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

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.Comment: To appear in: Journal of Operator Theor

Extreme environment effects on cognitive functions: a longitudinal study in high altitude in Antarctica

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