2. Preliminaries and notation

2.1. Notation

For a group $G$ , we denote by $DG$ the derived subgroup of $G$ , that is

(2.1) \begin{equation} DG \,:\!=\, \big\langle ghg^{-1}h^{-1} \mid g,h \in G \big\rangle. \end{equation}

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) \begin{equation} U_{\infty }(A) \,:\!=\, \underset{\to }{\lim }\, U_n(A), \end{equation}

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.Footnote ¹ We have general linear analogs 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 e.g., [Reference Rørdam, Larsen and Laustsen33, Chapter 11.4]). The set of tracial states on $A$ will be denoted $T(A)$ , which is a Choquet simplex [Reference Sakai36, Theorem 3.1.18], and we denote by $\textrm{Aff}\, T(A)$ the set of continuous affine functions $T(A) \to \mathbb{R}$ , which is an interpolation group with order unit (see [Reference Goodearl21, Chapter 11]). For unital $A$ , the pairing map $\rho _A\,:\, K_0(A) \to \textrm{Aff}\, T(A)$ 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) \begin{equation} \rho _A(x)(\tau ) \,:\!=\, \tau (p - q), \ \ \ \tau \in T(A). \end{equation}

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 [Reference de la Harpe and Skandalis12] (see [Reference de la Harpe11] 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) \begin{equation} \tilde{\Delta }_\tau ^n(\xi ) \,:\!=\, \int _0^1 \tau \left (\frac{1}{2\pi i}\xi^{\prime}(t)\xi (t)^{-1}\right )dt, \end{equation}

where this integral is just the Riemann integral in $E$ .Footnote ² We state the unitary variant of [Reference de la Harpe and Skandalis12, 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) \begin{equation} \tilde{\Delta }_\tau ^n(\xi _1\xi _2) = \tilde{\Delta }_\tau ^n(\xi _1) + \tilde{\Delta }_\tau ^n(\xi _2), \end{equation}
   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 $\|\xi (t) - 1\| \lt 1$ for all $t \in [0,1]$ , then

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

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 $\xi _p(t) \,:\!=\, pe^{2\pi i t} + (1-p)$ 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) \begin{equation} \Delta _\tau ^n\,:\, U_{\infty }^0(A) \to E/\tilde{\Delta }_\tau ^n\big(\pi _1\big(U_n^0(A)\big)\big) \end{equation}

given by $\Delta _\tau ^n(x) \,:\!=\, \left[\tilde{\Delta }_\tau ^n(\xi _x)\right]$ 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 $\text{Tr}_A\,:\, A_{sa} \to \textrm{Aff}\, T(A)$ , which is given by $\text{Tr}_A(a) \,:\!=\, \hat{a}$ , where $\hat{a} \in \textrm{Aff}\, T(A)$ 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 [Reference de la Harpe and Skandalis12, Lemme 3]. Take a continuous path $\xi \,:\, [0,1] \to U_n(A)$ and choose $k$ such that

(2.8) \begin{equation} \left \|\xi \left(\frac{j-1}{k}\right)^{-1}\xi \left(\frac{j}{k}\right) - 1\right \| \lt 1 \text{ for all } j=1,\dots,k. \end{equation}

Then taking

(2.9) \begin{equation} a_j \,:\!=\, \frac{1}{2\pi i}\log \left (\xi \left(\frac{j-1}{k}\right)^{-1}\xi \left(\frac{j}{k}\right)\right ), j=1,\dots,k, \end{equation}

$\xi$ will be homotopic to the path

(2.10) \begin{equation} \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. \end{equation}

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

(2.11) \begin{equation} X_j \,:\!=\, \xi \left (\frac{j-1}{k}\right )\text{ and }Y_j \,:\!=\, e^{2\pi i(kt - j + 1)a_j}. \end{equation}

Then

(2.12) \begin{equation} \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 i k a_jY_jY_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} \end{equation}

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 $A_{sa}/A_0 \simeq \textrm{Aff}\, T(A)$ sending an element $[a]$ to $\widehat{a}$ . Indeed, it is not hard to see that the map $A_{sa}/A_0 \to \textrm{Aff}\, T(A)$ given by $[a] \mapsto \hat{a}$ is a well-defined $\mathbb{R}$ -linear map. Moreover [Reference Cuntz and Pedersen8, 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 [Reference Goodearl21, Corollary 7.4] gives that the image is dense and therefore all of $\textrm{Aff}\, T(A)$ (since this is an isometry). We freely identify $A_{sa}/A_0$ with $\textrm{Aff}\, T(A)$ .

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) \begin{equation} \bar{\Delta }_\tau ^n\,:\, U_n^0(A) \to E/\overline{\tilde{\Delta }_\tau ^n\big(\pi _1\big(U_n^0(A)\big)\big)}, \end{equation}

given by $\bar{\Delta }_\tau ^n(x) \,:\!=\, \left[\tilde{\Delta }_\tau ^n(\xi _x)\right]$ 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 predeterminant (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, [Reference Thomsen39]). Let $A$ be a unital C*-algebra. Then

(2.14) \begin{equation} \ker \overline \Delta ^n = CU_n^0(A). \end{equation}

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 [Reference de la Harpe and Skandalis13, Reference Ng26, Reference Ng and Robert27, Reference Thomsen38].

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 $\textrm{Aff}\, T(A)$ .

Theorem 2.4 (Theorem 3.2, [Reference Thomsen39]). $\overline \Delta ^n$ gives a homeomorphic group isomorphism

(2.15) \begin{equation} U_n^0(A)/CU_n^0(A) \simeq \textrm{Aff}\, T(A)/\overline{\tilde{\Delta }^n\big(\pi _1\big(U_n^0(A)\big)\big)} \end{equation}

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

(2.16) \begin{equation} U_{\infty }^0(A)/CU_{\infty }^0(A) \simeq \textrm{Aff}\, T(A)/\overline{\rho _A(K_0(A))}. \end{equation}

2.4. The $KT_u$ invariant

Following [Reference Carrión, Gabe, Schafhauser, Tikuisis and White6], for a unital C*-algebra, we let

(2.17) \begin{equation} KT_u(A) \,:\!=\, (K_0(A),[1_A]_0,K_1(A),\rho _A,\textrm{Aff}\, T(A)) \end{equation}

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) \begin{equation} (\alpha _0,\alpha _1,\gamma )\,:\, KT_u(A) \to KT_u(B), \end{equation}

will be a triple $(\alpha _0,\alpha _1,\gamma )$ consisting of $\alpha _0\,:\, K_0(A) \to K_0(B)$ a group homomorphism such that $\alpha _0([1_A]_0) = [1_B]_0$ , $\alpha _1\,:\, K_1(A) \to K_1(B)$ a group homomorphism, and $\gamma \,:\, \textrm{Aff}\, T(A) \to \textrm{Aff}\, T(B)$ is a unital positive map such that

(2.19)

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 nonempty 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{equation} \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} \end{equation}

Thomsen’s isomorphism [Reference Thomsen39, Theorem 3] then brings about maps between quotients of $\textrm{Aff}\, T(A)$ and $\textrm{Aff}\, T(B)$ :

(3.2)

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)

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)

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

In the setting where $\pi _1\big(U^0(A)\big) \to K_0(A)$ and $\pi _1\big(U^0(B)\big) \to K_0(B)$ are surjections, we have induced maps between quotients

(3.5) \begin{equation} \begin{split} &\textrm{Aff}\, T(A)/\overline{\rho _A(K_0(A))} \to \textrm{Aff}\, T(B)/\overline{\rho _B(K_0(B))}, \\ &\textrm{Aff}\, T(A)/\rho _A(K_0(A)) \to \textrm{Aff}\, T(B)/\rho _B(K_0(B)) \end{split} \end{equation}

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 $\textrm{Aff}\, T(A) \to \textrm{Aff}\, T(B)$ . These spaces have further structure as dimension groups with order units [Reference Goodearl21, 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 $K_0(A) \simeq \pi _1\big(U^0(A)\big)$ and $K_0(B) \simeq \pi _1\big(U^0(B)\big)$ in the canonical way (which is true in the presence of $\mathcal{Z}$ -stability by [Reference Jiang25] or stable rank one [Reference Rieffel32]), we would like this map to be compatible with the group homomorphism

(3.6) \begin{equation} K_0(\theta )\,:\, K_0(A) \to K_0(B) \end{equation}

arising from the diagram

(3.7)

By compatible, we mean that

(3.8)

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 $\textrm{Aff}\, T(\cdot )$ respect the pairing.

Stone’s theorem [Reference Conway7, 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 1950s 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 [Reference Sakai35]. 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 [Reference Hatori and Molnár24].

Proof. Using the fact that $\theta$ is a norm-continuous homomorphism, $t \mapsto \theta \left(e^{2\pi i t a}\right)$ 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 $\theta \left(e^{2\pi i ta}\right) = e^{2\pi i tb}$ for all $t \in \mathbb{R}$ . The boundedness of $b$ follows from norm-continuity, and the uniqueness follows from the fact that $b = \frac{1}{2\pi i t_0} \log \theta \left(e^{2\pi i a t_0}\right)$ for sufficiently small $t_0 \gt 0$ . 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) \begin{equation} \theta \left(e^{2\pi i t a}\right) = e^{2\pi i t S_\theta (a)} \text{ for all } t \in \mathbb{R}. \end{equation}

Then $S_\theta$ is a bounded $\mathbb{R}$ -linear map (see [Reference Hatori and Molnár24, Reference Sakai35], 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.

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

(3.10) \begin{equation} \theta \left(e^{2\pi i t a}\right) = \theta \left(e^{2\pi i t b}\right) \end{equation}

for all $t \in \mathbb{R}$ . Injectivity of $\theta$ gives that $e^{2\pi i t a} = e^{2\pi i t b}$ 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 $(\theta ^{-1}\left(e^{2\pi i t b}\right))_{t \in \mathbb{R}} \subseteq U^0(A)$ is a norm-continuous one-parameter family of unitaries which we can write as $\left(e^{2\pi i t a}\right)_{t \in \mathbb{R}}$ for some $a \in A_{sa}$ . But then

(3.11) \begin{equation} \theta \left(e^{2\pi i t a}\right) = \theta \circ \theta ^{-1}(e^{\pi i t b}) = e^{2\pi i t b}. \end{equation}

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 [Reference Rørdam, Larsen and Laustsen33, 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 four 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 homomorphism, $E$ 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 $S_\theta$ induce a bounded $\mathbb{R}$ -linear map

(3.12) \begin{equation} \Lambda _\theta \,:\, \textrm{Aff}\, T(A) \to \textrm{Aff}\, T(B). \end{equation}

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

(3.13) \begin{equation} \begin{split} e^{2\pi i t S_\theta (uau^*)} &= \theta \left(e^{2\pi i t uau^*}\right) \\ &= \theta \left(ue^{2\pi i t a}u^*\right) \\\ &= \theta (u)e^{2\pi i t S_\theta (a)}\theta (u)^* \\ &= e^{2\pi i t \theta (u)S_\theta (a)\theta (u)^*} \end{split} \end{equation}

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

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

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) \begin{equation} \Lambda _\theta \,:\, \textrm{Aff}\, T(A) \simeq A_{sa}/A_0 \to B_{sa}/B_0 \simeq \textrm{Aff}\, T(B), \end{equation}

where we identify $A_{sa}/A_0 \simeq \textrm{Aff}\, T(A)$ and $B_{sa}/B_0 \simeq \textrm{Aff}\, T(B)$ .

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

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 [Reference Folland17, Chapter 4].

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

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) \begin{equation} \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 ] \end{equation}

and we have that

(3.25) \begin{equation} \tilde{\Delta }_\omega ^n(\xi ) = \tilde{\Delta }_\omega ^n(\eta ) = \sum _{j=1}^k \omega (a_j) \end{equation}

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 $t \in \left[\frac{j-1}{k},\frac{j}{k}\right]$ :

(3.26) \begin{equation} \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} \end{equation}

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) \begin{equation} \tilde{\Delta }_\tau ^1(\theta \circ \xi ) = \tilde{\Delta }_\tau ^1(\theta \circ \eta ) = \sum _{j=1}^k \tau \left(S_\theta (a_j)\right), \end{equation}

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) \begin{equation} q\,:\, E/\tilde{\Delta }_{\tau \circ S_\theta }^1\big(\pi \big(U^0(A)\big)\big) \to E/\tilde{\Delta }_\tau ^1\big(\pi \big(U^0(B)\big)\big) \end{equation}

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{equation} \begin{split} q\left(\Delta _{\tau \circ S_\theta }^1(u)\right) &= q\left (\tilde{\Delta }_{\tau \circ S_\theta }^1(\xi _u) + \tilde{\Delta }_{\tau \circ S_\theta }^1\big(\pi _1\big(U^0(A)\big)\big)\right ) \\ &= \tilde{\Delta }_{\tau \circ S_\theta }^1(\xi _u) + \tilde{\Delta }_\tau ^1\big(\pi _1\big(U^0(B)\big)\big) \\ &\overset{(1)}{=} \tilde{\Delta }_\tau ^1(\theta \circ \xi _u) + \tilde{\Delta }_\tau ^1\big(\pi _1\big(U^0(B)\big)\big) \\ &= \Delta _\tau ^1(\theta (u)). \end{split} \end{equation}

The remark about Thomsen’s variant is obvious.

Corollary 3.6. The following diagram commutes:

(3.30)

In particular, when the canonical maps

(3.31) \begin{equation} \pi _1\big(U^0(A)\big) \to K_0(A) \text{ and }\pi _1\big(U^0(B)\big) \to K_0(B) \end{equation}

are isomorphisms, we have that

(3.32)

commutes, where $K_0(\theta )\,:\, K_0(A) \to K_0(B)$ is the map induced from the diagram

(3.33)

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 haveFootnote ³

(3.34) \begin{align} \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 ) \nonumber\\ &= \int _0^1 \Lambda _\theta \circ \text{Tr}_A\left (\xi^{\prime}(t)\xi (t)^{-1}\right )dt\ \nonumber\\ &= \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 ).\nonumber \end{align}

Now we note that the diagram

(3.35)

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 $m \gt n$ , then

(3.36) \begin{equation} \tilde{\Delta }_\tau ^m\left(\xi \oplus 1_{m-n}\right) = \tilde{\Delta }_\tau ^n(\xi ). \end{equation}

Proposition 3.7. Let $A,B$ be unital C*-algebras. Then the map $\Lambda _\theta$ is a lift of the map

(3.37) \begin{equation} \textrm{Aff}\, T(A)/\tilde{\Delta }_A^1\big(\pi _1\big(U^0(A)\big)\big) \to \textrm{Aff}\, T(B)/\tilde{\Delta }_B^1\big(\pi _1\big(U^0(B)\big)\big) \end{equation}

as described in ( 3.4 ).

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

(3.38)

where $\tilde \theta ([u]) \,:\!=\, [\theta (u)]$ , $\delta _A^1\big(\big[e^{2\pi i a}\big]\big) \,:\!=\, [\hat{a}]$ , $\delta _B^1$ is defined similarly, and

(3.39) \begin{equation} P = \delta _B^1 \circ \tilde \theta \circ \left(\delta _A^1\right)^{-1}. \end{equation}

But then we have that

(3.40) \begin{equation} \begin{split} P([a]) &= \delta _B^1 \circ \tilde \theta \circ \left(\delta _A^1\right)^{-1}([\hat{a}]) \\ &= \delta _B^1 \circ \tilde \theta \left(\left[e^{2\pi i a}\right]\right) \\ &= \delta _B^1\left(\left[\theta \left(e^{2\pi i a}\right)\right]\right) \\ &= \delta _B^1\left(\left[e^{2\pi i S_\theta (a)}\right]\right) \\ &= \left[\widehat{S_\theta (a)}\right] \\ &= \left[\Lambda _\theta (\widehat{a})\right] \end{split} \end{equation}

In particular, the diagram

(3.41)

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, $\Lambda _\theta \,:\, \textrm{Aff}\, T(A) \to \textrm{Aff}\, T(B)$ is a lift of the map

(3.42) \begin{equation} \textrm{Aff}\, T(A)/\overline{\tilde{\Delta }_A^1\big(\pi _1\big(U^0(A)\big)\big)} \to \textrm{Aff}\, T(B)/\overline{\tilde{\Delta }_B^1\big(\pi _1\big(U^0(B)\big)\big)} \end{equation}

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)

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) \begin{equation} \pi _1\big(U^0(A)\big) \to K_0(A) \text{ and }\pi _1\big(U^0(B)\big) \to K_0(B) \end{equation}

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)

and

(3.46)

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)

C*-algebras satisfying the last condition arise naturally—for example, unital, separable, simple, pure C*-algebras of stable rank 1 such that every 2-quasitracial state on $A$ is a trace have this property [Reference Ng and Robert27].

4. The order on $\textbf{Aff}\, \boldsymbol{T}(\cdot )$

We now examine when the map induced on $\textrm{Aff}\, T(\cdot )$ 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 analog 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 [Reference Paulsen29].

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) \begin{equation} e^{2\pi i S_\theta (1)t} = \theta \left(e^{2\pi i t}\right) = e^{2\pi i nt} \end{equation}

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 $\textrm{Aff}\, T(A) \to \textrm{Aff}\, T(B)$ , 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 $\textrm{Aff}\, (\emptyset ) = C_{\mathbb{R}}(\emptyset )$ and with the complex analogs as well. However, the following lemma is true regardless of whether $T(A)$ is empty or not.

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 $a \in A_{sa}$ , $\hat{a} \in \textrm{Aff}\, T(A)$ 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 $\textrm{Aff}\, T(A) + i\textrm{Aff}\, T(A) \subseteq C(T(A))$ is an operator system and the canonical extension

(4.2) \begin{equation} \Lambda _\theta ^{\mathbb{C}}\,:\, \textrm{Aff}\, T(A) + i\textrm{Aff}\, T(A) \to \textrm{Aff}\, T(B) + i\textrm{Aff}\, T(B) \subseteq C(T(B)) \end{equation}

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

Proof. As $S_\theta (A_0) \subseteq B_0$ , we have

(4.3) \begin{equation} \begin{split} \|\Lambda _\theta (\hat{a})\|_{\textrm{Aff}\, T(B)} &= \|\widehat{S_\theta (a)}\|_{\textrm{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}\|_{\textrm{Aff}\, T(A)} \end{split} \end{equation}

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 [Reference Hatori and Molnár24]; we exemplify said argument to show the bound condition. We use the observation that

(4.4) \begin{equation} \frac{e^{2\pi i t a} - 1}{t} \to 2\pi i a \end{equation}

as $t \to 0$ . Since

(4.5) \begin{equation} \|e^{2\pi i t S_\theta (a)} - 1\| \leq K\|e^{2\pi i t a} - 1\| \end{equation}

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) \begin{equation} \|S_\theta (a)\| \leq K\|a\|. \end{equation}

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 $\textrm{Aff}\, T(A) \simeq A_{sa}/A_0$ and $\textrm{Aff}\, T(B) \simeq B_{sa}/B_0$ , noting that $S_\theta (A_0) = B_0$ , and that $\Lambda _\theta$ will preserve the quotient norms.

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.

Proof. This follows from the fact that the induced map $\Lambda _\theta \,:\, \textrm{Aff}\, T(A) \to \textrm{Aff}\, T(B)$ 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 [Reference Goodearl21, Chapter 7], there exists a continuous affine map $T_\theta \,:\, T(B) \to T(A)$ .

Proof. As $\theta (\mathbb{T}) = \mathbb{T}$ , $S_\theta (1) \in \{-1,1\}$ . Let $\pm \Lambda _\theta \,:\, \textrm{Aff}\, T(A) \to \textrm{Aff}\, T(B)$ 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)

Footnote ⁴ via the canonical maps, then there is an induced map

(4.8) \begin{equation} KT_u(\theta )\,:\, KT_u(A) \to \textrm{KT}_u(B). \end{equation}

Proof. Let

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

• $\theta _0\,:\, \pi _1\big(U^0(A)\big) \to \pi _1\big(U^0(B)\big)$ 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) \begin{equation} (\pm K_0(\theta ),K_1(\theta ),\pm \Lambda )\,:\, KT_u(A) \to KT_u(B) \end{equation}

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)

commutesFootnote ⁵ 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) \begin{equation} \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, \end{equation}

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

\begin{equation*} KT_u(\theta )\,:\, KT_u(A) \to KT_u(B). \end{equation*}

Proof. C*-algebras which are $\mathcal{Z}$ -stable or have stable rank one satisfy the hypotheses of Theorem 4.8 by [Reference Jiang25] and [Reference Rieffel32], respectively. So Theorem 4.8 applies.

Remark 4.11. The strict ordering on $\textrm{Aff}\, T(A)$ is given by $f \gg g$ if $f(\tau ) \gt g(\tau )$ 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 $\pm \Lambda _\theta \,:\, \textrm{Aff}\, T(A) \to \textrm{Aff}\, T(B)$ 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) \begin{equation} \pm \Lambda _\theta (f) \gg \pm \Lambda _\theta (g) \iff f \gg g. \end{equation}

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 $\sigma (a) \gt 0$ for all $\sigma \in T(A)$ , then $\tau (\pm S_\theta (a)) \gt 0$ 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 predeterminant applied a the loop. Precisely, a nonzero 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 $\tilde{\Delta }_\tau (\xi _x) \gt 0$ for all $\tau \in T(A)$ .

Although the following is known, for example, by very strong results in [Reference Ara and Mathieu1, Chapter 6] pertaining to certain prime C*-algebras, we give the following as a corollary by using $K$ -theoretic classification results.

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 [Reference Castillejos, Evington, Tikuisis, White and Winter5, Corollary D], [Reference Elliott, Guihua Gong and Niu16, Reference Gong, Lin and Niu18, Reference Gong, Lin and Niu19] and the references therein), $A \simeq B$ .

Using the state-of-the art classification of embeddings [Reference Carrión, Gabe, Schafhauser, Tikuisis and White6], 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 [Reference Carrión, Gabe, Schafhauser, Tikuisis and White6, 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 $\textrm{Aff}\, T(A) \to \textrm{Aff}\, T(B)$ 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.

Proof. Assuming such a $\theta$ exists, it gives rise to a $KT_u$ -morphism

(4.13) \begin{equation} KT_u(\theta )\,:\, KT_u(A) \to KT_u(B) \end{equation}

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 [Reference Carrión, Gabe, Schafhauser, Tikuisis and White6, 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)

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 [Reference de la Harpe and Skandalis12]), instead of closures of derived groups, there is a purely algebraic variant of the above diagram:

(5.2)

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) \begin{equation} A/\overline{[A,A]} \to B/\overline{[B,B]} \end{equation}

(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) \begin{equation} \tilde{G}_\theta \,:\, A/\overline{[A,A]} \to B/\overline{[B,B]} \end{equation}

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) \begin{equation} G_\theta (a) \,:\!=\, \lim _n \frac{n}{2\pi i}\log \theta \left(e^{2\pi i \frac{a}{n}}\right). \end{equation}

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

(5.6) \begin{equation} \|\theta \left(e^{2\pi i \frac{a}{n}}\right) - 1\| \lt 1. \end{equation}

We then have for $n \geq N$ ,

(5.7) \begin{align} \frac{n}{2\pi i}\log \theta \left(e^{2\pi i \frac{a}{n}}\right) &= \frac{n}{2\pi i}\log \theta \left(e^{2\pi i \frac{Na}{Nn}}\right) \nonumber\\[4pt] &= \frac{n}{2\pi i}\log \theta \left(e^{2\pi i \frac{a}{Nn}}\right)^N \nonumber\\[4pt] &= \frac{Nn}{2\pi i}\log \theta \left(e^{2\pi i \frac{a}{Nn}}\right) \nonumber\\[4pt] &= \frac{N}{2\pi i}\log \theta \left(e^{2\pi i \frac{a}{Nn}}\right)^n \\[4pt] &= \frac{N}{2\pi i}\log \theta \left(e^{2\pi i \frac{na}{Nn}}\right)\nonumber \\[4pt] &= \frac{N}{2\pi i}\log \theta \left(e^{2\pi i \frac{a}{N}}\right).\nonumber \end{align}

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

(5.8) \begin{equation} e^{a+b} = \lim _k \left (e^{\frac{a}{k}}e^{\frac{b}{k}}\right )^k \end{equation}

(see for example [Reference Hall23, 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) \begin{equation} 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} \end{equation}

(a variation of the argument given in the proof of [Reference Hall23, 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) \begin{equation} \tilde{G}_\theta \,:\, A/\overline{[A,A]} \to B/\overline{[B,B]}. \end{equation}

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) \begin{equation} \theta \left(e^{2\pi i t a}\right) = e^{2\pi i t G_\theta (a)}, t \in \mathbb{R}. \end{equation}

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{equation} \begin{split} \theta \left(e^{2\pi i t(a+bi)}\right) &= \left|e^{2\pi i t(a + bi)}\right|^{\alpha + \beta i}e^{2\pi i t n(a + bi)} \\ &= \left(e^{-2\pi t b}\right)^{\alpha + \beta i}e^{2\pi i t n(a + bi)} \\ &= e^{-2\pi t b\alpha }e^{-2\pi i t\beta i b}e^{2\pi i t n(a + bi)} \\ &= e^{2\pi i t((na - \alpha b) + i(n - \beta )b}. \end{split} \end{equation}

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) \begin{equation} G_\theta \begin{pmatrix} a \\[3pt] b \end{pmatrix} = \begin{pmatrix} n & \quad -\alpha \\[3pt] 0 & \quad n - \beta \end{pmatrix}\begin{pmatrix} a \\[3pt] b \end{pmatrix}. \end{equation}

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) \begin{equation} F(\tau )(a) \,:\!=\, \lim _{n \to \infty } \tau \left (\frac{n}{2\pi i}\log \theta \left(e^{2\pi i \frac{a}{n}}\right)\right ). \end{equation}

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, e.g., [Reference Cuntz and Pedersen8]), 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 [Reference Chand and Robert9].

   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 $KT_u$ [Reference Carrión, Gabe, Schafhauser, Tikuisis and White6]. 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 [Reference Ara and Mathieu1, 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 [Reference Carrión, Gabe, Schafhauser, Tikuisis and White6] contains $K$ -theory with coefficients (along with appropriate pairing maps—the Bosckstein maps discussed in [Reference Schochet37]). So we ask: do continuous group homomorphisms 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) \begin{equation} \|S_\theta \| \leq \inf \{ K \mid \theta \text{ is }K\text{-Lipschitz}\} \end{equation}
   by Lemma 4.4. Is this equality?

4. 4. For $A$ simple (or prime), is it true that any continuous injective homomorphism $\theta \,:\, U^0(A) \to U^0(B)$ 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.