On The Structure of A Commutative Banach Algebra Generated By Toeplitz Operators With Quasi-Radial Quasi-Homogeneous Symbols |
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Authors: | Wolfram Bauer Nikolai Vasilevski |
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Institution: | 1. Mathematisches Institut, Georg-August-Universit?t, Bunsenstr. 3-5, 37073, G?ttingen, Germany 2. Departamento de Matem??ticas, CINVESTAV del I.P.N, Av. IPN2508, Col. San Pedro Zacatenco, Mexico, D.F., 07360, Mexico
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Abstract: | Let ${\mathcal{A}_{\lambda}^2(\mathbb{B}^n)}$ denote the standard weighted Bergman space over the unit ball ${\mathbb{B}^n}$ in ${\mathbb{C}^n}$ . New classes of commutative Banach algebras ${\mathcal{T}(\lambda)}$ which are generated by Toeplitz operators on ${\mathcal{A}_{\lambda}^2(\mathbb{B}^n)}$ have been recently discovered in Vasilevski (Integr Equ Oper Theory 66(1):141?C152, 2010). These algebras are induced by the action of the quasi-elliptic group of biholomorphisms of ${\mathbb{B}^n}$ . 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 ${\mathcal{T}(\lambda)}$ . Since ${\mathcal{T}(\lambda)}$ is not invariant under the *-operation of ${\mathcal{L}(\mathcal{A}_{\lambda}^2(\mathbb{B}^n))}$ its inverse closedness is not obvious and is proved. We remark that the algebra ${\mathcal{T}(\lambda)}$ 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 ${\mathcal{T}(\lambda)}$ is always connected. |
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