Related papers: A Classification Theorem for Nuclear Purely Infini…

We show that two simple, separable, nuclear and $\mathcal{Z}_0$-stable $\mathrm{C}^\ast$-algebras are isomorphic if they are trace-preservingly homotopy equivalent. This result does not assume the UCT and can be viewed as a tracial stably…

Let $\mathcal{B}$ be a nonunital separable simple stable C*-algebra with strict comparison of positive elements and $T(\mathcal{B})$ having finite extreme boundary, and let $\mathcal{A}$ be a simple unital separable nuclear C*-algebra. We…

We consider a generalization $K_0^{\operatorname{gr}}(R)$ of the standard Grothendieck group $K_0(R)$ of a graded ring $R$ with involution. If $\Gamma$ is an abelian group, we show that $K_0^{\operatorname{gr}}$ completely classifies graded…

Given two unital C*-algebras $A$ and $B$, we study, when it exists, the universal unital $C^*$-algebra $\mathcal{U}(A,B)$ generated by the coefficients of a unital $*$-homomorphism $\rho\,:\, A\rightarrow B\otimes\mathcal{U}(A,B)$. When $B$…

We revisit a well-known "surjectivity onto quotient" type lemma of Kirchberg on the central sequence algebra of a separable unital ${\rm C}^*$-algebra, and use it to prove a "surjectivity onto quotient" result on approximately inner…

We determine the class of finite T_0-spaces allowing for a universal coefficient theorem computing equivariant KK-theory by filtrated K-theory.

Let $A$ be a unital simple C*-algebra with tracial rank zero and $X$ be a compact metric space. Suppose that $h_1, h_2: C(X)\to A$ are two unital monomorphisms. We show that $h_1$ and $h_2$ are approximately unitarily equivalent if and only…

Connectivity is a homotopy invariant property of separable C*-algebras which has three notable consequences: absence of nontrivial projections, quasidiagonality and a more geometric realization of KK-theory for nuclear C*-algebras using…

The class of simple separable KK-contractible (KK-equivalent to $\{0\}$) C*-algebras which have finite nuclear dimension is shown to be classified by the Elliott invariant. In particular, the class of C*-algebras $A\otimes \mathcal W$ is…

Reciprocality in Kirchberg algebras with finitely generated K-groups is regarded as a K-theoretic duality through K-groups and strong extension groups. We will prove that the reciprocal Kirchberg algebra has a universal property with…

The classification of homomorphisms between certain unital simple nuclear C*-algebras lead to the discovery of a natural transformation as part of the classifying invariant. We develop a uniqueness result and an abstract characterization of…

Let A be an approximately subhomogeneous (ASH) C*-algebra with slow dimension growth. We prove that if A is unital and simple, then the Cuntz semigroup of A agrees with that of its tensor product with the Jiang-Su algebra Z. In tandem with…

The Algebraic Kirchberg-Phillips Question for Leavitt path algebras asks whether unital $K$-theory is a complete isomorphism invariant for unital, simple, purely infinite Leavitt path algebras over finite graphs. Most work on this problem…

An example is given of a simple, unital C*-algebra which contains an infinite and a non-zero finite projection. This C*-algebra is also an example of an infinite simple C*-algebra which is not purely infinite. A corner of this C*-algebra is…

Suppose $A$ is a separable unital $C(X)$-algebra each fibre of which is isomorphic to the same strongly self-absorbing and $K_{1}$-injective $C^{*}$-algebra $D$. We show that $A$ and $C(X) \otimes D$ are isomorphic as $C(X)$-algebras…

In this paper we show that the $\mathrm{K}$-homology groups of a separable C*-algebra can be enriched with additional descriptive set-theoretic information, and regarded as definable groups. Using a definable version of the Universal…

We compute the nuclear dimension of extensions of C*-algebras involving commutative unital quotients and stable Kirchberg ideals. We identify the finite directed graphs whose C*-algebras are covered by this theorem.

The class of simple separable KK-contractible (KK-equivalent to $\{0\}$) C*-algebras which have finite nuclear dimension is shown to classified by the Elliott invariant. In particular, the class of C*-algebras $A\otimes \mathcal W$ is…

Motivated by advances in categorical probability, we introduce non-commutative almost everywhere (a.e.) equivalence and disintegrations in the setting of C*-algebras. We show that C*-algebras (resp. W*-algebras) and a.e. equivalence classes…

Given an arbitrary infinite 0--1 matrix A having no identically zero rows, we define an algebra OA as the universal C*-algebra generated by partial isometries subject to conditions that generalize, to the infinite case, those introduced by…