By Nikolai Vasilevski

ISBN-10: 3764387254

ISBN-13: 9783764387259

ISBN-10: 3764387262

ISBN-13: 9783764387266

This distinctive e-book is dedicated to the targeted research of the lately chanced on commutative C*-algebras of Toeplitz operators at the Bergman area over the unit disk. unusually, the major aspect to figuring out their constitution and classifying them lies within the hyperbolic geometry of the unit disk. The e-book develops a couple of very important difficulties whose profitable resolution used to be made attainable and is predicated at the particular beneficial properties of the Toeplitz operators from those commutative algebras.

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Then the mapping π ˜ : a ∈ A −→ a ˜ ∈ Γb (ξ) 8 Chapter 1. Preliminaries is a morphism of the C ∗ -algebras A and Γb (ξ), such that 1. ker π ˜= t∈T J(t), 2. Im π ˜ = A. In particular, the mapping π ˜ : A → A is an isometric ∗ -isomorphism if and only if J(t) = {0}. t∈T Proof. Indeed, π ˜ (a) = 0 if and only if a(t) = 0 for each t ∈ T , or a ∈ J(t) for each t ∈ T , or a ∈ t∈T J(t). 3). 8, gives in fact a general conception of the local principles in the theory of C ∗ -algebras: an isomorphic description of a C ∗ algebra A (or A/ t∈T J(t) ) collecting the information about “simpler objects”, the so-called local algebras.

Denote by ∆ = ∆(C1 , . . , Cn ) ⊂ Rn the joint spectrum of the operators C1 , . . , Cn . Then the C ∗ -algebra RC is naturally isomorphic to C(∆), and the coordinate function tk , k = 1, n, (t = (t1 , . . , tn ) ∈ ∆ ⊂ Rn ) is the image of Ck under this isomorphism. 20) qk (t) = diag (0, . . , 0, where t = (t1 , . . , tn ) ∈ ∆. 1 , 0, . . 21) 30 Chapter 1. Preliminaries To describe the algebra S(∆) we introduce first a sort of “maximal” algebra. Let n n ∆n−1 = {t = (t1 , . . , tn ) ∈ R : tk ≥ 0, k = 1, n, tk = 1} k=1 be the standard (n − 1) - dimensional simplex.

To do this we show that each point t0 ∈ V is interior. 12 there exist a function f ∈ Ft0 and neighborhoods u1 ⊂ u0 (u1 ⊂ u0 ), such that f |u1 ≡ 1 and f a < ε. Then for each point t ∈ Int u1 we have a(t) ≤ f a < ε, thus t0 ∈ Int u1 ⊂ V , and the point t0 is interior. Thus the ∗ -bundle topology is not stronger then the hull kernel topology. Now, the ∗ - bundle topology is the weakest one such that the sections a ˜ : t → a(t) are continuous for all a ∈ A, and hull kernel topology is the weakest one such that the sections of the form ˜b : t → b(t)e(t) are continuous for all b ∈ C(T ).

### Commutative Algebras of Toeplitz Operators on the Bergman Space by Nikolai Vasilevski

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