International Congress of Mathematicians

768 P. BianeObserve thatr(ai ... a n ) = R^(cti,..., a n ) + terms involving R^ for fc < nso that the R^ are well defined by (3.1) and can be computed by induction on n.They are called the noncrossing (or sometimes free) cumulant functionals on A.The formula (3.1) can be inverted to yieldR (n) (ai,...,a n )= Y Moeb([n,l n ])T[n](ai,...,a n ).TreNC(n)Here T[TT] (cti,..., a n ) = Yl V€n T~(aj x ... aj k ) where V = {ji,..., jk} are the classesof IT, and Moeb is the Möbius function of the lattice NC(n), see [20].For example, one hasfiW(ai) = r(ai); R^(ai,a 2 ) = r(aia 2 ) - r(ai)r(a 2 );R^(ai,a 2 ,a 3 ) = r(aia 2 a 3 ) - r(ai)r(a 2 az) - r(a 2 )r(aiaz)-r(az)r(aia 2 ) + 2r(ai)T(a 2 )r(az).Note that when the lattice of all partitions is used instead of noncrossing partitions,then one gets the usual family of cumulants (see Rota [16]), with another Alöbiusfunction.The connection between noncrossing cumulants and freeness is the followingresult from section 4 of [19].Theorem 2. Let {Afi £ 1} be a free family of subalgebras of (A,T), andai,...,a n £ A be such that aj belongs to some A tj for each j £ {1,2,..., n}. Thenone has R^ (ai,..., a n ) = 0 if there exists some j and k with ij ^ ik •This result leads to an explicit expression for r(ai.. .a n ), where ai,...,a nis an arbitrary sequence in A, such that each aj belongs to one of the algebrasAfi £ I. By Theorem 2, in the right hand side of (3.1), the terms correspondingto partitions n having a class containing two elements j, k such that aj and a^belong to distinct algebras give a zero contribution. Thus we have to sum overpartitions in which all j's belonging to a certain block of the partition are suchthat aj belongs to the same algebra. Since we can express noncrossing cumulantsin terms of moments we get the formula for r(ai ... a n ) in terms of the restrictionsof r to each of the subalgebras A t . Noncrossing cumulants are a powerful toolfor making computations in free probability, see [11], [12], [13], [14], [18], for someapplications. We give a simple illustration below.Let Xi and X 2 be two self-adjoint elements which are free, then the distributionof Xi + X 2 , depends only on the distributions of Xi and X 2 and can becomputed as follows. Let R^(Xi,... ,Xi) and R^(X 2 ,... ,X 2 ), for n > 1, be thenoncrossing cumulants of Xi and X 2 , then one can expand R^ (Xi + X 2 ,...,Xi +X 2 ) by multilinearity as ^. • R^ (X tl ,..., X in ) where the sum is over all sequencesof 1 and 2. By Theorem 2, all terms vanish except R^nfiXi,...,Xi) andRW (X 2 ,..., X 2 ). It follows thatR (n) (Xi + X 2 ,... ,Xi + X 2 ) = R (n) (Xi,.. .,Xi) + R (n) (X2,... ,X 2 )