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Commutators associated to a subfactor and its relative commutants

Hsiang-Ping Huang (2002)

Annales de l’institut Fourier

Let N M be an inclusion of I I 1 factors with finite Jones index. Then M = ( N ' M ) [ N , M ] as a vector space. Here [ N , M ] denotes the vector space spanned by the commutators of the form [ a , b ] where a N , b M .

Invariant subspaces for operators in a general II1-factor

Uffe Haagerup, Hanne Schultz (2009)

Publications Mathématiques de l'IHÉS

Let ℳ be a von Neumann factor of type II1 with a normalized trace τ. In 1983 L. G. Brown showed that to every operator T∈ℳ one can in a natural way associate a spectral distribution measure μ T (now called the Brown measure of T), which is a probability measure in ℂ with support in the spectrum σ(T) of T. In this paper it is shown that for every T∈ℳ and every Borel set B in ℂ, there is a unique closed T-invariant subspace 𝒦 = 𝒦 T ( B ) affiliated with ℳ, such that the Brown measure of T | 𝒦 is concentrated on B...

Quantum permutation groups: a survey

Teodor Banica, Julien Bichon, Benoît Collins (2007)

Banach Center Publications

This is a presentation of recent work on quantum permutation groups. Contains: a short introduction to operator algebras and Hopf algebras; quantum permutation groups, and their basic properties; diagrams, integration formulae, asymptotic laws, matrix models; the hyperoctahedral quantum group, free wreath products, quantum automorphism groups of finite graphs, graphs having no quantum symmetry; complex Hadamard matrices, cocycle twists of the symmetric group, quantum groups acting on 4 points; remarks...

Truncation and Duality Results for Hopf Image Algebras

Teodor Banica (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

Associated to an Hadamard matrix H M N ( ) is the spectral measure μ ∈ [0,N] of the corresponding Hopf image algebra, A = C(G) with G S N . We study a certain family of discrete measures μ r [ 0 , N ] , coming from the idempotent state theory of G, which converge in Cesàro limit to μ. Our main result is a duality formula of type 0 N ( x / N ) p d μ r ( x ) = 0 N ( x / N ) r d ν p ( x ) , where μ r , ν r are the truncations of the spectral measures μ,ν associated to H , H t . We also prove, using these truncations μ r , ν r , that for any deformed Fourier matrix H = F M Q F N we have μ = ν.

Unitaires multiplicatifs en dimension finie et leurs sous-objets

Saad Baaj, Étienne Blanchard, Georges Skandalis (1999)

Annales de l'institut Fourier

On appelle pré-sous-groupe d’un unitaire multiplicatif V agissant sur un espace hilbertien de dimension finie une droite vectorielle L de telle que V ( L L ) = L L . Nous montrons que les pré-sous-groupes sont en nombre fini, donnons un équivalent du théorème de Lagrange et généralisons à ce cadre la construction du “bi-produit croisé”. De plus, nous établissons des bijections entre pré-sous-groupes et sous-algèbres coïdéales de l’algèbre de Hopf associée à V , et donc, d’après Izumi, Longo, Popa, avec les...

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