Conditionally compact semitopological one-parameter inverse semigroups of partial isometries

Author:
M. O. Bertman

Journal:
Trans. Amer. Math. Soc. **240** (1978), 263-275

MSC:
Primary 22A20; Secondary 47D05

DOI:
https://doi.org/10.1090/S0002-9947-1978-0476906-7

MathSciNet review:
0476906

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Abstract | References | Similar Articles | Additional Information

Abstract: The algebraic structure of one-parameter inverse semigroups has been completely described. Furthermore, if *B* is the bicyclic semigroup and if *B* is contained in any semitopological semigroup, the relative topology on *B* is discrete. We show that if *F* is an inverse semigroup generated by an element and its inverse, and *F* is contained in a compact semitopological semigroup, then the relative topology is discrete; in fact, if *F* is any one-parameter inverse semigroup contained in a compact semitopological semigroup, then the multiplication on *F* is jointly continuous if and only if the inversion is continuous on *F*, and we describe in that case. We also show that if is a one-parameter semigroup of bounded linear operators on a (separable) Hilbert space, then generates a one-parameter inverse semigroup *T* with if and only if is a one-parameter semigroup of partial isometries, and we describe the weak operator closure of *T* in that case.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1978-0476906-7

Keywords:
Partial isometries,
one-parameter semigroups,
inverse semigroups,
compact semitopological semigroups

Article copyright:
© Copyright 1978
American Mathematical Society