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Analysis on Locally Compact Semitopological Semigroups

  • Author / Creator
    Huang, Qianhong
  • This thesis focuses on the measure algebra M(S) of a locally compact
    semitopological semigroup S. In particular, we consider the analog of the
    group algebra L1(G) of a locally compact group G on S and the topological
    amenability of S. Among other results which shall be explained further in the
    introduction, the thesis answers the following open problems.

    1. Baker 90' and Dzinotyiweyi 84' [6, 18] Let L(S) = {μ ∈ M(S); s →δs*|μ| is weakly continuous}. It is known that if S = G, then L(S) = L1(G). Is L(S) a norm closed ideal of M(S) that closed under absolute continuity in general? We shall answer this question in the positive in Section 3.3 and 3.4.
    2. Day 82' [15] We say S is strong topological left amenable if there is a net of probability measure (μα) such that ||ν* μα- μα|| → 0 uniformly for all probability measures _ supported on a compact subset K of S. Does strong topological amenability implies non-trivial L(S)? The background for this question will be explained fully in Section 4.1, along with a counterexample that answers this problem in the negative.
    3. Wong 79' [50] We say S is topological left amenable if there is a net of probability measure (μα) such that ||ν* μα- μα|| → 0 for any probability measure ν on S. It was shown that when S is a discrete semigroup or a locally compact group, a locally compact Borel subsemigroup T is topological left amenable if and only if (1) S is topological left T-amenable, that is, there is a net of probability measures (μα) on S, such that ||ν* μα- μα|| → 0 for any probability measure _ on S that is supported on T, and (2) limα μα (T) > 0. Does similar result hold for locally compact semitopological semigroups? We shall prove this result in Section 4.2.

  • Subjects / Keywords
  • Graduation date
    Fall 2019
  • Type of Item
    Thesis
  • Degree
    Doctor of Philosophy
  • DOI
    https://doi.org/10.7939/r3-07rt-ev93
  • License
    Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.