Abstract
Every locally compact group, G, has defined on it a semigroup of continuous, positive definite functions,P(G).This semigroup, additionally equipped with a normalized, partially ordered, convex structure is a complete invariant of the underlying group. This semigroup has an identity and we investigate what it means to differentiate in the classical calculus sense at this identity. This leads us to the concept of a semiderivation. We are also naturally led to consider the cohomology of continuous, unitary representations of G, as well as the “screw functions” of J. von Neumann and I. J. Schoenberg, a Lévy-Khinchin formula, and a characterization of groups with property (T).
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Walter, M.E. (1998). Semigroups of Positive Definite Functions and Related Topics. In: Ross, K.A., Singh, A.I., Anderson, J.M., Sunder, V.S., Litvinov, G.L., Wildberger, N.J. (eds) Harmonic Analysis and Hypergroups. Trends in Mathematics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4348-5_13
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DOI: https://doi.org/10.1007/978-0-8176-4348-5_13
Publisher Name: Birkhäuser, Boston, MA
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