# 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 $\text{\Big\{\rho(\si)\,\big|\,\si\in\ast\limits_{i=1}^n P_i\Big\}^\prime=:\L^\infty\Big(\ast\limits_{i=1}^n P_i\Big)}$ of the right regular 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 $\text{\Big\{\lambda(\si)\,\big|\,\si\in \ast\limits_{i=1}^n P_i\Big\}}$ extending Beurling's theorem, and show that the analytic Toeplitz algebra $\text{\L^\infty\Big(\ast\limits_{i=1}^n P_i\Big)}$ is reflexive $\text{(resp. hyper-reflexive)}$ and has property $\text{\bbb A_1}$ 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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