On the problem of classifying simple compact quantum groups
We review the notion of simple compact quantum groups and examples, and discuss the problem of construction and classification of simple compact quantum groups.
On the relationship between the reversibility of dynamics and balance conditions
On the structural theory of factors of negatively curved groups
Ozawa showed in [21] that for any i.c.c. hyperbolic group, the associated group factor is solid. Developing a new approach that combines some methods of Peterson [29], Ozawa and Popa [27, 28], and Ozawa [25], we strengthen this result by showing that is strongly solid. Using our methods in cooperation with a cocycle superrigidity result of Ioana [12], we show that profinite actions of lattices in , , are virtually -superrigid.
On the transient and recurrent parts of a quantum Markov semigroup
We define the transient and recurrent parts of a quantum Markov semigroup 𝓣 on a von Neumann algebra 𝓐 and we show that, when 𝓐 is σ-finite, we can write 𝓣 as the sum of such semigroups. Moreover, if 𝓣 is the countable direct sum of irreducible semigroups each with a unique faithful normal invariant state ρₙ, we find conditions under which any normal invariant state is a convex combination of ρₙ's.
On two quantum versions of the detailed balance condition
Quantum detailed balance conditions are often formulated as relationships between the generator of a quantum Markov semigroup and the generator of a dual semigroup with respect to a certain scalar product defined by an invariant state. In this paper we survey some results describing the structure of norm continuous quantum Markov semigroups on ℬ(h) satisfying a quantum detailed balance condition when the duality is defined by means of pre-scalar products on ℬ(h) of the form (s ∈ [0,1]) in order...
Operator algebras associated with polymorphisms.
Order Structures, Traces and Weights on Morita Equivalent C*-Algebras.
Orthogonal harmonic analysis of fractal measures.
Permanence properties of -exact groups.
Permanence properties of the Baum-Connes conjecture.
Pointwise Convergence Theorems in L2 over a von Neumann Algebra.
Poisson boundaries of discrete quantum groups
This is a survey article about a theory of a Poisson boundary associated with a discrete quantum group. The main problem of the theory, that is, the identification problem is explained and solved for some examples.
Positive C...-semigroups on C*-algebras.
Positive energy representations for quantum spin models in 1+1 dimensions
We present recent results on positive energy representations of quantum spin models.
Powers's Property and Simple C*-Algebras.
Principe d'Oka, K-théorie et systèmes dynamiques non commutatifs.
Purely infinite -algebras arising from dynamical systems
Purely infinite C*-algebras from boundary actions of discrete groups.
Quantum detailed balance conditions with time reversal: the finite-dimensional case
We classify generators of quantum Markov semigroups on (h), with h finite-dimensional and with a faithful normal invariant state ρ satisfying the standard quantum detailed balance condition with an anti-unitary time reversal θ commuting with ρ, namely for all x,y ∈ and t ≥ 0. Our results also show that it is possible to find a standard form for the operators in the Lindblad representation of the generators extending the standard form of generators of quantum Markov semigroups satisfying the usual...
Quantum dynamical entropy revisited
We define a new quantum dynamical entropy for a C*-algebra automorphism with an invariant state (and for an appropriate 'approximating' subalgebra), which entropy is a 'hybrid' of the two alternative definitions by Connes, Narnhofer and Thirring resp. by Alicki and Fannes (and earlier, Lindblad). We report on this entropy's properties and on three examples.