It is proved that the free product state, in the reduced free product of C*-algebras, is faithful if the initial states are faithful.
1998 Academic Press
The reduced free product of C*-algebras was introduced by Voiculescu as the appropriate construction for C*-algebras in the setting of his theory of freeness [1]. (See also the book [2]). Given a set I and for each @ # I a unital C*-algebra A @ with a state , @ whose GNS representation is faithful, the reduced free product construction yields the unique, unital C*-algebra A with unital embeddings A @ / ร A and a state , on A such that (i) ,| A @ =, @ , (ii) (A @ ) @ # I is free in (A, ,), (iii) A=C*( @ # I A @ ), (iv) the GNS representation of , is faithful on A.
We denote the reduced free product by
and , is called the free product state. In this note, we prove that if , @ is faithful on A @ for every @ # I then , is faithful on A.
Voiculescu [1] proved that the free product state in the analogous construction for von Neumann algebras is faithful under the hypothesis that each , @ is faithful. This implies that for the reduced free product of C*-algebras in (1), , is faithful if, for each @ # I, letting (? @ , H @ , ! @ ) denote the GNS construction for (A @ , , @ ), the vector ! @ is cyclic for the commutant, ? @ (A @