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Operator algebra
About: Operator algebra is a research topic. Over the lifetime, 5783 publications have been published within this topic receiving 165303 citations.
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61 citations
TL;DR: In this paper, an analytic approach to spectral flow along paths of self-adjoint Breuer-Fredholm operators in a type $I$ or $II_\infty$ von Neumann algebra is given.
Abstract: We give a comprehensive account of an analytic approach to spectral flow along paths of self-adjoint Breuer-Fredholm operators in a type $I_{\infty}$ or $II_\infty$ von Neumann algebra ${\mathcal N}$ The framework is that of {\it odd unbounded} $\theta$-{\it summable} {\it Breuer-Fredholm modules} for a unital Banach *-algebra, $\mathcal A$ In the type $II_{\infty}$ case spectral flow is real-valued, has no topological definition as an intersection number and our formulae encompass all that is known We borrow Ezra Getzler's idea (suggested by I M Singer) of considering spectral flow (and eta invariants) as the integral of a closed one-form on an affine space Applications in both the type I and type II cases include a general formula for the relative index of two projections, representing truncated eta functions as integrals of one forms and expressing spectral flow in terms of the JLO cocycle to give the pairing of the $K$-homology and $K$-theory of $\mathcal A$
61 citations
07 Feb 2006
TL;DR: A soundness and finiteness theorem is proved for this interpretation of the multiplicative and exponential fragment of linear logic (MELL) and it is shown that Girard's original approach to GoI 1 via operator algebras is exactly captured in this categorical framework.
Abstract: We consider the multiplicative and exponential fragment of linear logic (MELL) and give a geometry of interaction (GoI) semantics for it based on unique decomposition categories. We prove a soundness and finiteness theorem for this interpretation. We show that Girard's original approach to GoI 1 via operator algebras is exactly captured in this categorical framework.
61 citations
TL;DR: The Shilov idempotent theorem can be viewed as a characterization of the zero-dimensional Cech cohomology group H(AA, Z) as discussed by the authors, which implies immediately that N(AA, Z) is isomorphic to the additive subgroup of a commutative Banach algebra with identity.
61 citations
TL;DR: In this article, a pseudo-multiplicative unitary construction of Hopf bimodules is proposed, which leads to two Hopf structures dual to each other, and an action of one of these structures on the algebra M 0 ⊂M 1 such that M 0 is the fixed point subalgebra.
61 citations