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C * -algebras of multiplication operators on Bergman spaces. (English) Zbl 0583.46049

Let Ω be bounded domain in n (n1), let A 2 (Ω) be the Bergman space of Ω, and let C * (Ω) be the unital C * -algebra generated by the operators of multiplication by the coordinate functions on Ω. The authors present the first example of a domain Ω in 2 for which C * (Ω) is not type I, in sharp contrast with the n=1 situation and with many previously known instances in several variables where C * (Ω) has a composition series of finite length. The domain in question is of Reinhardt type, and therefore the generators of C * (Ω) are multivariable weighted shifts. The authors associate to such operators a groupoid, and they then analyze the structure of the groupoid in terms of the behavior of the weight sequences. A careful analysis of that behavior for the domain

Ω δ 1 δ 2 :={(z 1 ,z 2 ) 2 :|z 1 |<δ 1 and|z 2 |<1,or|z 1 |<1and|z 2 |<δ 2 }

(0<δ 1 ,δ 2 <1), allows the authors to conclude that C * (Ω δ 1 ,δ 2 ) is type I if and only if ln δ 1 /lnδ 2 is a rational number, thus providing a whole collection of Ω ’s with non-type I C * -algebra.

MSC:
46L35Classifications of C * -algebras
46J15Banach algebras of differentiable or analytic functions, H p -spaces
47B38Operators on function spaces (general)
47B37Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47B35Toeplitz operators, Hankel operators, Wiener-Hopf operators
32A07Special domains in n (Reinhardt, Hartogs, circular, tube)