Toeplitz operators and solvable 𝐶*-algebras on Hermitian symmetric spaces

Abstract

Bounded symmetric domains (Cartan domains and exceptional domains) are higher-dimensional generalizations of the open unit disc. In this note we give a structure theory for the C*-algebra T generated by all Toeplitz operators Tf(h) := P{fh) with continuous symbol function ƒ G C(S) on the Shilov boundary 5 of a bounded symmetric domain D of arbitrary rank r. Here h belongs to the Hardy space H(S), and P : L(S) —• H(S) is the Szego projection. For domains of rank 1 and tube domains of rank 2, the structure of T has been determined in [1, 2]. In these cases Toeplitz operators are closely related to pseudodifferential operators. For the open unit disc, T is the C*-algebra generated by the unilateral shift. The structure theory for the general case [12] is based on the fact that D can be realized as the open unit ball of a unique Jordan triple system Z [7, Theorem 4.1]. Denoting the Jordan triple product by {uv*w}, a tripotent e G Z satisfies {ee*e} = e. Tripotents generalize the partial isometries of matrix algebras and determine the boundary structure of D C Z (cf. [7, Theorem 6.3]). Our principal result ([12]; cf. also [3, 4, 8]) is the following;