Algebras Generated by the Bergman Projection and Operators of Multiplication by Piecewise Continuous Functions

Abstract

Let D be the unit disk. For A ⊂ L∞(D) containing piecewise continuous functions, we study the C * − algebras RA generated by the Bergman projection for D and operators of multiplication by functions of A. These algebras are related to the algebra generated by more than two projections depending on how many limits a function has at a boundary point. We find the description of the symbol algebra of RA, denoted here by RA. Interesting facts about representations of RA are found and we construct a special family of coefficients such that the algebra RA has irreducible representations of predefined dimensions. Mathematics Subject Classification (2000). 47B34, 47L25, 44A15.

C*-Algebras of Bergman Type Operators with Piecewise Continuous Coefficients

“…For every k ∈ N and every λ ∈ R, the functions g λ,k = S k−1 Π (λ)g λ,1 and g λ,k = (S * Π ) k−1 (λ) g λ,1 (5.28) are norm one generators of the spaces Im B (k) (λ) and Im B (k) (λ), respectively, and Proof. By [13] and [9, Lemma 9.3], for every λ ∈ R the functions…”

“…A generalization of this paper to piecewise continuous coefficients admitting more than two one-sided limits at points of ∂G was elaborated in [13]. In [9] we constructed a symbol calculus for the C * -algebra A 1,1 generated by the Bergman projection B Π , by the anti-Bergman projection B Π , and by the operators of multiplication by piecewise continuous functions in P C(L) admitting two or more one-sided limits at the points z ∈ L∩Ṙ, and also obtained a Fredholm criterion for the operators A ∈ A 1,1 .…”

“…Making use of the Allan-Douglas local principle [5, Theorem 1.34], limit operators techniques (see, e.g., [4], [18]), and the Plamenevsky results [17] (also see [8]) on two-dimensional convolution operators with symbols admitting homogeneous discontinuities, we reduce the study to simpler C * -algebras associated with the points z ∈ Π and pairs (z, λ) ∈ (Ṙ ∩ L) × R. While in [13] the study of local algebras is based on a symbol calculus constructed in [20] for the unital C * -algebras generated by N orthogonal projections sum of which equals the unit and by 1 one-dimensional orthogonal projection, we apply a symbol calculus for the unital C * -algebras generated by N orthogonal projections sum of which equals the unit and by M one-dimensional orthogonal projections, which was constructed in [9]. Such algebras are models of local algebras for A n,m at points z ∈ ∂Π being the discontinuity points of coefficients.…”

“…In Section 3 we state the Allan-Douglas local principle and apply it to the quotient C * -algebra A π n,m = A n,m /K, reducing the study to local C * -algebras A π n,m,z associated with the points z ∈ Π. In Section 4, by analogy with [13] and [9], we study the local algebras A π n,m,z 524 Karlovich and Pessoa IEOT projections B Π,(j) of L 2 (Π) onto A 2 (j) (Π), and χ k are the characteristic functions of the arcs T + ∩ R k . The remaining part of this section is devoted to the verification of applicability of the abstract symbol calculus established in [9,Theorem 8.1,Lemma 8.3] to the C * -algebras A n,m,N,λ ⊂ B(L 2 (T + )) (λ ∈ R) generated by the operators B (i) (λ), B (j) (λ), χ k I (i = 1, .…”

Algebras Generated by the Bergman and Anti-Bergman Projections and by Multiplications by Piecewise Continuous Functions

“…To calculate the images of the operators B(λ) and B(λ) we follow in general the scheme of [10]. For the sake of completeness we give the corresponding proofs.…”