Formal Haagerup standard form on infinite index morphisms of factors

Abstract: The Haagerup $L^2$-space construction, introduced by Haagerup in the 70's, associates a standard form to every von Neumann algebra, without any reference to weights, and is thus regarded as a coordinate free version of the GNS construction. The Haagerup standard form and the Connes fusion tensor product organize von Neumann algebras and their representations into a bicategory. The main interest on this bicategory comes from the fact that it encodes weak Morita equivalence as a formal homotopy relation.

Bicategories are a specific type of categorical structure of second order, corresponding to globular sets. The second order categorical structures corresponding to cubical sets are double categories. Results studying relations between cubical and globular categories have been obtained continually since the 60's, mainly in nonabelian homotopy theory, but more recently in areas ranging from algebraic geometry to network theory. I will explain results of this type regarding the existence of two non-equivalent double categories of representations of factors, and how these structures relate to questions of functoriality of the Haagerup standard form and the Connes fusion tensor product.

This program builds on work of Bartels, Douglas and Hénriques on the theory of coordinate free conformal nets and their relation to the Stolz-Teichner program.

functional analysisgroup theoryoperator algebras

Audience: researchers in the topic

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