Journal of Operator Theory
Volume 41, Issue 2, Spring 1999 pp. 391-420.
A generalization of Beurling's theorem and a class of reflexive algebrasAuthors: Gelu Popescu
Author institution: Division of Mathematics and Statistic, The University of Texas at San Antonio, San Antonio, TX 78249, U.S.A.
Summary: $\def\si{\sigma}$ $\def\ast{\mathop*}$ We study the commutant $\Big\{\rho(\si)\,\big|\,\si\in\ast\limits_{i=1}^n P_i\Big\}^\prime=:\mathcal L^\infty\Big(\ast\limits_{i=1}^n P_i\Big)$ of the right regular representation of the free product semigroup $\ast\limits_{i=1}^n P_i$, where $P_i$, $i=1,2,\ldots, n$, $n\ge 2$, are discrete semigroups with involution, no divisors of the identity, and the cancellation property. We obtain a description of the invariant subspace structure of the left regular representation $\Big\{\lambda(\si)\,\big|\,\si\in \ast\limits_{i=1}^n P_i\Big\}$ extending Beurling's theorem, and show that the analytic Toeplitz algebra $\mathcal L^\infty\Big(\ast\limits_{i=1}^n P_i\Big)$ is reflexive $\text{(resp. hyper-reflexive)}$ and has property $\mathbb A_1$ if $n\ge 2$. This leads also to an inner-outer factorization and Szegö type theorem in this algebra when $P_i$ $(i=1,2,\ldots, n)$ are certain totally ordered semigroups.
Keywords: Reflexive algebra, free product semigroup, regular representation, inner-outer factorization
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