# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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## Conditionally compact semitopological one-parameter inverse semigroups of partial isometriesHTML articles powered by AMS MathViewer

by M. O. Bertman
Trans. Amer. Math. Soc. 240 (1978), 263-275 Request permission

## 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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