Let $A$ be a unital separable simple $C^*$-algebra with tracial rank zero and let $x, \, y\in A$ be two normal elements. We show that $x$ is in the closure of the convex full of the unitary obit of $y$ if and only if there exists a sequence of unital completely positive linear maps $ϕ_n$ from $A$ to $A$ such that the sequence $ϕ_n(y)$ convergent to $x$ in norm and also approximately preserves the trace values. A purely measure theoretical description for normal elements in the closure of convex hull of unitary orbit of $y$ is also given. In the case that $A$ has a unique tracial state some classical results about the closure of the convex hull of the unitary orbits in von Neumann algebras are proved to be hold in $C^*$-algebras setting.
In this paper, we study closed convex hulls of unitary orbits in various C$^*$-algebras. For unital C$^*$-algebras with real rank zero and a faithful tracial state determining equivalence of projections, a notion of majorization describes the closed convex hulls of unitary orbits for self-adjoint operators. Other notions of majorization are examined in these C$^*$-algebras. Combining these ideas with the Dixmier property, we demonstrate unital, infinite dimensional C$^*$-algebras of real rank zero and strict comparison of projections with respect to a faithful tracial state must be simple and …
In this paper, we characterize the closures of convex hulls of unitary orbits of self-adjoint operators in unital, separable, simple C$^*$-algebras with non-trivial tracial simplex, real rank zero, stable rank one, and strict comparison of projections with respect to tracial states. In addition, an upper bound for the number of unitary conjugates in a convex combination needed to approximate a self-adjoint are obtained.
Let $C$ be a unital AH-algebra and let $A$ be a unital separable simple C*-algebra with tracial rank no more than one. Suppose that $ϕ, ψ: C\to A$ are two unital monomorphisms. With some restriction on $C,$ we show that $ϕ$ and $ψ$ are approximately unitarily equivalent if and only if [ϕ]=[ψ] in KL(C,A) τ\circ ϕ=τ\circ ψfor all tracial states of A and ϕ^‡=ψ^‡, here ϕ^‡ and ψ^‡ are homomorphisms from $U(C)/CU(C)\to U(A)/CU(A) induced by ϕand ψ, respectively, and where CU(C) and CU(A) are closures of the subgroup generated by commutators of the unitary groups of C and B.
We show that a tracial state on a unital C*-algebra admits a Haar unitary if and only if it is diffuse, if and only if it does not dominate a tracial functional that factors through a finite-dimensional quotient. It follows that a unital C*-algebra has no finite-dimensional representations if and only if each of its tracial states admits a Haar unitary. More generally, we study when nontracial states admit Haar unitaries. In particular, we show that every state on a unital, simple, infinite-dimensional C*-algebra admits a Haar unitary. We obtain applications to the structure of reduced free pr…
We consider the tracial crossed product algebra $M=A\rtimesΛ$ arising from a trace preserving action $σ:Λ\curvearrowright A$ of a discrete group $Λ$ on a tracial von Neumann algebra $A$. For a unitary subgroup $\mathcal{G}\subset \mathcal{U}(M)$, we study when this $\mathcal{G}$ can be conjugated into $\mathcal{U}(A)\cdotΛ$ in $M$. We provide a general sufficient condition for this to happen. Our result generalizes a result of Ioana, Popa and Vaes, which treats the case when $M$ is the group von Neumann algebra $L(Λ)$.
Let $A$ be a unital separable simple infinite dimensional \CA with tracial rank no more than one and with the tracial state space $T(A)$ and let $U(A)$ be the unitary group of $A.$ Suppose that $u\in U_0(A),$ the connected component of $U(A)$ containing the identity. We show that, for any $\ep>0,$ there exists a selfadjoint element $h\in A_{s.a}$ such that $$ \|u-\exp(ih)\|<\ep. $$ We also study the problem when $u$ can be approximated by unitaries in $A$ with finite spectrum. Denote by $CU(A)$ the closure of the subgroup of unitary group of $U(A)$ generated by its commutators. It is known tha…
Let $C=C(X)$ be the unital $C^*$-algebra of all continuous functions on a finite CW complex $X$ and let $A$ be a unital simple $C^*$-algebra with tracial rank at most one. We show that two unital monomorphisms $ϕ, ψ: C\to A$ are asymptotically unitarily equivalent, i.e., there exists a continuous path of unitaries $\{u_t: t\in [0,1)\}\subset A$ such that $$ \lim_{t\to 1} u_t^*ϕ(f)u_t=ψ(f) {\rm for all} \in C(X), $$ if and only if \beq [ϕ]&=&[ψ] {\rm in} KK(C, A), τ\circ ϕ&=&τ\circ ψ{\rm for all} τ\in T(A), and ϕ^†&=&ψ^†, \eneq where $T(A)$ is the simplex of tracial states of $A$ and $ϕ^†, ψ^†:…
Let $A$ and $C$ be two unital simple C*-algebas with tracial rank zero. Suppose that $C$ is amenable and satisfies the Universal Coefficient Theorem. Denote by ${KK}_e(C,A)^{++}$ the set of those $κ$ for which $κ(K_0(C)_+\setminus\{0\})\subset K_0(A)_+\setminus\{0\}$ and $κ([1_C])=[1_A]$. Suppose that $κ\in {KK}_e(C,A)^{++}.$ We show that there is a unital monomorphism $ϕ: C\to A$ such that $[ϕ]=κ.$ Suppose that $C$ is a unital AH-algebra and $λ: \mathrm{T}(A)\to \mathrm{T}_{\mathtt{f}}(C)$ is a continuous affine map for which $τ(κ([p]))=λ(τ)(p)$ for all projections $p$ in all matrix algebras …
We define the notion of self-tracial stability for tracial von Neumann algebras and show that a tracial von Neumann algebra satisfying the Connes Embedding Problem is self-tracially stable if and only if it is amenable. We then generalize a result of Jung by showing that a separable tracial von Neumann algebra that satisfies the Connes Embedding Problem is amenable if and only if any two embeddings into $R^\mathcal{U}$ are ucp-conjugate. Moreover we show that for a II$_1$ factor $N$ satisfying CEP, the space $\mathbb{H}$om$(N, \prod_{k\to \mathcal{U}}M_k)$ of unitary equivalence classes of emb…
Let $C$ be a unital AH-algebra and let $A$ be a unital separable simple \CA with tracial rank zero. Suppose that $ϕ_1, ϕ_2: C\to A$ are two unital monomorphisms. We show that there is a continuous path of unitaries $\{u_t: t\in [0, \infty)\}$ of $A$ such that $$ \lim_{t\to\infty}u_t^*ϕ_1(a)u_t=ϕ_2(a)\tforal a\in C $$ if and only if $[ϕ_1]=[ϕ_2]$ in $KK(C,A),$ $τ\circ ϕ_1=τ\circ ϕ_2$ for all $τ\in T(A)$ and the rotation map ${\tildeη}_{ϕ_1,ϕ_2}$ associated with $ϕ_1$ and $ϕ_2$ is zero. In particular, an automorphism $\af$ on a unital separable simple \CA $A$ in ${\cal N}$ with tracial rank zero…
We classify unital monomorphisms into certain simple Z-stable C^*-algebras up to approximate unitary equivalence. The domain algebra C is allowed to be any unital separable commutative C^*-algebra, or any unital simple separable nuclear Z-stable C^*-algebra satisfying the UCT such that C\otimes B is of tracial rank zero for a UHF algebra B. The target algebra A is allowed to be any unital simple separable Z-stable C^*-algebra such that A\otimes B has tracial rank zero for a UHF algebra B, or any unital simple separable exact Z-stable C^*-algebra whose projections separate traces and whose extr…
We investigate the closed convex hull of unitary orbits of selfadjoint elements in arbitrary unital C*-algebras. Using a notion of majorization against unbounded traces, a characterization of these closed convex hulls is obtained. Furthermore, for C*-algebras satisfying Blackadar's strict comparison of positive elements by traces or for collections of C*-algebras with a uniform bound on their nuclear dimension, an upper bound for the number of unitary conjugates in a convex combination required to approximate an element in the closed convex hull within a given error is shown to exist. This pro…
Motivated by the recent resolution of Asymptotic Quantum Birkhoff Conjecture (AQBC), we attempt to estimate the distance between a given unital quantum channel and the convex hull of unitary channels. We provide two lower bounds on this distance by employing techniques from quantum information and operator algebras, respectively. We then show how to apply these results to construct some explicit counterexamples to AQBC. We also point out an interesting connection between the Grothendieck's inequality and AQBC.
Let $C$ and $A$ be two unital separable amenable simple C*-algebras with tracial rank no more than one. Suppose that $C$ satisfies the Universal Coefficient Theorem and suppose that $ϕ_1, ϕ_2: C\to A$ are two unital monomorphisms. We show that there is a continuous path of unitaries $\{u_t: t\in [0, \infty)\}$ of $A$ such that $$ \lim_{t\to\infty}u_t^*ϕ_1(c)u_t=ϕ_2(c)\tforal c\in C $$ if and only if $[ϕ_1]=[ϕ_2]$ in $KK(C,A),$ $ϕ_1^‡=ϕ_2^‡,$ $(ϕ_1)_T=(ϕ_2)_T$ and a rotation related map $\bar{R}_{ϕ_1,ϕ_2}$ associated with $ϕ_1$ and $ϕ_2$ is zero. Applying this result together with a result of W…
We found that if $u$ and $v$ are any two unitaries in a unital $C^*$-algebra with $\|uv-vu\|<2$ such that $uvu^*v^*$ commutes with $u$ and $v,$ then the \SCA\, $A_{u,v}$ generated by $u$ and $v$ is isomorphic to a quotient of the rotation algebra $A_θ$ provided that $A_{u,v}$ has a unique tracial state. We also found that the Exel trace formula holds in any unital $C^*$-algebra. Let $θ\in (-1/2, 1/2)$ be a rational number. We prove the following: For any $\ep>0,$ there exists $\dt>0$ satisfying the following: if $u$ and $v$ are two unitary matrices such that $$ \|uv-e^{2πiθ}vu\|<\dt\andeqn {1\…
Let $C$ be a unital AH-algebra and $A$ be a unital simple C*-algebra with tracial rank zero. It has been shown that two unital monomorphisms $ϕ, ψ: C\to A$ are approximately unitarily equivalent if and only if $$ [ϕ]=[ψ] {\rm in} KL(C,A) and τ\circ ϕ=τ\circ ψ\tforal τ\in T(A), $$ where $T(A)$ is the tracial state space of $A.$ In this paper we prove the following: Given $κ\in KL(C,A)$ with $κ(K_0(C)_+\setminus \{0\})\subset K_0(A)_+\setminus \{0\}$ and with $κ([1_C])=[1_A]$ and a continuous affine map $λ: T(A)\to T_{\mathtt{f}}(C)$ which is compatible with $κ,$ where $T_{\mathtt{f}}(C)$ is the…
Let $A$ be a separable simple exact ${\cal Z}$-stable $C^*$-algebra. We show that the unitay group of ${\tilde A}$ has the cancellation property. If $A$ has continuous scale, the Cuntz semigroup of $\tilde A$ has the strict comparison property and a weak cancellation property. Let $C$ be a 1-dimensional non-commutative CW complex with $K_1(C)=\{0\}.$ Suppose that $λ: {\rm Cu}^\sim(C)\to {\rm Cu}^\sim(A)$ is a morphism in Cuntz semigroups which is strictly positive. Then there exists a sequence of homomorphisms $ϕ_n: C\to A$ such that $\lim_{n\to\infty}{\rm Cu}^\sim(ϕ_n)=λ.$ This result leads t…
Let $ε>0$ be a positive number. Is there a number $δ>0$ satisfying the following? Given any pair of unitaries $u$ and $v$ in a unital simple $C^*$-algebra $A$ with $[v]=0$ in $K_1(A)$ for which $$ \|uv-vu\|<\dt, $$ there is a continuous path of unitaries $\{v(t): t\in [0,1]\}\subset A$ such that $$ v(0)=v, v(1)=1 \and \|uv(t)-v(t)u\|<ε\forall t\in [0,1]. $$ An answer is given to this question when $A$ is assumed to be a unital simple $C^*$-algebra with tracial rank no more than one. Let $C$ be a unital separable amenable simple $C^*$-algebra with tracial rank no more than one which also satisf…
Two areas of mathematics which have received substantial attention in recent years are the theory of optimal transport and the Elliott classification programme for C*-algebras. We combine these two seemingly unrelated disciplines to make progress on a classical problem of Weyl. In particular, we show how results from the Elliott classification programme can be used to translate continuous transport of spectral measures into optimal unitary conjugation in C*-algebras. As a consequence, whenever two normal elements of a sufficiently well-behaved C*-algebra share a spectrum amenable to such conti…
Quantum Science Center of Guangdong-Hong Kong-Macao Greater Bay Area（粤港澳大湾区量子科学中心）；QMATH, Department of Mathematical Sciences, University of Copenhagen（哥本哈根大学数学科学学院 QMATH）
For $n\geq 3,$ let $Γ=SL_n(\mathbb Z).$ We prove the following superridigity result for $Γ$ in the context of operator algebras. Let $L(Γ)$ be the von Neumann algebra generated by the left regular representation of $Γ.$ Let $M$ be a finite factor and let $U(M)$ be its unitary group. Let $π: Γ\to U(M)$ be a group homomorphism such that $π(Γ)''=M.$ Then \begin{itemize} \item[(i)] either $M$ is finite dimensional, or \item [(ii)] there exists a subgroup of finite index $Λ$ of $Γ$ such that $π|_Λ$ extends to a homomorphism $U(L(Λ))\to U(M).$ \end{itemize} The result is deduced from a complete desc…
Spectral gap, internality of relative commutants, and non-hyperlinear groups
Institut für Geometrie, TU Dresden（德累斯顿工业大学几何研究所）；Department of Mathematics and Beijing International Center for Mathematical Research, Peking University（北京大学数学科学学院与北京国际数学研究中心）
We investigate the tracial and ideal structures of $C^*$-algebras of quasi-regular representations of stabilizers of boundary actions. Our main tool is the notion of boundary maps, namely $Γ$-equivariant unital completely positive maps from $Γ$-$C^*$-algebras to $C(\partial_FΓ)$, where $\partial_FΓ$ denotes the Furstenberg boundary of a group $Γ$. For a unitary representation $π$ coming from the groupoid of germs of a boundary action, we show that there is a unique boundary map on $C^*_π(Γ)$. Consequently, we completely describe the tracial structure of the $C^*$-algebras $C^*_π(Γ)$, and for a…
Let $A$ and $B$ be unital separable simple amenable \CA s which satisfy the Universal Coefficient Theorem. Suppose {that} $A$ and $B$ are $\mathcal Z$-stable and are of rationally tracial rank no more than one. We prove the following: Suppose that $ϕ, ψ: A\to B$ are unital {monomorphisms}. There exists a sequence of unitaries $\{u_n\}\subset B$ such that $$ \lim_{n\to\infty} u_n^*ϕ(a) u_n=ψ(a)\tforal a\in A, $$ if and only if $$ [ϕ]=[ψ]\,\,\,\text{in}\,\,\, KL(A,B), ϕ_{\sharp}=ψ_{\sharp}\andeqnϕ^‡=ψ^‡, $$ where $ϕ_{\sharp}, ψ_{\sharp}: \aff(T(A))\to \aff(T(B))$ and $ϕ^‡, ψ^‡: U(A)/CU(A)\to U(B…
We study variants of the Dixmier property that apply to elements of a unital C*-algebra, rather than to the C*-algebra itself. By a Dixmier element in a C*-algebra we understand one that can be averaged into a central element by means of a sequence of unitary mixing operators. Examples include all self-commutators and all quasinilpotent elements. We do a parallel study of an element-wise version of weak centrality, where the averaging to the centre is done using unital completely positive elementary operators (as in Magajna's characterization of weak centrality). We also obtain complete descri…
We define and study large and stably large subalgebras of simple unital C*-algebras. The basic example is the orbit breaking subalgebra of a crossed product by Z, as follows. Let X be an infinite compact metric space, let h be a minimal homeomorphism of X, and let Y be a closed subset of X. Let u be the standard unitary in C* (Z, X, h). The Y-orbit breaking subalgebra is the subalgebra of C* (Z, X, h) generated by C (X) and all elements f u for f in C (X) such that f vanishes on Y. If intersects each orbit of h at most once, then the Y-orbit breaking subalgebra is large in C* (Z, X, h). Large …
Let $A$ be a unital separable simple ${\cal Z}$-stable C*-algebra which has rational tracial rank at most one and let $u\in U_0(A),$ the connected component of the unitary group of $A.$ We show that, for any $ε>0,$ there exists a self-adjoint element $h\in A$ such that $$ |u-\exp(ih)|<ε. $$ The lower bound of $|h|$ could be as large as one wants. If $u\in CU(A),$ the closure of the commutator subgroup of the unitary group, we prove that there exists a self-adjoint element $h\in A$ such that $$ |u-\exp(ih)| <εand |h|\le 2π. $$ Examples are given that the bound $2π$ for $|h|$ is the optimal in g…
We introduce the concept of finitely coloured equivalence for unital *-homomorphisms between C*-algebras, for which unitary equivalence is the 1-coloured case. We use this notion to classify *-homomorphisms from separable, unital, nuclear C*-algebras into ultrapowers of simple, unital, nuclear, Z-stable C*-algebras with compact extremal trace space up to 2-coloured equivalence by their behaviour on traces; this is based on a 1-coloured classification theorem for certain order zero maps, also in terms of tracial data. As an application we calculate the nuclear dimension of non-AF, simple, separ…
We prove that the C*-algebra of a minimal diffeomorphism satisfies Blackadar's Fundamental Comparability Property for positive elements. This leads to the classification, in terms of K-theory and traces, of the isomorphism classes of countably generated Hilbert modules over such algebras, and to a similar classification for the closures of unitary orbits of self-adjoint elements. We also obtain a structure theorem for the Cuntz semigroup in this setting, and prove a conjecture of Blackadar and Handelman: the lower semicontinuous dimension functions are weakly dense in the space of all dimensio…
The principal result in this note is a strengthened version of Kadison's transitivity theorem for unital JB$^*$-algebras, showing that for each minimal tripotent $e$ in the bidual, $\mathfrak{A}^{**}$, of a unital JB$^*$-algebra $\mathfrak{A}$, there exists a self-adjoint element $h$ in $\mathfrak{A}$ satisfying $e\leq \exp(ih)$, that is, $e$ is bounded by a unitary in the principal connected component of the unitary elements in $\mathfrak{A}$. This new result opens the way to attack new geometric results, for example, a Russo--Dye type theorem for maximal norm closed proper faces of the close…
Let $X$ be a finite CW complex and let $h_1, h_2: C(X)\to A$ be two unital \hm s, where $A$ is a unital C*-algebra. We study the problem when $h_1$ and $h_2$ are approximately homotopic. We present a $K$-theoretical necessary and sufficient condition for them to be approximately homotopic under the assumption that $A$ is a unital separable simple C*-algebra of tracial rank zero, or $A$ is a unital purely infinite simple C*-algebra. When they are approximately homotopic, we also give a bound for the length of the homotopy. Suppose that $h: C(X)\to A$ is a monomorphism and $u\in A$ is a unitary …
Let $A$ be an $AH$ algebra $A=\lim\limits_{n\to \infty}(A_{n}=\bigoplus\limits_{i=1}\limits^{t_{n}}P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}, ϕ_{n,m})$, where $X_{n,i}$ are compact metric spaces, $t_{n}$ and $[n,i]$ are positive integers, and $P_{n,i}\in M_{[n,i]}(C(X_{n,i}))$ are projections. Suppose that $A$ has the ideal property: each closed two-sided ideal of $A$ is generated by the projections inside the ideal, as a closed two sided ideal. In this article, we will classify all $AH$ algebras with ideal property of no dimension growth---that is, $sup_{n,i}dim(X_{n,i})<+\infty$. This result gener…
