Free independence of unital subalgebras 
in an algebra with a state 
means that
 |
whenever each 
has 
and consecutive indices differ. It is a notion of independence for noncommuting variables.
Variables are freely independent when their generated unital subalgebras have this
property. The condition controls alternating centered products, rather than requiring
commutativity. Free independence is used when semicircular
noise is added in the definition of free entropy.
Factors.