Download Commutative Algebras of Toeplitz Operators on the Bergman by Nikolai Vasilevski PDF

By Nikolai Vasilevski

This designated e-book is dedicated to the exact examine of the lately found commutative C*-algebras of Toeplitz operators at the Bergman house over the unit disk. strangely, the major element to realizing their constitution and classifying them lies within the hyperbolic geometry of the unit disk. The ebook develops a couple of vital difficulties whose winning answer used to be made attainable and is predicated at the particular positive factors of the Toeplitz operators from those commutative algebras.

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Let P ∈ Prim A be a primitive ideal of the algebra A. Show first, that there exists a point t ∈ T such that P ∩ Z = Jt . Indeed, let π be the irreducible representation of the C ∗ -algebra A with ker π = P ; then C · I ⊂ π(Z) ⊂ π(A)′ = C · I. Thus C∼ = π(Z) = (Z + P )/P ∼ = Z/(Z ∩ P ), and therefore Z ∩ P is a maximal ideal of the algebra Z, thus equal to Jt for some point t ∈ T . It is easy to see that the conditions P ∩ Z = Jt and P ⊃ J(t) are equivalent for each primitive ideal P . 7, J(t) = P.

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.

Cn of the algebra A, such that n ck = e and C ∗ (c1 , . . , cn ) = A. k=1 Of course, there are many possibilities to construct such a family. For example one may take, for α ∈ [0, 1], ck = ck (α) = 1 1+α sk ), (e + 4n0 sk k = 1, n0 , n 0 1 ck = ck (α) = (e − cj ), n − n0 j=1 k = n0 + 1, n. Finally, the family of the following systems of self-adjoint idempotents (projections) satisfies all the necessary properties, 1/2 1/2 p = p(α) = (cj (α) · ck (α))nj,k=1 , qk = qk (α) = diag (0, . . , 0, where α ∈ [0, 1].

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