Functional Analysis Seminar

The problem of stability of approximate homomorphisms was first posed by S. Ulam in the context of groups equipped with a metric. If $G$ and $H$ are groups and $H$ is equipped with a metric $d$, then $\varphi\colon G\to H$ is an $\varepsilon$-homomorphism if $d(\varphi(xy), \varphi(x)\varphi(y))\leq \varepsilon$ for all $x,y\in G$. Ulam’s well-studied problem asks how closely such a map can be approximated by a true homomorphism.
Analogous questions have been investigated in many algebraic and analytic settings. For C*-algebras, the notion of an $\varepsilon$-*-homomorphism admits several possible formalizations. The variant I will discuss, while perhaps not the most immediate, turns out to be particularly interesting, because its associated Ulam stability problem is closely related to rigidity for corona C*-algebras. Namely, Ulam stability of $\varepsilon$-*-isomorphisms between C*-algebras in a certain class (e.g., AF algebras) is equivalent to the rigidity question for coronas of direct sums of C*-algebras in this class.

 

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It is known for a long time, due to a celebrated theorem of Ornstein and Weiss, that (classical/plain) orbit equivalence offers no information about ergodic probability measure preserving actions of amenable groups. On the other hand, conjugacy is too intractable, and effectively hopeless to study in full generality. Quantitative orbit equivalence aims to bridge this gap by adding intermediate layers of rigidity— a strategy that has borne fruit already in the late 1960s but was used as a general framework only semi-recently. In this talk, Spyridon Petrakos will introduce aspects of quantitative orbit equivalence and present a complete picture of it for integer odometers. This is joint work with Petr Naryshkin.

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