Conditionally compact semitopological one-parameter inverse semigroups of partial isometries

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