International Congress of Mathematicians

788 Liming GèThis independence gives a "tensor-product" relation among subalgebras Aj : ifA is generated by Aj, then A — ®jAj (in the case of C*- or W*-probability spaces,the tensor-product shall reflect the corresponding topological structures on A andAfi. _Distributions and moments: Given (A, T), for A in A, we define a map PA '•C[x] —¥ C by PA(P(X)) = r(p(Aj). Then PA is the distribution of A. For Ai,..., A nin A, the joint distribution pA 1: ...,A n '• C(#i,..., x n ) —¥ C is given byPA u ...,A n (p(xi,- • -,x n j) = r(p(Ai,.. .,A„fi).Ifp is a monomial, r(p(Ai,... ,A n j) is called a (p-)moment. When random variablesare non self-adjoint, one also considers (joint) * distributions of random variables,that can be defined in a similar way. In this case, there is a natural identificationof C(#i,... ,x n ,x*,... ,x* n ) with the semigroup algebra CS 2n , where S 2n is the freesemigroup on 2n generators. Alonomials are given by words in S 2n .Conditional Expectations: Suppose B is a subalgebra of A. A conditional expectationfrom A onto B is a B-bimodule map (a projection of norm one in the case of C*-algebras) of A onto B so that the restriction on B is the identity map.Alany other concepts in probability theory and measure theory can be generalizedto operator algebras, especially von Neumann algebras which can be regardedas non-commutative measure spaces. For basic operator algebra theory, we refer to[KR] and [T].2. GNS representation and von Neumann algebrasGiven a C*-probability space (A,T), one defines an inner product (A,B) =T(B*A) on A. Yet L 2 (A,T) be the Hilbert space obtained by the completion ofA under the L 2 -norm given by this inner product. Then A acts on L 2 (A,T) byleftmultiplication. This representation of A on the Hilbert space L 2 (A,T) is calledthe GNS representation. In a similar way, one can define L P (A,T), where ||A|| P =r(|A| p ) 1 / p = T((A*A) p / 2 ) 1 / p . The von Neumann algebra generated by A (or thestrong-operator closure of Â) is sometimes denoted by L°°(A, T) (C L P (A,T), p >1). All von Neumann algebras admit such a form. Any von Neumann algebra is a(possibly, continuous) direct sum of "simple" algebras, or factors (algebras with atrivial center). Von Neumann algebras that admit a faithful (finite) trace are saidto be finite. The classification of (infinite-dimensional) finite factors has becomethe central problem in von Neumann algebras.Alurray and von Neumann [A1N] also separate factors into three types:Type I: Factors contain a minimal projection. They are isomorphic to fullmatrix algebras M n (C) or B(7fi).Type II: Factors contain a "finite" projection but without minimal projections:it is said to be of type Hi when the identity J is a finite projection; of type J/QOwhen J is infinite. Every type IIQO is the tensor product of B(fii) with a factor oftype Hi.