 
Summary: Free Processes via Matrix Theory
Concentration week on free Probability
Texas A&M, July 913.
Nizar Demni
Paris VI University
July 10, 2007
Nizar Demni Paris VI University Free Processes via Matrix Theory Concentration week on free Pr
Definitions
Non commutative probability space :
Unital Algebra A + linear functional : A C (state).
Additional requirements :
1 Involutive Banach algebras (A , ,  · ).
2 C algebras : aa  = a2
,
B(H) (H : Hilbert space), (a) =< a, >,  = 1 (pure
state) GNS representation.
3 W algebras: weakly closed subalgebra of B(H) A = A .
p1Lp(, F, P) Mm(C) : the space of random matrices of
all order moments with the normalized trace expectation:
(a) =
