Finite dimensional irreducible representations and the uniqueness of the Lebesgue decomposition of positive functionals

Zsolt Szűcs; Balázs Takács

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arXiv · 2105.12199

Finite dimensional irreducible representations and the uniqueness of the Lebesgue decomposition of positive functionals

Zsolt Szűcs·Balázs Takács

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Abstract

We prove for an arbitrary complex $^*$-algebra $A$ that every topologically irreducible $^*$-representation of $A$ on a Hilbert space is finite dimensional precisely when the Lebesgue decomposition of representable positive functionals over $A$ is unique. In particular, the uniqueness of the Lebesgue decomposition of positive functionals over the $L^1$-algebras of locally compact groups provides a new characterization of Moore groups.

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Zsolt Szűcs, Balázs Takács. 2021-05-25. Finite dimensional irreducible representations and the uniqueness of the Lebesgue decomposition of positive functionals. https://arxiv.org/abs/2105.12199

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