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

Let denote the standard weighted Bergman space over the unit ball in . New classes of commutative Banach algebras which are generated by Toeplitz operators on 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 . 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 . Since is not invariant under the *-operation of its inverse closedness is not obvious and is proved. We remark that the algebra 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 is always connected.