. FA ] 2 6 Ju l 2 00 5 A new kind of index theorem ∗
نویسنده
چکیده
Index theory has had profound impact on many branches of mathematics. In this note we discuss the context for a new kind of index theorem. We begin, however, with some operator-theoretic results. In [11] Berger and Shaw established that finitely cyclic hyponormal operators have trace-class self-commutators. In [9], [31] Berger and Voiculescu extended this result to operators whose self-commutators can be expressed as the sum of a positive and a trace-class operator. In this note we show this result can't be extended to operators whose self-commutators can be expressed as the sum of a positive and a S p-class operator. Then we discuss a conjecture of Arveson [4] on homogeneous submodules of the m-shift Hilbert space H 2 m and propose some refinements of it. Further, we show how a positive solution would enable one to define K-homology elements for subvarieties in a strongly pseudo-convex domain with smooth boundary using submodules of the corresponding Bergman module. Finally, we discuss how the Chern character of these classes in cyclic cohomology could be defined and indicate what we believe the index to be. The complex Hilbert space H is said to be a Hilbert module over the algebra A if H is a unital module over A. This is equivalent to a representation of A on H. In the last two decades, there has been considerable interest in the study of Hilbert modules for various classes of algebras, in part as an approach to multivariate operator theory. In [20] A was assumed to be a function algebra and module multiplication to be bounded.
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