Journal of Operator Theory

Volume 32, Issue 1, Summer 1994 pp. 157-183.

Solutions of the operator-valued integrated Cauchy functional equation

Authors: Cho-Ho Chu (1) and Ka-Sing Lau (2)
Author institution:Goldsmiths’ College, University of London, London SE14, ENGLAND
Department of Mathematics, University of Pittsburgh, Pittsburgh PA 15260, U.S.A.

Summary: Let G be a separable, metrizable locally compact abelian group and let $\sigma$ be a vector measure on G taking values in the centre of a von Neumann algebra $\mathcal A$. Given an $\mathcal A$-valued measure $\mu$ on G, we define the convolution $\mu * \sigma$ and study the equation $\mu = \mu * \sigma$, using Choquet’s integral representation theory as in [7] where the same equation for scalar measures was studied.

Keywords: Convolution equation, locally compact group, exponential function, operator-valued measure, von Neumann algebra, conditional expectation, Choquet’s integral representation.

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