Journal of Operator Theory

Volume 54, Issue 2, Fall 2005 pp. 305-316.

A large weak operator closure for the algebra generated by two isometries

Authors: C.J. Read
Author institution: Faculty of Mathematics, University of Leeds, Leeds LS2 9JT, UK

Summary: Given two (or $n$) isometries on a Hilbert space $\mathcal{H}$, such that their ranges are mutually orthogonal, one can use them to generate a $C^*$-algebra. If the ranges sum to $\mathcal{H}$, then this $C^*$-algebra, the Cuntz $C^*$-algebra ${\mathcal O}_n$, is unique up to $*$-isomorphism. However if we omit to close under the $*$ operation and merely consider the norm closed algebra generated by such isometries, that is certainly not unique; even if we take the weak operator topology (WOT) closure of such an algebra, it is still not unique. Among the possible WOT closures that one might conceivably get, the largest is the von Neumann algebra that will be obtained if it should chance to happen that the WOT closed algebra generated is, in fact, closed under hermitian conjugation after all. In this paper we show that this largest possible case can indeed happen (a problem which was posed by Davidson); we exhibit pairs of isometries $S_0$, $S_1$ with disjoint ranges, such that the WOT closed algebra generated by $S_0$ and $S_1$ is the whole of $\mathcal{B(H)}$.

Keywords: Isometry, Cuntz algebra, $C^*$-algebra, operator algebra, free semigroup

Contents Full-Text PDF