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

Abstract

The C * -algebra A generated by the Bergman and anti-Bergman projections and by the operators of multiplication by piecewise continuous functions on the Lebesgue space L 2 (Π) over the upper half-plane is studied. Making use of a local principle, limit operators techniques, and the Plamenevsky results on two-dimensional singular integral operators with coefficients admitting homogeneous discontinuities we reduce the study to simpler C * -algebras associated with points z ∈ Π ∪ ∂Π and pairs (z, λ) ∈ ∂Π × R. We construct a symbol calculus for unital C * -algebras generated by n orthogonal projections sum of which equals the unit and by m one-dimensional orthogonal projections. Such algebras are models of local algebras at points z ∈ ∂Π being the discontinuity points of coefficients. A symbol calculus for the C * -algebra A and a Fredholm criterion for the operators A ∈ A are obtained. Finally, a C * -algebra isomorphism between the quotient algebra A π = A/K, where K is the ideal of compact operators, and its analogue A π D for the unit disk is constructed. Mathematics Subject Classification (2000). Primary 47A53, 47L15; Secondary 47G10, 47L30.

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

“…Given λ ∈ R, we introduce the C * -algebra Ω λ ⊂ B(L 2 (T)) generated by the operators a(t)I and E(λ) −1 b(w)E(λ) where a, b ∈ L ∞ (T) and E(λ) ∈ B(L 2 (T)) are unitary operators defined in [17] (see also [8] and [9]). Note that below we do not use the explicit form of the operators E(λ) ±1 .…”

“…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 . The present paper generalizes the results of [9] to much more complicated C * -algebras A n,m .…”

“…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, .…”

Banach algebra generated by a finite number of bergman polykernel operators, continuous coefficients, and a finite group of shifts

“…32). In recent years, C * -algebras generated by a finite number of polykernel and antipolykernel Bergman operators and by a broad class of piecewise-continuous coefficients (only for p = 2) have been described (see [5,6]). A C * -algebra (for p = 2) generated by a Bergman operator, continuous coefficients, and the second-order Carleman shift was described in [7].…”