arXiv subjects

Roman Sasyk

Publications and source records attributed to Roman Sasyk.

7 recordsLinked to original sources

arXiv2019

An ultrapower construction of the multiplier algebra of a $C^{\ast}$-algebra and an application to boundary amenability of groups

Using ultrapowers of $C^{\ast}$-algebras we provide a new construction of the multiplier algebra of a $C^{\ast}$-algebra. This extends the work of Avsec and Goldbring [Houston J. Math., to appear, arXiv:1610.09276.] to the setting of noncommutative and nonseparable $C^{\ast}$-algebras. We also extend their work to give a new proof of the fact that groups that act transitively on locally finite trees with boundary amenable stabilizers are boundary amenable.

math.OA↗

arXiv2019

Permanence properties of the second nilpotent product of groups

We show that amenability, the Haagerup property, the Kazhdan's property (T) and exactness are preserved under taking second nilpotent product of groups. We also define the restricted second nilpotent wreath product of groups, this is a semi-direct product akin to the restricted wreath product but constructed from the second nilpotent product. We then show that if two discrete groups have the Haagerup property, the restricted second nilpotent wreath product of them also has the Haagerup property. We finally show that if a discrete group is abelian, then the restricted second nilpotent wreath product constructed from it is unitarizable if and only if the acting group is amenable.

math.GR↗

arXiv2009

Turbulence and Araki-Woods factors

Using Baire category techniques we prove that Araki-Woods factors are not classifiable by countable structures. As a result, we obtain a far reaching strengthening as well as a new proof of the well-known theorem of Woods that the isomorphism problem for ITPFI factors is not smooth. We derive as a consequence that the odometer actions of Z that preserve the measure class of a finite non-atomic product measure are not classifiable up to orbit equivalence by countable structures.

math.OA↗

arXiv2009

Borel reducibility and classification of von Neumann algebras

We announce some new results regarding the classification problem for separable von Neumann algebras. Our results are obtained by applying the notion of Borel reducibility and Hjorth's theory of turbulence to the isomorphism relation for separable von Neumann algebras.

math.LO↗

arXiv2009

The classification problem for von Neumann factors

We prove that it is not possible to classify separable von Neumann factors of types $\II_1$, $\II_\infty$ or $\III_λ$, $0\leq λ\leq1$, up to isomorphism by a Borel measurable assignment of "countable structures" as invariants. In particular the isomorphism relation of type $\II_1$ factors is not smooth. We also prove that the isomorphism relation for von Neumann $\II_1$ factors is analytic, but is not Borel.

math.OA↗

arXiv2006

On II$_1$ factors arising from 2-cocycles of w-rigid groups

We consider $\text{\rm II}_1$ factors $L_μ(G)$ arising from 2-cocyles $μ\in \text{\rm H}^2(G,\Bbb T)$ on groups $G$ containing infinite normal subgroups $H \subset G$ with the relative property $\text{\rm(T)}$ (i.e. $G$ {\it w-rigid}). We prove that given any separable $\text{\rm II}_1$ factor $M$, the set of 2-cocycles $μ_{|H}\in \text{\rm H}^2(H,\Bbb T)$ with the property that $L_μ(G)$ is embeddable into $M$ is at most countable. We use this result, the relative property (T) of $\Bbb Z^2 \subset \Bbb Z^2 \rtimes Γ$ for $Γ\subset SL(2,\Bbb Z)$ non-amenable and the fact that every cocycle $μ_α\in {\text{\rm H}}^2(\Bbb Z^2,\Bbb T)\simeq \Bbb T$ extends to a cocycle on $\Bbb Z^2 \rtimes SL(2,\Bbb Z)$, to show that the one parameter family of II$_1$ factors $M_α(Γ)=L_{μ_α}(\Bbb Z^2 \rtimes Γ)$, $α\in \Bbb T$, are mutually non-isomorphic, modulo countable sets, and cannot all be embedded into the same separable II$_1$ factor. Other examples and applications are discussed.

math.OA↗

arXiv2003

On the Cohomology of Actions of Groups by Bernoulli Shifts

We prove that if $G$ is a countable, discrete group having infinite, normal subgroups with the relative property (T), then the Bernoulli shift action of $G$ on ${\underset g \in G \to Π} (X_0, μ_0)_g$ for $(X_{0},μ_{0})$ an arbitrary probability space, has first cohomology group isomorphic to the character group of $G$.

math.OA↗