# Mini-workshop in Operator Algebras

Speakers: Tatiana Shulman, Mattias Byléhn, Alexey Kuzmin, Lyudmila Turowska, Ulrik Enstad, Sanaz Pooya, Eusebio Gardella## Tuesday January 11

**15.30-16.15, Tatiana Shulman: Central sequence algebras via nilpotent elements **

Abstract: Central sequence algebras play an important role in C*-algebra and von Neumann algebra theory. A central sequence in a C*-algebra is a sequence (x_n) of elements such that [x_n, a] converges to zero, for any element a of the C*-algebra. In von Neumann algebra setting one typically means the convergence with respect to tracial norms, while in C*-theory it is with respect to the C*-norm.

In this talk we will consider the C*-theory version of central sequences. We will discuss properties of central sequence algebras and in particular address a question of J. Phillips and of Ando and Kirchberg of which separable C*-algebras have abelian central sequence algebras. Joint work with Dominic Enders.**16.15-17.00, Mattias Byléhn: Bochner-type Theorems for Non-Tempered Distributions**

Abstract: The classical Bochner-Schwartz theorem states that a positive-definite tempered distribution on Euclidean space R^d is the Fourier transform of a unique tempered positive Radon measure on R^d. We will consider a slight generalization of this setting when the distribution in question is non-tempered and positive-definite with respect to test functions that are invariant under finitely many reflections. Such distributions turn out to be complex Fourier transforms of tempered positive Radon measures on certain tubes in complex space C^d. The proofs involved make use of the classical Godement-Plancherel theorem for commutative *-algebras. If time permits, I will also describe how to construct examples of such distributions using the Abel transform on non-compact symmetric spaces.

Based on joint work with Michael Björklund.

## Wednesday January 12

**9.15-10.00, Alexey Kuzmin: The C*-isomorphism class of the q-CCR algebra is independent of q**

Abstract: In this talk we consider the question of independence of C*-isomorphism class of the q-canonical commutation relations (q-CCR) from the deformation parameter q. We prove that q-CCR is isomorphic to the Cuntz-Toeplitz algebra (0-CCR) for every q such that |q| < 1.**10.30-11.15, Lyudmila Turowska: Beurling-Fourier algebras and Complexification**

Abstract: Fourier algebra A(G) of a locally compact group G, introduced by Eymard, is one of the favourite objects in abstract harmonic analysis. It has an advantage to be commutative that allows one to examine its Gelfand spectrum, which is known to be topologically isomorphic to G; the fact makes a non-trivial connection between Banach algebras and groups. We will discuss a weighted variant of Fourier algebra and show its connection with complexification of the underlying group. For compact groups this was done thanks to abstract complexification due to McKennon [Crelle, 79'] and Cartwright/McMullen [Crelle, 82']. We extended this theory to general locally compact groups and use the model to describe the Gelfand spectrum of weighted Fourier algebras, showing that the latter is a part of the complexification for a wide class of locally compact groups and weights. I shall also present different examples of weights and determine the spectrum of the corresponding algebras. **11.15-12.00, Ulrik Enstad: On sufficient density conditions for lattice orbits of relative discrete series**

Abstract: Let π be a discrete series representation of a locally compact group G on a Hilbert space 𝓗π and let Γ be a lattice in G. Using von Neumann algebraic techniques, we study the existence of vectors η ∈ 𝓗π$ such that the system π(Γ) η = { π(γ) η : γ ∈ Γ } has desirable spanning or linear independence properties for 𝓗π. We give new sufficient conditions on G and π which ensure that the existence of such vectors η is equivalent to inequalities involving the number dπ·vol(G/Γ), where dπ is the formal dimension of π and vol(G/Γ) is the covolume of Γ in G. The talk is based on joint work with Jordy T. van Velthoven.

### MV:L15

**13.30-14.15, Sanaz Pooya: The Baum-Connes assembly map for certain subgroups of Z^2 \rtimes GL(2, Z)**

Abstract: The Baum-Connes conjecture suggests a link between operator algebras and topology/geometry. The link is provided via the so-called assembly map and the conjecture is that this map is an isomorphism of two abelian groups; equivariant K-homology and K-theory of specific objects constructed from a group. It is known that the conjecture holds true for large classes of groups, including a-T-menable groups, however it is still open for linear groups. In this talk, I will discuss the assembly maps of ℤ2 ⋊ H, where the subgroup H is either the Sanov subgroup of SL(2,ℤ), its commutator subgroup or the principal subgroup of level 2. The subgroup H is these cases is abstractly a free group. Further I will report on the cases where H is either SL(2,ℤ) or GL(2,ℤ). It is known that these groups satisfy the conjecture. This work in progress will provide an alternative (and explicit) proof of this fact on the one hand, and on the other hand will elucidate on the assembly maps of abstractly isomorphic groups. This is joint work with R. Flores, A. Valette, and A. Zumbrunnen.**14.15-15.00, Eusebio Gardella: Amenable actions of nonamenable groups and classifiability of crossed products**

Abstract: For a large class of nonamenable groups containing all hyperbolic groups, we show that every amenable and minimal action on a compact space X has the following form of paradoxicality: for any closed set K and any open set U in X, there is an open cover of K whose members can be transported by the action into disjoint subsets of U. This has strong implications on the structure of the crossed product $C(X)\rtimes G$, most notably the fact that if the action is in addition topologically free, then $C(X)\rtimes G$ is classifiable by K-theory, regardless of the dimension of X. This is an unexpected phenomenon in the nonamenable setting, and shows that classifiability of $C(X)\rtimes G$ does not require any version of mean dimension zero or the small boundary property, unlike in the amenable setting.

This is joint work with Shirly Geffen, Julian Kranz, and Petr Naryshkin.

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