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 states one started with are faithful. 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 ι ∈ I a unital C–algebra Aι with a state φι whose GNS representation is faithful, the reduced free product construction yields the unique, unital C–algebra A with unital embeddings Aι →֒ A and a state φ on A such that (i) φ|Aι = φι, (ii) (Aι)ι∈I is free in (A,φ), (iii) A = C( ⋃ ι∈I Aι), (iv) the GNS representation of φ is faithful on A. We denote the reduced free product by (A,φ) = ∗ ι∈I (Aι, φι) (1) and φ is called the free product state. In this note, we prove that if φι is faithful on Aι for every ι ∈ I then φ 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 φι is faithful. This implies that for the reduced free product of C–algebras in (1), φ is faithful if, for each ι ∈ I, letting (πι,Hι, ξι) 1991 Mathematics Subject Classification. 46L30. Typeset by AMS-TEX 1