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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
MathSciNet review: 0476906
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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 $ \bar F$ in that case. We also show that if $ \{ {J_t}\} $ is a one-parameter semigroup of bounded linear operators on a (separable) Hilbert space, then $ \{ {J_t}\} \cup \{ J_t^\ast\} $ generates a one-parameter inverse semigroup T with $ J_t^{ - 1} = J_t^\ast$ if and only if $ \{ {J_t}\} $ is a one-parameter semigroup of partial isometries, and we describe the weak operator closure of T in that case.

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Keywords: Partial isometries, one-parameter semigroups, inverse semigroups, compact semitopological semigroups
Article copyright: © Copyright 1978 American Mathematical Society

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