Real valued spectral flow in a type II(,infinity) factor

Vicumpriya Sriyantha Perera, Purdue University

Abstract

Let ${\cal F}\sb{\rm II}$ be the space of Fredholm elements in a type II$\sb\infty$ von Neumann algebra factor in the sense of Breuer. Let ${\cal F}\sbsp{\rm II}{\rm sa}$ be its subset of self adjoint elements. It is shown that ${\cal F}\sbsp{\rm II}{\rm sa}$ has three path components and two of them are contractible. The non trivial path component is denoted by ${\cal F}\sbsp{\rm II,*}{\rm sa}$. In this thesis the homotopy groups of ${\cal F}\sbsp{\rm II,*}{\rm sa}$ are calculated. In particular it is shown that the fundamental group of ${\cal F}\sbsp{\rm II,*}{\rm sa}$ is isomorphic to $\IR$ via a map called spectral flow. The choice of this notion is justified by the fact that the fundamental group of the Fredholm operators in an algebra of bounded operators on a separable complex Hilbert space, ${\cal L}({\cal H})$ is isomorphic to the integers, $\doubz$ via spectral flow ( (AS), (APS1-2), (BW)). It is also shown that the notion of essential codimension introduced in (BDF) can be generalized to type II$\sb\infty$ factors and to Hilbert C*-modules. We obtain a relation between spectral flow and essential codimension in a type II$\sb\infty$ factor generalizing a result from bounded operators case. We also generalize the notion of spectral flow to Hilbert C*-modules and extend the relation between spectral flow and essential codimension. We also study an application of spectral flow to index theory on infinite covers of compact manifolds.

Degree

Ph.D.

Advisors

Kaminker, Purdue University.

Subject Area

Mathematics

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