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

TL;DR

This paper establishes a connection between finite dimensional irreducible representations of complex *-algebras and the uniqueness of Lebesgue decompositions of positive functionals, offering new insights into the structure of Moore groups.

Contribution

It proves that finite dimensional irreducible *-representations are equivalent to the uniqueness of Lebesgue decompositions for positive functionals on any complex *-algebra.

Findings

Finite dimensional irreducible representations characterize the uniqueness of Lebesgue decompositions.

The result provides a new characterization of Moore groups.

The paper links representation theory with measure-theoretic decompositions.

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.