arXiv subjects

Eberhard Kirchberg

Publications and source records attributed to Eberhard Kirchberg.

13 recordsLinked to original sources

arXiv2024

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 $\mathcal{O}_2$.

math.OA↗

arXiv2023

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.

math.OA↗

arXiv2016

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 $n\geq 2$.

math.LO↗

arXiv2016

Non-commutativity of the central sequence algebra for separable non-type I C$^{\ast}$-algebras

We show that if $A$ is a separable, simple and non-type I C$^{\ast}$ algebra, then for every properly infinite hyperfinite von Neumann algebra $M$ with separable predual, its Ocneanu ultrapower $M'\cap M^ω$ arises as a sub-quotient of the central sequence algebra $F(A)$ defined by the second named author. In particular, this answers affirmatively the question of the second named author (Abel Symposium '04): the central sequence C$^{\ast}$-algebra of the reduced free group C$^{\ast}$-algebra $C_{\rm{red}}^*(\mathbb{F}_2)$ is non-commutative.

math.OA↗

arXiv2016

Filling families and strong pure infiniteness

We introduce filling families with matrix diagonalization as a refinement of the work by Rørdam and the first named author. As an application we improve a result on local pure infiniteness and show that the minimal tensor product of a strongly purely infinite $C^*$-algebra and a exact $C^*$-algebra is again strongly purely infinite. Our results also yield a sufficient criterion for the strong pure infiniteness of crossed products $A\rtimes_φ\mathbb{N}$ by an endomorphism $φ$ of $A$ (cf. Theorem 7.6). Our work confirms that the special class of nuclear Cuntz-Pimsner algebras constructed by Harnisch and the first named author consist of strongly purely infinite $C^*$-algebras, and thus absorb $\mathcal{O}_\infty$ tensorially.

math.OA↗

arXiv2015

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 $A\rtimes_{σ, λ} G$ 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 $σ_{g_j}(C_j)\subseteq U_j$, but we do not require that $σ_{g_j}(U_j)\subseteq U_j$. 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.

math.OA↗

arXiv2014

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.

math.OA↗

arXiv2013

Central sequence C*-algebras and tensorial absorption of the Jiang-Su algebra

We study properties of the central sequence algebra of a C*-algebra, and we present an alternative approach to a recent result of Matui and Sato. They prove that every unital separable simple nuclear C*-algebra, whose trace simplex is finite dimensional, tensorially absorbs the Jiang-Su algebra if and only if it has the strict comparison property. We extend their result to the case where the extreme boundary of the trace simplex is closed and of finite topological dimension. We are also able to relax the assumption on the C*-algebra of having the strict comparison property to a weaker property, that we call local weak comparison. Namely, we prove that a unital separable simple nuclear C*-algebra, whose trace simplex has finite dimensional closed extreme boundary, tensorially absorbs the Jiang-Su algebra if and only if it has the local weak comparison property. We can also eliminate the nuclearity assumption, if instead we assume the (SI) property of Matui and Sato, and, moreover, that each II_1-factor representation of the C*-algebra is a McDuff factor.

math.OA↗

arXiv2012

Decomposable approximations of nuclear C*-algebras

We show that nuclear C*-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 C*-algebra A which is closely contained in a C*-algebra B embeds into B.

math.OA↗

arXiv2007

Irreducible representations of inner quasidiagonal C*-algebras

It is shown that a separable C*-algebra is inner quasidiagonal if and only if it has a separating family of quasidiagonal irreducible representations. As a consequence, a separable C*-algebra is a strong NF algebra if and only if it is nuclear and has a separating family of quasidiagonal irreducible representations. We also obtain some permanence properties of the class of inner quasidiagonal C*-algebras.

math.OA↗

arXiv2003

Purely infinite C*-algebras: ideal-preserving zero homotopies

We show that if A is a separable, nuclear, O_infty-absorbing (or strongly purely infinite) C*-algebra, which is homotopic to zero in an ideal-system preserving way, then A is the inductive limit of C*-algebras of the form M_k(C_0(G,v)), where G is a finite graph (and C_0(G,v) is the algebra of continuous functions on G that vanish at a distinguished point v in G). We show further that any separable, nuclear, stable, O_2-absorbing C*-algebra is isomorphic to a crossed product of a C*-algebra D with the integers by an action alpha, where D is an inductive limit of C*-algebras of the form M_k(C_0(G,v)) (and D is O_2-absorbing and homotopic to zero in an ideal-system preserving way).

math.OA↗

arXiv2002

Covering dimension and quasidiagonality

We introduce the decomposition rank, a notion of covering dimension for nuclear C^*-algebras. The decomposition rank generalizes ordinary covering dimension and has nice permanence properties; in particular, it behaves well with respect to direct sums, quotients, inductive limits, unitization and quasidiagonal extensions. Moreover, it passes to hereditary subalgebras and is invariant under stabilization. It turns out that the decomposition rank can be finite only for strongly quasidiagonal C^*-algebras and that it is closely related to the classification program.

math.OA↗

arXiv1997

Embedding of exact C*-algebras and continuous fields in the Cuntz algebra O_2

We prove that any separable exact C*-algebra is isomorphic to a subalgebra of the Cuntz algebra ${\cal O}_2.$ We further prove that if $A$ is a simple separable unital nuclear C*-algebra, then ${\cal O}_2 \otimes A \cong {\cal O}_2,$ and if, in addition, $A$ is purely infinite, then ${\cal O}_{\infty} \otimes A \cong A.$ The embedding of exact C*-algebras in $\OA{2}$ is continuous in the following sense. If $A$ is a continuous field of C*-algebras over a compact manifold or finite CW complex $X$ with fiber $A (x)$ over $x \in X,$ such that the algebra of continuous sections of $A$ is separable and exact, then there is a family of injective homomorphisms $ϕ_x : A (x) \to {\cal O}_2$ such that for every continuous section $a$ of $A$ the function $x \mapsto ϕ_x (a (x))$ is continuous. Moreover, one can say something about the modulus of continuity of the functions $x \mapsto ϕ_x (a (x))$ in terms of the structure of the continuous field. In particular, we show that the continuous field $θ\mapsto A_θ$ of rotation algebras posesses unital embeddings $ϕ_θ$ in ${\cal O}_2$ such that the standard generators $u (θ)$ and $v (θ)$ are mapped to $\operatorname{Lip}^{1/2}$ functions.

funct-an↗