Let be bounded domain in , let be the Bergman space of , and let be the unital -algebra generated by the operators of multiplication by the coordinate functions on . The authors present the first example of a domain in for which is not type I, in sharp contrast with the situation and with many previously known instances in several variables where has a composition series of finite length. The domain in question is of Reinhardt type, and therefore the generators of 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
, allows the authors to conclude that is type I if and only if ln is a rational number, thus providing a whole collection of ’s with non-type I -algebra.