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Marius Dadarlat

Publications and source records attributed to Marius Dadarlat.

At least 19 recordsLinked to original sources
arXiv2025

On Chern classes of almost representations

For a discrete group $\Gamma$, we study vector bundles $E_\rho$ on compact subsets of $B\Gamma$ associated to almost representations $\rho:\Gamma \to U(n)$. We compute the first Chern class of $E_\rho$ in terms of $\rho$. When $\rho$ is both projective and almost multiplicative, we determine its Chern character. These invariants yield obstructions to perturbing almost representations to those arising from projective representations. For residually finite amenable groups, the $K$-theory classes of $E_\rho$ classify almost representations up to stable equivalence. Finally, for $\mathbb{Z}^d$, $\mathbb{Z}\times \mathbb{H}_3$, and $\mathbb{H}_3\times \mathbb{H}_3$, we construct explicit almost representations with prescribed Chern classes.

math.KT↗
arXiv2025

Central extensions and almost representations

For a sequence of unital tracial $C^*$-algebras $(A_n,\tau_n),$ we construct a canonical central extension of the unitary group $U(\ell^\infty (\mathbb{N},A_n)/c_0(\mathbb{N},A_n))$ by $Q(\mathbb{R})=c_0(\mathbb{N},\mathbb{R})/\mathbb{R}^\infty,$ using de la Harpe-Skandalis pre-determinant. For an asymptotic group homomorphism $\rho_n : \Gamma \to U(A_n),$ the corresponding pullback of the canonical central extension gives a 2-cohomology class in $H^2(\Gamma,Q(\mathbb{R}))$ which obstructs the perturbation of $(\rho_n)$ to a sequence of true homomorphisms of groups $\pi_n:\Gamma \to GL(A_n)$. The pairing of the obstruction class with elements of $H_2(\Gamma,\mathbb{Z})$ yields numerical invariants in $\tau_{n\,*} (K_0(A_n))$ that subsume the winding number invariants of Kazhdan, Exel and Loring. For generality, we allow bounded asymptotic homomorphisms to map the group $\Gamma$ into the general linear group of any sequence of tracial unital Banach algebras. In that case, the obstruction class belongs to $H^2(\Gamma,Q(\mathbb{C})),$ where $Q(\mathbb{C})=c_0(\mathbb{N},\mathbb{C})/\mathbb{C}^\infty.$ As an application, we show that 2-cohomology obstructs various stability properties under weaker assumptions than those found in existing literature. In particular we show that the full group $C^*$-algebra $C^*(\Gamma)$ of a discrete group $\Gamma$ is not $C^*$-stable if $H^2(\Gamma,\mathbb{R})\neq 0$.

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arXiv2023

Non-stable groups

In this article we discuss cohomological obstructions to two kinds of group stability. In the first part, we show that residually finite groups $Γ$ which arise as fundamental groups of compact Riemannian manifolds with strictly negative sectional curvature are not uniform-to-local stable with respect to the operator norm if their even Betti numbers $b_{2i}(Γ)$ do not vanish. In the second part, we show that non-vanishing of Betti numbers $b_{i}(Γ)$ in dimension $i>1$ obstructs $C^*$-algebra stability for groups approximable by unitary matrices that admit a coarse embedding in a Hilbert space.

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arXiv2023

Computing cohomology groups that classify bundles of strongly self-absorbing $C^*$-algebras

Locally trivial bundles of $C^*$-algebras with fibre $D \otimes \mathcal{K}$ for a strongly self-absorbing $C^*$-algebra $D$ over a finite CW-complex $X$ form a group $E^1_D(X)$ that is the first group of a cohomology theory $E^*_D(X)$. In this paper we compute these groups by expressing them in terms of ordinary cohomology and connective $K$-theory. To compare the $C^*$-algebraic version of $gl_1(KU)$ with its classical counterpart we also develop a uniqueness result for the unit spectrum of complex periodic topological $K$-theory.

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arXiv2022

Bundles of strongly self-absorbing $C^*$-algebras with a Clifford grading

We extend our previous results on generalized Dixmier-Douady theory to graded $C^*$-algebras, as means for explicit computations of the invariants arising for bundles of ungraded $C^*$-algebras. For a strongly self-absorbing $C^*$-algebra $D$ and complex Clifford algebras $\mathbb{C}\ell_{n}$ we show that the classifying spaces of the groups of graded automorphisms $\mathrm{Aut}_{\text{gr}}(\mathbb{C}\ell_{n}\otimes \mathcal{K }\otimes D)$ admit compatible infinite loop space structures giving rise to a cohomology theory $\hat{E}^*_D(X)$. For $D$ stably finite and $X$ a finite CW-complex, we show that the tensor product operation defines a group structure on the isomorphism classes of locally trivial bundles of graded $C^*$-algebras with fibers $ \mathbb{C}\ell_{k}\otimes D \otimes \mathcal{K}$ and that this group is isomorphic to $H^0(X,\mathbb{Z}/2)\oplus \hat{E}^1_{D}(X)$. Moreover, we establish isomorphisms $\hat{E}^1_{D}(X)\cong H^1(X;\mathbb{Z}/2) \times_{_{tw}} E^1_{D}(X)$ and $\hat{E}^1_{D}(X)\cong E^1_{D\otimes \mathcal{O}_\infty}(X)$, where $E^1_{D}(X)$ is the group that classifies the locally trivial bundles with fibers $D\otimes \mathcal{K}$. In particular $E^1_{\mathcal{O}_\infty}(X)\cong H^1(X;\mathbb{Z}/2) \times_{_{tw}} E^1_{\mathcal{Z}}(X)$ where $\mathcal{Z}$ is the Jiang-Su algebra and the multiplication on the last two factors is twisted similarly to the Brauer theory for bundles with fibers the graded compact operators on a finite and respectively infinite dimensional Hilbert space.

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arXiv2021

Quasi-representations of groups and two-homology

The Exel-Loring formula asserts that two topological invariants associated to a pair of almost commuting unitary matrices coincide. Such a pair can be viewed as a quasi-representation of $\mathbb{Z}^2$. We give a generalization of this formula for countable discrete groups. We also show the nontriviality of the corresponding invariants for quasidiagonal groups which are coarsely embeddable in a Hilbert space and have nonvanishing second Betti number.

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arXiv2021

Connective Bieberbach groups

We prove that a Bieberbach group with trivial center is not connective and use this property to show that a Bieberbach group is connective if and only if it is poly-Z.

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arXiv2021

Obstructions to matricial stability of discrete groups and almost flat K-theory

A discrete countable group G is matricially stable if the finite dimensional approximate unitary representations of G are perturbable to genuine representations in the point-norm topology. For large classes of groups G, we show that matricial stability implies the vanishing of the rational cohomology of G in all nonzero even dimensions. We revisit a method of constructing almost flat K-theory classes of BG which involves the dual assembly map and quasidiagonality properties of G. The existence of almost flat K-theory classes of BG which are not flat represents an obstruction to matricial stability of G due to continuity properties of the approximate monodromy correspondence.

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arXiv2019

Connective C*-algebras

Connectivity is a homotopy invariant property of separable C*-algebras which has three notable consequences: absence of nontrivial projections, quasidiagonality and a more geometric realization of KK-theory for nuclear C*-algebras using asymptotic morphisms. The purpose of this paper is to further explore the class of connective C*-algebras. We give new characterizations of connectivity for exact and for nuclear separable C*-algebras and show that an extension of connective separable nuclear C*-algebras is connective. We establish connectivity or lack of connectivity for C*-algebras associated to certain classes of groups: virtually abelian groups, linear connected nilpotent Lie groups and linear connected semisimple Lie groups.

math.OA↗
arXiv2018

Localization C*-algebras and K-theoretic duality

Based on the localization algebras of Yu, and their subsequent analysis by Qiao and Roe, we give a new picture of KK-theory in terms of time-parametrized families of (locally) compact operators that asymptotically commute with appropriate representations.

math.KT↗
arXiv2016

Deformations of Wreath Products

Connectivity is a homotopy invariant property of a separable C*-algebra A which has three important consequences: absence of nontrivial projections, quasidiagonality and realization of the Kasparov group KK(A,B) as homotopy classes of asymptotic morphisms from A to the stabilization of B if A is nuclear. Here we give a new characterization of connectivity for separable exact C*-algebras and use this characterization to show that the class of discrete countable amenable groups whose augmentation ideals are connective is closed under generalized wreath products. In a related circle of ideas, we give a result on quasidiagonality of reduced crossed-product C*-algebras associated to noncommutative Bernoulli actions.

math.OA↗
arXiv2016

Simple nuclear C*-algebras not equivariantly isomorphic to their opposites

We exhibit examples of simple separable nuclear C*-algebras, along with actions of the circle group and outer actions of the integers, which are not equivariantly isomorphic to their opposite algebras. In fact, the fixed point subalgebras are not isomorphic to their opposites. The C*-algebras we exhibit are well behaved from the perspective of structure and classification of nuclear C*-algebras: they are unital C*-algebras in the UCT class, with finite nuclear dimension. One is an AH-algebra with unique tracial state and absorbs the CAR algebra tensorially. The other is a Kirchberg algebra.

math.OA↗
arXiv2015

Deformations of nilpotent groups and homotopy symmetric $C^*$-algebras

The homotopy symmetric $C^*$-algebras are those separable $C^*$-algebras for which one can unsuspend in E-theory. We find a new simple condition that characterizes homotopy symmetric nuclear $C^*$-algebras and use it to show that the property of being homotopy symmetric passes to nuclear $C^*$-subalgebras and it has a number of other significant permanence properties. As an application, we show that if $I(G)$ is the kernel of the trivial representation $ι:C^*(G)\to \mathbb{C}$ for a countable discrete torsion free nilpotent group $G$, then $I(G)$ is homotopy symmetric and hence the Kasparov group $KK(I(G),B)$ can be realized as the homotopy classes of asymptotic morphisms $[[I(G),B \otimes \mathcal{K}]]$ for any separable $C^*$-algebra $B$.

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arXiv2015

A Dixmier-Douady theory for strongly self-absorbing C*-algebras

We show that the Dixmier-Douady theory of continuous field $C^*$-algebras with compact operators $\mathbb{K}$ as fibers extends significantly to a more general theory of fields with fibers $A\otimes \mathbb{K}$ where $A$ is a strongly self-absorbing C*-algebra. The classification of the corresponding locally trivial fields involves a generalized cohomology theory which is computable via the Atiyah-Hirzebruch spectral sequence. An important feature of the general theory is the appearance of characteristic classes in higher dimensions. We also give a necessary and sufficient $K$-theoretical condition for local triviality of these continuous fields over spaces of finite covering dimension.

math.OA↗
arXiv2015

A Dixmier-Douady Theory for strongly self-absorbing C*-algebras II: the Brauer group

We have previously shown that the isomorphism classes of orientable locally trivial fields of $C^*$-algebras over a compact metrizable space $X$ with fiber $D\otimes \mathbb{K}$, where $D$ is a strongly self-absorbing $C^*$-algebra, form an abelian group under the operation of tensor product. Moreover this group is isomorphic to the first group $\bar{E}^1_D(X)$ of the (reduced) generalized cohomology theory associated to the unit spectrum of topological K-theory with coefficients in $D$. Here we show that all the torsion elements of the group $\bar{E}^1_D(X)$ arise from locally trivial fields with fiber $D \otimes M_n(\mathbb{C})$, $n\geq 1$, for all known examples of strongly self-absorbing $C^*$-algebras $D$. Moreover the Brauer group generated by locally trivial fields with fiber $D\otimes M_n(\mathbb{C})$, $n\geq 1$ is isomorphic to ${\rm Tor}(\bar{E}^1_D(X))$.

math.OA↗
arXiv2015

Almost flat K-theory of classifying spaces

We give a rigorous account and prove continuity properties for the correspondence between almost flat bundles on a triangularizable compact connected space and the quasi-representations of its fundamental group. For a discrete countable group $Γ$ with finite classifying space $BΓ$, we study a correspondence between between almost flat K-theory classes on $BΓ$ and group homomorphism $K_0(C^*(Γ))\to \mathbb{Z}$ that are implemented by pairs of discrete asymptotic homomorphisms from $C^*(Γ)$ to matrix algebras.

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