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How Non-Crossing Partitions Occur in Free Probability Theory

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How Non-Crossing Partitions Occur in Free Probability Theory How non-crossing partitions occur in free probability theory Alexandru Nica (talk at A.I.M. on January 10, 2005) This is one of the survey talks at the workshop on \Braid groups, clusters, and free probability" at the American Institute of Mathematics, January 10-14, 2005. The goal of the talk is to give a quick and elementary introduction to some combinatorial aspects of free probability, addressed to an audience who is not familiar with free probability but is acquainted to lattices of non-crossing partitions. The (attempted) line of approach is to select a cross-section of relevant material, and increase its rate of absorption by working through a set of exercises (the exercises are embedded in the lecture, and are straightforward applications of the material presented). While this is hopefully an efficient approach, it is quite far from being comprehensive; for a more detailed treatment of the subject, one can check the references [3, 5, 8, 9] listed at the end of the lecture. (Among these references, the one which is closest to the spirit of the present lecture is [3] { but, of course, there is a lot of material from [3] that could not be even mentioned here.) The plan of the lecture is as follows. The first section reviews a minimal amount of general terminology, and then the Sections 2 and 3 present some basic tools used in the combinatorics of free probability { the non-crossing cumulants and the R-transform. The main Section 4 is devoted to explaining how convolution in lattices of non-crossing partitions is intimately related to the multiplication of free random variables. This connection can in principle be used both ways, but (to my knowledge) its applications up to present have been focused on the direction of doing computations with free random variables via the combinatorics of NC(n). We give some illustrative examples of how this is done, and we explain how the S-transform of Voiculescu (a device which is also used in connection to the multiplication of free random variables) fits in this picture. 1. Basic free probabilistic terminology The framework used in free probability is the one of a non-commutative probability space. We will work with a version of this concept which goes under the name of \ -probability space". ∗ 1.1 Definition. By a -probability space we will understand a system ( ;') where: is a unital algebra∗ over C. A •A is a endowed with a -operation (an antilinear map a a such that •A ∗ A 3 7! ∗ 2 A (a∗)∗ = a and such that (ab)∗ = b∗a∗, for a; b ). ' : C is a linear functional which is2normalized A by the condition that '(1 ) = 1 • A! A (where 1 is the unit of ), and is strictly positive definite in the sense that '(a∗a) 0 for A A ≥ 1 all a , with equality holding if and only if a = 0. 2 A 1.2 Remarks (and disclaimers). (a) A more suggestive name for an ( ;') as above would probably be \generalized algebra of integrable functions" { the elementsA of are viewed as some sort of integrable functions, and applying ' to an a is viewedA like an operation of taking an integral. This comment is just made to explain2 theA terminology, we will not encounter the functional analysis side of free probability in the present lecture. (b) There are various alternative “flavours" for the definition of a non-commutative prob- ability space. For instance one sometimes requires that ' is a trace { i.e. that '(ab) = '(ba) for all a; b . On the other hand one sometimes drops the non-degeneracy assumption 2 A that '(a∗a) = 0 implies a = 0, or renounces to the -operation and to the positivity of ' altogether (but in this lecture we will stick to the definition∗ as given above). (c) Another useful property of ' which could have been mentioned in Definition 1.1 is that '(a∗) = '(a); a : (1.1) 8 2 A The reason for not mentioning (1.1) explicitly in the definition is that it follows from the positivity of '. Indeed, it is easily seen that (1.1) is equivalent to the fact that '(a) R 2 whenever a is such that a = a∗, and the latter fact follows from the positivity by writing 2 A 1 a = (a + 1 )2 (a 1 )2 : 4h A − − A i 1.3 Examples. (a) One can take to be the algebra of polynomials C[t], and then let ' : C A1 A! be defined by '(f) = 0 f(t) dt, for f . This is an example where the elements of really are functions, andR applying ' really2 A amounts to taking an integral. (Of course, inA this example one could replace C[t] by some larger algebra of integrable functions on [0; 1].) (b) One can take to be the algebra n(C) of n n matrices, and then let ' : C be the normalized trace,A M × A! n 1 1 n '(A) = Tr(A) = αi;i; for A = [αi;j] . n n X i;j=1 2 A i=1 (c) Let G be a group (not necessarily finite). One can take to be the associated group algebra, A = C[G] = span χg g G ; A f j 2 g where the elements of the linear basis χg g G for are multiplied according to the f j 2 g A rule χ χ = χ , and the -operation is determined by the condition χ = χ 1 , g G. g h gh g∗ g− Then one· can take ' : ∗C to be the so-called canonical trace on C[G], i.e. to be2 the linear functional determinedA! by the fact that 1; if a = e, the unit of G, '(χ ) = g 0; otherwise. 2 1.4 Remark (how to think of free independence). Let ( ;') be a -probability space, A ∗ and let 1;:::; s be unital -subalgebras of (where by \ -subalgebra" we understand a subalgebraA of Awhich is also∗ closed under theA -operation).∗ Consider the algebra (equiva- A ∗ lently -algebra) generated by 1;:::; s together, ∗ A A := Alg( 1 s) : U A [···[A ⊆ A Then is spanned linearly by products of elements from 1 s; since ' is not assumed U A [· · ·[A to be (and typically is not) multiplicative, the knowledge of the restrictions ' 1;:::;' s isn't usually sufficient for determining what is ' . Well, free independence is ajA recipe whichjA jU is allowing us to compute ' { if we know ' 1;:::;' s, and we know that 1;:::; s are freely independent. jU jA jA A A So once again, for the purpose of this lecture, the free independence of 1;:::; s is a recipe which allows us to compute any value A A '(a1a2 an); with n 1 and a1 i ; : : : ; an i (and where 1 i1; : : : in s), ··· ≥ 2 A 1 2 A n ≤ ≤ in terms of the restrictions ' 1;:::;' s. The recipe is usually stated in a special case, to which the computations canjA always bejA reduced. More precisely, we have: 1.5 Definition. Let ( ;') be a -probability space, and let 1;:::; s be unital A ∗ A A -subalgebras of . We say that 1;:::; s are freely independent if the following implication∗ holds:A A A 1 i1; : : : ; in s ≤ ≤ 9 such that i1 = i2; i2 = i3; : : : ; in 1 = in 6 6 − 6 > > '(a1 an) = 0: (1.2) = ) ··· a1 i1 ; : : : ; an in 2 A 2 A > such that '(a1) = = '(an) = 0 > ··· ;> In order to see how the Equation (1.2) can be bootstrapped to a recipe which computes any value '(a1 an), it is probably best to work out explicitly a few monomials of small length. ··· Exercise 1. Let ( ;') be a -probability space, and let ; be two unital -subalgebras of which are freelyA independent.∗ Verify that: B C ∗ A(a) '(bc) = '(b) '(c), b , c . · 8 2 B 2 C (b) '(b1cb2) = '(b1b2) '(c), b1; b2 , c . · 8 2 B 2 C (c) '(b1c1b2c2) = '(b1b2) '(c1) '(c2) + '(b1) '(b2) '(c1c2) · · · · '(b1) '(b2) '(c1) '(c2); b1; b2 , c1; c2 : − · · · 8 2 B 2 C [Comment: A less tedious derivation for the expression in the Exercise 1(c) appears in Section 2, Exercise 6.] 3 For practicing the concept of free independence, here are a couple more exercises. Exercise 2. Let ( ;') be a -probability space. (a) Verify that theA unital -subalgebra∗ C1 of is freely independent from any unital -subalgebra . ∗ A A ∗ (b) Let Bbe ⊆ a A unital -subalgebra of , such that is freely independent from itself. Prove that B = C1 . ∗ A B (c) Let B and Abe unital -subalgebras of such that is freely independent from , and such thatB commutesC with∗ (i.e. bc = cbA, for every bB and c ). Prove that atC least one of theB algebras , is equalC to C1 . 2 B 2 C B C A [Comments: The non-degeneracy of ' is important for the parts (b) and (c) of this exercise { indeed, when we aim to show an equality x = y in it may be more convenient to do that by checking that '((x y) (x y) ) = 0. Note thatA the Exercise 2(b) implies in particular − ∗ − that if the unital -subalgebras 1;:::; s of are freely independent, then i j = C1 for every 1 i <∗ j s.] A A A A \A A ≤ ≤ Exercise 3. Let ( ;') be a -probability space, and let 1;:::; s be unital - subalgebras of which areA freely independent.∗ Let be the subalgebraA ofA generated by∗ A U A 1;:::; s together. Prove that if ' i is a trace for every 1 i s, then ' is a trace asA well. A jA ≤ ≤ jU 2. Using non-crossing cumulants to describe free independence 2.1 Remark (how to think of non-crossing cumulants).
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