The notion of index for inclusions of von Neumann algebras goes back to a seminal work of Jones on subfactors of type II 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 finitedimensional centers. Our results generalize some aspects of the Jones index for multimatrix inclusions (finite direct sums of matrix algebras), e.g., the minimal index always equals the squared norm of a matrix, that we call 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. MotivationOne motivation for studying von Neumann algebras with non-trivial centers and inclusions, or better bimodules, between them comes from the theory of Quantum Information.In an operator-algebraic description of quantum systems, observables are described by the self-adjoint part of a non-commutative von Neumann algebra M (with separable predual), while states correspond to normal faithful positive functionals ϕ : M → C normalized such that ϕ(1) = 1, where 1 denotes the identity operator. Keep in mind as an example the most commonly studied case of finite quantum systems [OP93, Part I] where the algebra generated by the observables is finite-dimensional, thus a multi-matrix algebra. Namely, M ∼ = i=1,...,m M k i (C), where m, k i ∈ N and M k i (C) is the algebra of k i × k i matrices over C, realized on the finite dimensional Hilbert space C N , N = k 1 +. . .+k m . More generally, the center Z(M) = M ∩ M ′ is the classical part of the system, in the previous case Z(M) ∼ = C m , while each factor in the central decomposition of M, in the previous case M k i (C), is a purely *

We develop a semi-classical field method for the study of the weakly interacting Bose gas at finite temperature, which, contrarily to the usual classical field model, does not suffer from an ultraviolet cut-off dependence. We apply the method to the study of thermal vortices in spatially homogeneous, two-dimensional systems. We present numerical results for the vortex density and the vortex pair distribution function. Insight in the physics of the system is obtained by comparing the numerical results with the predictions of simple analytical models. In particular, we calculate the activation energy required to form a vortex pair at low temperature.

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