Conditionally compact semitopological one-parameter inverse semigroups of partial isometries

Bertman, M. O.

doi:10.1090/S0002-9947-1978-0476906-7

Conditionally compact semitopological one-parameter inverse semigroups of partial isometries
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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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