Generalized Berezin Transform and Commutator Ideals

For a quasi-free module over a function algebra A(Ω), we define an analogue of the Berezin transform and relate this to the quotient of the C*-algebra it generates modulo the commutator ideal. Certainly the best understood commutative Banach algebras are those that consist of all the continuous complex-valued functions on a compact Hausdorff space. Indeed, most self-adjoint phenomena involving them have been thoroughly investigated. In particular, the study of their representation theory as operators on a Hilbert space, which is essentially the spectral theory for normal operators, shows that such representations are defined by multiplication on L-spaces. Over the past few decades, other classes of operators have been introduced that are defined by functions, but which involve more complicated methods. One example is Toeplitz operators while another example is the class of pseudo-differential operators. In both cases, one shows that the operators so defined behave like the functions used to define them, up to operators of lower order. To be more precise, let H(D) be the Hardy space of functions in L(T) consisting of the functions with zero negative Fourier coefficients and P be the projection of L(T) onto H(D). The Toeplitz operator Tφ for the function φ in L (T) is defined on H(D) to be pointwise multiplication by φ followed by P . If T denotes the C*-algebra generated by the Toeplitz operators Tφ for φ a continuous function on T, then T contains the algebra of compact operators K and the quotient algebra T/K is isometrically isomorphic to C(T). If instead of H(D) 1991 Mathematics Subject Classification. 46L06, 47L05, 47L15, 47L20, 47L80.