{p}_{k}$ is the
$k^{\prime}$th spectral subspace of the action $\alpha^{p}$. It is follows
from the definitions and from the fact that each canonical morphism $\nu^{m}$
is injective that the spectral subspaces of the action $\alpha$ are the
closures in $\mathfrak{C}$ of the unions over $m$ of the images through
$\nu^{m}$ of the subspaces $\mathbf{M}^{p_{m}}_{k}$.

######  Theorem 3.6.

The Banach ∗-algebra

$\ell^{1}(\mathfrak{C})=\bigoplus^{1,\alpha}_{k\in\mathbb{N}}\overline{\bigcup_{m\in\mathbb{Z}}\nu^{m}\big{(}\mathbf{M}^{p_{m}}_{k}\big{)}}$

is reduced, symmetric and inverse closed in $\mathfrak{C}$ .

######  Remark 3.7.

In [32] Takesaki (see also [11, VIII.17]) provided an approach to UHF algebras
which would lead to a different grading, and consequently also to a different
symmetric $\ell^{1}$-type Banach ∗-algebra, this time over the infinite
restricted product ${\sf G}:=\prod^{\prime}_{m\in\mathbb{N}}{\sf G}_{m}$ with
the discrete topology, where each ${\sf G}_{m}$ is a cyclic group of a certain
order $q_{m}$ . Actually, the UHF algebra $\mathfrak{C}$ is isomorphic to an
infinite tensor product $\otimes_{m}\mathbf{M}^{q_{m}}$, where
$q_{m}:=p_{m+1}/p_{m}$ . In its turn, this one is shown to be isomorphic to
the full crossed product $C(\widehat{\sf G}\,)\\!\rtimes{\sf G}$ . The dual
$\widehat{\sf G}$ can be identified with the product group
$\prod_{m\in\mathbb{N}}{\sf G}_{m}$ .

###  3.4 CAR algebras

Let $\big{(}\mathcal{R},(\cdot|\cdot)\big{)}$ be an infinitely dimensional
separable Hilbert space and $a:\mathcal{R}\to\mathbb{B}(\mathcal{H})$ a
representation of the canonical anticommutation relations in the Hilbert space
$\mathcal{H}$ , generating the unital $C^{*}$-algebra ${\rm
CCR}(\mathcal{R})\subset\mathbb{B}(\mathcal{H})$ . Thus $a$ is lineal and for
every $r,s\in\mathcal{R}$ one has

$a(r)a(s)+a(s)a(r)=0\,,\quad a^{*}(r)a(s)+a^{*}(s)a(r)=(r|s)\,.$

It is known that ${\rm CCR}(\mathcal{R})$ is isomorphic to the UHF algebra
$\mathbf{M}(2^{\infty}):=\underset{m\in\mathbb{N}}{\rm
lim\,ind}\,\mathbf{M}^{2^{m}}$ as well as with the infinite tensor product
$\bigotimes_{m\in\mathbb{N}}\mathbf{M}^{2}$.

Let also $V:\widehat{\sf G}\to\mathbb{B}(\mathcal{R})$ be a strongly
continuous unitary representation of the compact Abelian group $\widehat{\sf
G}$ . Associated to $V$ there is a continuous action $\upsilon:\widehat{\sf
G}\to{\rm Aut}\big{[}{\rm CCR}(\mathcal{R})\big{]}$ , uniquely defined on
generators by

$\upsilon_{\chi}\big{[}a(r)\big{]}=a\big{[}V_{\chi}(r)\big{]}\,,\quad\forall\,r\in\mathcal{R}\,,\,\chi\in\widehat{\sf
G}\,.$

We denote by ${\sf G}$ the dual of $\widehat{\sf G}$ ; it is a discrete
Abelian group. We already have the grading

${\rm CCR}(\mathcal{R})=\widetilde{\bigoplus}_{g\in{\sf G}}{\rm
CCR}(\mathcal{R})^{\upsilon}_{g}$

in terms of spectral subspaces of the action $\upsilon$ . The next corollary
follows directly from Theorem 2.4. Note the generality: any representation
$\big{(}\widehat{\sf G},V\big{)}$ provides a result.

######  Corollary 3.8.

The corresponding $\ell^{1}$-algebra

$\ell^{1}\big{(}{\rm CCR}(\mathcal{R})\big{)}=\bigoplus_{g\in{\sf
G}}^{1,\upsilon}{\rm CCR}(\mathcal{R})^{\upsilon}_{g}$

is symmetric and inverse closed in ${\rm CCR}(\mathcal{R})$ and in
$\mathbb{B}(\mathcal{H})$ .

It is clear that $a(r)\in{\rm CCR}(\mathcal{R})^{\upsilon}_{g}$ if and only if
$r\in\mathcal{R}^{V}_{g}$, meaning by definition that $V_{\chi}(r)=\chi(g)r$
for all $\chi\in\widehat{\sf G}$ . Similarly, $a^{*}(r)\in{\rm
CCR}(\mathcal{R})^{\upsilon}_{g}$ if and only if
$r\in\mathcal{R}^{V}_{g^{-1}}$ .

######  Example 3.9.

If $V:\mathbb{T}\to\mathbb{B}(\mathcal{R})$ is given by $V_{\tau}(r):=e^{2\pi
i\tau}r$ and the group duality is implemented by $\tau(k):=e^{2\pi ik\tau}$
($k\in\mathbb{Z}$) , then $\mathcal{R}^{V}_{1}=\mathcal{R}$ , while
$\mathcal{R}^{V}_{k}=\\{0\\}$ for any $k\neq 1$ . Thus $a(r)\in{\rm
CCR}(\mathcal{R})^{\upsilon}_{1}$ and $a^{*}(r)\in{\rm
CCR}(\mathcal{R})^{\upsilon}_{-1}$ , from which it follows that

$a^{*}(r_{1})\dots a^{*}(r_{n})a(s_{1})\dots a(s_{m})\in{\rm
CCR}(\mathcal{R})^{\upsilon}_{m-n}\,,\quad\forall\,r_{1},\dots,r_{n},s_{1},\dots
s_{m}\in\mathcal{R}\,.$

###  3.5 Wiener-Hopf algebras associated to quasi-lattice ordered groups

One can treat this subject by using partial crossed products, as in Subsection
2.2, due to results from [27, 10, 9]. We found it easier for our purposes to
use the initial article [23], combined with Theorem 2.4.

We fix a sub-monoid ${\sf P}$ of a discrete amenable group ${\sf G}$ such that
${\sf P}\cap{\sf P}^{-1}=\\{{\sf e}\\}$ . One defines a left-invariant order
relation in ${\sf G}$ by $g\leq h$ iff $g^{-1}h\in{\sf P}$. The ordered group
$({\sf G},{\sf P})$ is quasi-lattice ordered if any $g\in{\sf P}{\sf P}^{-1}$
has a least upper bound in ${\sf P}$. Many examples may be found at [23,
pag.23].

Consider the Hilbert space $\ell^{2}({\sf P})$ with its usual orthonormal
basis $\\{e_{q}\\}_{q\in{\sf P}}$. For $p\in{\sf P}$, consider the bounded
linear operator $W_{p}\in\mathbb{B}(\ell^{2}({\sf P}))$ defined by

$W_{p}(e_{q})=e_{pq}\,,\quad\forall\,q\in{\sf P}.$

We refer to $W$ as the regular semigroup of isometries of ${\sf P}$. The
$C^{*}$-algebra of operators on $\ell^{2}({\sf P})$ generated by the range of
$W$ is called the Wiener-Hopf algebra of $\,{\sf P}$ and denoted by
$\mathfrak{W}({\sf P})$ .

For every $g\in{\sf P}{\sf P}^{-1}$ we define the closed subspace

$\mathfrak{W}({\sf
P})_{g}:=\overline{\textrm{span}}\,\big{\\{}W_{p}W_{q}^{*}\,\big{|}\,pq^{-1}\\!=g\big{\\}}$

and set $\mathfrak{W}({\sf P})_{g}=\\{0\\}$ if $g\notin{\sf P}{\sf P}^{-1}$. A
characterization of the Abelian $C^{*}$-subalgebra $\mathfrak{W}({\sf P})_{\sf
e}$ is

$\mathfrak{W}({\sf P})_{\sf e}=\left\\{T\in\mathfrak{W}({\sf
P})\\!\mid\\!T\textrm{ has a diagonal matrix relatively to the canonical basis
of }\ell^{2}({\sf P})\right\\}.$

In Section 3 of Nica’s paper [23] it is shown that that the collection
$\big{\\{}\mathfrak{W}({\sf P})_{g}\,\big{|}\,g\in{\sf G}\big{\\}}$ provides a
topologically graded structure of the Wiener-Hopf algebra. We can apply now
Theorem 2.4 and get

######  Corollary 3.10.

The Banach ∗-algebra $\ell^{1}\Big{(}\widehat{\bigoplus}_{g\in{\sf
G}}\,\mathfrak{W}({\sf P})_{g}\Big{)}$ is symmetric and inverse closed in
$\mathbb{B}\big{[}\ell^{2}({\sf P})\big{]}$ .

######  Remark 3.11.

Suppose that ${\sf P}$ is finitely generated (a slightly more general
assumption is possible). Then $\ell^{1}\Big{(}\widehat{\bigoplus}_{g\in{\sf
G}}\,\mathfrak{W}({\sf P})_{g}\Big{)}$ is also Fredholm inverse closed. We
only sketch a proof. As in [27, 10, 9], one has an isomorphism
$\mathfrak{W}({\sf P})\cong\mathfrak{W}({\sf P})_{\sf
e}\rightthreetimes_{\theta}{\sf G}$ for a certain partial action $\theta$ of
${\sf G}$ on the diagonal Abelian algebra. This one is induced from a
(Bernoulli-type) partial topological action $\Theta$ on its Gelfand spectrum
$\Omega_{\sf e}$ , which is a compactification of the monoid ${\sf P}$. This
compactification is regular (${\sf P}$ is an open dense orbit) under the
stated hypothesis on ${\sf P}$. It is shown in [23, 6.3] that this regularity
is equivalent with the fact that the ideal of all compact operators,
identified with $C({\sf P})\rightthreetimes_{\theta}{\sf G}$ , is contained in
$\mathfrak{W}({\sf P})\,.$ Then the result follows easily from our Corollary
2.18.

###  3.6 Higher rank graph algebras

In this subsection we are going to use constructions from [18] and [28].

######  Definition 3.12.

A ${\rm k}$-graph (higher rank graph) is a countable category $\Lambda$
endowed with a functor ${\rm d}:\Lambda\to\mathbb{N}^{\rm k}$ (the degree
functor) satisfying the factorization property: For every $\lambda\in\Lambda$
such that ${\rm d}(\lambda)=\mathfrak{m}+\mathfrak{n}$ , there exist unique
elements $\mu,\nu\in\Lambda$ such that ${\rm d}(\mu)=\mathfrak{m}$ , ${\rm
d}(\nu)=\mathfrak{n}$ and $\lambda=\mu\nu$ .

If ${\rm k}=1$ , one identifies a $1$-graph with the path category
$E^{*}\\!:=\bigsqcup_{n\in\mathbb{N}}E^{n}$ of a directed graph
$\big{(}s,r:E^{1}\to E^{0}\big{)}$ and ${\rm d}(\lambda)$ is the usual length
of a path $\lambda:=e_{1}\dots e_{{\rm d}(\lambda)}$ . Accordingly, for the
${\rm k}$-graph $\big{(}{\rm d}:\Lambda\to\mathbb{N}^{\rm k}\big{)}$ , one
denotes by ${\rm s},{\rm r}:\Lambda\to\Lambda^{0}\equiv{\rm Obj}(\Lambda)$ the
source and the range map in the category. One has the decomposition

$\Lambda=\bigsqcup_{\mathfrak{n}\in\mathbb{N}^{\rm
k}}\Lambda^{\mathfrak{n}}\equiv\bigsqcup_{\mathfrak{n}\in\mathbb{N}^{\rm
k}}{\rm d}^{-1}(\mathfrak{n})\,.$

For elements from the object set $\Lambda^{0}$ (called vertices) one prefers
notations as $v,w$ . We also set set
$\Lambda^{\mathfrak{n}}(v):=\Lambda^{\mathfrak{n}}\cap{\rm r}^{-1}(v)$
(”paths” of ”length” $\mathfrak{n}\in\mathbb{N}^{\rm k}$ ending in
$v\in\Lambda^{0}$ ).

######  Definition 3.13.

One says that the ${\rm k}$-graph is admissible if every
$\Lambda^{\mathfrak{n}}(v)$ is finite (raw-finite ${\rm k}$-graph) and non-
void (no sources) and $\Lambda^{0}$ is finite (insuring that the
$C^{*}$-algebra below is unital, with unit $\sum_{v\in\Lambda^{0}}{\sf
S}_{v}$).

######  Definition 3.14.

Let $\Lambda\overset{{\rm d}}{\longrightarrow}\mathbb{N}^{\rm k}$ be an
admissible ${\rm k}$-graph. A Cuntz-Krieger $\Lambda$-family is a set $\\{{\sf
T}_{\lambda}\\!\mid\\!\lambda\in\Lambda\\}$ of partial isometries in a
$C^{*}$-algebra $\mathfrak{D}$ such that

  1. (a)

the elements $\big{\\{}{\sf T}_{v}\\!\mid\\!v\in\Lambda^{0}\big{\\}}$ are
mutually orthogonal projections,

  2. (b)

if ${\rm s}(\lambda)={\rm r}(\mu)$ , then ${\sf T}_{\lambda}{\sf T}_{\mu}={\sf
T}_{\lambda\mu}$ ,

  3. (c)

${\sf T}^{*}_{\lambda}{\sf T}_{\lambda}={\sf T}_{{\rm s}(\lambda)}$ for every
$\lambda\in\Lambda$ ,

  4. (d)

${\sf T}_{v}=\sum_{\lambda\in\Lambda^{\mathfrak{n}}(v)}{\sf T}_{\lambda}{\sf
T}^{*}_{\lambda}$ for every $v\in\Lambda^{0}$ and
$\mathfrak{n}\in\mathbb{N}^{\rm k}$.

We define $C^{*}(\Lambda,{\rm d})\equiv\mathfrak{C}(\Lambda)$ to be the
universal $C^{*}$-algebra generated by a Cuntz-Krieger $\Lambda$-family
$\\{{\sf S}_{\lambda}\\!\mid\\!\lambda\in\Lambda\\}$ . Universality means that
for every Cuntz-Krieger $\Lambda$-family $\\{{\sf
T}_{\lambda}\\!\mid\\!\lambda\in\Lambda\\}\subset\mathfrak{D}$ , there exists
a unique $C^{*}$-morphism $\pi_{\sf T}:\mathfrak{C}(\Lambda)\to\mathfrak{D}$
such that $\pi_{\sf T}({\sf S}_{\lambda})={\sf T}_{\lambda}$ for every
$\lambda$ .

We identify now families of symmetric Banach ∗-algebras, starting with
functors ${\sf F}:\Lambda\to{\sf G}$ , where ${\sf G}$ is supposed to be a
discrete Abelian group. One particular case is ${\sf G}=\mathbb{N}^{\rm k}$
and ${\sf F}={\rm d}$ , but many others are possible. For every $g\in{\sf G}$
one sets

$\mathfrak{C}(\Lambda)_{g}:=\overline{\rm span}\big{\\{}{\sf S}_{\lambda}{\sf
S}^{*}_{\mu}\,{\big{|}\,\sf F}(\lambda){\sf F}(\mu)^{-1}=g\big{\\}}\,.$

It turns out that these are spectral subspaces of a dual action $\alpha^{\sf
F}:\widehat{\sf G}\to{\rm Aut}\big{[}\mathfrak{C}(\Lambda)\big{]}$ , uniquely
determined (use the universal property of $\mathfrak{C}(\Lambda)$) by

$\alpha^{\sf F}_{\chi}({\sf S}_{\lambda})=\chi\big{[}{\sf
F}(\lambda)\big{]}{\sf
S}_{\lambda}\,,\quad\forall\,\lambda\in\Lambda\,,\,\chi\in\widehat{\sf G}\,.$

Applying Theorem 2.8 one gets

######  Corollary 3.15.

Let $\Lambda\overset{{\rm d}}{\longrightarrow}\mathbb{N}^{\rm k}$ be an
admissible ${\rm k}$-graph and ${\sf F}:\Lambda\to{\sf G}$ a functor with
values in an Abelian discrete group. Then

$\ell^{1}\big{(}\mathfrak{C}(\Lambda)\big{)}:=\bigoplus_{g\in{\sf
G}}^{1,\alpha^{\sf F}}\,\overline{\rm span}\big{\\{}{\sf S}_{\lambda}{\sf
S}^{*}_{\mu}\,{\big{|}\,\sf F}(\lambda){\sf F}(\mu)^{-1}=g\big{\\}}$

is a reduced and symmetric Banach ∗-algebra, which is inverse closed in
$\mathfrak{C}(\Lambda)$ .

######  Definition 3.16.

On $\mathbb{N}^{\sf k}$ one considers the order
$\mathfrak{m}\leq\mathfrak{n}\,\Leftrightarrow\,\mathfrak{m}_{i}\leq\mathfrak{n}_{i}\,,\forall\,i$
. Let $\Lambda\overset{{\rm d}}{\longrightarrow}\mathbb{N}^{\rm k}$ be an
admissible ${\rm k}$-graph.

  1. (a)

We say that $\rho$ is a minimal common extension of $\mu,\nu\in\Lambda$ , and
we write $\rho\in{\rm MCE}(\mu,\nu)$ , if ${\rm d}(\rho)=\max\\{{\rm
d}(\mu),{\rm d}(\nu)\\}$ and $\rho=\mu\mu^{\prime}=\nu\nu^{\prime}$ for some
$\mu^{\prime},\nu^{\prime}\in\Lambda$ .

  2. (b)

The ${\sf k}$-graph is called aperiodic if for every $\mu,\nu\in\Lambda$ with
${\rm s}(\mu)={\rm s}(\nu)$ there exists $\lambda\in\Lambda$ with ${\rm
r}(\lambda)={\rm s}(\mu)$ and ${\rm MCE}(\mu\lambda,\nu\lambda)=\emptyset$ .

The two notions are easier to visualize for ${\sf k}=1$ . In this case, for
instance, ${\rm MCE}(\mu,\nu)$ is either void, or a singleton (one of the two
elements) and aperiodicity boils down to the well-known condition (L).

######  Proposition 3.17.

Let $\Lambda\overset{{\rm d}}{\longrightarrow}\mathbb{N}^{\rm k}$ be an
admissible and aperiodic ${\rm k}$-graph and ${\sf F}:\Lambda\to{\sf G}$ a
functor with values in an Abelian discrete group. Let $\\{{\sf
T}_{\lambda}\\!\mid\\!\lambda\in\Lambda\\}$ be a Cuntz-Krieger
$\Lambda$-family of partial isometries in the Hilbert space $\mathcal{H}$ such
that ${\sf T}_{v}\neq 0$ for every $v\in\Lambda^{0}$. Then (the quite obvious
action $\beta^{\sf F}$ is explained in the proof)

$\bigoplus_{g\in{\sf G}}^{1,\beta^{\sf F}}\,\overline{\rm span}\big{\\{}{\sf
T}_{\lambda}{\sf T}^{*}_{\mu}\,{\big{|}\,\sf F}(\lambda){\sf
F}(\mu)^{-1}\\!=g\big{\\}}$ (3.5)

is inverse closed in $\mathbb{B}(\mathcal{H})$ .

###### Proof.

Under the stated assumptions, ”the Cuntz-Krieger unicity theorem” (see [18,
31] for example) states that the unique representation $\pi_{\sf
T}:\mathfrak{C}(\Lambda)\to\mathbb{B}(\mathcal{H})$ satisfying

$\pi_{\sf T}({\sf S_{\lambda}})={\sf
T}_{\lambda}\,,\quad\forall\lambda\in\Lambda$

(provided by the universal property) is injective. We denote by
$\mathfrak{C}({\sf T})$ its range, so that $\mathfrak{C}(\Lambda)$ and
$\mathfrak{C}({\sf T})$ are isomorphic. Theorem 2.4 states then that $\pi_{\sf
T}\Big{[}\ell^{1}\big{(}\mathfrak{C}(\Lambda)\big{)}\Big{]}$ is inverse closed
in $\mathfrak{C}({\sf T})$ and in $\mathbb{B}(\mathcal{H})$ . There is an
action $\beta^{\sf F}\\!:\widehat{\sf G}\to{\rm Aut}\big{[}\mathfrak{C}({\sf
T})\big{]}$ such that

$\beta^{\sf F}_{\chi}({\sf T}_{\lambda})=\chi[{\sf F}(\lambda)]{\sf
T}_{\lambda}\,,\quad\forall\,\chi\in\widehat{\sf G}\,,\,\lambda\in\Lambda$

and $\pi_{T}$ intertwines the two actions. It follows that $\pi_{\sf
T}\Big{[}\ell^{1}\big{(}\mathfrak{C}(\Lambda)\big{)}\Big{]}$ coincides with
(3.5). ∎

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D. Jauré and M. Măntoiu:

Facultad de Ciencias, Departamento de Matemáticas, Universidad de Chile

Las Palmeras 3425, Casilla 653 , Santiago, Chile.

E-mails<EMAIL_ADDRESS>and<EMAIL_ADDRESS>