# Unitary groups, $K$-theory and traces

Pawel Sarkowicz<EMAIL_ADDRESS>Department of Mathematics and Statistics,
University of Ottawa, 75 Laurier Ave. East, Ottawa, ON, K1N 6N5 Canada

###### Abstract.

We show that continuous group homomorphisms between uni-tary groups of unital
C*-algebras induce maps between spaces of continuous real-valued affine
functions on the trace simplices. Under certain $K$-theoretic regularity
conditions, these maps can be seen to commute with the pairing between $K_{0}$
and traces. If the homomorphism is contractive and sends the unit circle to
the unit circle, the map between spaces of continuous real-valued affine
functions can further be shown to be unital and positive (up to a minus sign).

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries and notation
    1. 2.1 Notation
    2. 2.2 The de la Harpe–Skandalis determinant and Thomsen’s variant
    3. 2.3 Thomsen’s variant
    4. 2.4 The $KT_{u}$-invariant
  3. 3 Continuous unitary group homomorphisms and traces
  4. 4 The order on $\operatorname{Aff}T(\cdot)$
  5. 5 A slight general linear variant
  6. 6 Final remarks and open questions

##  1\. Introduction

Unitary groups of C*-algebras have been long studied, and for many classes of
operator algebras they form a complete invariant. In [Dye53], Dye studied the
unitary group isomorphism problem between non-atomic W*-algebras, with the
assumption of _weak bicontinuity_ of the isomorphism. He later showed that the
unitary group, this time as an algebraic object, determined the type of a
factor [Dye55] (except for type $\text{I}_{2n}$). He showed that such group
isomorphisms were the restrictions of a *-isomorphism or a conjugate linear
*-isomorphism multiplied by a possibly discontinuous character ([Boo98,
Appendix A] gives exposition). Sakai generalized Dye’s results to show that
any uniformly continu-ous unitary group isomorphism between AW*-factors comes
from a *-isomor-phism or conjugate-linear *-isomorphism [Sak55] (see also
[Yen56] for general AW*-algebras which have no component of type $I_{n}$).

Dye’s method was generalized to large classes of real rank zero C*-algebras by
Al-Rawashdeh, Booth and Giordano in [ARBG12], where they applied the method to
obtain induced maps between $K$-theory, with a general linear variant being
done by Giordano and Sierakowski in [GS16]. The stably finite and purely
infinite cases were handled separately. The unital, simple AH-algebras of slow
dimension growth and of real rank zero were classified by the topological
group isomorphism class of their unitary groups (or general linear groups),
and the unital, simple, purely infinite UCT algebras were classified via the
algebraic isomorphism classes of their unitary groups (or general linear
groups). These results made use of the abundance of projections in real rank
zero C*-algebras (at least to show there were isomorphic $K_{0}$-groups), and
made use of the Dadarlat-Elliott-Gong [Dad95, Gon97] and Kirchberg-Phillips
[Phi00] classification theorems respectively (see Theorems 3.3.1 and 8.4.1 of
[Rør02] for each respective case).

In [Pat83], it was proven by Paterson that two unital C*-algebras are iso-
morphic if and only there is an isometric isomorphism of the unitary groups
which acts as the identity on the circle. In a similar vein, the metric
structure of the unitary group has also played a role in determining the
Jordan *-algebra structure on C*-algebras. In [HM14], Hatori and Molnár showed
that two unital C*-algebras are Jordan *-isomorphic if and only if their
unitary groups are isometric as metric spaces, not taking into account any
algebraic structure.

Chand and Robert have shown in [CR23] that if $A$ and $B$ are prime traceless
C*-algebras with full square zero elements such that $U^{0}(A)$, the subgroup
of unitaries which are path connected to the identity, is algebraically
isomorphic to $U^{0}(B)$, then $A$ is either isomorphic or anti-isomorphic to
$B$. In fact, the group isomorphism is the restriction of a *-isomorphism or
anti-*-isomorphism which follows from the fact that unitary groups associated
to these C*-algebras have certain automatic continuity properties that allow
one to use character-izations of _commutativity preserving maps_ [Bre93] (see
[AM03]). Chand and Robert also show that if $A$ is a unital separable
C*-algebra with at least one tracial state, then $U^{0}(A)$ admits
discontinuous automorphisms. Thus the existence of traces is an obstruction to
classification via algebraic structure on the unitary groups – at least an
obstruction to unitary group homomorphisms being the restrictions of
*-homomorphisms or anti-*-homomorphisms.

In this paper, we show that uniformly continuous unitary group homo-morphisms
yield maps between traces which have several desirable $K$-theoretic
properties – especially under stricter continuity assumptions. Namely, that
the homomorphism sends the circle to the circle and is contractive, which
would be automatic if it had a lift to a *-homomorphism or conjugate-linear
*-homomorphism.

We state our main results. Recall that if $A$ is a unital C*-algebra, $T(A)$
denotes the simplex of tracial states and $\operatorname{Aff}T(A)$ is the real
Banach space of continuous affine functions $T(A)\to\mathbb{R}$.

######  Theorem A (Corollary 3.6).

Let $A,B$ be unital C*-algebras. If $\theta:U^{0}(A)\to U^{0}(B)$ is a
uniformly continuous group homomorphism, then there exists a bounded
$\mathbb{R}$-linear map
$\Lambda_{\theta}:\operatorname{Aff}T(A)\to\operatorname{Aff}T(B)$ such that

(1.1)
${\pi_{1}(U^{0}(A))}$${\operatorname{Aff}T(A)}$${\pi_{1}(U^{0}(B))}$${\operatorname{Aff}T(B)}$$\scriptstyle{\tilde{\Delta}^{1}_{A}}$$\scriptstyle{\pi_{1}(\theta)}$$\scriptstyle{\Lambda_{\theta}}$$\scriptstyle{\tilde{\Delta}^{1}_{B}}$

commutes.

Here $\pi_{1}(\theta)$ is the map between fundamental groups induced by
$\theta$, and, for a C*-algebra $A$, $\tilde{\Delta}^{1}_{A}$ is the _pre-
determinant_ (used in the definition of the de la Harpe-Skandalis determinant
associated to the universal trace) that takes a piece-wise smooth path in
$U^{0}(A)$ beginning at the unit to an element of $\operatorname{Aff}T(A)$.
See Section 2.2 for details.

Recall that the $K_{0}$-group of a unital C*-algebra can be identified with
the fundamental group $\pi_{1}(U_{\infty}^{0}(A))$. Restricting to C*-algebras
with sufficient $K_{0}$-regularity – by this we mean C*-algebras whose
$K_{0}$-group can be realized as loops in the connected component of its
unitary group – we get a map between $K_{0}$-groups and a map between spaces
of continuous real-valued affine functions on the trace simplex which commute
with the pairing.

######  Corollary B (Corollary 3.6).

Let $A,B$ be unital C*-algebras such that the canonical maps

(1.2) $\pi_{1}(U^{0}(A))\to K_{0}(A)\text{ and }\pi_{1}(U^{0}(B))\to K_{0}(B)$

are isomorphisms. If $\theta:U^{0}(A)\to U^{0}(B)$ is a continuous group
homomorphism then there exists a bounded linear map
$\Lambda_{\theta}:\operatorname{Aff}T(A)\to\operatorname{Aff}T(B)$ such that

(1.3)
${K_{0}(A)}$${\operatorname{Aff}T(A)}$${K_{0}(B)}$${\operatorname{Aff}T(B)}$$\scriptstyle{\rho_{A}}$$\scriptstyle{K_{0}(\theta)}$$\scriptstyle{\Lambda_{\theta}}$$\scriptstyle{\rho_{B}}$

commutes, where $K_{0}(\theta)$ is the map induced by $\pi_{1}(\theta)$
together with the isomorphisms of (1.2).

C*-algebras satisfying the above hypothesis are quite common – for example
C*-algebras having stable rank one [Rie87] or that are $\mathcal{Z}$-stable
[Jia97] have this property. Viewing $\operatorname{Aff}T(A)$ and
$\operatorname{Aff}T(B)$ as partially ordered real Banach spaces (under the
uniform norm) with order units, it is not however true that $\Lambda_{\theta}$
is unital or positive (see Example 3.4). This is remedied by adding stricter
continuity assumptions on the homomorphism $\theta$ (and possibly by replacing
$\Lambda_{\theta}$ with $-\Lambda_{\theta}$).

When $\theta:U(A)\to U(B)$ is contractive, injective and sends the circle to
the circle, then we show (Lemma 4.3) that either $\Lambda_{\theta}$ or
$-\Lambda_{\theta}$ is unital and positive, and therefore $\theta$ induces a
map between $K$-theory and traces in such a manner that respects the pairing
(which in turn gives a map between Elliott invariants for certain simple
C*-algebras). As a consequence, we can identify certain unitary subgroups with
C*-subalgebras by using $K$-theoretic classification of embeddings [CGS+23].

######  Theorem C (Corollary 4.13).

Let $A$ be a unital, separable, simple, nuclear C*-algebra satisfying the UCT
which is either $\mathcal{Z}$-stable or has stable rank one, and $B$ be a
unital, separable, simple, nuclear $\mathcal{Z}$-stable C*-algebra. If there
is a contractive injective group homomorphism $U(A)\to U(B)$ which maps the
circle to the circle, then there is a unital embedding $A\hookrightarrow B$.

This paper is structured as follows. In Section 3 we use a continuous unitary
group homomorphism to construct a map between spaces of continuous affine
functions on the trace simplices, and use the de la Harpe-Skandalis
determinant to show that this map has several desirable properties with
respect to the map induced on the fundamental groups of the unitary groups. In
Section 4 we discuss how our map between spaces of affine functions respects
or flips the order under certain continuity assumptions on the unitary group
homomorphism. In Section 5 we discuss a slight general linear variant. We
finish in Section 6 with some open questions.

## Acknowledgements

Many thanks to my PhD supervisors Thierry Giordano and Aaron Tikuisis for many
helpful discussions. Thanks to the authors of [CGS+23] for sharing a draft of
their paper. Finally, thanks to the referee for asking for clarification on
the general linear variant, which led me to consider a counter-example to
$\mathbb{C}$-linearity.

##  2\. Preliminaries and notation

###  2.1. Notation

For a group $G$, we denote by $DG$ the derived subgroup of $G$, i.e.,

(2.1) $DG:=\langle ghg^{-1}h^{-1}\mid g,h\in G\rangle.$

If $G$ has an underlying topology, we denote by $CG$ the closure of $DG$ and
$G^{0}$ the connected component of the identity.

For a unital C*-algebra $A$, $U(A)$ denotes the unitary group of $A$, while
$U^{0}(A)$ denotes the connected component of the identity in $U(A)$. For
$n\in\mathbb{N}$, we write $U_{n}(A):=U(M_{n}(A))$,
$U_{n}^{0}(A):=U^{0}(M_{n}(A))$, and we set

(2.2) $U_{\infty}(A):=\underset{\to}{\lim}\,U_{n}(A),$

to be the inductive limit with connecting maps $U_{n}(A)\ni u\mapsto u\oplus
1\in U_{n+1}(A)$. This makes $U_{\infty}(A)$ both a topological space (with
the inductive limit topology) and a group.111As pointed out in footnote 58 of
[CGS+23], $U_{\infty}(A)$ is not in general a topological group since
multiplication is not in general jointly continuous in this topology. We have
general linear analogues by replacing $U$ with $GL$, where $GL(A)$ denotes the
group of invertible elements of $A$. Similarly, we define
$M_{\infty}(A)=\underset{\to}{\lim}\,M_{n}(A)$ (as an algebraic direct limit)
with connecting maps $x\mapsto x\oplus 0$. If $E$ is real Banach space and
$\tau:A_{sa}\to E$ is a linear map that is tracial (i.e.,
$\tau(a^{*}a)=\tau(aa^{*})$ for all $a\in A$), we extend this canonically to
$(M_{\infty}(A))_{sa}$ by setting $\tau((a_{ij})):=\sum_{i}\tau(a_{ii})$ for
$(a_{ij})\in(M_{n}(A)_{sa}$.

We write $\pi_{0}(X)$ for the space of connected components of a topological
space $X$, and $\pi_{1}(X)$ for the fundamental group of $X$ with
distinguished base point. In our case, we will usually have $X=U_{n}(A)$ or
$X=U_{n}^{0}(A)$, for $n\in\mathbb{N}\cup\\{\infty\\}$, with the base point
being the unit.

For a unital C*-algebra $A$, we let $K_{0}(A),K_{1}(A)$ be the topological
$K$-groups of $A$. We will often use the identification of $K_{0}(A)$ with the
fundamental group $\pi_{1}(U_{\infty}^{0}(A))$ (see for example [RLL00,
Chapter 11.4]). The set of tracial states on $A$ will be denoted $T(A)$, which
is a Choquet simplex ([Sak12, Theorem 3.1.18]), and we denote by
$\operatorname{Aff}T(A)$ the set of continuous affine functions
$T(A)\to\mathbb{R}$, which is an interpolation group with order unit (see
[Goo86, Chapter 11]). For unital $A$, the pairing map
$\rho_{A}:K_{0}(A)\to\operatorname{Aff}T(A)$ is defined as follows: if $x\in
K_{0}(A)$, we can write $x=[p]-[q]$ where $p,q\in M_{n}(A)$ are projections,
and then

(2.3) $\rho_{A}(x)(\tau):=\tau(p-q),\ \ \ \tau\in T(A).$

###  2.2. The de la Harpe–Skandalis determinant and Thomsen’s variant

We recall the definition of the unitary variant of the de la Harpe–Skandalis
determinant [dlHS84a] (see [dlH13] for a more in-depth exposition). By a
bounded trace we mean a bounded linear map $\tau:A_{sa}\to E$, where $E$ is a
real Banach space, such that $\tau(a^{*}a)=\tau(aa^{*})$ for all $a\in A$. For
$n\in\mathbb{N}\cup\\{\infty\\}$, a bounded trace $\tau:A_{sa}\to E$, and a
piece-wise smooth path $\xi:[0,1]\to U_{n}(A)$, set

(2.4) $\tilde{\Delta}_{\tau}^{n}(\xi):=\int_{0}^{1}\tau\left(\frac{1}{2\pi
i}\xi^{\prime}(t)\xi(t)^{-1}\right)dt,$

where this integral is just the Riemann integral in $E$.222Note that
$\xi^{\prime}(t)\xi(t)$ is skew-adjoint by [Phi92, Proposition 1.4] so that
$\frac{1}{2\pi i}\xi^{\prime}(t)\xi(t)^{-1}$ is self-adjoint. We state the
unitary variant of [dlHS84a, Lemme 1].

######  Proposition 2.1.

Let $\tau:A_{sa}\to E$ be a bounded trace and
$n\in\mathbb{N}\cup\\{\infty\\}$. The map $\tilde{\Delta}_{\tau}^{n}$, which
takes a piece-wise smooth path in $U_{n}^{0}(A)$ to an element in $E$, has the
following four properties:

  1. (1)

it takes pointwise products to sums: if $\xi_{1},\xi_{2}$ are two piece-wise
smooth paths, then

(2.5)
$\tilde{\Delta}_{\tau}^{n}(\xi_{1}\xi_{2})=\tilde{\Delta}_{\tau}^{n}(\xi_{1})+\tilde{\Delta}_{\tau}^{n}(\xi_{2}),$

where $\xi_{1}\xi_{2}$ is the piece-wise smooth path
$t\mapsto\xi_{1}(t)\xi_{2}(t)$ from $\xi_{1}(0)\xi_{2}(0)$ to
$\xi_{1}(1)\xi_{2}(1)$;

  2. (2)

if $\|\xi(t)-1\|<1$ for all $t\in[0,1]$, then

(2.6) $2\pi
i\tilde{\Delta}_{\tau}^{n}(\xi)=\tau\big{(}\log\xi(1)-\log\xi(0)\big{)};$

  3. (3)

$\tilde{\Delta}_{\tau}^{n}(\xi)$ depends only on the continuous homotopy class
of $\xi$;

  4. (4)

if $p\in M_{n}(A)$ is a projection, then the path $\xi_{p}:[0,1]\to
U_{n}^{0}(A)$ given by $\xi_{p}(t):=pe^{2\pi it}+(1-p)$ satisfies
$\tilde{\Delta}_{\tau}^{n}(p)=\tau(p)$.

The de la Harpe–Skandalis determinant associated to $\tau$ (at the
$n^{\text{th}}$ level) is then the map

(2.7) $\Delta_{\tau}^{n}:U_{\infty}^{0}(A)\to
E/\tilde{\Delta}_{\tau}^{n}(\pi_{1}(U_{n}^{0}(A)))$

given by $\Delta_{\tau}^{n}(x):=[\tilde{\Delta}_{\tau}^{n}(\xi_{x})]$ where
$\xi_{x}$ is any piece-wise smooth path $\xi_{x}:[0,1]\to U_{n}^{0}(A)$ from 1
to $x$. This is a well-defined group homomorphism (using Proposition 2.1) to
an abelian group and therefore factors through the derived group, i.e.,
$DU_{n}^{0}(A)\subseteq\ker\Delta_{\tau}^{n}$. For the $n=\infty$ case, we
just write $\tilde{\Delta}_{\tau}$ and $\Delta_{\tau}$ for
$\tilde{\Delta}_{\tau}^{\infty}$ and $\Delta_{\tau}^{\infty}$ respectively.

We will often be interested in the _universal trace_
$\text{Tr}_{A}:A_{sa}\to\operatorname{Aff}T(A)$, which is given by
$\text{Tr}_{A}(a):=\hat{a}$, where $\hat{a}\in\operatorname{Aff}T(A)$ is the
function given by $\hat{a}(\tau):=\tau(a)$ for $\tau\in T(A)$. We note that in
this case, for $[x]\in K_{0}(A)$, we have that $\text{Tr}(x)=\rho_{A}([x])$.
When considering the universal trace, we will write $\Delta^{n}$ and $\Delta$
for $\Delta_{\text{Tr}}^{n}$ and $\Delta_{\text{Tr}}^{\infty}$ respectively.
If the C*-algebra needs to be specified, we write $\Delta_{A}^{n}$ or
$\Delta_{A}$.

######  Proposition 2.2.

Let $n\in\mathbb{N}\cup\\{\infty\\}$. Every continuous path $\xi:[0,1]\to
U_{n}(A)$ is homotopic to a piece-wise smooth path (even a piece-wise smooth
exponential path if we are in $U_{n}^{0}(A)$). Moreover, there exists $a\in
A_{sa}$ such that $\tilde{\Delta}_{\omega}^{n}(\xi)=\omega(a)$ whenever
$\omega:A_{sa}\to E$ is a bounded trace.

In particular, as $\tilde{\Delta}^{n}$ is homotopy-invariant, it makes sense
to apply $\tilde{\Delta}^{n}$ to any continuous path.

###### Proof.

This is essentially [dlHS84a, Lemme 3]. Take a continuous path $\xi:[0,1]\to
U_{n}(A)$ and choose $k$ such that

(2.8) $\left\|\xi(\frac{j-1}{k})^{-1}\xi(\frac{j}{k})-1\right\|<1\text{ for
all }j=1,\dots,k.$

Then taking

(2.9) $a_{j}:=\frac{1}{2\pi
i}\log\left(\xi(\frac{j-1}{k})^{-1}\xi(\frac{j}{k})\right),j=1,\dots,k,$

$\xi$ will be homotopic to the path

(2.10) $\eta(t)=\xi\left(\frac{j-1}{k}\right)e^{2\pi
i(kt-j+1)a_{j}},t\in\left[\frac{j-1}{k},\frac{j}{k}\right],j=1,\dots,k.$

We note that $\tilde{\Delta}_{\tau}^{n}(\eta)=\sum_{j=1}^{k}\tau(a_{j})$.
Indeed, for simplicity denote by

(2.11) $X_{j}:=\xi\left(\frac{j-1}{k}\right)\text{ and }Y_{j}:=e^{2\pi
i(kt-j+1)a_{j}}.$

Then

(2.12)
$\begin{split}\tilde{\Delta}_{\omega}^{n}(\xi)&=\tilde{\Delta}_{\omega}^{n}(\eta)\\\
&=\sum_{j=1}^{k}\int_{\frac{j-1}{k}}^{\frac{j}{k}}\tau\left(\frac{1}{2\pi
i}X_{j}2\pi ika_{j}Y_{j}Y_{j}^{*}X_{j}^{*}\right)dt\\\
&=\sum_{j=1}^{k}\int_{\frac{j-1}{k}}^{\frac{j}{k}}k\tau(a_{j})dt\\\
&=\sum_{j=1}^{k}\tau(a_{j}).\\\ \end{split}$

If we take $a:=\text{tr}(a^{\prime})$, where $a^{\prime}=\sum_{j=1}^{k}a_{j}$
(here $\text{tr}:M_{\infty}(A)\to A$ is the unnormalized trace), we see that
$\tilde{\Delta}_{\omega}^{n}(\xi)=\omega(a)$. ∎

Let $A_{0}$ consist of elements $a\in A_{sa}$ satisfying $\tau(a)=0$ for all
$\tau\in T(A)$. This is a norm-closed real subspace of $A_{sa}$ such that
$A_{0}\subseteq\overline{[A,A]}$, and there is an isometric identification
$A_{sa}/A_{0}\simeq\operatorname{Aff}T(A)$ sending an element $[a]$ to
$\widehat{a}$. Indeed, it is not hard to see that the map
$A_{sa}/A_{0}\to\operatorname{Aff}T(A)$ given by $[a]\mapsto\hat{a}$ is a
well-defined $\mathbb{R}$-linear map. Moreover [CP79, Theorem 2.9], together
with a convexity argument, gives that this is isometric identification. To see
that we have all the real-valued affine functions, we note that the image of
this map contains constant functions and separates points, so [Goo86,
Corollary 7.4] gives that the image is dense and therefore all of
$\operatorname{Aff}T(A)$ (since this is an isometry). We freely identify
$A_{sa}/A_{0}$ with $\operatorname{Aff}T(A)$.

###  2.3. Thomsen’s variant

Thomsen’s variant of the de la Harpe–Skandalis determinant is the
Hausdorffized version, taking into account the closure of the image of the
homotopy groups. For a bounded trace $\tau:A_{sa}\to E$, we consider the map

(2.13) $\bar{\Delta}_{\tau}^{n}:U_{n}^{0}(A)\to
E/\overline{\tilde{\Delta}_{\tau}^{n}(\pi_{1}(U_{n}^{0}(A)))},$

given by $\bar{\Delta}_{\tau}^{n}(x):=[\tilde{\Delta}_{\tau}^{n}(\xi_{x})]$
where $\xi_{x}:[0,1]\to U_{n}^{0}(A)$ is any piece-wise smooth path from 1 to
$x\in U_{n}^{0}(A)$. This is similar to the map $\Delta_{\tau}^{n}$, except
the codomain is now the quotient by the closure of the image of the
fundamental group under the pre-determinant (i.e., the Hausdorffization of the
codomain). When considering the universal trace, we just write
$\overline{\Delta}^{n}$ for $\overline{\Delta}_{\text{Tr}}^{n}$ and
$\overline{\Delta}$ for $\overline{\Delta}_{\text{Tr}}^{\infty}$. If the
C*-algebra needs to be specified, we write $\overline{\Delta}_{A}^{n}$ or
$\overline{\Delta}_{A}$.

If one considers the universal trace, the kernel of $\overline{\Delta}^{n}$
can be identified with $CU_{n}^{0}(A)$ (where the closure is taken with
respect to the inductive limit topology in the $n=\infty$ case).

######  Lemma 2.3 (Lemma 3.1, [Tho95]).

Let $A$ be a unital C*-algebra. Then

(2.14) $\ker\overline{\Delta}^{n}=CU_{n}^{0}(A).$

It is not in general true that the kernel of $\Delta^{n}$ can be identified
with the derived group $DU_{n}^{0}(A)$, although there are several positive
results [dlHS84b, Tho93, Ng14, NR17, NgRobert15].

It immediately follows that the quotient of $U_{n}^{0}(A)$ by the closure of
the commutator subgroup (under the inductive limit topology in the $n=\infty$
case) can be identified with a quotient of $\operatorname{Aff}T(A)$.

######  Theorem 2.4 (Theorem 3.2, [Tho95]).

$\overline{\Delta}^{n}$ gives a homeomorphic group iso-morphism

(2.15)
$U_{n}^{0}(A)/CU_{n}^{0}(A)\simeq\operatorname{Aff}T(A)/\overline{\tilde{\Delta}^{n}(\pi_{1}(U_{n}^{0}(A)))}$

for every $n\in\mathbb{N}\cup\\{\infty\\}$. In particular,

(2.16)
$U_{\infty}^{0}(A)/CU_{\infty}^{0}(A)\simeq\operatorname{Aff}T(A)/\overline{\rho_{A}(K_{0}(A))}.$

###  2.4. The $KT_{u}$-invariant

Following [CGS+23], for a unital C*-algebra, we let

(2.17)
$KT_{u}(A):=(K_{0}(A),[1_{A}]_{0},K_{1}(A),\rho_{A},\operatorname{Aff}T(A))$

be the invariant consisting of the $K_{0}$-group, the position of the unit in
$K_{0}$, the $K_{1}$ group, the pairing between $K_{0}$ and traces, and the
continuous real-valued affine functions on the trace simplex (viewed as a
partially ordered Banach space with order unit). For two unital C*-algebras
$A,B$, a $KT_{u}$-morphism

(2.18) $(\alpha_{0},\alpha_{1},\gamma):KT_{u}(A)\to KT_{u}(B),$

will be a triple $(\alpha_{0},\alpha_{1},\gamma)$ consisting of
$\alpha_{0}:K_{0}(A)\to K_{0}(B)$ a group homomor-phism such that
$\alpha_{0}([1_{A}]_{0})=[1_{B}]_{0}$, $\alpha_{1}:K_{1}(A)\to K_{1}(B)$ a
group homomorphism, and
$\gamma:\operatorname{Aff}T(A)\to\operatorname{Aff}T(B)$ is a unital positive
map such that

(2.19)
${K_{0}(A)}$${\operatorname{Aff}T(A)}$${K_{0}(B)}$${\operatorname{Aff}T(B)}$$\scriptstyle{\rho_{A}}$$\scriptstyle{\alpha_{0}}$$\scriptstyle{\gamma}$$\scriptstyle{\rho_{B}}$

commutes.

We note that for large classes of unital simple C*-algebras – for example the
class of unital, simple, separable, nuclear $\mathcal{Z}$-stable C*-algebras
satisfying the UCT – $KT_{u}(\cdot)$ recovers the Elliott invariant.

##  3\. Continuous unitary group homomorphisms and traces

Throughout, $A$ and $B$ will be unital C*-algebras with non-empty trace
simplices, and $\theta:U^{0}(A)\to U^{0}(B)$ will denote a uniformly
continuous group homomorphism between the connected components of the
identities in the respective unitary groups. We will specify any additional
assumptions as we go along. As $\theta$ is a continuous group homomorphism, it
send commutators to commutators and limits of commutators to limits of
commutators. Thus there are induced group homomorphisms

(3.1) $\begin{split}&U^{0}(A)/CU^{0}(A)\to U^{0}(B)/CU^{0}(B)\text{ and }\\\
&U^{0}(A)/DU^{0}(A)\to U^{0}(B)/DU^{0}(B).\end{split}$

Thomsen’s isomorphism [Tho95, Theorem 3] then brings about maps between
quotients of $\operatorname{Aff}T(A)$ and $\operatorname{Aff}T(B)$:

(3.2)
${U^{0}(A)/CU^{0}(A)}$${\operatorname{Aff}T(A)/\overline{\tilde{\Delta}_{A}^{1}(\pi_{1}(U^{0}(A)))}}$${U^{0}(B)/CU^{0}(B)}$${\operatorname{Aff}T(B)/\overline{\tilde{\Delta}_{B}^{1}(\pi_{1}(U^{0}(B)))}.}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

In a similar vein, when $DU^{0}(A)=\ker\Delta_{A}^{1}$ and
$DU^{0}(B)=\ker\Delta_{B}^{1}$, there is a purely algebraic variant of the
above diagram:

(3.3)
${U^{0}(A)/DU^{0}(A)}$${\operatorname{Aff}T(A)/\tilde{\Delta}_{A}^{1}(\pi_{1}(U^{0}(A)))}$${U^{0}(B)/DU^{0}(B)}$${\operatorname{Aff}T(B)/\tilde{\Delta}_{B}^{1}(\pi_{1}(U^{0}(B))).}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

In fact, there is always a diagram as above with $DU^{0}(A)$ and $DU^{0}(B)$
replaced with $\ker\Delta_{A}^{1}$ and $\ker\Delta_{B}^{1}$, respectively, by
Proposition 3.5(3). That is, whether or not the kernel of the determinant
agrees with the commutator subgroup of the connected component of the
identity, we have:

(3.4)
${U^{0}(A)/\ker\Delta_{A}^{1}}$${\operatorname{Aff}T(A)/\tilde{\Delta}_{A}^{1}(\pi_{1}(U^{0}(A)))}$${U^{0}(B)/\ker\Delta_{B}^{1}}$${\operatorname{Aff}T(B)/\tilde{\Delta}_{B}^{1}(\pi_{1}(U^{0}(B))).}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

The diagram in (3.3) is just a special case of (3.4).

In the setting where $\pi_{1}(U^{0}(A))\to K_{0}(A)$ and $\pi_{1}(U^{0}(B))\to
K_{0}(B)$ are surjections, we have induced maps between quotients

(3.5)
$\begin{split}&\operatorname{Aff}T(A)/\overline{\rho_{A}(K_{0}(A))}\to\operatorname{Aff}T(B)/\overline{\rho_{B}(K_{0}(B))},\\\
&\operatorname{Aff}T(A)/\rho_{A}(K_{0}(A))\to\operatorname{Aff}T(B)/\rho_{B}(K_{0}(B))\end{split}$

in the respective Hausdorffized and non-Hausdorffized settings.

One question to be answered is whether or not we can lift the maps on the
right of (3.2) and (3.4) to maps
$\operatorname{Aff}T(A)\to\operatorname{Aff}T(B)$. These spaces have further
structure as dimension groups with order units [Goo86, Chapter 7], so we would
like to be able to alter the lift to get a map which is unital and positive.
We show that we can always lift this map, and altering it to be unital and
positive is possible under a certain continuity assumption on $\theta$.

If we further assume that $K_{0}(A)\simeq\pi_{1}(U^{0}(A))$ and
$K_{0}(B)\simeq\pi_{1}(U^{0}(B))$ in the canonical way (which is true in the
presence of $\mathcal{Z}$-stability by [Jia97] or stable rank one [Rie87]), we
would like this map to be compatible with the group homomorphism

(3.6) $K_{0}(\theta):K_{0}(A)\to K_{0}(B)$

arising from the diagram

(3.7)
${K_{0}(A)}$${\pi_{1}(U^{0}(A))}$${K_{0}(B)}$${\pi_{1}(U^{0}(B)).}$$\scriptstyle{\simeq}$$\scriptstyle{K_{0}(\theta)}$$\scriptstyle{\pi_{1}(\theta)}$$\scriptstyle{\simeq}$

By compatible, we mean that

(3.8)
${K_{0}(A)}$${\operatorname{Aff}T(A)}$${K_{0}(B)}$${\operatorname{Aff}T(B)}$$\scriptstyle{\pi_{1}(\theta)}$$\scriptstyle{\rho_{A}}$$\scriptstyle{\rho_{B}}$

commutes, where the map on the right is the lift coming from maps induced by
(3.2) and (3.4). If our map between spaces of affine continuous functions is
not unital and positive, but we can alter it accordingly, we must do the same
to our map between $K_{0}$. We would still have a commuting diagram as above,
but it would give that maps induced on $K_{0}(\cdot)$ and
$\operatorname{Aff}T(\cdot)$ respect the pairing.

Stone’s theorem [Con85, Section X.5] allows one to recover from a strongly
continuous one parameter family $U(t)$ of unitaries a (possibly unbounded)
self-adjoint operator $X$ such that $U(t)=e^{itX}$ for all $t\in\mathbb{R}$.
If it is a norm-continuous one parameter family of unitaries, one can recover
a bounded self-adjoint operator $X$, and $X$ will lie in the C*-algebra
generated by the unitaries. The use of Stone’s theorem to deduce that
continuous group homomorphisms between unitary groups send exponentials to
exponentials is not new. Sakai used it in the 1950’s in order to show that a
norm-continuous group isomorphism between unitary groups of AW*-algebras are
induced by a *-isomorphism or conjugate-linear *-isomorphism between the
algebras themselves [Sak55]. More recently, this sort of idea has been used to
understand how the metric structure of the unitary groups can be related to
the Jordan *-algebra structure of the algebras [HM14].

######  Lemma 3.1.

Let $A,B$ be unital C*-algebras and $\theta:U^{0}(A)\to U^{0}(B)$ be a
continuous group homomorphism. Suppose that $a\in A_{sa}$ and represent
$B\subseteq\mathcal{B}(\mathcal{H})$ faithfully. Then $(\theta(e^{2\pi
ita}))_{t\in\mathbb{R}}$ is a one-parameter norm-continuous family of
unitaries, and consequently is of the form $(e^{2\pi itb})_{t\in\mathbb{R}}$
for a unique $b\in B_{sa}$.

###### Proof.

Using the fact that $\theta$ is a norm-continuous homomorphism,
$t\mapsto\theta(e^{2\pi ita})$ is a norm-continuous one-parameter family of
unitaries. Stone’s theorem gives that there is a unique self-adjoint
$b\in\mathcal{B}(\mathcal{H})$ such that $\theta(e^{2\pi ita})=e^{2\pi itb}$
for all $t\in\mathbb{R}$. The boundedness of $b$ follows from norm-continuity,
and the uniqueness follows from the fact that $b=\frac{1}{2\pi
it_{0}}\log\theta(e^{2\pi iat_{0}})$ for sufficiently small $t_{0}>0$. This
also gives that $b\in B$ by functional calculus. ∎

Let $S_{\theta}:A_{sa}\to B_{sa}$ be defined via the correspondence given
above:

(3.9) $\theta(e^{2\pi ita})=e^{2\pi itS_{\theta}(a)}\text{ for all
}t\in\mathbb{R}.$

Then $S_{\theta}$ is a bounded $\mathbb{R}$-linear map (see [Sak55, HM14], or
note that it is easy to see that its kernel is closed). It is also easily
checked to respect commutation, and that its canonical extension to a map from
$A$ to $B$ actually sends commutators to commutators, although we will not
explicitly use this. Recall that for a C*-algebra $A$, $A_{0}$ denotes the set
of self-adjoint elements that vanish on every tracial state.

######  Lemma 3.2.

If $\theta:U^{0}(A)\to U^{0}(B)$ is a continuous group homomorphism, then
$S_{\theta}$ is a bounded linear map and the following hold.

  1. (1)

If $\theta$ is injective, then $S_{\theta}$ is injective.

  2. (2)

If $\theta$ is a homeomorphism, then $S_{\theta}$ is bijective.

###### Proof.

As already remarked, $S_{\theta}$ is bounded. Assuming that $\theta$ is
injective, suppose that $S_{\theta}(a)=S_{\theta}(b)$. Then

(3.10) $\theta(e^{2\pi ita})=\theta(e^{2\pi itb})$

for all $t\in\mathbb{R}$. Injectivity of $\theta$ gives that that $e^{2\pi
ita}=e^{2\pi itb}$ for all $t\in\mathbb{R}$. But this implies that $a=b$ since
we can take $t$ appropriately close to 0 and take logarithms.

Now if we further assume that $\theta$ is a homeomorphism, then
$(\theta^{-1}(e^{2\pi itb}))_{t\in\mathbb{R}}\subseteq U^{0}(A)$ is a norm-
continuous one-parameter family of unitaries which we can write as $(e^{2\pi
ita})_{t\in\mathbb{R}}$ for some $a\in A_{sa}$. But then

(3.11) $\theta(e^{2\pi ita})=\theta\circ\theta^{-1}(e^{\pi itb})=e^{2\pi
itb}.$

Thus $S_{\theta}(a)=b$ by the uniqueness in Lemma 3.1. ∎

Recall that a linear map $\tau:A_{sa}\to E$, where $E$ is a real Banach space,
is a bounded trace is if it is a bounded $\mathbb{R}$-linear map such that
$\tau(a^{*}a)=\tau(aa^{*})$ for all $a\in A$. Note that this is equivalent to
$\tau\circ\text{Ad}_{u}=\tau$ for all $u\in U(A)$ – to see this, follow the
steps in [RLL00, Exercise 3.6] together with a complexification argument. This
is further equivalent to $\tau\circ\text{Ad}_{u}=\tau$ for all $u\in U^{0}(A)$
(as one can write every element in a C*-algebra as the linear combination of 4
unitaries, each of which can be made to not have full spectrum).

######  Proposition 3.3.

Let $\theta:U^{0}(A)\to U^{0}(B)$ be a continuous group homo-morphism, $E$ be
a real Banach space and $\tau:B_{sa}\to E$ a bounded trace. Then $\tau\circ
S_{\theta}:A_{sa}\to E$ is a bounded trace. In particular,
$S_{\theta}(A_{0})\subseteq B_{0}$ and $S_{\theta}$ induces a bounded
$\mathbb{R}$-linear map

(3.12) $\Lambda_{\theta}:\operatorname{Aff}T(A)\to\operatorname{Aff}T(B).$

###### Proof.

Observe that, for $a\in A_{sa}$ and $u\in U^{0}(A)$, we have

(3.13) $\begin{split}e^{2\pi itS_{\theta}(uau^{*})}&=\theta(e^{2\pi
ituau^{*}})\\\ &=\theta(ue^{2\pi ita}u^{*})\\\ \ &=\theta(u)e^{2\pi
itS_{\theta}(a)}\theta(u)^{*}\\\ &=e^{2\pi
it\theta(u)S_{\theta}(a)\theta(u)^{*}}\end{split}$

for all $t\in\mathbb{R}$. Therefore
$S_{\theta}(uau^{*})=\theta(u)S_{\theta}(a)\theta(u)^{*}$, and applying $\tau$
yields

(3.14) $\begin{split}\tau\circ
S_{\theta}(uau^{*})&=\tau\left(\theta(u)S_{\theta}(a)\theta(u)^{*}\right)\\\
&=\tau(S_{\theta}(a)),\end{split}$

i.e., $\tau\circ S_{\theta}$ is tracial.

Thus if $a\in A_{0}$, it vanishes on every tracial state (hence on every
tracial functional), and so $\tau\circ S_{\theta}(a)=0$ for all $\tau\in
T(B)$. Therefore $S_{\theta}(A_{0})\subseteq B_{0}$ and so $S_{\theta}$
factors through a map

(3.15) $\Lambda_{\theta}:\operatorname{Aff}T(A)\simeq A_{sa}/A_{0}\to
B_{sa}/B_{0}\simeq\operatorname{Aff}T(B),$

where we identify $A_{sa}/A_{0}\simeq\operatorname{Aff}T(A)$ and
$B_{sa}/B_{0}\simeq\operatorname{Aff}T(B)$. ∎

One cannot expect $\Lambda_{\theta}$ (or $S_{\theta}$) to be unital or
positive, as the following examples show.

######  Example 3.4.

  1. (1)

Consider a continuous homomorphism
$\theta:\mathbb{T}\to\mathbb{T}=U^{0}(\mathbb{C})=U(\mathbb{C})$. By
Pontryagin duality, $\theta(z)=z^{n}$ for some $n\in\mathbb{Z}$. We then have
that $S_{\theta}:\mathbb{R}\to\mathbb{R}$ is given by $S_{\theta}(x)=nx$. If
$n\neq 1$, clearly $S_{\theta}$ is not unital. If $n<0$, then $S_{\theta}$ is
not positive since it sends $1$ to $n<0$. An important observation, however,
is that if $n<0$, $-S_{\theta}:\mathbb{R}\to\mathbb{R}$ is positive, and
$-\frac{1}{n}S$ is unital and positive.

  2. (2)

Consider $\theta:\mathbb{T}^{3}\to\mathbb{T}$ given by
$\theta(z,w,v)=\overline{z}wv$. The corresponding map
$S_{\theta}:\mathbb{R}^{3}\to\mathbb{R}$ is given by

(3.16) $S_{\theta}(a,b,c)=-a+b+c.$

Clearly $(1,0,0)\in\mathbb{C}^{3}$ is a positive element, however
$S_{\theta}(1,0,0)=-1<0$. This map is however unital.

  3. (3)

Let $\theta:U_{2}\to\mathbb{T}$ be defined by $\theta(u)=\det(u)$. Then
$S_{\theta}:(M_{2})_{sa}\to\mathbb{R}$ is defined by
$S_{\theta}(A)=\text{tr}A$, where tr is the unnormalized trace. Clearly this
map is not unital, but it is positive.

  4. (4)

Let $\theta:U_{2}\to U_{3}$ be defined by $\theta(u)=u\oplus 1$. Then
$S_{\theta}:(M_{2})_{sa}\to(M_{3})_{sa}$ is given by $S_{\theta}(A)=A\oplus
0$, which is again not unital, but is positive. The induced map
$\Lambda_{\theta}:\mathbb{R}\to\mathbb{R}$ is given by
$\Lambda_{\theta}(x)=\frac{2}{3}x$ for $x\in\mathbb{R}$.

  5. (5)

Let $\theta:\mathbb{T}\hookrightarrow U_{2}$ be defined by

(3.17) $\theta(\lambda)=\begin{pmatrix}\lambda\\\ &\lambda\end{pmatrix}.$

Then $S_{\theta}$ is a unital, positive isometry and $\Lambda_{\theta}$ gives
rise the identity map

(3.18)
$\mathbb{R}=\operatorname{Aff}T(\mathbb{C})\to\operatorname{Aff}T(M_{2})=\mathbb{R}.$

  6. (6)

Let $\theta:\mathbb{T}\hookrightarrow U_{2}$ be defined by

(3.19) $\theta(\lambda)=\begin{pmatrix}\lambda&\\\
&\overline{\lambda}\end{pmatrix}.$

Then $S_{\theta}:\mathbb{R}\to(M_{2})_{sa}$ is defined by

(3.20) $S_{\theta}(x)=\begin{pmatrix}x&\\\ &-x\end{pmatrix}$

and $\Lambda_{\theta}$ is identically zero.

The above examples are important. If $\theta(\mathbb{T})\subseteq\mathbb{T}$,
which is a moderate assumption (e.g., if $\theta$ was the restriction of a
unital *-homomorphism or an anti-*-homomorphism), we can restrict the
homomorphism to the circle to get a continuous group homomorphism
$\mathbb{T}\to\mathbb{T}$. We understand such homomorphisms by Pontryagin
duality [Fol16, Chapter 4].

We now use (pre-)determinant techniques in order to show desirable relation-
ships between our maps.

######  Proposition 3.5.

Let $A,B$ be unital C*-algebras, $\theta:U^{0}(A)\to U^{0}(B)$ be a uniformly
continuous homomorphism, $E$ a real Banach space and $\tau:B_{sa}\to E$ a
bounded trace.

  1. (1)

Let $\xi:[0,1]\to U^{0}(A)$ be a piece-wise smooth path with $\xi(0)=1$. Then

(3.21) $\tilde{\Delta}_{\tau\circ
S_{\theta}}^{1}(\xi)=\tilde{\Delta}_{\tau}^{1}(\theta\circ\xi).$

In particular, $\tilde{\Delta}_{\tau\circ
S_{\theta}}^{1}(\pi_{1}(U^{0}(A)))\subseteq\tilde{\Delta}_{\tau}^{1}(\pi_{1}(U^{0}(B)))$.

  2. (2)

The following diagram commutes:

(3.22)
${\pi_{1}(U^{0}(A))}$${E}$${\pi_{1}(U^{0}(B))}$${E.}$$\scriptstyle{\pi_{1}(\theta)}$$\scriptstyle{\tilde{\Delta}_{\tau\circ
S_{\theta}}^{1}}$id$\scriptstyle{\tilde{\Delta}_{\tau}^{1}}$

  3. (3)

The following diagram commutes:

(3.23) ${U^{0}(A)}$${E/\tilde{\Delta}_{\tau\circ
S_{\theta}}^{1}(\pi_{1}(U^{0}(A)))}$${U^{0}(B)}$${E/\tilde{\Delta}_{\tau}^{1}(\pi_{1}(U^{0}(B))),}$$\scriptstyle{\Delta_{\tau\circ
S_{\theta}}^{1}}$$\scriptstyle{\theta}$$\scriptstyle{\Delta_{\tau}^{1}}$

where the map on the right is the canonical map induced from the inclusion
$\tilde{\Delta}_{\tau\circ
S_{\theta}}^{1}(\pi_{1}(U^{0}(A)))\subseteq\tilde{\Delta}_{\tau}^{1}(\pi_{1}(U^{0}(B)))$
coming from (1). In particular, $\theta(\ker\Delta_{\tau\circ
S_{\theta}}^{1})\subseteq\ker\Delta_{\tau}^{1}$.

The analogous diagram commutes if we consider Thomsen’s variant of the de la
Harpe-Skandalis determinant associated to $\tau$ and $\tau\circ S_{\theta}$ in
(3.23).

###### Proof.

By the proof of Proposition 2.2, we can find $k\in\mathbb{N}$ and
$a_{1},\dots,a_{k}\in(M_{n}(A))_{sa}$ such that $\xi$ is homotopic to the path
$\eta:[0,1]\to U(A)$ given by

(3.24) $\eta(t)=\xi\left(\frac{j-1}{k}\right)e^{2\pi
i(kt-j+1)a_{j}},t\in\left[\frac{j-1}{k},\frac{j}{k}\right]$

and we have that

(3.25)
$\tilde{\Delta}_{\omega}^{n}(\xi)=\tilde{\Delta}_{\omega}^{n}(\eta)=\sum_{j=1}^{k}\omega(a_{j})$

whenever $\omega:A_{sa}\to F$ is a bounded trace to a real Banach space $F$.

Now $\theta\circ\xi$ is homotopic to $\theta\circ\eta$, which has the
following form, for $j=1,\dots,k$ and $t\in[\frac{j-1}{k},\frac{j}{k}]$:

(3.26)
$\begin{split}\theta\circ\eta(t)&=\theta\circ\xi\left(\frac{j-1}{k}\right)\theta\left(e^{2\pi
i(kt-j+1)a_{j}}\right)\\\ &=\theta\circ\xi\left(\frac{j-1}{k}\right)e^{2\pi
i(kt-j+1)S_{\theta}(a_{j})}.\end{split}$

By taking $X_{j}:=\theta\circ\xi\left(\frac{j-1}{k}\right)$ and
$Y_{j}:=e^{2\pi i(kt-j+1)S_{\theta}(a_{j})}$ in (2.12), we see that

(3.27)
$\tilde{\Delta}_{\tau}^{1}(\theta\circ\xi)=\tilde{\Delta}_{\tau}^{1}(\theta\circ\eta)=\sum_{j=1}^{k}\tau(S_{\theta}(a_{j})),$

and this last quantity is just $\tilde{\Delta}_{\tau\circ
S_{\theta}}^{1}(\xi)$ by (3.25).

As for parts (2) and (3), they follow from (1). Indeed, (2) is obvious and to
see (3) we let

(3.28) $q:E/\tilde{\Delta}_{\tau\circ S_{\theta}}^{1}(\pi(U^{0}(A)))\to
E/\tilde{\Delta}_{\tau}^{1}(\pi(U^{0}(B)))$

be the canonical surjection. If $\xi_{u}:[0,1]\to U^{0}(A)$ is a path from 1
to $u$, then we have that

(3.29) $\begin{split}q(\Delta_{\tau\circ
S_{\theta}}^{1}(u))&=q\left(\tilde{\Delta}_{\tau\circ
S_{\theta}}^{1}(\xi_{u})+\tilde{\Delta}_{\tau\circ
S_{\theta}}^{1}(\pi_{1}(U^{0}(A)))\right)\\\ &=\tilde{\Delta}_{\tau\circ
S_{\theta}}^{1}(\xi_{u})+\tilde{\Delta}_{\tau}^{1}(\pi_{1}(U^{0}(B)))\\\
&\overset{(1)}{=}\tilde{\Delta}_{\tau}^{1}(\theta\circ\xi_{u})+\tilde{\Delta}_{\tau}^{1}(\pi_{1}(U^{0}(B)))\\\
&=\Delta_{\tau}^{1}(\theta(u)).\end{split}$

The remark about Thomsen’s variant is obvious. ∎

######  Corollary 3.6.

The following diagram commutes:

(3.30)
${\pi_{1}(U^{0}(A))}$${\operatorname{Aff}T(A)}$${\pi_{1}(U^{0}(B))}$${\operatorname{Aff}T(B).}$$\scriptstyle{\pi_{1}(\theta)}$$\scriptstyle{\tilde{\Delta}_{A}^{1}}$$\scriptstyle{\Lambda_{\theta}}$$\scriptstyle{\tilde{\Delta}_{B}^{1}}$

In particular, when the canonical maps

(3.31) $\pi_{1}(U^{0}(A))\to K_{0}(A)\text{ and }\pi_{1}(U^{0}(B))\to
K_{0}(B)$

are isomorphisms, we have that

(3.32)
${K_{0}(A)}$${\operatorname{Aff}T(A)}$${K_{0}(B)}$${\operatorname{Aff}T(B)}$$\scriptstyle{K_{0}(\theta)}$$\scriptstyle{\rho_{A}}$$\scriptstyle{\Lambda_{\theta}}$$\scriptstyle{\rho_{B}}$

commutes, where $K_{0}(\theta):K_{0}(A)\to K_{0}(B)$ is the map induced from
the diagram

(3.33)
${K_{0}(A)}$${\pi_{1}(U^{0}(A))}$${K_{0}(B)}$${\pi_{1}(U^{0}(B)).}$$\scriptstyle{\simeq}$$\scriptstyle{K_{0}(\theta)}$$\scriptstyle{\pi_{1}(\theta)}$$\scriptstyle{\simeq}$

###### Proof.

By definition, we have that
$\Lambda_{\theta}\circ\text{Tr}_{A}=S_{\theta}\circ\text{Tr}_{B}$ and so if
$\xi:[0,1]\to U^{0}(A)$ is a (piece-wise smooth) path, we have

(3.34)
$\begin{split}\Lambda_{\theta}\left(\tilde{\Delta}_{\text{Tr}_{A}}^{1}(\xi)\right)&=\Lambda_{\theta}\left(\int_{0}^{1}\text{Tr}_{A}\left(\xi^{\prime}(t)\xi(t)^{-1}\right)dt\right)\\\
&=\int_{0}^{1}\Lambda_{\theta}\circ\text{Tr}_{A}\left(\xi^{\prime}(t)\xi(t)^{-1}\right)dt\
\\\ &=\int_{0}^{1}\text{Tr}_{B}\circ
S_{\theta}\left(\xi^{\prime}(t)\xi(t)^{-1}\right)dt\\\
&=\tilde{\Delta}_{\text{Tr}_{B}\circ S_{\theta}}(\xi).\end{split}$

Now we note that the diagram

(3.35)
${\operatorname{Aff}T(A)}$${\pi_{1}(U^{0}(A))}$${\operatorname{Aff}T(B)}$${\pi_{1}(U^{0}(B))}$${\operatorname{Aff}T(B)}$$\scriptstyle{\Lambda_{\theta}}$$\scriptstyle{\tilde{\Delta}_{A}^{1}}$$\scriptstyle{\pi_{1}(\theta)}$$\scriptstyle{\tilde{\Delta}_{\text{Tr}_{B}\circ
S_{\theta}}^{1}}$id$\scriptstyle{\tilde{\Delta}_{B}^{1}}$

commutes, where the top triangle commutes by the (3.34) and the bottom square
commutes by Proposition 3.5(2). This gives that (3.30) commutes.

The second part follows since if $\xi$ is a (piece-wise smooth) path in
$U_{n}(A)$ and $m>n$, then

(3.36) $\tilde{\Delta}_{\tau}^{m}(\xi\oplus
1_{m-n})=\tilde{\Delta}_{\tau}^{n}(\xi).$

∎

######  Proposition 3.7.

Let $A,B$ be unital C*-algebras. Then the map $\Lambda_{\theta}$ is a lift of
the map

(3.37)
$\operatorname{Aff}T(A)/\tilde{\Delta}_{A}^{1}(\pi_{1}(U^{0}(A)))\to\operatorname{Aff}T(B)/\tilde{\Delta}_{B}^{1}(\pi_{1}(U^{0}(B)))$

as described in (3.4).

###### Proof.

Let us label the maps in the diagram (3.4):

(3.38)
${U^{0}(A)/\ker\Delta^{1}_{A}}$${\operatorname{Aff}T(A)/\tilde{\Delta}_{A}^{1}(\pi_{1}(U^{0}(A)))}$${U^{0}(B)/\ker\Delta^{1}_{B}}$${\operatorname{Aff}T(B)/\tilde{\Delta}_{B}^{1}(\pi_{1}(U^{0}(B)))}$$\scriptstyle{\tilde{\theta}}$$\scriptstyle{\delta_{A}^{1}}$$\scriptstyle{P}$$\scriptstyle{\delta_{B}^{1}}$

where $\tilde{\theta}([u]):=[\theta(u)]$, $\delta_{A}^{1}([e^{2\pi
ia}]):=[\hat{a}]$, $\delta_{B}^{1}$ is defined similarly, and

(3.39) $P=\delta_{B}^{1}\circ\tilde{\theta}\circ(\delta_{A}^{1})^{-1}.$

But then we have that

(3.40)
$\begin{split}P([a])&=\delta_{B}^{1}\circ\tilde{\theta}\circ(\delta_{A}^{1})^{-1}([\hat{a}])\\\
&=\delta_{B}^{1}\circ\tilde{\theta}([e^{2\pi ia}])\\\
&=\delta_{B}^{1}([\theta(e^{2\pi ia})])\\\ &=\delta_{B}^{1}([e^{2\pi
iS_{\theta}(a)}])\\\ &=[\widehat{S_{\theta}(a)}]\\\
&=[\Lambda_{\theta}(\widehat{a})]\end{split}$

In particular, the diagram

(3.41)
${\operatorname{Aff}T(A)}$${\operatorname{Aff}T(A)/\tilde{\Delta}_{A}^{1}(\pi_{1}(U^{0}(A)))}$${\operatorname{Aff}T(B)}$${\operatorname{Aff}T(B)/\tilde{\Delta}_{B}^{1}(\pi_{1}(U^{0}(B)))}$$\scriptstyle{\Lambda_{\theta}}$$\scriptstyle{q_{A}^{1}}$$\scriptstyle{P}$$\scriptstyle{q_{B}^{1}}$

commutes, where $q_{A}^{1}$ and $q_{B}^{1}$ are the respective quotient maps.
∎

######  Proposition 3.8.

Let $A,B$ be unital C*-algebras and $\theta:U^{0}(A)\to U^{0}(B)$ be a
continuous group homomorphism. Then
$\Lambda_{\theta}:\operatorname{Aff}T(A)\to\operatorname{Aff}T(B)$ is a lift
of the map

(3.42)
$\operatorname{Aff}T(A)/\overline{\tilde{\Delta}_{A}^{1}(\pi_{1}(U^{0}(A)))}\to\operatorname{Aff}T(B)/\overline{\tilde{\Delta}_{B}^{1}(\pi_{1}(U^{0}(B)))}$

as described in (3.2).

###### Proof.

One can mimic the proof above or apply the above result and appeal to the
commuting diagram

(3.43)
${\operatorname{Aff}T(A)/\tilde{\Delta}_{A}^{1}(\pi_{1}(U^{0}(A)))}$${\operatorname{Aff}T(A)/\overline{\tilde{\Delta}_{A}^{1}(\pi_{1}(U^{0}(A)))}}$${\operatorname{Aff}T(B)/\tilde{\Delta}_{B}^{1}(\pi_{1}(U^{0}(B)))}$${\operatorname{Aff}T(B)/\overline{\tilde{\Delta}_{B}^{1}(\pi_{1}(U^{0}(B)))},}$

where the vertical maps are defined via the diagrams from (3.4) and (3.2)
respectively, and the horizontal maps are the canonical surjections. ∎

In particular, assuming some $K_{0}$-regularity gives that $\Lambda_{\theta}$
is a lift of a map between quotients of spaces of continuous real-valued
affine functions by images of $K_{0}$.

######  Corollary 3.9.

Let $A,B$ be unital C*-algebras such that the canonical maps

(3.44) $\pi_{1}(U^{0}(A))\to K_{0}(A)\text{ and }\pi_{1}(U^{0}(B))\to
K_{0}(B)$

are surjections. If $\theta:U^{0}(A)\to U^{0}(B)$ is a continuous
homomorphism, then $\Lambda_{\theta}$ is a lift of the maps on the right of
the following two commutative diagrams:

(3.45)
${U^{0}(A)/CU^{0}(A)}$${\operatorname{Aff}T(A)/\overline{\rho_{A}(K_{0}(A))}}$${U^{0}(B)/CU^{0}(B)}$${\operatorname{Aff}T(B)/\overline{\rho_{B}(K_{0}(B))}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

and

(3.46)
${U^{0}(A)/\ker\Delta_{A}^{1}}$${\operatorname{Aff}T(A)/\rho_{A}(K_{0}(A))}$${U^{0}(B)/\ker\Delta_{B}^{1}}$${\operatorname{Aff}T(B)/\rho_{B}(K_{0}(B)).}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

Further, if $\ker\Delta_{A}^{1}=DU^{0}(A)$ and $\ker\Delta_{B}^{1}=DU^{0}(B)$,
then $\Lambda_{\theta}$ is a lift of the map induced by the diagram

(3.47)
${U^{0}(A)/DU^{0}(A)}$${\operatorname{Aff}T(A)/\rho_{A}(K_{0}(A))}$${U^{0}(B)/DU^{0}(B)}$${\operatorname{Aff}T(B)/\rho_{B}(K_{0}(B)).}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

C*-algebras satisfying the last condition arise naturally – for example
unital, separable, simple, pure C*-algebras of stable rank 1 such that every
2-quasi-tracial state on $A$ is a trace have this property [NR17].

##  4\. The order on $\operatorname{Aff}T(\cdot)$

We now examine when the map induced on $\operatorname{Aff}T(\cdot)$ is
positive in order to compare $K$-theory, traces, and the pairing. As we saw in
Example 3.4, the map we get between spaces of affine functions on the trace
simplices need not be positive nor unital in general. In this section, we will
be able to use the map $\Lambda_{\theta}$ to construct a unital positive map,
under some extra assumptions on $\theta$.

We record the following results as they give us necessary and sufficient
conditions for the $\Lambda_{\theta}$ to be positive. We use the C*-algebra-
valued analogue of the fact that any unital, contractive linear functional is
positive, along with the fact that completely positive maps are (completely)
bounded with the norm determined by the image of the unit. Recall that an
operator system is a self-adjoint unital subspace of a C*-algebra. The
following is a combination of Proposition 2.11, Theorem 3.9 and Proposition
3.6 in [Pau02].

######  Proposition 4.1.

Let $\mathcal{S}$ be an operator system and $B$ a unital C*-algebra.

  1. (1)

If $\phi:\mathcal{S}\to B$ is a unital contraction, then $\phi$ is positive.

  2. (2)

If $B=C(X)$ and $\phi:\mathcal{S}\to B$ is positive, then it is bounded with
$\|\phi\|=\|\phi(1)\|$.

######  Lemma 4.2.

Suppose $A,B$ are unital C*-algebras and $\theta:U^{0}(A)\to U^{0}(B)$ be a
continuous group homomorphism such that $\theta(\mathbb{T})=\mathbb{T}$. If
$\theta|_{\mathbb{T}}$ is injective, then $S_{\theta}(1)\in\\{1,-1\\}$.

###### Proof.

The restriction $\theta|_{\mathbb{T}}:\mathbb{T}\to\mathbb{T}$ is a continuous
group homomorphism, hence by Pontryagin duality is of the form
$\theta(z)=z^{n}$ for some $n$. Injectivity implies that $n\in\\{1,-1\\}$. We
then have that

(4.1) $e^{2\pi iS_{\theta}(1)t}=\theta(e^{2\pi it})=e^{2\pi int}$

for all $t\in\mathbb{R}$. This implies that $S_{\theta}(1)=n\cdot
1\in\\{1,-1\\}$. ∎

In the sequel, we will be interested in the case that $T(A)\neq\emptyset$. The
map $S_{\theta}$ will still descend to a map $\Lambda_{\theta}:A_{sa}/A_{0}\to
B_{sa}/B_{0}$ regardless of their being tracial states on either C*-algebra,
but these quotients are zero if there are no traces. If we take the zero
Banach space to be partially ordered with order unit trivially, then one can
speak of unital and positive maps
$\operatorname{Aff}T(A)\to\operatorname{Aff}T(B)$, regardless of whether
$T(B)$ is empty or not (any map from a partially ordered Banach space with
order unit to the zero Banach space will be unital and positive). However, if
$T(A)=\emptyset$ and $T(B)\neq\emptyset$, then our map $\Lambda_{\theta}$
would have no chance of being unital. We identify the zero Banach space with
$\operatorname{Aff}(\emptyset)=C_{\mathbb{R}}(\emptyset)$ and with the complex
analogues as well. However, the following lemma is true regardless of whether
$T(A)$ is empty or not.

######  Lemma 4.3.

Suppose $A,B$ are unital C*-algebras and $\theta:U^{0}(A)\to U^{0}(B)$ is a
continuous group homomorphism such that $\theta(\mathbb{T})=\mathbb{T}$. If
$\theta$ is injective, the following are equivalent:

  1. (1)

one of $\Lambda_{\theta}$ or $-\Lambda_{\theta}$ is positive;

  2. (2)

$\Lambda_{\theta}$ is contractive.

###### Proof.

First suppose that $T(A),T(B)\neq\emptyset$. By Lemma 4.2, we know that
$S_{\theta}(1)\in\\{1,-1\\}$ and consequently
$\Lambda_{\theta}(\hat{1})\in\\{\hat{1},\widehat{-1}\\}$ (where we recall
that, for $a\in A_{sa}$, $\hat{a}\in\operatorname{Aff}T(A)$ is the affine
function $\hat{a}(\tau)=\tau(a)$). By replacing $\Lambda_{\theta}$ with
$-\Lambda_{\theta}$, we can without loss of generality assume that
$\Lambda_{\theta}$ is unital. Using the fact that
$\operatorname{Aff}T(A)+i\operatorname{Aff}T(A)\subseteq C(T(A))$ is an
operator system and the canonical extension

(4.2)
$\Lambda_{\theta}^{\mathbb{C}}:\operatorname{Aff}T(A)+i\operatorname{Aff}T(A)\to\operatorname{Aff}T(B)+i\operatorname{Aff}T(B)\subseteq
C(T(B))$

is a unital linear map with abelian target algebra, this is an easy
consequence of the two parts of Proposition 4.1.

Finally, if $T(A)=\emptyset$, then any map from $\operatorname{Aff}T(A)$ is
both contractive and positive trivially. The same is true of any map with
codomain $\operatorname{Aff}T(B)$ if $T(B)=\emptyset$. ∎

######  Lemma 4.4.

Suppose $A,B$ are unital C*-algebras and $\theta:U^{0}(A)\to U^{0}(B)$ is a
continuous group homomorphism. Suppose that $T(A)\neq\emptyset$.

  1. (1)

$\|\Lambda_{\theta}\|\leq\|S_{\theta}\|$.

  2. (2)

If $K>0$ is such that $\|\theta(u)-\theta(v)\|\leq K\|u-v\|$ for all $u,v\in
U^{0}(A)$, then $\|S_{\theta}\|\leq K$ and $\|\Lambda_{\theta}\|\leq K$.

  3. (3)

If $\theta$ is a homeomorphism, then $\Lambda_{\theta}$ is bijective with
bounded inverse being $\Lambda_{\theta^{-1}}$.

  4. (4)

If $\theta$ is isometric, then so is $S_{\theta}$. If $\theta$ is a surjective
isometry, then $\Lambda_{\theta}$ is a surjective isometry.

###### Proof.

As $S_{\theta}(A_{0})\subseteq B_{0}$, we have

(4.3)
$\begin{split}\|\Lambda_{\theta}(\hat{a})\|_{\operatorname{Aff}T(B)}&=\|\widehat{S_{\theta}(a)}\|_{\operatorname{Aff}T(B)}\\\
&=\inf_{b\in B_{0}}\|S_{\theta}(a)+b\|\\\
&\leq\inf_{b\in\theta(A_{0})}\|S_{\theta}(a)+b\|\\\ &=\inf_{a^{\prime}\in
A_{0}}\|S_{\theta}(a)+S_{\theta}(a^{\prime})\|\\\
&\leq\|S_{\theta}\|\inf_{a^{\prime}\in A_{0}}\|a+a^{\prime}\|\\\
&=\|S_{\theta}\|\|\hat{a}\|_{\operatorname{Aff}T(A)}\end{split}$

whenever $a\in A_{sa}$. This gives that
$\|\Lambda_{\theta}\|\leq\|S_{\theta}\|$.

The fact that $S_{\theta}$ is an isometry whenever $\theta$ is an isometry
follows from an argument in [HM14]; we exemplify said argument to show the
bound condition. We use the observation that

(4.4) $\frac{e^{2\pi ita}-1}{t}\to 2\pi ia$

as $t\to 0$. Since

(4.5) $\|e^{2\pi itS_{\theta}(a)}-1\|\leq K\|e^{2\pi ita}-1\|$

for all $t\in\mathbb{R}$, we can divide both sides by $\frac{1}{2\pi}|t|$ and
take $t\to 0$ to get that

(4.6) $\|S_{\theta}(a)\|\leq K\|a\|.$

Thus $\|S_{\theta}\|\leq K$. It then follows from (1) that
$\|\Lambda_{\theta}\|\leq K$ as well.

If $\theta$ is a homeomorphism, $S_{\theta}$ and $S_{\theta^{-1}}$ are both
defined and its clear that $S_{\theta}^{-1}=S_{\theta^{-1}}$.

The surjectivity of $\theta$ implies the surjectivity of $S_{\theta}$ and thus
if $b\in B_{sa}$, we can find $a\in A_{sa}$ with $S_{\theta}(a)=b$. But then
$\Lambda_{\theta}(\hat{a})=\widehat{S_{\theta}(a)}=\hat{b}$. In particular,
$\Lambda_{\theta}$ is surjective. Now if $\theta$ is a surjective isometry, we
identify $\operatorname{Aff}T(A)\simeq A_{sa}/A_{0}$ and
$\operatorname{Aff}T(B)\simeq B_{sa}/B_{0}$, noting that
$S_{\theta}(A_{0})=B_{0}$, and that $\Lambda_{\theta}$ will preserve the
quotient norms. ∎

######  Corollary 4.5.

Suppose $A,B$ are unital C*-algebras and $\theta:U^{0}(A)\to U^{0}(B)$ is a
continuous group homomorphism. Suppose that $T(A)\neq\emptyset$. If
$S_{\theta}(1)=n$ and $\|S_{\theta}\|=|n|$, then $\frac{1}{n}S_{\theta}$ is a
unital contraction, hence positive. In particular, if
$\theta(\mathbb{T})=\mathbb{T}$ and $\theta|_{\mathbb{T}}$ is an injection,
then either $\Lambda_{\theta}$ or $-\Lambda_{\theta}$ is unital and positive.

###### Proof.

The first part follows from the above lemma. If $\theta$ is an injection with
$\theta(\mathbb{T})=\mathbb{T}$, we have that
$S_{\theta}(\hat{1})\in\\{\hat{1},\widehat{-1}\\}$ and that $\Lambda_{\theta}$
is contractive, so one of $\Lambda_{\theta}$ or $-\Lambda_{\theta}$ is a
unital contraction, hence positive by part (1) of Proposition 4.1. ∎

######  Theorem 4.6.

Suppose $A,B$ are unital C*-algebras and $\theta:U^{0}(A)\to U^{0}(B)$ is a
contractive injective homomorphism such that $\theta(\mathbb{T})=\mathbb{T}$.
Suppose that $T(A)\neq\emptyset$. Then there is a continuous affine map
$T_{\theta}:T(B)\to T(A)$ such that $\Lambda_{\theta}(f)=f\circ T_{\theta}$ or
$-\Lambda_{\theta}(f)=f\circ T_{\theta}$, depending on whether
$\Lambda_{\theta}$ or $-\Lambda_{\theta}$ is positive.

###### Proof.

This follows from the fact that the induced map
$\Lambda_{\theta}:\operatorname{Aff}T(A)\to\operatorname{Aff}T(B)$ will have
the property that $\Lambda_{\theta}$ or $-\Lambda_{\theta}$ will be a unital
positive map. Therefore by contravariant identification of compact convex sets
(of locally convex Hausdorff linear spaces) with the state space of the space
of continuous real-valued affine valued functions on them ([Goo86, Chapter
7]), there exists a continuous affine map $T_{\theta}:T(B)\to T(A)$. ∎

######  Theorem 4.7.

Suppose $A,B$ be unital C*-algebras and $\theta:U^{0}(A)\to U^{0}(B)$ is a
contractive topological group isomorphism such that
$\theta(\mathbb{T})=\mathbb{T}$. Suppose that $T(A)\neq\emptyset$. Then the
map $T_{\theta}:T(B)\to T(A)$ induced by $\Lambda_{\theta}$ is an affine
homeomorphism.

###### Proof.

As $\theta(\mathbb{T})=\mathbb{T}$, $S_{\theta}(1)\in\\{-1,1\\}$. Let
$\pm\Lambda_{\theta}:\operatorname{Aff}T(A)\to\operatorname{Aff}T(B)$ be
either $\Lambda_{\theta}$ or $-\Lambda_{\theta}$, depending on which is
unital, positive, contractive and surjective by combining Lemmas 4.4, 4.3 and
3.2(2). By the duality of (compact) simplices and continuous affine functions
on them, the map $T_{\theta}:T(B)\to T(A)$ is an affine homeomorphism. ∎

######  Theorem 4.8.

Let $A,B$ be unital C*-algebras and $\theta:U(A)\to U(B)$ be a contractive
injective homomorphism such that $\theta(\mathbb{T})=\mathbb{T}$. Suppose that
$T(A)\neq\emptyset$. If

(4.7) $\pi_{i}(U(A))\simeq K_{i-1}(A)\text{ and }\pi_{i}(U(B))\simeq
K_{i-1}(B)\text{ for }i=0,1,$

via the canonical maps, then there is an induced map

(4.8) $KT_{u}(\theta):KT_{u}(A)\to\operatorname{KT}_{u}(B).$

###### Proof.

Let

  * •

$\Lambda:=\Lambda_{\theta|_{U^{0}(A)}}:\operatorname{Aff}T(A)\to\operatorname{Aff}T(B)$,

  * •

$\theta_{0}:\pi_{1}(U^{0}(A))\to\pi_{1}(U^{0}(B))$ be the map induced on
fundamental groups by $\theta|_{U^{0}(A)}$,

  * •

$K_{0}(\theta)$ be the map induced on $K_{0}$ by $\theta_{0}$ together with
(4.7) for $i=1$,

  * •

$\theta_{1}:\pi_{0}(U(A))\to\pi_{0}(U(B))$ be the map induced by $\theta$ on
connected components (so that
$\theta_{1}([u]_{\sim_{h}})=[\theta_{1}(u)]_{\sim_{h}}$) and

  * •

$K_{1}(\theta)$ be the map induced by $\theta_{1}$ together with (4.7) for
$i=0$.

Then

(4.9) $(\pm K_{0}(\theta),K_{1}(\theta),\pm\Lambda):KT_{u}(A)\to KT_{u}(B)$

is a $KT_{u}$-morphism, where $\pm\Lambda$ is either $\Lambda$ or $-\Lambda$
depending on which one is unital and positive, and $\pm K_{0}(\theta)$ is
either $K_{0}(\theta)$ if $\Lambda$ is positive or $-K_{0}(\theta)$ if
$-\Lambda$ is positive. Indeed, $\pm K_{0}(\theta),\theta_{1},\pm\Lambda$ are
all appropriate morphisms, and we have that

(4.10)
${K_{0}(A)}$${\operatorname{Aff}T(A)}$${K_{0}(B)}$${\operatorname{Aff}T(B)}$$\scriptstyle{\pm
K_{0}(\theta)}$$\scriptstyle{\rho_{A}}$$\scriptstyle{\pm\Lambda}$$\scriptstyle{\rho_{B}}$

commutes555The map $-K_{0}(\theta)$ will take a piece-wise smooth loop $\xi$
to the loop $-\theta\circ\xi$ defined by
$(-\theta\circ\xi)(t)=\theta(\xi(-t))$. From here its obvious that the diagram
commutes. by Corollary 3.6. ∎

######  Corollary 4.9.

Let $A,B$ be unital C*-algebras, $\theta:U^{0}(A)\to U^{0}(B)$ is a
contractive topological group isomorphism such that
$\theta(\mathbb{T})=\mathbb{T}$, and suppose that $T(A)\neq\emptyset$. If

(4.11) $\pi_{i}(U(A))\simeq K_{i-1}(A)\text{ and }\pi_{i}(U(B))\simeq
K_{i-1}(B)\text{ for }i=0,1,$

via the canonical maps, then $KT_{u}(A)\simeq KT_{u}(B)$.

###### Proof.

By Corollary 4.8, we have an induced $KT_{u}$-morphism. This map is
necessarily an isomorphism since $\theta$ is. ∎

######  Corollary 4.10.

Let $A,B$ be unital C*-algebras which are either $\mathcal{Z}$-stable or of
stable rank one and suppose that $T(A)\neq\emptyset$. Let $\theta:U(A)\to
U(B)$ be a contractive injective homomorphism such that
$\theta(\mathbb{T})=\mathbb{T}$. Then there is an induced map

$KT_{u}(\theta):KT_{u}(A)\to KT_{u}(B).$

###### Proof.

C*-algebras which are $\mathcal{Z}$-stable or have stable rank one satisfy the
hypotheses of Theorem 4.8 by [Jia97] and [Rie87] respectively. So Theorem 4.8
applies. ∎

######  Remark 4.11.

The strict ordering on $\operatorname{Aff}T(A)$ is given by $f\gg g$ if
$f(\tau)>g(\tau)$ for all $\tau\in T(A)$. If $A,B$ are unital with
$T(A)\neq\emptyset$ and $\theta:U^{0}(A)\to U^{0}(B)$ is a contractive
injective homomorphism such that $\theta(\mathbb{T})=\mathbb{T}$, then
$\pm\Lambda_{\theta}:\operatorname{Aff}T(A)\to\operatorname{Aff}T(B)$ is a
unital positive contraction by Lemma 4.3 (again $\pm\Lambda_{\theta}$ is
$\Lambda_{\theta}$ or $-\Lambda_{\theta}$ depending on which is positive). We
moreover have that

(4.12) $\pm\Lambda_{\theta}(f)\gg\pm\Lambda_{\theta}(g)\iff f\gg g.$

Indeed, let us show that $f\gg 0$ if and only if its image is $\gg 0$. As
$\pm\Lambda_{\theta}$ has the form $\pm\Lambda_{\theta}(\hat{a})=\widehat{\pm
S_{\theta}(a)}$, it suffices to show that if $\sigma(a)>0$ for all $\sigma\in
T(A)$, then $\tau(\pm S_{\theta}(a))>0$ for all $\tau\in T(B)$. But this is
trivial because $\tau\circ\pm S_{\theta}:A_{sa}\to\mathbb{R}$ extends
canonically to a tracial state $A\to\mathbb{C}$ by Proposition 3.3, so
evaluating it against $a$ must give that it is strictly positive.

The above says the following: for certain C*-algebras, we can read off
positivity in $K_{0}$, thinking of it as the fundamental group of the unitary
group, from the strict positivity of the pre-determinant applied a the loop.
Precisely, a non-zero element $x\in K_{0}(A)$, where $A$ is a unital, simple
C*-algebra with strict comparison, is in the positive cone if and only if the
corresponding loop $\xi_{x}$ satisfies $\tilde{\Delta}_{\tau}(\xi_{x})>0$ for
all $\tau\in T(A)$.

Although the following is known, for example by very strong results in [AM03,
Chapter 6] pertaining to certain prime C*-algebras, we give the follow-ing as
a corollary by using $K$-theoretic classification results.

######  Corollary 4.12.

Let $A,B$ be unital, separable, simple, nuclear $\mathcal{Z}$-stable
C*-algebras satisfying the UCT. Then $A\simeq B$ if and only if there is a
contractive isomorphism $U(A)\simeq U(B)$.

###### Proof.

Its clear that two isomorphic C*-algebras have isomorphic unitary groups. On
the other hand, if $U(A)\simeq U(B)$, then since these C*-algebras are
$\mathcal{Z}$-stable, Corollary 4.9 applies. As $KT_{u}(\cdot)$ recovers the
Elliott invariant, which is a complete invariant for the C*-algebras as in the
statement of the theorem (by [CET+21, Corollary D], [EGLN15, GLN20a, GLN20b]
and the references therein), $A\simeq B$. ∎

Using the state of the art classification of embeddings [CGS+23], there is an
enlarged invariant of $KT_{u}(\cdot)$ which is able to classify morphisms
between certain C*-algebras. Any $KT_{u}$-morphisms automatically has a lift
to this larger invariant [CGS+23, Theorem 3.9], and so under the assumption
that the $KT_{u}$-morphism is faithful (i.e., the map $T(B)\to T(A)$ induced
by the map $\operatorname{Aff}T(A)\to\operatorname{Aff}T(B)$ sends traces on
$B$ to faithful traces on $A$), there is a *-homomorphism witnessing the
$KT_{u}$-morphism. Therefore as a corollary of their main theorem, we have
that for an abundance of C*-algebras, there is an (contractive) embedding of
unitary groups if and only if there is an embedding of C*-algebras.

######  Corollary 4.13.

Let $A$ be a unital, separable, simple nuclear C*-algebra sat-isfying the UCT
which is either $\mathcal{Z}$-stable or of stable rank one, and $B$ be a
unital, separable, simple, nuclear $\mathcal{Z}$-stable C*-algebra. If there
is a contractive injective homomorphism $\theta:U(A)\to U(B)$ such that
$\theta(\mathbb{T})=\mathbb{T}$, then there is an embedding $A\hookrightarrow
B$.

###### Proof.

Assuming such a $\theta$ exists, it gives rise to a $KT_{u}$-morphism

(4.13) $KT_{u}(\theta):KT_{u}(A)\to KT_{u}(B)$

by Corollary 4.10. As $A,B$ are simple, the map $T_{\theta}:T(B)\to T(A)$
necessarily maps traces on $B$ to faithful traces on $A$, and thus the
$KT_{u}$-morphism $KT_{u}(\theta)$ is “faithful”. Therefore $KT_{u}(\theta)$
induces an embedding $A\hookrightarrow B$ by [CGS+23, Theorem B]. ∎

##  5\. A slight general linear variant

Here we briefly describe a slight general linear variant of some of the
results above. Unfortunately, the maps we get at the level of $A,B$ and
complex-valued continuous affine functions are not $\mathbb{C}$-linear in
general (see Example 5.3). In the presence of a continuous homomorphism
$\theta:GL^{0}(A)\to GL^{0}(B)$, we have corresponding maps

(5.1)
${GL^{0}(A)/CGL^{0}(A)}$${\left(A/\overline{[A,A]}\right)/\overline{\tilde{\Delta}_{A}^{1}(\pi_{1}(U^{0}(A)))}}$${GL^{0}(B)/CGL^{0}(B)}$${\left(A/\overline{[A,A]}\right)/\overline{\tilde{\Delta}_{B}^{1}(\pi_{1}(U^{0}(B)))}.}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

Again, by modding out by algebraic commutator subgroups when
$DGL^{0}(A)=\ker\Delta_{A}^{1}$ and $DGL^{0}(B)=\ker\Delta_{B}^{1}$ (both with
respect to the general linear variant of the de la Harpe-Skandalis
determinant, as originally introduced in [dlHS84a]), instead of closures of
derived groups, there is a purely algebraic variant of the above diagram:

(5.2)
${GL^{0}(A)/DGL^{0}(A)}$${\left(A/\overline{[A,A]}\right)/\tilde{\Delta}_{A}^{1}(\pi_{1}(GL^{0}(A)))}$${GL^{0}(B)/DGL^{0}(B)}$${\left(A/\overline{[B,B]}\right)/\tilde{\Delta}_{B}^{1}(\pi_{1}(GL^{0}(B))).}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

Thinking of $K_{0}(A)$ as the Grothendieck group of the semigroup of
equivalence classes of idempotents and
$K_{0}(A)\simeq\pi_{1}(GL_{\infty}^{0}(A))$, we would like to lift the maps on
the right of (5.1) and (5.2) to a map

(5.3) $A/\overline{[A,A]}\to B/\overline{[B,B]}$

(the latter holding true when $A,B$ are C*-algebras whose determinant has
appropriate kernel).

We can always achieve a bounded $\mathbb{R}$-linear map.

######  Proposition 5.1.

Let $A,B$ be unital C*-algebras and $\theta:GL^{0}(A)\to GL^{0}(B)$ be a
continuous group homomorphism. Then there is a continuous $\mathbb{R}$-linear
map

(5.4) $\tilde{G}_{\theta}:A/\overline{[A,A]}\to B/\overline{[B,B]}$

which lifts the maps on the right of (5.1) and (5.2) (the latter holding when
$DGL^{0}(A)=\ker\Delta_{A}^{1}$ and $DGL^{0}(B)=\ker\Delta_{B}^{1}$).

###### Proof.

We define $G_{\theta}:A\to B$ given by

(5.5) $G_{\theta}(a):=\lim_{n}\frac{n}{2\pi i}\log\theta(e^{2\pi
i\frac{a}{n}}).$

We note that the sequence on the right is eventually constant: choose $N$
large enough such that $n\geq N$ implies that

(5.6) $\|\theta(e^{2\pi i\frac{a}{n}})-1\|<1.$

We then have for $n\geq N$,

(5.7) $\begin{split}\frac{n}{2\pi i}\log\theta(e^{2\pi
i\frac{a}{n}})&=\frac{n}{2\pi i}\log\theta(e^{2\pi i\frac{Na}{Nn}})\\\
&=\frac{n}{2\pi i}\log\theta(e^{2\pi i\frac{a}{Nn}})^{N}\\\ &=\frac{Nn}{2\pi
i}\log\theta(e^{2\pi i\frac{a}{Nn}})\\\ &=\frac{N}{2\pi i}\log\theta(e^{2\pi
i\frac{a}{Nn}})^{n}\\\ &=\frac{N}{2\pi i}\log\theta(e^{2\pi
i\frac{na}{Nn}})\\\ &=\frac{N}{2\pi i}\log\theta(e^{2\pi
i\frac{a}{N}}).\end{split}$

To see that the map is additive, one can use the Lie product formula

(5.8) $e^{a+b}=\lim_{k}\left(e^{\frac{a}{k}}e^{\frac{b}{k}}\right)^{k}$

(see for example [Hal15, Theorem 2.11]), along with the fact that $\theta$ is
a continuous homomorphism. From here, it is clear that $G_{\theta}$ is
continuous and $\mathbb{Q}$-linear, hence $\mathbb{R}$-linear. Moreover, one
can use the formula

(5.9)
$e^{[a,b]}=\lim_{k}\left(e^{\frac{-a}{k}}e^{\frac{-b}{k}}e^{\frac{a}{k}}e^{\frac{b}{k}}\right)^{k^{2}}$

(a variation of the argument given in the proof of [Hal15, Theorem 2.11] will
give the desired formula), together with $\theta$ being a continuous group
homomorphism, to show that it respects commutation. Note that the same proof
indeed shows that $S_{\theta}$ respects commutation, although we never
explicitly used this. From here, $G_{\theta}([A,A])\subseteq[B,B]$, and
consequently $G_{\theta}(\overline{[A,A]})\subseteq\overline{[B,B]}$ by
continuity. Thus there is an induced $\mathbb{R}$-linear map

(5.10) $\tilde{G}_{\theta}:A/\overline{[A,A]}\to B/\overline{[B,B]}.$

The fact that $\tilde{G}_{\theta}$ is a lift of the maps on the right of (5.1)
and (5.2) follows from the same arguments as in Propositions 3.7 and 3.8. ∎

######  Remark 5.2.

We note that $G_{\theta}$ could have also been defined in the same manner as
$S_{\theta}$. In particular, there is a correspondence

(5.11) $\theta(e^{2\pi ita})=e^{2\pi itG_{\theta}(a)},t\in\mathbb{R}.$

One can use this to show that
$\theta(\ker\Delta_{A}^{1})\subseteq\ker\Delta_{B}^{1}$ as in Proposition 3.5
by using $G_{\theta}$ in place of $S_{\theta}$, along with the fact that
$G_{\theta}(\overline{[A,A]})\subseteq\overline{[B,B]}$. So there will always
be a general linear variant of the commuting diagram (3.4).

Let us use (5.11) to show that the maps $G_{\theta}$ and $\tilde{G}_{\theta}$
in Proposition 5.1 are not always $\mathbb{C}$-linear.

######  Example 5.3.

Consider $A=B=\mathbb{C}$ and
$\theta:\mathbb{C}^{\times}\to\mathbb{C}^{\times}$ given by
$\theta(z)=|z|^{(\alpha+\beta i)}z^{n}$ where $\alpha,\beta\in\mathbb{R}$ and
$n\in\mathbb{Z}$. It is easy to see that $\theta$ is a continuous group
homomorphism. However, the map $G_{\theta}$ is not $\mathbb{C}$-linear.
Indeed, we have that

(5.12) $\begin{split}\theta(e^{2\pi it(a+bi)})&=|e^{2\pi
it(a+bi)}|^{\alpha+\beta i}e^{2\pi itn(a+bi)}\\\ &=(e^{-2\pi
tb})^{\alpha+\beta i}e^{2\pi itn(a+bi)}\\\ &=e^{-2\pi tb\alpha}e^{-2\pi
it\beta ib}e^{2\pi itn(a+bi)}\\\ &=e^{2\pi it((na-\alpha
b)+i(n-\beta)b}.\end{split}$

In particular, thinking of $\mathbb{C}$ as $\mathbb{R}^{2}$ with 1 and $i$
corresponding to the basis vectors $(1,0)$ and $(0,1)$ respectively, we have
that $G_{\theta}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ is the map

(5.13) $G_{\theta}\begin{pmatrix}a\\\
b\end{pmatrix}=\begin{pmatrix}n&-\alpha\\\
0&n-\beta\end{pmatrix}\begin{pmatrix}a\\\ b\end{pmatrix}.$

In this example, the map $G_{\theta}$ is $\mathbb{C}$-linear if and only if
$\alpha=\beta=0$. We do note, however, that $\theta$ sends unitaries to
unitaries and $\theta|_{\mathbb{T}}(z)=z^{n}$.

In general, it is clear that if $\theta:GL(A)\to GL(B)$ sends unitaries to
unitaries, then we can use techniques in Section 4 to get maps between spaces
of continuous affine functions on the trace simplices. If one had that
$\theta$ was the restriction of a *-homomorphism or a conjugate-linear
*-homomorphism, then this would be true.

##  6\. Final remarks and open questions

An alternate way to construct the map $\Lambda_{\theta}$, using duality of
traces, is as follows. Denote by $\mathfrak{T}_{s}(A)$ the set of all tracial
functionals on $A$. Suppose that $A,B$ are unital C*-algebras with
$T(A),T(B)\neq\emptyset$ and $\theta:U^{0}(A)\to U^{0}(B)$ is a continuous
homomorphism. Define, for $a\in A_{sa}$ and $\tau\in\mathfrak{T}_{s}(B)$,

(6.1) $F(\tau)(a):=\lim_{n\to\infty}\tau\left(\frac{n}{2\pi
i}\log\theta(e^{2\pi i\frac{a}{n}})\right).$

######  Proposition 6.1.

For $\tau\in\mathfrak{T}_{s}(B)$, the map $F(\tau):A_{sa}\to\mathbb{R}$ is a
well-defined, bounded, self-adjoint, tracial functional. Moreover,
$F:\mathfrak{T}_{s}(B)\to\mathfrak{T}_{s}(A)$ given by $\tau\mapsto F(\tau)$
is a bounded $\mathbb{R}$-linear map.

###### Proof.

Using the same arguments as in Proposition 5.1, it is clear the formula (6.1)
is well-defined and gives rise to a bounded $\mathbb{R}$-linear map
$F:\mathfrak{T}_{s}(B)\to\mathfrak{T}_{s}(A)$. ∎

One can identify $\left(A_{sa}/A_{0}\right)^{*}\simeq\mathfrak{T}_{s}(A)$
(see, for example [CP79]), and so we can use duality to get a map
$\tilde{\Lambda}_{\theta}:=F^{*}:\mathfrak{T}_{s}(A)^{*}\to\mathfrak{T}_{s}(B)^{*}$
and restrict it to the dense set $A_{sa}/A_{0}$. One can check that the image
lies in $B_{sa}/B_{0}$ and that the restriction is just the map
$\Lambda_{\theta}$ that we got before.

We finish by listing some open problems.

  1. (1)

There are classes where topological isomorphisms between $U(A)$ and $U(B)$ (or
even $U^{0}(A)$ and $U^{0}(B)$) come from *-isomorphisms or
anti-*-isomorphisms. For example, if $A,B$ are prime traceless C*-algebras
containing full square zero elements, this is true by results in [CR23].

If $A$ is a unital, separable, nuclear C*-algebra satisfying the UCT and $B$
is a unital simple separable nuclear $\mathcal{Z}$-stable C*-algebra, then
unital embeddings $A\hookrightarrow B$ are classified by an invariant
$\underline{K}T_{u}(\cdot)$ which is an enlargement of $KT_{u}$ [CGS+23]. Thus
any isometric unitary group homomorphism $U(A)\to U(B)$ will give a
$KT_{u}$-morphism $KT_{u}(\theta)$ and therefore there will be an embedding
$\phi:A\hookrightarrow B$ such that $KT_{u}(\phi)=KT_{u}(\theta)$. However it
is not clear that $\phi$ satisfies $\phi|_{U(A)}=\theta$. More generally
though – in the tracial setting – are there continuous group homomorphisms
which do not have lifts to *-homomorphisms or anti-*-homomorphisms?

Note that in [AM03, Chapter 6], Lie isomorphisms between certain C*-algebras
are shown to be the sum of a Jordan *-isomorphism and a center-valued trace.
Is there a result for general (injective) Lie homomorphisms between certain
classes of C*-algebras?

  2. (2)

This enlargement of $KT_{u}$ discussed in [CGS+23] contains $K$-theory with
coefficients (along with appropriate pairing maps – the Bosckstein maps
discussed in [Sch84]). So we ask: do continuous group homo-morphisms induce
maps between $K$-theory with coefficients?

  3. (3)

For a general continuous homomorphism $\theta:U^{0}(A)\to U^{0}(B)$, does the
norm $\|S_{\theta}\|$ determine a Lipschitz constant for $\theta$? We clearly
have that

(6.2) $\|S_{\theta}\|\leq\inf\\{K\mid\theta\text{ is }K\text{-Lipschitz}\\}$

by Lemma 4.4. Is this equality?

  4. (4)

For $A$ simple (or prime), is it true that any continuous injective homo-
morphism $\theta:U^{0}(A)\to U^{0}(B)$ is isometric? Contractive? What if $B$
is simple (or prime)?

  5. (5)

In the initial draft of this paper, we claimed that any continuous group
homomorphism in Proposition 5.1 gave rise to a $\mathbb{C}$-linear
$\tilde{G}_{\theta}$. This is clearly false by Example 5.3. Is there a way to
guarantee that the map $\tilde{G}_{\theta}$ is $\mathbb{C}$-linear? Or can one
alter it accordingly for this to happen? Or alter it to get a map between
unitary groups, which would then allow one to use the results in Section 4?
Maybe if one starts with an injective, contractive group homomorphism
$GL^{0}(A)\to GL^{0}(B)$ which sends $\mathbb{C}^{\times}$ to
$\mathbb{C}^{\times}$, one can say something.

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