Journal of Operator Theory

Volume 91, Issue 1, Winter 2024 pp. 55-95.

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

Authors: Zsolt Szucs (1), Balazs Takacs (2)
Author institution:(1) Department of Differential Equations, Institute of Mathematics, Budapest Univ. of Technology and Economics, Muegyetem rkp. 3., H-1111 Budapest, Hungary
(2) Department of Analysis, Institute of Mathematics, Budapest University of Technology and Economics, Muegyetem rkp. 3., H-1111 Budapest, Hungary

Summary: For an arbitrary complex $*$-algebra $A$, we prove 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.

DOI: http://dx.doi.org/10.7900/jot.2022jan28.2395
Keywords: irreducible representation, Lebesgue decomposition, positive functional, enveloping von Neumann algebra, Moore group, locally $C^*$-algebra

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