We present a fairly general construction of unbounded representatives for the
interior Kasparov product. As a main tool we develop a theory of
C^1-connections on operator * modules; we do not require any smoothness
assumptions; our sigma-unitality assumptions are minimal. Furthermore, we use
work of Kucerovsky and our recent Local Global Principle for regular operators
in Hilbert C*-modules.
As an application we show that the Spectral Flow Theorem and more generally
the index theory of Dirac-Schr\"odinger operators can be nicely explained in
terms of the interior Kasparov product.Comment: 40 page
Abstract. A self Morita equivalence over an algebra B, given by a B-bimodule E, is thought of as a line bundle over B. The corresponding Pimsner algebra O E is then the total space algebra of a noncommutative principal circle bundle over B. A natural Gysin-like sequence relates the KK-theories of O E and of B. Interesting examples come from O E a quantum lens space over B a quantum weighted projective line (with arbitrary weights). The KK-theory of these spaces is explicitly computed and natural generators are exhibited.
We present a definition of spectral flow for any norm closed ideal J in any von Neumann algebra N . Given a path of selfadjoint operators in N which are invertible in N=J , the spectral flow produces a class in K 0 .J /.Given a semifinite spectral triple .A;H;D/ relative to .N; / with A separable, we construct a class OED 2 KK 1 .A;K.N //. For a unitary u 2 A, the von Neumann spectral flow between D and u Du is equal to the Kasparov product OEu Ő A OED, and is simply related to the numerical spectral flow, and a refined C -spectral flow.
We study the spectral metric aspects of the standard Podleś sphere, which is a homogeneous space for quantum SU (2). The point of departure is the real equivariant spectral triple investigated by Da ֒ browski and Sitarz. The Dirac operator of this spectral triple interprets the standard Podleś sphere as a 0-dimensional space and is therefore not isospectral to the Dirac operator on the 2-sphere. We show that the seminorm coming from commutators with this Dirac operator provides the Podleś sphere with the structure of a compact quantum metric space in the sense of Rieffel.2010 Mathematics Subject Classification. 58B32,58B34;46L89,46L30.
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