On The Structure of A Commutative Banach Algebra Generated By Toeplitz Operators With Quasi-Radial Quasi-Homogeneous Symbols (original) (raw)

Let A 2 λ (B n) denote the standard weighted Bergman space over the unit ball B n in C n. New classes of commutative Banach algebras T (λ) which are generated by Toeplitz operators on A 2 λ (B n) have been recently discovered in Vasilevski (Integr Equ Oper Theory 66(1):141-152, 2010). These algebras are induced by the action of the quasi-elliptic group of biholomorphisms of B n. In the present paper we analyze in detail the internal structure of such an algebra in the lowest dimensional case n = 2. We explicitly describe the maximal ideal space and the Gelfand map of T (λ). Since T (λ) is not invariant under the *-operation of L(A 2 λ (B n)) its inverse closedness is not obvious and is proved. We remark that the algebra T (λ) is not semi-simple and we derive its radical. Several applications of our results are given and, in particular, we conclude that the essential spectrum of elements in T (λ) is always connected.