Related papers: The UCT problem for nuclear $C^\ast$-algebras

In the 1970s Alain Connes identified the appropriate notion of amenabilty for von Neumann algebras, and used it to obtain a deep internal finite dimensional approximation structure for these algebras. This structure is exactly what is…

I give an overview of recent developments in the structure and classification theory of separable, simple, nuclear C*-algebras. I will in particular focus on the role of quasidiagonality and amenability for classification, and on the…

A universal coefficient theorem in the setting of Kirchberg's ideal-related KK-theory was obtained in the fundamental case of a C*-algebra with one specified ideal by Bonkat and proved there to split, unnaturally, under certain conditions.…

It is shown that the coloured isomorphism class of a unital, simple, $\mathcal{Z}$-stable, separable amenable C$^*$-algebra satisfying the Universal Coefficient Theorem (UCT) is determined by its tracial simplex.

We define an easily verifiable notion of an atomic formula having uniformly bounded arrays in a structure $M$. We prove that if $T$ is a complete $L$-theory, then $T$ is mutually algebraic if and only if there is some model $M$ of $T$ for…

We compute the nuclear dimension of separable, simple, unital, nuclear, Z-stable C*-algebras. This makes classification accessible from Z-stability and in particular brings large classes of C*-algebras associated to free and minimal actions…

We construct uncountably many mutually nonisomorphic simple separable stably finite unital exact C$^\ast$-algebras which are not isomorphic to their opposite algebras. In particular, we prove that there are uncountably many possibilities…

We introduce a notion of a uniform structure on the set of all representations of a given separable, not necessarilly commutative $C^*$-algebra $\mathfrak{A}$ by introducing a suitable family of metrics on the set of representations of…

We show that group C*-algebras of finitely generated, nilpotent groups have finite nuclear dimension. It then follows, from a string of deep results, that the C*-algebra $A$ generated by an irreducible representation of such a group has…

We present a stable uniqueness theorem for non-unital C*-algebras. Generalized tracial rank one is defined for stably projectionless simple C*-algebras. Let $A$ and $B$ be two stably projectionless separable simple amenable C*-algebras with…

It was recently shown that each C*-algebra generated by a faithful irreducible representation of a finitely generated, torsion free nilpotent group is classified by its ordered K-theory. For the three step nilpotent group $UT(4,\mathbb{Z})$…

We define a categorical framework in which we build a systematic construction that provides generic invariants for C*-algebras. The benefit is significant as we show that any invariant arising this way automatically enjoys nice properties…

We write arbitrary separable nuclear C*-algebras as limits of inductive systems of finite-dimensional C*-algebras with completely positive connecting maps. The characteristic feature of such CPC*-systems is that the maps become more and…

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…

Let $\A$ be a unital separable nuclear $C^*$--algebra which belongs to the bootstrap category $\N$ and $\B$ be a separable stable $C^*$--algebra. In this paper, we consider the group $\Ext_u(\A,\B)$ consisting of the unitary equivalence…

We calculate the Cuntz semigroup of the tensor product A with A. We restrict our attention to C*-algebras A which are unital, simple, nuclear, stably finite, have stable rank one, absorbs the Jiang-Su algebra tensorially and satisfy the…

The unification problem in algebras capable of describing sets has been tackled, directly or indirectly, by many researchers and it finds important applications in various research areas--e.g., deductive databases, theorem proving, static…

A trace on a C*-algebra is amenable (resp. quasidiagonal) if it admits a net of completely positive, contractive maps into matrix algebras which approximately preserve the trace and are approximately multiplicative in the 2-norm (resp.…

We present a generalization of the notion of an algebra norm relevant to real finite-dimensional unital associative algebras. Among other things, this leads to a novel set of algebra isomorphism invariants, some of which are computationally…

We define united KK-theory for real C*-algebras A and B such that A is separable and B is sigma-unital, extending united K-theory in the sense that KK\crt(\R, B) = K\crt(B). United KK-theory contains real, complex, and self-conjugate…