The Bi-free Extension of Free Probability
Free probability is a noncommutative probability theory adapted to variables with the highest degree of noncommutativity. The theory has connections with random matrices, combinatorics, and operator algebras. Recently, we realized that the theory has an e
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Abstract Free probability is a noncommutative probability theory adapted to variables with the highest degree of noncommutativity. The theory has connections with random matrices, combinatorics, and operator algebras. Recently, we realized that the theory has an extension to systems with left and right variables, based on a notion of bi-freeness. We provide a look at the development of this new direction. The paper is an expanded version of the plenary lecture at the 10th ISAAC Congress in Macau. Keywords Two-faced pair · Bi-free probability · Bi-free convolution 2000 Mathematics Subject Classification. Primary: 46L54 · Secondary: 46L53
1 Introduction Free probability is now in its early thirties. After such a long time I became aware that the theory has a natural extension to a theory with two kinds of variables: left and right. This is not the same as passing from a theory of modules to a theory of bimodules, since our left and right variables will not commute in general, a noncommutation which will appear already when we shall take a look at what the Gaussian variables of the theory are. We call the theory with left and right variables bi-free probability and the independence relation that underlies it is called bi-freeness. This new type of independence does not contradict the theorems of Muraki [15] and Speicher [20] about the possible types of independence in noncommutative probability with all the nice properties (“classical”, free, Boolean and if we give up symmetry also monotonic and antimonotonic), the reason being that we play a new game here, by replacing the usual sets of random variables by sets with two types of variables. D.-V. Voiculescu (B) Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720-3840, USA e-mail: [email protected] © Springer International Publishing Switzerland 2016 T. Qian and L.G. Rodino (eds.), Mathematical Analysis, Probability and Applications – Plenary Lectures, Springer Proceedings in Mathematics & Statistics 177, DOI 10.1007/978-3-319-41945-9_8
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The observation about the possibility of left and right variables could have been made at the very beginning of free probability. At present it becomes necessary to look back and think about the problems which would have appeared earlier had we been aware of the possibility. Several advances on this road have been made. Developments are happening faster since the lines along which free probability developed are serving often as a guide.
2 Free Probability Background Free probability is a noncommutative probability framework adapted for variables with the highest degree of noncommutativity. By “highest” noncommutativity we mean the kind of noncommutativity one encounters, for instance, in free groups, free semigroups or in the creation and destruction operators on a full Fock space. At this heuristic level, Bosonic and Fermionic creation and destruction operators are “less noncommutative” because the commutation or anticommutation relations they satisfy represent restric
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