Summary: Free convolution
Michael Anshelevich
March 5, 2004
µ, probability measures on R. Their convolution
µ (S) =
R
µ(S - x) d(x).
Symmetrically,
R
f(z) d(µ )(z) =
R×R
f(x + y) dµ(x) d(y).
Other definitions below.
EQUIVALENT DEFINITIONS OF FREE CONVOLUTION
1) Let 1, 2 be discrete groups (1 = 2 = Z works).
1 1 = their free product.
L(1), L(2), L(1 2) = group von Neumann alge-
bras.
All with distinguished (tracial) states.
1