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Register a new userMinimal index and dimension for 2-$C^*$-categories with finite-dimensional centers
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In the first part of this paper, we give a new look at inclusions of von Neumann algebras with finite-dimensional centers and finite Jones index. The minimal conditional expectation is characterized by means of a canonical state on the relative commutant, that we call the spherical state; the minimal index is neither additive nor multiplicative (it is submultiplicative), contrary to the subfactor case. So we introduce a matrix dimension with the good functorial properties: it is always additive and multiplicative. The minimal index turns out to be the square of the norm of the matrix dimension, as was known in the multi-matrix inclusion case. In the second part, we show how our results are valid in a purely 2-$C^*$-categorical context, in particular they can be formulated in the framework of Connes bimodules over von Neumann algebras.
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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.
Let $Vsubseteq A$ be a conformal inclusion of vertex operator algebras and let $mathcal{C}$ be a category of grading-restricted generalized $V$-modules that admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. We give conditions under which $mathcal{C}$ inherits semisimplicity from the category of grading-restricted generalized $A$-modules in $mathcal{C}$, and vice versa. The most important condition is that $A$ be a rigid $V$-module in $mathcal{C}$ with non-zero categorical dimension, that is, we assume the index of $V$ as a subalgebra of $A$ is finite and non-zero. As a consequence, we show that if $A$ is strongly rational, then $V$ is also strongly rational under the following conditions: $A$ contains $V$ as a $V$-module direct summand, $V$ is $C_2$-cofinite with a rigid tensor category of modules, and $A$ has non-zero categorical dimension as a $V$-module. These results are vertex operator algebra interpretations of theorems proved for general commutative algebras in braided tensor categories. We also generalize these results to the case that $A$ is a vertex operator superalgebra.
We construct a new class of finite-dimensional C^*-quantum groupoids at roots of unity q=e^{ipi/ell}, with limit the discrete dual of the classical SU(N) for large orders. The representation category of our groupoid turns out to be tensor equivalent to the well known quotient C^*-category of the category of tilting modules of the non-semisimple quantum group U_q({mathfrak sl}_N) of Drinfeld, Jimbo and Lusztig. As an algebra, the C^*-groupoid is a quotient of U_q({mathfrak sl}_N). As a coalgebra, it naturally reflects the categorical quotient construction. In particular, it is not coassociative, but satisfies axioms of the weak quasi-Hopf C^*-algebras: quasi-coassociativity and non-unitality of the coproduct. There are also a multiplicative counit, an antipode, and an R-matrix. For this, we give a general construction of quantum groupoids for complex simple Lie algebras {mathfrak g} eq E_8 and certain roots of unity. Our main tools here are Drinfelds coboundary associated to the R-matrix, related to the algebra involution, and certain canonical projections introduced by Wenzl, which yield the coproduct and Drinfelds associator in an explicit way. Tensorial properties of the negligible modules reflect in a rather special nature of the associator. We next reduce the proof of the categorical equivalence to the problems of establishing semisimplicity and computing dimension of the groupoid. In the case {mathfrak g}={mathfrak sl}_N we construct a (non-positive) Haar-type functional on an associative version of the dual groupoid satisfying key non-degeneracy properties. This enables us to complete the proof.
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.
Given a connected and locally compact Hausdorff space X with a good base K we assign, in a functorial way, a C(X)-algebra to any precosheaf of C*-algebras A defined over K. Afterwards we consider the representation theory and the Kasparov K-homology of A, and interpret them in terms, respectively, of the representation theory and the K-homology of the associated C(X)-algebra. When A is an observable net over the spacetime X in the sense of algebraic quantum field theory, this yields a geometric description of the recently discovered representations affected by the topology of X.
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