Journal of Operator Theory

Volume 41, Issue 2, Spring 1999 pp. 391-420.

A generalization of Beurling's theorem and a class of reflexive algebras

Authors: Gelu Popescu
Author institution: Division of Mathematics and Statistic, The University of Texas at San Antonio, San Antonio, TX 78249, U.S.A.

Summary: We study the commutant

$\Big\{\rho(\si)\,\big|\, \si\in \ast\limits_{i=1}^n P_i\ Big\}^\prime=:\L^\infty\Big(\ast\limits_{i=1}^n P_i\Big)$ of the right regular r epresentation 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 ca ncellation 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 Beur\-ling's theorem, and show that the analytic Toeplitz algebra

$\L^\infty\Big(\ast\limits_{i=1}^n P_i\Big)$ is reflexive (resp. hyper-reflexive) and has property

$\bbb A_1$ if

$n\ge 2$ . This leads also to an inner-outer factorization and Szeg\"o type theorem in this algebra when

$P_i$

$(i=1,2,\ldots, n)$ are certain totally ordered semigroups.

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