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UBC Theses and Dissertations

Topological diffuse semigroups (systems with a stochastic product). Armstrong, William Ward


A generalization of the concept of a locally compact Hausdorff topological semigroup and its associated convolution semigroup of regular Borel probabilities is defined and studied. Let S be a locally compact Hausdorff space, ℬ be the σ-algebra of all Borel subsets of S , and ℘ be the set of all regular Borel probability measures on S. Suppose P*Q is an associative binary operation defined for P, Q in ℘ with values in ℘ which is affine and continuous (in the topology of weak convergence) in each variable P, Q separately. Then the structure (S,ℬ,℘,*) so defined is termed a topological diffuse semigroup. S is considered to be embedded in ℘ as the set of all point-masses. If the operation * is restricted to S , then S will not be a semigroup in general, for the product x*y for x, y in S will be a probability measure on S . If this is a point-measure for all x, y in S , then S will be a semigroup, and ℘ will be the usual convolution semigroup of regular probabilities. Topological diffuse semigroups are also generalizations of the "generalized convolutions" studied by K. Urbanik. In order to study the structure of topological diffuse semigroups, a product of subsets of S is defined: EF is defined to be the closure of the union of the supports of the measures x*y for x in E and y in F . A right ideal R is then defined to be a subset of S such that RS is contained in R . Left ideals and two-sided ideals are defined similarly. S is defined to be a diffuse group if {X}S = S{x} = S for all x in S . Using these concepts one can partially carry over to compact diffuse semigroups the Rees-Suschkewitsch structure theorem. There are interesting deviations from semigroup behaviour however. For example, it may happen that the union of the minimal right ideals is merely dense in the minimal two-sided ideal of a compact diffuse semigroup. If the minimal one-sided ideals of a compact diffuse semigroup are assumed to be open, then the minimal two-sided ideal is the disjoint union of the finite number of minimal right (or left) ideals, and also the finite disjoint union of diffuse groups. Idempotent probability measures are studied and in some cases expressed in terms of "normed Haar measures" on compact diffuse groups. A limit theorem of Rosenblatt for Cesàro averages of *-powers is generalized to compact diffuse semigroups with jointly continuous multiplication. As an application of the theory of topological diffuse semigroups, it is shown that the "generalized convolutions" of Urbanik are jointly continuos operations.

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