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Truncation and Duality Results for Hopf Image Algebras

100%
Banica T.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2014
|
tom 62
|
nr 2
161-179
EN
Associated to an Hadamard matrix $H ∈ M_N(ℂ)$ is the spectral measure μ ∈ 𝓟[0,N] of the corresponding Hopf image algebra, A = C(G) with $G ⊂ S⁺_N$. We study a certain family of discrete measures $μ^r ∈ 𝓟[0,N]$, coming from the idempotent state theory of G, which converge in Cesàro limit to μ. Our main result is a duality formula of type $∫_0^N (x/N)^p dμ^r(x) = ∫_0^N (x/N)^r dν^p(x)$, 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.
2
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Quantum permutations, Hadamard matrices, and the search for matrix models

100%
Banica T.
Banach Center Publications
|
2012
|
tom 98
|
nr 1
11-42
EN
This is a presentation of recent work on quantum permutation groups, complex Hadamard matrices, and the connections between them. A long list of problems is included. We include as well some conjectural statements about matrix models.
3
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Idempotent States and the Inner Linearity Property

51%
Banica T., Franz U., Skalski A.
Bulletin of the Polish Academy of Sciences. Mathematics
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2012
|
tom 60
|
nr 2
123-132
EN
We find an analytic formulation of the notion of Hopf image, in terms of the associated idempotent state. More precisely, if π:A → Mₙ(ℂ) is a finite-dimensional representation of a Hopf C*-algebra, we prove that the idempotent state associated to its Hopf image A' must be the convolution Cesàro limit of the linear functional φ = tr ∘ π. We then discuss some consequences of this result, notably to inner linearity questions.
4
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Quantum permutation groups: a survey

51%
Banica T., Bichon J., Collins B.
Banach Center Publications
|
2007
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tom 78
|
nr 1
13-34
EN
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 and comments.
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