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C*-categories are essentially norm-closed *-categories of bounded linear operators between Hilbert spaces. The purpose of this work is to identify suitable axioms defining Krein C*-categories, i.e. those categories that play the role of C*-categories whenever Hilbert spaces are replaced by more general indefinite inner product Krein spaces, and provide some basic examples. Finally we provide a Gelfand-Naimark representation theorem for Krein C*-algebras and Krein C*-categories.
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We discuss a number of general constructions concerning additive $ C^* $-categories, focussing in particular on establishing the existence of bicolimits. As an illustration of our results we show that balanced tensor products of module categories over $ C^* $-tensor categories exist without any finiteness assumptions.
We provide definitions for strict involutive higher categories (a vertical categorification of dagger categories), strict higher C*-categories and higher Fell bundles (over arbitrary involutive higher topological categories). We put forward a proposal for a relaxed form of the exchange property for higher (C*)-categories that avoids the Eckmann-Hilton collapse and hence allows the construction of explicit non-trivial non-commutative examples arising from the study of hypermatrices and hyper-C*-algebras, here defined. Alternatives to the usual globular and cubical settings for strict higher categories are also explored. Applications of these non-commutative higher C*-categories are envisaged in the study of morphisms in non-commutative geometry and in the algebraic formulation of relational quantum theory.
We introduce a notion of Krein C*-module over a C*-algebra and more generally over a Krein C*-algebra. Some properties of Krein C*-modules and their categories are investigated.
A Banach involutive algebra is called a Krein C*-algebra if there is a fundamental symmetry (an involutive automorphism of period 2) such that the C*-property is satisfied when the original involution is replaced with the new one obtained by composing the automorphism with the old involution. For a given fundamental symmetry, a Krein C*-algebra decomposes as a direct sum of an even part (a C*-algebra) and an odd part (a Hilbert C*-bimodule on the even part). Our goal here is to develop a spectral theory for commutative unital Krein C*-algebras when the odd part is a symmetric imprimitivity C*-bimodule over the even part and there exists an additional suitable exchange symmetry between the odd and even parts.
A bicommutant category is a higher categorical analog of a von Neumann algebra. We study the bicommutant categories which arise as the commutant $mathcal{C}$ of a fully faithful representation $mathcal{C}tooperatorname{Bim}(R)$ of a unitary fusion category $mathcal{C}$. Using results of Izumi, Popa, and Tomatsu about existence and uniqueness of representations of unitary (multi)fusion categories, we prove that if $mathcal{C}$ and $mathcal{D}$ are Morita equivalent unitary fusion categories, then their commutant categories $mathcal{C}$ and $mathcal{D}$ are equivalent as bicommutant categories. In particular, they are equivalent as tensor categories: [ Big(,,mathcal{C},,simeq_{text{Morita}},,mathcal{D},,Big) qquadLongrightarrowqquad Big(,,mathcal{C},,simeq_{text{tensor}},,mathcal{D},,Big). ] This categorifies the well-known result according to which the commutants (in some representations) of Morita equivalent finite dimensional $rm C^*$-algebras are isomorphic von Neumann algebras, provided the representations are `big enough. We also introduce a notion of positivity for bi-involutive tensor categories. For dagger categories, positivity is a property (the property of being a $rm C^*$-category). But for bi-involutive tensor categories, positivity is extra structure. We show that unitary fusion categories and $operatorname{Bim}(R)$ admit distinguished positive structures, and that fully faithful representations $mathcal{C}tooperatorname{Bim}(R)$ automatically respect these positive structures.
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