We prove a kind of logarithmic Sobolev inequality claiming that the mutual free Fisher information dominates the microstate free entropy adapted to projections in the case of two projections.
A randomized q-central or q-commutative limit theorem on a family of bialgebras with one complex parameter is shown.
We introduce a notion of Morita equivalence for Hilbert C*-modules in terms of the Morita equivalence of the algebras of compact operators on Hilbert C*-modules. We investigate the properties of the new Morita equivalence. We apply our results to study continuous actions of locally compact groups on full Hilbert C*-modules. We also present an extension of Green's theorem in the context of Hilbert C*-modules.
The paper presents several combinatorial properties of the boolean cumulants. A consequence is a new proof of the multiplicative property of the boolean cumulant series that can be easily adapted to the case of boolean independence with amalgamation over an algebra.
Surjective isometries between unital C*-algebras were classified in 1951 by Kadison [K]. In 1972 Paterson and Sinclair [PS] handled the nonunital case by assuming Kadison’s theorem and supplying some supplementary lemmas. Here we combine an observation of Paterson and Sinclair with variations on the methods of Yeadon [Y] and the author [S1], producing a fundamentally new proof of the structure of surjective isometries between (nonunital) C*-algebras. In the final section we indicate how our techniques...
S. L. Woronowicz's theory of C*-algebras generated by unbounded elements is applied to q-normal operators satisfying the defining relation of the quantum complex plane. The unique non-degenerate C*-algebra of bounded operators generated by a q-normal operator is computed and an abstract description is given by using crossed product algebras. If the spectrum of the modulus of the q-normal operator is the positive half line, this C*-algebra will be considered as the algebra of continuous functions...
It is shown that every monocompact submeasure on an orthomodular poset is order continuous. From this generalization of the classical Marczewski Theorem, several results of commutative Measure Theory are derived and unified.
Let ℳ be a type II₁ von Neumann algebra, τ a trace in ℳ, and L²(ℳ,τ) the GNS Hilbert space of τ. If L²(ℳ,τ)₊ is the completion of the set of selfadjoint elements, then each element ξ ∈ L²(ℳ,τ)₊ gives rise to a selfadjoint unbounded operator on L²(ℳ,τ). In this note we show that the exponential exp: L²(ℳ,τ)₊ → L²(ℳ,τ), , is continuous but not differentiable. The same holds for the Cayley transform . We also show that the unitary group with the strong operator topology is not an embedded submanifold...
We show that for the t-deformed semicircle measure, where 1/2 < t ≤ 1, the expansions of functions with respect to the associated orthonormal polynomials converge in norm when 3/2 < p < 3 and do not converge when 1 ≤ p < 3/2 or 3 < p. From this we conclude that natural expansions in the non-commutative spaces of free group factors and of free commutation relations do not converge for 1 ≤ p < 3/2 or 3 < p.
A generalisation of quantum principal bundles in which a quantum structure group is replaced by a coalgebra is proposed.