Harmonic functions, crossed products and approximation properties

Abstract: The space of harmonic functions on a locally compact group $G$ is the fixed point space of a certain Markov operator. Its `quantization', the corresponding fixed point space of operators on $L^2(G)$, coincides with the weak-* closed bimodule over the group von Neumann algebra generated by this space. We examine the analogous spaces of jointly harmonic functions and their quantized operator bimodules. This leads to two different notions of crossed product of operator spaces by actions of $G$, which coincide when $G$ satisfies a certain approximation property. The corresponding (dual) notions of crossed products of (co-) actions by the von Neumann algebra of $G$ always coincide. This gives a new approach to the correspondence between spectral synthesis and operator synthesis.

The talk is a survey of joint work with M. Anoussis and I.G. Todorov, and of recent work by D. Andreou.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( slides | video )

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Noncommutative Geometry in NYC

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