A new kind of index theorem
By Ronald G. Douglas. 2005
53 citations recorded by the source corpus.
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-commutator 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.
Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/0e0f69771d8752497cebdbbe5d025a76121f9b2c
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Citations in the corpus
- Essentially Reductive Hilbert Modules II (2006, background): “Finally, it seems likely that this K-homology class agrees with the fundamental class defined by ∂Ωφ ∩ Z(M) as conjectured in [11].”
- Essentially normal Hilbert modules and K-homology (2008, methodology): “Concerning the study of p-smoothness of extensions arising from quotient submodules, in scalar-valued version, Douglas [ 17 ] raised a refinement for Arveson’s conjecture as follows: Let I be an ideal of C[z1 ,..., zd ], then the quotient module L 2(Bd )/[I ] is p-essentially normal for p > dimC Z (”
- A harmonic analysis approach to essential normality of principal submodules (2011, background): “However, in [10], it was conjectured that this result on the commutators actually holds for p > dimZ(p).”