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Topological diffuse semigroups (systems with a stochastic product). Armstrong, William Ward
Abstract
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 σalgebra 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 pointmasses. 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 pointmeasure 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 twosided 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 ReesSuschkewitsch 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 twosided ideal of a compact diffuse semigroup. If the minimal onesided ideals of a compact diffuse semigroup are assumed to be open, then the minimal twosided 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.
Item Metadata
Title 
Topological diffuse semigroups (systems with a stochastic product).

Creator  
Publisher 
University of British Columbia

Date Issued 
1966

Description 
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 σalgebra 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 pointmasses. 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 pointmeasure 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 twosided 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 ReesSuschkewitsch 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 twosided ideal of a compact diffuse semigroup. If the minimal onesided ideals of a compact diffuse semigroup are assumed to be open, then the minimal twosided 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.

Genre  
Type  
Language 
eng

Date Available 
20110810

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0080555

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

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Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.