64 research outputs found
Faithfulness of free product states
It is proved that the free product state, in the reduced free product of
C*-algebras, is faithful if the initial states are faithful
On the S-transform over a Banach algebra
The S-transform is shown to satisfy a specific twisted multiplicativity
property for free random variables in a B-valued Banach noncommutative
probability space, for an arbitrary unital complex Banach algebra B. Also, a
new proof of the additivity of the R-transform in this setting is given.Comment: 16 pages. The revised version includes a new proof of additivity of
the R-transform, as well as some minor correction
Projections in free product C*-algebras
Consider the reduced free product of C*-algebras,
(A,\phi)=(A_1,\phi_1)*(A_2,\phi_2), with respect to states \phi_1 and \phi_2
that are faithful. If \phi_1 and \phi_2 are traces, if the so-called Avitzour
conditions are satisfied, (i.e. A_1 and A_2 are not ``too small'' in a specific
sense) and if A_1 and A_2 are nuclear, then it is shown that the positive cone
of the K_0-group of A consists of those elements g in K_0(A) for which g=0 or
K_0(\phi)(g)>0. Thus, the ordered group K_0(A) is weakly unperforated.
If, on the other hand, \phi_1 or \phi_2 is not a trace and if a certain
condition weaker than the Avitzour conditions hold, then A is properly
infinite
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