On the Structure of Commutative Banach Algebras Generated by Toeplitz Operators on the Unit Ball. Quasi-Elliptic Case. II: Gelfand Theory

Abstract

Extending our results in Bauer and Vasilevski (J Funct Anal 265(11):2956-2990, 2013) the present paper gives a detailed structural analysis of a class of commutative Banach algebras generated by Toeplitz operators on the standard weighted Bergman spaces over the complex unit ball in . In the most general situation we explicitly determine the set of maximal ideals of and we describe the Gelfand transform on a dense subalgebra. As an application to the spectral theory we prove the inverse closedness of algebras in the full algebra of bounded operators on for certain choices of . Moreover, it is remarked that is not semi-simple. In the case of we explicitly describe the radical of the algebra . This result generalizes and simplifies the characterization of , which was given in Bauer and Vasilevski (Integr Equ Oper Theory 74:199-231, 2012).