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$C\sp*$-algebras of multiplication operators on Bergman spaces. (English) Zbl 0583.46049
Let $\Omega$ be bounded domain in ${\bbfC}\sp n$ $(n\ge 1)$, let $A\sp 2(\Omega)$ be the Bergman space of $\Omega$, and let $C\sp*(\Omega)$ be the unital $C\sp*$-algebra generated by the operators of multiplication by the coordinate functions on $\Omega$. The authors present the first example of a domain $\Omega$ in ${\bbfC}\sp 2$ for which $C\sp*(\Omega)$ is not type I, in sharp contrast with the $n=1$ situation and with many previously known instances in several variables where $C\sp*(\Omega)$ has a composition series of finite length. The domain in question is of Reinhardt type, and therefore the generators of $C\sp*(\Omega)$ 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 $$ \Omega\sb{\delta\sb 1\delta\sb 2}:=\{(z\sb 1,z\sb 2)\in {\bbfC}\sp 2:\quad \vert z\sb 1\vert <\delta\sb 1\quad and\quad \vert z\sb 2\vert <1,\quad or\quad \vert z\sb 1\vert <1\quad and\quad \vert z\sb 2\vert <\delta\sb 2\} $$ $(0<\delta\sb 1,\delta\sb 2<1)$, allows the authors to conclude that $C\sp*(\Omega\sb{\delta\sb 1,\delta\sb 2})$ is type I if and only if ln $\delta\sb 1/\ln \delta\sb 2$ is a rational number, thus providing a whole collection of $\Omega$ ’s with non-type I $C\sp*$-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 ${\Bbb C}^n$ (Reinhardt, Hartogs, circular, tube)
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Full Text: DOI
References:
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