arXiv subjects

Fyodor A. Sukochev

Publications and source records attributed to Fyodor A. Sukochev.

4 recordsLinked to original sources

arXiv2006

The Chern Character of Semifinite Spectral Triples

In previous work we generalised both the odd and even local index formula of Connes and Moscovici to the case of spectral triples for a *-subalgebra \A of a general semifinite von Neumann algebra. Our proofs are novel even in the setting of the original theorem and rely on the introduction of a function valued cocycle (called the resolvent cocycle) which is `almost' a (b,B)-cocycle in the cyclic cohomology of \A. In this paper we show that this resolvent cocycle `almost' represents the Chern character, and assuming analytic continuation properties for zeta functions, we show that the associated residue cocycle, which appears in our statement of the local index theorem does represent the Chern character.

math.OA↗

arXiv2004

The Local Index Formula in Semifinite von Neumann Algebras I: Spectral Flow

We generalise the local index formula of Connes and Moscovici to the case of spectral triples for a *-subalgebra \A of a general semifinite von Neumann algebra. In this setting it gives a formula for spectral flow along a path joining an unbounded self adjoint Breuer-Fredholm operator, affiliated to the von Neumann algebra, to a unitarily equivalent operator. Our proof is novel even in the setting of the original theorem and relies on the introduction of a function valued cocycle which is `almost' a (b,B)-cocycle in the cyclic cohomology of \A.

math.OA↗

arXiv2004

The Local Index Formula in Semifinite von Neumann Algebras II: The Even Case

We generalise the even local index formula of Connes and Moscovici to the case of spectral triples for a *-subalgebra \A of a general semifinite von Neumann algebra. The proof is a variant of that for the odd case which appears in Part I. To allow for algebras with a non-trivial centre we have to establish a theory of unbounded Fredholm operators in a general semifinite von Neumann algebra and in particular prove a generalised McKean-Singer formula.

math.OA↗

arXiv1993

A Uniform Kadec-klee Property For Symmetric Operator Spaces

We show that if a rearrangement invariant Banach function space $E$ on the positive semi-axis satisfies a non-trivial lower $q-$ estimate with constant $1$ then the corresponding space $E(\nm)$ of $τ-$measurable operators, affiliated with an arbitrary semi-finite von Neumann algebra $\nm$ equipped with a distinguished faithful, normal, semi-finite trace $τ$, has the uniform Kadec-Klee property for the topology of local convergence in measure. In particular, the Lorentz function spaces $L_{q,p}$ and the Lorentz-Schatten classes ${\cal C}_{q,p}$ have the UKK property for convergence locally in measure and for the weak-operator topology, respectively. As a partial converse , we show that if $E$ has the UKK property with respect to local convergence in measure then $E$ must satisfy some non-trivial lower $q$-estimate. We also prove a uniform Kadec-Klee result for local convergence in any Banach lattice satisfying a lower $q$-estimate.

math.FA↗