Summary: Non-commutative probability I:
Operator algebras background
November 21, 2002
Let (A, ) be a non-commutative probability space.
A =-algebra with identity,
(a + b) = ¯a + ¯b; (ab) = ba.
= linear functional A C, a state, i.e. positive
 = 1.
A is a C-algebra, has a norm · such that
ab a b , a = a ,
aa = a 2
Example . Let be a compact set, then A = C(, C)
is a C-algebra with complex conjugation f = ¯f and