Toeplitz C ∗ -Algebras on Bounded Symmetric Domains

Abstract

In this paper, we develop a structure theory for the Toeplitz C*-algebra Y for an arbitrary bounded symmetric domain D in C' (classical or exceptional). As the main result, it is shown that 5Y is a C *-algebra of type I which is solvable, i.e., admits a finite composition sequence of closed ideals Sk in $7 such that the subquotients fk? i/ k are continuous fields of elementary C *-algebras (of compact operators). The length of this composition sequence coincides with the rank r of D as a hermitian symmetric space. For 0 < k < r, the spectral components Sk of AY, parametrizing the irreducible representations of Y, are realized in terms of the boundary components (complex analytic faces) of D. The structure theory for Y is based on the fact that the tangent space Z C C' of D carries a Jordan algebraic structure: Z is a Jordan triple system with triple product { uv *w } describing the geometry of D in algebraic terms. For example, D is the open unit ball with respect to the so-called spectral norm on Z and the boundary components of D are in 1-1 correspondence with the tripotents e E Z (partial isometries satisfying { ee*e 4 = e). In particular, the Shilov boundary S of D consists of all tripotents of maximal rank r. The Jordan theoretic description of D was used in [17] to determine an explicit Peter-Weyl decomposition of the Hardy space H2(S) (cf. [15]) in terms of the norm functions (generalized determinants) associated with the Jordan triple