Submission date: 4 November 2021

Abstract:

For a locally compact group G, we show that it is possible to present the class of continuous unitary representations of G as an elementary class of metric structures, in the sense of continuous logic. More precisely, we show how non-degenerate *-representations of a general *-algebra A (with some mild assumptions) can be viewed as an elementary class, in a many-sorted language, and use the correspondence between continuous unitary representations of G and non-degenerate *-representations of L^1(G). We relate the notion of ultraproduct of logical structures, under this presentation, with other notions of ultraproduct of representations appearing in the literature, and characterise property (T) for G in terms of the definability of the sets of fixed points of L^1 functions on G.

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Full text arXiv 2111.02835: pdf, ps.