arXiv subjects

Tatiana Shulman

Publications and source records attributed to Tatiana Shulman.

At least 19 recordsLinked to original sources

arXiv2026

The MF property for amalgamated free products

A C*-algebra (or a group) is called MF (matricial field) if it admits finite dimensional approximate unitary representations which are approximately injective, where approximately is meant with respect to the operator norm. It is proved that for any MF C*-algebra $A$ and its C*-subalgebra $C$, $A\ast_C A$ is MF. For general amalgamated free products, $A\ast_C B$, a necessary and sufficient condition for being MF is given. It is shown that the following groups -- amalgamated free products of amenable groups, semidirect products of amenable groups by free groups, and $\mathbb Z^2\rtimes SL_2(\mathbb Z)$ -- all have MF full group C*-algebra. It is shown that the class of MF C*-algebras is closed under maximal tensor products with $C^*(\mathbb F_n)$.

math.OA↗

arXiv2026

Homotopy lifting, asymptotic homomorphisms, and traces

The following homotopy lifting theorem is proved: Let $ϕ, ψ: B \to D/I$ be homotopic $\ast$-homomorphisms and suppose $ψ$ lifts to a (discrete) asymptotic homomorphism. Then $ϕ$ lifts to a (discrete) asymptotic homomorphism. Moreover the whole homotopy lifts. We also prove a cp version of this theorem and a version where $ϕ$ is replaced by an asymptotic homomorphism. We obtain a lifting characterization of several important properties of C*-algebras and use them together with the lifting theorem to get the following applications: 1) MF-property is homotopy invariant; 2) If either $A$ or $B$ is exact, $A$ is homotopy dominated by $B$ and all amenable traces on $B$ are quasidiagonal, then all amenable traces on $A$ are quasidiagonal; 3) If a C*-algebra $A$ is homotopy dominated by a nuclear C*-algebra $B$ and all (hyperlinear) traces on $B$ are MF, then all hyperlinear traces on $A$ are MF. 4) Some of the extension groups introduced by Manuilov and Thomsen coincide. 5) The C*-algebra $qA$ from Cuntz's picture of KK-theory is always quasidiagonal. 6) Every homotopy symmetric C*-algebra is MF.

math.OA↗

arXiv2026

Decomposition theorems for unital graph C*-algebras

We prove that unital graph C*-algebras often admit a convenient decomposition into amalgamated free products. We use this to give a complete characterization of when a unital graph C*-algebra is residually finite-dimensional and when it is operator norm stable (that is, matricially semiprojective).

math.OA↗

arXiv2026

On the (Local) Lifting Property

The (Local) Lifting Property ((L)LP) is introduced by Kirchberg and deals with lifting completely positive maps. We give a characterization of the (L)LP in terms of lifting $\ast$-homomorphisms. We use it to prove that if $A$ and $B$ have the LP and $F$ is their finite-dimensional C*-subalgebra, then $A\ast_F B$ has the LP. This answers a question of Ozawa. We prove that Exel's soft tori have the LP. As a consequence we obtain that $C^*(F_n\times F_n)$ is inductive limit of RFD C*-algebras with the LP. We prove that for a class of C*-algebras including $C^*(F_n\times F_n)$, all contractible C*-algebras and all suspensions, the LLP is equivalent to Ext being a group. As byproduct of methods developed in the paper we generalize Kirchberg's theorem about extensions with the WEP, give short proofs of several, old and new, facts about soft tori, new unified proofs of Li and Shen's characterization of RFD property of free products amalgamated over a finite-dimensional subalgebra and Blackadar's characterization of semiprojectivity of them.

math.OA↗

arXiv2026

Sections and cones

By Bartle-Graves theorem every surjective map between C*-algebras has a continuous section, and Loring proved that that there exists a continuous section of norm arbitrary close to 1. Here we prove that there exists a continuous section of norm exactly 1. This result is used in the second part of the paper which is devoted to properties of cone C*-algebras. It is proved that any $\ast$-homomorphism from the cone over a separable C*-algebra to a quotient C*-algebra always lifts to a contractive asymptotic homomorphism. As an application we give a short proof and strengthen the result of Forough-Gardella-Thomsen that states that any cpc (order zero) map has an asymptotically cpc (order zero, respectively) lift. As another application we give unified proofs of Voiculescu's result that cones are quasidiagonal and Brown-Carrion-White's result that all amenable traces on cones are quasidiagonal. We also prove that all hyperlinear traces on cones are MF.

math.OA↗

arXiv2026

RFD property for groupoid C*-algebras of amenable groupoids and for crossed products by amenable actions

By Bekka's theorem the group C*-algebra of an amenable group $G$ is residually finite dimensional (RFD) if and only if $G$ is maximally almost periodic (MAP). We generalize this result in two directions of dynamical flavour. Firstly, we completely characterize the RFD property for crossed products by amenable actions of discrete groups on C*-algebras in terms of the action. The characterisation can be formulated in various terms, such as primitive ideals, (pure) states and approximations of representations, and the latter can be viewed as a dynamical version of Exel-Loring characterization of RFD C*-algebras. %The result leads among other consequences to a characterization of when a semidirect product by an amenable group has RFD full C*-algebra. As byproduct of our methods we characterize the property FD of Lubotzky and Shalom for semidirect products by amenable groups and obtain characterizations of the properties MAP and RF for general semidirect products of groups. These descriptions allow us to obtain the properties MAP, RF, RFD and FD for various new examples and generalize some results of Lubotzky and Shalom. Secondly, as another generalization of Bekka's theorem, we provide a sufficient condition and a necessary condition for the C*-algebra of an amenable étale groupoid to be RFD.

math.OA↗

arXiv2023

On amenable Hilbert-Schmidt stable groups

We examine Hilbert-Schmidt stability (HS-stability) of discrete amenable groups from several angles. We give a short, elementary proof that finitely generated nilpotent groups are HS-stable. We investigate the permanence of HS-stability under central extensions by showing HS-stability is preserved by finite central quotients, but is not preserved in general. We give a characterization of HS-stability for semidirect products $G\rtimes_γ\mathbb{Z}$ with $G$ abelian. We use it to construct the first example of a finitely generated amenable HS-stable group which is not permutation stable. Finally, it is proved that for amenable groups flexible HS-stability is equivalent to HS-stability, and very flexible HS stability is equivalent to maximal almost periodicity. There is some overlap of our work with the very recent and very nice preprint of Levit and Vigdorovich. We detail this overlap in the introduction. Where our work overlaps it appears that we take different approaches to the proofs and we feel the two works compliment each other.

math.GR↗

arXiv2023

Almost commuting matrices, cohomology, and dimension

We investigate which relations for families of commuting matrices are stable under small perturbations, or in other words, which commutative $C^*$-algebras $C(X)$ are matricially semiprojective. Extending the works of Davidson, Eilers-Loring-Pedersen, Lin and Voiculescu on almost commuting matrices, we identify the precise dimensional and cohomological restrictions for finite-dimensional spaces $X$ and thus obtain a complete characterization: $C(X)$ is matricially semiprojective if and only if $\dim(X)\leq 2$ and $H^2(X;\mathbb{Q})=0$. We give several applications to lifting problems for commutative $C^*$-algebras, in particular to liftings from the Calkin algebra and to $l$-closed $C^*$-algebras in the sense of Blackadar.

math.OA↗

arXiv2022

Commutativity of central sequence algebras

The question of which separable C*-algebras have abelian central sequence algebras was raised and studied by Phillips ([Ph88]) and Ando-Kirchberg ([AK14]). In this paper we give a complete answer to their question: A separable C*-algebra $A$ has abelian central sequence algebra if and only if A satisfies Fell's condition. Moreover, we introduce a higher-dimensional analogue of Fell's condition and show that it completely characterizes subhomogeneity of central sequence algebras. In contrast, we show that any non-trivial extension by compact operators has not only non-abelian but not even residually type I central sequence algebra. In particular its central sequence algebra is not type I and not residually finite-dimensional (RFD). Our techniques extensively use properties of nilpotent elements in C*-algebras.

math.OA↗

arXiv2022

Central amalgamation of groups and the RFD property

It is an old and challenging topic to investigate for which discrete groups G the full group C*-algebra C*(G) is residually finite-dimensional (RFD). In particular not much is known about how the RFD property behaves under fundamental constructions, such as amalgamated free products and HNN-extensions. In [CS19] it was proved that central amalgamated free products of virtually abelian groups are RFD. In this paper we prove that this holds much beyond this case. Our method is based on showing a certain approximation property for characters induced from central subgroups. In particular it allows us to prove that free products of polycyclic-by-finite groups amalgamated over finitely generated central subgroups are RFD. On the other hand we prove that the class of RFD C*-algebras (and groups) is not closed under central amalgamated free products. Namely we give an example of RFD groups (in fact finitely generated amenable RF groups) whose central amalgamated free product is not RFD, moreover it is not even maximally almost periodic. This answers a question of Khan and Morris [KM82].

math.OA↗

arXiv2021

C*-stability of discrete groups

A group may be considered $C^*$-stable if almost representations of the group in a $C^*$-algebra are always close to actual representations. We initiate a systematic study of which discrete groups are $C^*$-stable or only stable with respect to some subclass of $C^*$-algebras, e.g. finite dimensional $C^*$-algebras. We provide criteria and invariants for stability of groups and this allows us to completely determine stability/non-stability of crystallographic groups, finitely generated torsion-free step-2 nilpotent groups, surface groups, virtually free groups and certain Baumslag-Solitar groups.

math.OA↗

arXiv2019

Free products with amalgamation over central C*-subalgebras

Let A and B be C*-algebras whose quotients are all RFD, and let C be a central C*-subalgebra in both A and B. We prove that the full amalgamated free product of A and B over C is then RFD. This generalizes Korchagin's result that amalgamated free products of commutative C*-algebras are RFD. When applied to the case of a trivial amalgam, our methods recover the result of Exel-Loring for separable C*-algebras. As corollaries to our theorem, we give sufficient conditions for amalgamated free products of maximally almost periodic (MAP) groups to have RFD C*-algebras and hence to be MAP.

math.OA↗

arXiv2019

Stability of group relations under small Hilbert-Schmidt perturbations

If matrices almost satisfying a group relation are close to matrices exactly satisfying the relation, then we say that a group is matricially stable. Here "almost" and "close" are in terms of the Hilbert-Schmidt norm. Using tracial 2-norm on $II_1$-factors we similarly define $II_1$-factor stability for groups. Our main result is that all 1-relator groups with non-trivial center are $II_{1}$-factor stable. Many of them are also matricially stable and RFD. For amenable groups we give a complete characterization of matricial stability in terms of the following approximation property for characters: each character must be a pointwise limit of traces of finite-dimensional representations. This allows us to prove matricial stability for the discrete Heisenberg group $\mathbb H_3$ and for all virtually abelian groups. For non-amenable groups the same approximation property is a necessary condition for being matricially stable. We study this approximation property and show that RF groups with character rigidity have it.

math.OA↗

arXiv2018

Continuity of spectral radius and type I $C^*$-algebras

It is shown that the spectral radius is continuous on a $C^*$-algebra if and only if the $C^*$-algebra is type I. This answers a question of V. Shulman and Yu.~Turovskii [10]. It is shown also that the closure of nilpotents in a $C^*$-algebra contains an element with non-zero spectrum if and only if the $C^*$-algebra is not type I.

math.OA↗

arXiv2018

Variations of projectivity for C*-algebras

We consider various lifting problems for C*-algebras. As an application of our results we show that any commuting family of order zero maps from matrices to a von Neumann central sequence algebra can be lifted to a commuting family of order zero maps to the C*-central sequence algebra.

math.OA↗

arXiv2017

Geometry of quantum dynamics in infinite dimension

We develop a geometric approach to quantum mechanics based on the concept of the Tulczyjew triple. Our approach is genuinely infinite-dimensional and including a Lagrangian formalism in which self-adjoint (Schroedinger) operators are obtained as Lagrangian submanifolds associated with the Lagrangian. As a byproduct we obtain also results concerning coadjoint orbits of the unitary group in infinite dimension, embedding of the Hilbert projective space of pure states in the unitary group, and an approach to self-adjoint extensions of symmetric relations.

math-ph↗

arXiv2017

Elements of $C^*$-algebras Attaining Their Norm in a Finite-Dimensional Representation

We characterize the class of RFD $C^*$-algebras as those containing a dense subset of elements that attain their norm under a finite-dimensional representation. We show further that this subset is the whole space precisely when every irreducible representation of the $C^*$-algebra is finite-dimensional, which is equivalent to the $C^*$-algebra having no simple infinite-dimensional AF subquotient. We apply techniques from this proof to show the existence of elements in more general classes of $C^*$-algebras whose norms in finite-dimensional representations fit certain prescribed properties.

math.OA↗

arXiv2017

Tracial stability for C*-algebras

We consider tracial stability, which requires that tuples of elements of a C*-algebra with a trace that nearly satisfy the relation are close to tuples that actually satisfy the relation. Here both "near" and "close" are in terms of the associated 2-norm from the trace, e.g., the Hilbert-Schmidt norm for matrices. Precise definitions are stated in terms of liftings from tracial ultraproducts of C*-algebras. We completely characterize matricial tracial stability for nuclear C*-algebras in terms of certain approximation properties for traces. For non-nuclear $C^{\ast}$-algebras we find new obstructions for stability by relating it to Voiculescu's free entropy dimension. We show that the class of C*-algebras that are stable with respect to tracial norms on real-rank-zero C*-algebras is closed under tensoring with commutative C*-algebras. We show that $C(X)$ is tracially stable with respect to tracial norms on all $C^{\ast}$-algebras if and only if $X$ is approximately path-connected.

math.OA↗