Related papers: Tracial approximation and ${\cal Z}$-stability

Let $\Theta=(\theta_{j,k})_{3\times 3}$ be a non-degenerate real skew-symmetric $3\times 3$ matrix, where $\theta_{j,k}\in [0,1).$ For any $\varepsilon>0$, we prove that there exists $\delta>0$ satisfying the following: if $v_1,v_2,v_3$ are…

We develop the concept of weak tracial Rokhlin property for finite group actions on simple (not necessarily unital) C*-algebras and study its properties systematically. In particular, we show that this property is stable under restriction…

We prove that a unital simple approximately homogeneous (AH) C*-algebra with no dimension growth absorbs the Jiang-Su algebra tensorially without appealing to the classification theory of these algebras. Our main result continues to hold…

In this paper, we introduce a class of generalized tracial approximation ${\rm C^*}$-algebras. Let $\mathcal{P}$ be a class of unital ${\rm C^*}$-algebras which have tracially $\mathcal{Z}$-absorbing (tracial nuclear dimension at most $n$,…

We introduce a concept of the bounded rank (with respect to a positive constant) for unital C*-algebras as a modification of the usual real rank and present a series of conditions insuring that bounded and real ranks coincide. These…

It is shown that if a C*-algebra has nuclear dimension $n$ then its Cuntz semigroup has the property of $n$-comparison. It then follows from results by Ortega, Perera, and Rordam that $\sigma$-unital C*-algebras of finite nuclear dimension…

Let $G$ be an inductive limit of finite cyclic groups and let $A$ be a unital simple projectionless C*-algebra with $K_1(A) \cong G$ and with a unique tracial state, as constructed based on dimension drop algebras by Jiang and Su. First, we…

We consider algebras in a modular tensor category C. If the trace pairing of an algebra A in C is non-degenerate we associate to A a commutative algebra Z(A), called the full centre, in a doubled version of the category C. We prove that two…

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 $\kappa$ for which…

We introduce diagonal comparison, a regularity property of diagonal pairs where the sub-C*-algebra has totally disconnected spectrum, and establish its equivalence with the concurrence of strict comparison of the ambient C*-algebra and…

Let $\mathcal{A}$ be the class of unital separable simple amenable $C$*-algebras $A$ which satisfy the Universal Coefficient Theorem for which $A\otimes M_{\texttt{P}}$ has tracial rank zero for some supernatural number $\texttt{p}$ of…

We study the automorphism group of a unital, simple, $\mathcal{Z}$-stable $C^{*}$-algebra. In this paper, we generalize the results by the authors in \cite{pr_auto} to $\mathcal{Z}$-stable $C^{*}$-algebras $\mathfrak{A}$ such that…

We say that a unital C*-algrebra A has the approximate positive factorization property (APFP) if every element of A is a norm limit of products of positive elements of A. (There is also a definition for the nonunital case.) T. Quinn has…

Let $\epsilon>0$ be a positive number. Is there a number $\delta>0$ 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…

We characterize when the reduced C*-algebra of a group has unique tracial state, respectively, is simple, in terms of Dixmier-type properties of the group C*-algebra. We also give a simple proof of the recent result by Breuillard, Kalantar,…

In this note we state a conjecture that characterizes unital C*-algebras for which the unitary group is amenable as a topological group in the norm topology. We prove the conjecture for simple, separable, stably finite, unital, $\mathcal…

The uniform tracial completion of a C*-algebra A with compact non-empty trace space T(A) is obtained by completing the unit ball with respect to the uniform 2-seminorm $\|a\|_{2,T(A)}=\sup_{\tau \in T(A)} \tau(a^*a)^{1/2}$. The trace…

We study the general and connected stable ranks for C*-algebras. We estimate these ranks for pullbacks of C*-algebras, and for tensor products by commutative C*-algebras. Finally, we apply these results to determine these ranks for certain…

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…

We construct a unital pre-C*-algebra $A_0$ which is stably finite, in the sense that every left invertible square matrix over $A_0$ is right invertible, while the C*-completion of $A_0$ contains a non-unitary isometry, and so it is…