When central sequence C*-algebras have characters

We investigate C*-algebras whose central sequence algebra has no characters, and we raise the question if such C*-algebras necessarily must absorb the Jiang-Su algebra (provided that they also are separable). We relate this question to a question of Dadarlat and Toms if the Jiang-Su algebra always embeds into the infinite tensor power of any unital C*-algebra without characters. We show that absence of characters of the central sequence algebra implies that the C*-algebra has the so-called strong Corona Factorization Property, and we use this result to exhibit simple nuclear separable unital C*-algebras whose central sequence algebra does admit a character. We show how stronger divisibility properties on the central sequence algebra imply stronger regularity properties of the underlying C*-algebra.Comment: 28 page

Strong pure infiniteness of crossed products

Consider an exact action of discrete group $G$ on a separable $C^*$-algebra $A$. It is shown that the reduced crossed product $A\rtimes_{\sigma, \lambda} G$ is strongly purely infinite - provided that the action of $G$ on any quotient $A/I$ by a $G$-invariant closed ideal $I\neq A$ is element-wise properly outer and that the action of $G$ on $A$ is $G$-separating (cf. Definition 4.1). This is the first non-trivial sufficient criterion for strong pure infiniteness of reduced crossed products of $C^*$-algebras $A$ that are not $G$-simple. In the case $A=\mathrm{C}_0(X)$ the notion of a $G$-separating action corresponds to the property that two compact sets $C_1$ and $C_2$, that are contained in open subsets $C_j\subseteq U_j \subseteq X$, can be mapped by elements of $g_j\in G$ onto disjoint sets $\sigma_{g_j}(C_j)\subseteq U_j$, but we do not require that $\sigma_{g_j}(U_j)\subseteq U_j$. A generalization of strong boundary actions on compact spaces to non-unital and non-commutative $C^*$-algebras $A$ (cf. Definition 6.1) is also introduced. It is stronger than the notion of $G$-separating actions by Proposition 6.6, because $G$-separation does not imply $G$-simplicity and there are examples of $G$-separating actions with reduced crossed products that are stably projection-less and non-simple.Comment: 30 pages, parts were taken out and included elsewher

The inverse problem for primitive ideal spaces

A pure topological characterization of primitive ideal spaces of separable nuclear C*-algebras is given. We show that a $T_0$-space $X$ is a primitive ideal space of a separable nuclear C*-algebra $A$ if and only if $X$ is point-complete second countable, and there is a continuous pseudo-open and pseudo-epimorphic map from a locally compact Polish space $P$ into $X$. We use this pseudo-open map to construct a Hilbert bi-module $\mathcal{H}$ over $C_0(X)$ such that $X$ is isomorphic to the primitive ideal space of the Cuntz--Pimsner algebra $\mathcal{O}_\mathcal{H}$ generated by $\mathcal{H}$. Moreover, our $\mathcal{O}_\mathcal{H}$ is $KK(X;.,.)$-equivalent to $C_0(P)$ (with the action of $X$ on $C_0(P)$ given be the natural map from $\mathbb{O}(X)$ into $\mathbb{O}(P)$, which is isomorphic to the ideal lattice of $C_0(P)$. Our construction becomes almost functorial in $X$ if we tensor $\mathcal{O}_\mathcal{H}$ with the Cuntz algebra $\mathcal{O}_2$.Comment: This paper was written in 2005 and is now uploaded to the arXiv on the recommendation of several colleagues. The second named author passed away August, 202

Quantifier elimination in C*-algebras

The only C*-algebras that admit elimination of quantifiers in continuous logic are $\mathbb{C}, \mathbb{C}^2$, $C($Cantor space$)$ and $M_2(\mathbb{C})$. We also prove that the theory of C*-algebras does not have model companion and show that the theory of $M_n(\mathcal {O_{n+1}})$ is not $\forall\exists$-axiomatizable for any $n\geq 2$.Comment: More improvements and bug fixes. To appear in IMR

Minimal dynamical systems on prime C*-algebras

We give a number of examples of exotic actions of locally compact groups on separable nuclear C*-algebras. In particular, we give examples of the following: (1) Minimal effective actions of ${\mathbb{Z}}$ and $F_n$ on unital nonsimple prime AF algebras. (2) For any second countable noncompact locally compact group, a minimal effective action on a separable nuclear nonsimple prime C*-algebra. (3) For any amenable second countable noncompact locally compact group, a minimal effective action on a separable nuclear nonsimple prime C*-algebra (unital when the group is ${\mathbb{Z}}$ or ${\mathbb{R}}$) such that the crossed product is $K \otimes {\mathcal{O}}_2$ (${\mathcal{O}}_2$ when the group is ${\mathbb{Z}}$). (4) For any second countable locally compact abelian group which is not discrete, an action on $K \otimes {\mathcal{O}}_2$ such that the crossed product is a nonsimple prime C*-algebra. In most of these situations, we can specify the primitive ideal space of the C*-algebra (of the crossed product in the last item) within a class of spaces.Comment: This paper was nearly done about 10 years ago, but was pushed aside under the press of other projects, and then forgotten. It is now being posted, after Kirchberg's death. The key preprint of Harnisch and Kirchberg was never published and has disappeared from the website of the Universitaet Muenster. It is temporarily at: https://pages.uoregon.edu/ncp/Research/Misc/heft399.pd