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Register a new userCategorical Non-commutative Geometry
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The purpose of this short note was to outline the current status, then in 2011, of some research programs aiming at a categorification of parts of A.Connes non-commutative geometry and to provide an outlook on some possible subsequent developments in categorical non-commutative geometry.
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After an introduction to some basic issues in non-commutative geometry (Gelfand duality, spectral triples), we present a panoramic view of the status of our current research program on the use of categorical methods in the setting of A.Connes non-commutative geometry: morphisms/categories of spectral triples, categorification of Gelfand duality. We conclude with a summary of the expected applications of categorical non-commutative geometry to structural questions in relativistic quantum physics: (hyper)covariance, quantum space-time, (algebraic) quantum gravity.
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.
In this thesis we give obstructions for Drinfeld twist deformation quantization on several classes of symplectic manifolds. Motivated from this quantization procedure, we further construct a noncommutative Cartan calculus on any braided commutative algebra, as well as an equivariant Levi-Civita covariant derivative for any non-degenerate equivariant metric. This generalizes and unifies the Cartan calculus on a smooth manifold and the Cartan calculus on twist star product algebras. We prove that the Drinfeld functor leads to equivalence classes in braided commutative geometry and commutes with submanifold algebra projection.
The notion of index for inclusions of von Neumann algebras goes back to a seminal work of Jones on subfactors of type ${I!I}_1$. In the absence of a trace, one can still define the index of a conditional expectation associated to a subfactor and look for expectations that minimize the index. This value is called the minimal index of the subfactor. We report on our analysis, contained in [GL19], of the minimal index for inclusions of arbitrary von Neumann algebras (not necessarily finite, nor factorial) with finite-dimensional centers. Our results generalize some aspects of the Jones index for multi-matrix inclusions (finite direct sums of matrix algebras), e.g., the minimal index always equals the squared norm of a matrix, that we call emph{matrix dimension}, as it is the case for multi-matrices with respect to the Bratteli inclusion matrix. We also mention how the theory of minimal index can be formulated in the purely algebraic context of rigid 2-$C^*$-categories.
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.
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