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Limit Theorems for the Hierarchy of Freeness

100%

Franz U. , Lenczewski R.

Probability and Mathematical Statistics

1999

tom Vol. 19, Fasc. 1

23--41

The central limit theorem, the invariance principle and the Poisson limit theorem for the hierarchy of freeness are studied. We show that for given m ϵ N the limit laws can be expressed in terms of non-crossing partitions of depth smaller than or equal to m. For A = C[x], we solve the associated moment problems and find explicitly the discrete limit measures.

Idempotent States and the Inner Linearity Property

80%

Banica T. , Franz U. , Skalski A.

Bulletin of the Polish Academy of Sciences. Mathematics

2012

tom Vol. 60, no 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.