Inverse *-semigroups *-generated by families of isometries

Szymański, Wacław

doi:10.1090/S0002-9939-1990-0982408-2

Inverse $$-semigroups $$-generated by families of isometries
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Proc. Amer. Math. Soc. 108 (1990), 101-106 Request permission

Abstract:

It is shown that if a *-semigroup *-generated by a family of commuting Hilbert space isometries that commute each other, none of which commutes with the adjoint of another one, and none of which is a nonzero power of another one, consists of partial isometries, then it is singly *-generated. Also, the following result on algebraic semigroups is proved: If $S$ is an inverse *semigroup *-generated by a set $X$ satisfying the generating relations: ${a^ * }a = 1, ab = ba$, for all $a,b \in X$, then $S$ is the bicyclic semigroup. Both results follow from the special behavior of inverse *-semigroups *-generated by analytic Toeplitz operators.

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