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On two possible constructions of the quantum semigroup of all quantum permutations of an infinite countable set

100%

Goswami D. , Skalski A.

nr 1

199-214

Two different models for a Hopf-von Neumann algebra of bounded functions on the quantum semigroup of all (quantum) permutations of infinitely many elements are proposed, one based on projective limits of enveloping von Neumann algebras related to finite quantum permutation groups, and the second on a universal property with respect to infinite magic unitaries.

On ergodic properties of convolution operators associated with compact quantum groups

100%

Franz U. , Skalski A.

Colloquium Mathematicum

2008

tom 113

nr 1

13-23

Recent results of M. Junge and Q. Xu on the ergodic properties of the averages of kernels in noncommutative $L^{p}$-spaces are applied to the analysis of almost uniform convergence of operators induced by convolutions on compact quantum groups.

Idempotent States and the Inner Linearity Property

80%

Banica T. , Franz U. , Skalski A.

tom 60

nr 2

123-132

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.