1996/12/10 by Nica, Alexandru, Speicher, Roland · 1 citation
#FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.funct-an/9612001
Let A be a unital C^*-algebra, given together with a specified state ϕ:A → C. Consider two selfadjoint elements a,b of A, which are free with respect to ϕ (in the sense of the free probability theory of Voiculescu). Let us denote c:=i(ab-ba), where the i in front of the commutator is introduced to make c selfadjoint. In this paper we show how the spectral distribution of c can be calculated from the spectral distributions of a and b. Some properties of the corresponding operation on probability measures are also discussed. The methods we use are combinatorial, based on the description of freeness in terms of non-crossing partitions; an important ingredient is the notion of R-diagonal pair, introduced and studied in our previous paper funct-an/9604012.