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Marius Dadarlat

Publications and source records attributed to Marius Dadarlat.

At least 37 records · Page 2Linked to original sources
arXiv2013

On groups with quasidiagonal C*-algebras

We examine the question of quasidiagonality for C*-algebras of discrete amenable groups from a variety of angles. We give a quantitative version of Rosenberg's theorem via paradoxical decompositions and a characterization of quasidiagonality for group C*-algebras in terms of embeddability of the groups. We consider several notable examples of groups, such as topological full groups associated with Cantor minimal systems and Abels' celebrated example of a finitely presented solvable group that is not residually finite, and show that they have quasidiagonal C*-algebras. Finally, we study strong quasidiagonality for group C*-algebras, exhibiting classes of amenable groups with and without strongly quasidiagonal C*-algebras.

math.OA↗
arXiv2013

Quasi-representations of surface groups

By a quasi-representation of a group $G$ we mean an approximately multiplicative map of $G$ to the unitary group of a unital $C^*$-algebra. A quasi-representation induces a partially defined map at the level $K$-theory. In the early 90s Exel and Loring associated two invariants to almost-commuting pairs of unitary matrices $u$ and $v$: one a $K$-theoretic invariant, which may be regarded as the image of the Bott element in $K_0(C(\mathbb{T}^2))$ under a map induced by quasi-representation of $\mathbb{Z}^2$ in U(n); the other is the winding number in $\mathbb{C}\setminus \{0\}$ of the closed path $t\mapsto \det(tvu + (1-t)uv)$. The so-called Exel-Loring formula states that these two invariants coincide if $\|uv - vu\|$ is sufficiently small. A generalization of the Exel-Loring formula for quasi-representations of a surface group taking values in U(n) was given by the second-named author. Here we further extend this formula for quasi-representations of a surface group taking values in the unitary group of a tracial unital $C^*$-algebra.

math.OA↗
arXiv2013

Unit spectra of K-theory from strongly self-absorbing C*-algebras

We give an operator algebraic model for the first group of the unit spectrum $gl_1(KU)$ of complex topological K-theory, i.e. $[X, BGL_1(KU)]$, by bundles of stabilized infinite Cuntz C*-algebras $O_{\infty} \otimes \K$. We develop similar models for the localizations of $KU$ at a prime $p$ and away from $p$. Our work is based on the $\mathcal{I}$-monoid model for the units of $K$-theory by Sagave and Schlichtkrull and it was motivated by the goal of finding connections between the infinite loop space structure of the classifying space of the automorphism group of stabilized strongly self-absorbing C*-algebras that arose in our generalization of the Dixmier-Douady theory and classical spectra from algebraic topology.

math.AT↗
arXiv2011

Recovering the Elliott invariant from the Cuntz semigroup

Let $A$ be a simple, separable C$^*$-algebra of stable rank one. We prove that the Cuntz semigroup of $\CC(\T,A)$ is determined by its Murray-von Neumann semigroup of projections and a certain semigroup of lower semicontinuous functions (with values in the Cuntz semigroup of $A$). This result has two consequences. First, specializing to the case that $A$ is simple, finite, separable and $\mathcal Z$-stable, this yields a description of the Cuntz semigroup of $\CC(\T,A)$ in terms of the Elliott invariant of $A$. Second, suitably interpreted, it shows that the Elliott functor and the functor defined by the Cuntz semigroup of the tensor product with the algebra of continuous functions on the circle are naturally equivalent.

math.OA↗
arXiv2010

The C*-algebra of a vector bundle

We prove that the Cuntz-Pimsner algebra O(E) of a vector bundle E over a compact metrizable space X is determined up to an isomorphism of C(X)-algebras by the ideal (1-[E])K(X) of the K-theory ring K(X). Moreover, if E and F are vector bundles of rank >1, then a unital embedding of C(X)-algebras of O(E) into O(F) exists if and only if 1-[E] is divisible by 1-[F] in the ring K(X). We introduce related, but more computable K-theory and cohomology invariants for O(E) and study their completeness. As an application we classify the unital separable continuous fields with fibers isomorphic to the Cuntz algebra O(m+1) over a finite connected CW complex X of dimension d< 2m+4 provided that the cohomology of X has no m-torsion.

math.OA↗
arXiv2009

Ranks of operators in simple C*-algebras

Let A be a unital simple separable C*-algebra with strict comparison of positive elements. We prove that the Cuntz semigroup of A is recovered functorially from the Murray-von Neumann semigroup and the tracial state space T(A) whenever the extreme boundary of T(A) is compact and of finite covering dimension. Combined with a result of Winter, we obtain Z \otimes A isomorphic to A whenever A moreover has locally finite decomposition rank. As a corollary, we confirm Elliott's classification conjecture under reasonably general hypotheses which, notably, do not require any inductive limit structure. These results all stem from our investigation of a basic question: what are the possible ranks of operators in a unital simple C*-algebra?

math.OA↗
arXiv2009

E-Theory for C*-algebras over topological spaces

We define E-theory for separable C*-algebras over second countable topological spaces and establish its basic properties. This includes an approximation theorem that relates the E-theory over a general space to the E-theories over finite approximations to this space. We obtain effective criteria for determining the invertibility of E-theory elements over possibly infinite-dimensional spaces. Furthermore, we prove a Universal Multicoefficient Theorem for C*-algebras over totally disconnected metrisable compact spaces.

math.KT↗
arXiv2009

Continuous fields of C*-algebras over finite dimensional spaces

Let $X$ be a finite dimensional compact metrizable space. We study a technique which employs semiprojectivity as a tool to produce approximations of $C(X)$-algebras by $C(X)$-subalgebras with controlled complexity. The following applications are given. All unital separable continuous fields of C*-algebras over $X$ with fibers isomorphic to a fixed Cuntz algebra $\mathcal{O}_n$, $n\in\{2,3,...,\infty\}$ are locally trivial. They are trivial if $n=2$ or $n=\infty$. For $n\geq 3$ finite, such a field is trivial if and only if $(n-1)[1_A]=0$ in $K_0(A)$, where $A$ is the C*-algebra of continuous sections of the field. We give a complete list of the Kirchberg algebras $D$ satisfying the UCT and having finitely generated K-theory groups for which every unital separable continuous field over $X$ with fibers isomorphic to $D$ is automatically (locally) trivial. In a more general context, we show that a separable unital continuous field over $X$ with fibers isomorphic to a $KK$-semiprojective is trivial if and only if it satisfies a K-theoretical Fell type condition.

math.OA↗
arXiv2009

Strongly Self-Absorbing C*-algebras which contain a nontrivial projection

It is shown that a strongly self-absorbing C*-algebra is of real rank zero and absorbs the Jiang-Su algebra if it contains a nontrivial projection. We also consider cases where the UCT is automatic for strongly self-absorbing C*-algebras, and K-theoretical ways of characterizing when Kirchberg algebras are strongly self-absorbing.

math.OA↗
arXiv2008

A direct proof of Z-stability for AH algebras of bounded topological dimension

We prove that a unital simple approximately homogeneous (AH) C*-algebra with no dimension growth absorbs the Jiang-Su algebra tensorially without appealing to the classification theory of these algebras. Our main result continues to hold under the slightly weaker hypothesis of exponentially slow dimension growth.

math.OA↗
arXiv2007

A universal property for the Jiang-Su algebra

We prove that the infinite tensor power of a unital separable C*-algebra absorbs the Jiang-Su algebra Z tensorially if and only if it contains, unitally, a subhomogeneous algebra without characters. This yields a succinct universal property for Z in a category so large that there are no unital separable C*-algebras without characters known to lie outside it. This category moreover contains the vast majority of our stock-in-trade separable amenable C*-algebras, and is closed under passage to separable superalgebras and quotients, and hence to unital tensor products, unital direct limits, and crossed products by countable discrete groups. One consequence of our main result is that strongly self-absorbing ASH algebras are Z-stable, and therefore satisfy the hypotheses of a recent classification theorem of W. Winter. One concludes that Z is the only projectionless strongly self-absorbing ASH algebra, completing the classification of strongly self-absorbing ASH algebras.

math.OA↗
arXiv2007

Trivialization of C(X)-algebras with strongly self-absorbing fibres

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 provided the compact Hausdorff space $X$ is finite-dimensional. This statement is known not to extend to the infinite-dimensional case.

math.OA↗
arXiv2007

On the KK-theory of strongly self-absorbing C*-algebras

Let $\Dh$ and $A$ be unital and separable $C^{*}$-algebras; let $\Dh$ be strongly self-absorbing. It is known that any two unital $^*$-homomorphisms from $\Dh$ to $A \otimes \Dh$ are approximately unitarily equivalent. We show that, if $\Dh$ is also $K_{1}$-injective, they are even asymptotically unitarily equivalent. This in particular implies that any unital endomorphism of $\Dh$ is asymptotically inner. Moreover, the space of automorphisms of $\Dh$ is compactly-contractible (in the point-norm topology) in the sense that for any compact Hausdorff space $X$, the set of homotopy classes $[X,\Aut(\Dh)]$ reduces to a point. The respective statement holds for the space of unital endomorphisms of $\Dh$. As an application, we give a description of the Kasparov group $KK(\Dh, A\ot \Dh)$ in terms of $^*$-homomorphisms and asymptotic unitary equivalence. Along the way, we show that the Kasparov group $KK(\Dh, A\ot \Dh)$ is isomorphic to $K_0(A\ot \Dh)$.

math.OA↗
arXiv2006

Fiberwise KK-equivalence of continuous fields of C*-algebras

Let A and B be separable nuclear continuous C(X)-algebras over a finite dimensional compact metrizable space X. It is shown that an element $σ$ of the parametrized Kasparov group KK_X(A,B) is invertible if and only if all its fiberwise components $σ_x\in KK(A(x),B(x))$ are invertible. This criterion does not extend to infinite dimensional spaces since there exist nontrivial unital separable continuous fields over the Hilbert cube with all fibers isomorphic to the Cuntz algebra O_2. Several applications to continuous fields of Kirchberg algebras are given. It is also shown that if each fiber of a separable nuclear continuous C(X)-algebra A over a finite dimensional locally compact space X satisfies the UCT, then A satisfies the UCT.

math.OA↗
arXiv2005

On the topology of the Kasparov groups and its applications

In this paper we establish a direct connection between stable approximate unitary equivalence for $*$-homomorphisms and the topology of the KK-groups which avoids entirely C*-algebra extension theory and does not require nuclearity assumptions. To this purpose we show that a topology on the Kasparov groups can be defined in terms of approximate unitary equivalence for Cuntz pairs and that this topology coincides with both Pimsner's topology and the Brown-Salinas topology. We study the generalized Rørdam group $KL(A,B)=KK(A,B)/\bar{0}$, and prove that if a separable exact residually finite dimensional C*-algebra satisfies the universal coefficient theorem in KK-theory, then it embeds in the UHF algebra of type $2^\infty$. In particular such an embedding exists for the C*-algebra of a second countable amenable locally compact maximally almost periodic group.

math.OA↗
arXiv2005

Uniform embeddability of relatively hyperbolic groups

Let $Γ$ be a finitely generated group which is hyperbolic relative to a finite family $\{H_1,...,H_n\}$ of subgroups. We prove that $Γ$ is uniformly embeddable in a Hilbert space if and only if each subgroup $H_i$ is uniformly embeddable in a Hilbert space.

math.GR↗
arXiv2004

Continuous Fields of Kirchberg C*-algebras

In this paper we study the C*-algebras associated to continuous fields over locally compact metrisable zero dimensional spaces whose fibers are Kirchberg C*-algebras satisfying the UCT. We show that these algebras are inductive limits of finite direct sums of Kirchberg algebras and they are classified up to isomorphism by topological invariants.

math.OA↗