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In this article, a theorem is proved asserting that any linear functional defined on a JBW-algebra admits a Lebesque decomposition with respect to any normal state defined on the algebra. Then we show that the positivity (and the unicity) of this decomposition is insured for the trace states defined on the algebra. In fact, this property can be used to give a new characterization of the trace states amoungst all the normal states.
The Banach-Lie algebras ℌκ of all holomorphic infinitesimal isometries of the classical symmetric complex Banach manifolds of compact type (κ = 1) and non compact type (κ = −1) associated with a complex JB*-triple Z are considered and the Lie ideal structure of ℌκ is studied.
The Lukacs property of the free Poisson distribution is studied. We prove that if free and are free Poisson distributed with suitable parameters, then + and are free. As an auxiliary result we compute the joint cumulants of and for free Poisson distributed . We also study the Lukacs property of the free Gamma distribution.
In this article, two results regarding the Moore-Penrose inverse in the frame of C*-algebras are considered. In first place, a characterization of the so-called reverse order law is given, which provides a solution of a problem posed by M. Mbekhta. On the other hand, Moore-Penrose hermitian elements, that is C*-algebra elements which coincide with their Moore-Penrose inverse, are introduced and studied. In fact, these elements will be fully characterized both in the Hilbert space and in the C*-algebra...
The main result of the paper is that for a circular element c in a C*-probability space, is an R-diagonal pair in the sense of Nica and Speicher for every n = 1,2,... The coefficients of the R-series are found to be the generalized Catalan numbers of parameter n-1.
We review the notion of simple compact quantum groups and examples, and discuss the problem of construction and classification of simple compact quantum groups.
We define algebraic families of (all) morphisms which are purely algebraic analogs of quantum families of (all) maps introduced by P. M. Sołtan. Also, algebraic families of (all) isomorphisms are introduced. By using these notions we construct two classes of Hopf-algebras which may be interpreted as the quantum group of all maps from a finite space to a quantum group, and the quantum group of all automorphisms of a finite noncommutative (NC) space. As special cases three classes of NC objects are...
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.
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