Decomposable approximations of nuclear <i>C</i>^*-algebras

We show that nuclear &lt;i&gt;C&lt;/i&gt;&lt;sup&gt;*&lt;/sup&gt;-algebras have a refined version of the completely positive approximation property, in which the maps that approximately factorize through finite dimensional algebras are convex combinations of order zero maps. We use this to show that a separable nuclear &lt;i&gt;C&lt;/i&gt;&lt;sup&gt;*&lt;/sup&gt;-algebra A which is closely contained in a &lt;i&gt;C&lt;/i&gt;&lt;sup&gt;*&lt;/sup&gt;-algebra B embeds into B

A Simple Separable Exact C*-Algebra not Anti-isomorphic to Itself

We give an example of an exact, stably finite, simple. separable C*-algebra D which is not isomorphic to its opposite algebra. Moreover, D has the following additional properties. It is stably finite, approximately divisible, has real rank zero and stable rank one, has a unique tracial state, and the order on projections over D is determined by traces. It also absorbs the Jiang-Su algebra Z, and in fact absorbs the 3^{\infty} UHF algebra. We can also explicitly compute the K-theory of D, namely K_0 (D) = Z[1/3] with the standard order, and K_1 (D) = 0, as well as the Cuntz semigroup of D.Comment: 16 pages; AMSLaTeX. The material on other possible K-groups for such an algebra has been moved to a separate paper (1309.4142 [math.OA]

Relative commutants of strongly self-absorbing C*-algebras

The relative commutant $A'\cap A^{\mathcal{U}}$ of a strongly self-absorbing algebra $A$ is indistinguishable from its ultrapower $A^{\mathcal{U}}$. This applies both to the case when $A$ is the hyperfinite II$_1$ factor and to the case when it is a strongly self-absorbing C*-algebra. In the latter case we prove analogous results for $\ell_\infty(A)/c_0(A)$ and reduced powers corresponding to other filters on $\bf N$. Examples of algebras with approximately inner flip and approximately inner half-flip are provided, showing the optimality of our results. We also prove that strongly self-absorbing algebras are smoothly classifiable, unlike the algebras with approximately inner half-flip.Comment: Some minor correction

Locally Trivial W*-Bundles

We prove that a tracially continuous W$^*$-bundle $\mathcal{M}$ over a compact Hausdorff space $X$ with all fibres isomorphic to the hyperfinite II$_1$-factor $\mathcal{R}$ that is locally trivial already has to be globally trivial. The proof uses the contractibility of the automorphism group $\mathrm{Aut}({\mathcal{R}})$ shown by Popa and Takesaki. There is no restriction on the covering dimension of $X$.Comment: 20 pages, this version will be published in the International Journal of Mathematic

Decomposable approximations of nuclear C*-algebras

We show that nuclear C*-algebras have a re ned version of the completely positive approximation property, in which the maps that approximately factorize through finite dimensional algebras are convex combinations of order zero maps. We use this to show that a separable nuclear C*-algebra A which is closely contained in a C*-algebra B embeds into B

Perturbations of nuclear C*-algebras

Kadison and Kastler introduced a natural metric on the collection of all C*-subalgebras of the bounded operators on a separable Hilbert space. They conjectured that sufficiently close algebras are unitarily conjugate. We establish this conjecture when one algebra is separable and nuclear. We also consider one-sided versions of these notions, and we obtain embeddings from certain near inclusions involving separable nuclear C*-algebras. At the end of the paper we demonstrate how our methods lead to improved characterisations of some of the types of algebras that are of current interest in the classification programme.Comment: 45 page

Classification of graph C*-algebras with no more than four primitive ideals

We describe the status quo of the classification problem of graph C*-algebras with four primitive ideals or less

C*-algebras associated with endomorphisms and polymorphisms of compact abelian groups

A surjective endomorphism or, more generally, a polymorphism in the sense of \cite{SV}, of a compact abelian group $H$ induces a transformation of $L^2(H)$. We study the C*-algebra generated by this operator together with the algebra of continuous functions $C(H)$ which acts as multiplication operators on $L^2(H)$. Under a natural condition on the endo- or polymorphism, this algebra is simple and can be described by generators and relations. In the case of an endomorphism it is always purely infinite, while for a polymorphism in the class we consider, it is either purely infinite or has a unique trace. We prove a formula allowing to determine the $K$-theory of these algebras and use it to compute the $K$-groups in a number of interesting examples.Comment: 25 page

Rokhlin Dimension for Flows

This research was supported by GIF Grant 1137/2011, SFB 878 Groups, Geometry and Actions and ERC Grant No. 267079. Part of the research was conducted at the Fields institute during the 2014 thematic program on abstract harmonic analysis, Banach and operator algebras, and at the Mittag–Leffler institute during the 2016 program on Classification of Operator Algebras: Complexity, Rigidity, and Dynamics.Peer reviewedPostprin