ON THE TENSOR PRODUCTS OF IC-ALGEBRAS By FATMAH B. JAMJOOM [Received 12 November 1991; in revised form 18 May 1992] Introduction Our standard references of tensor product of C*-algebras are [3,7,9,10]. Given a fC-algebra, A, we let 1J1A denote the canonical embedding of A into C*(A) and we let <PA denote the canonical involutary *-antiautomorphism on C*(A) which fixes each point of 1J1A(A). Often we simply identify A with 1J1A(A), when there is no danger of confusion [5, Chapter 7]. The real C*-algebra R*(A) = {x E C*(A): <PAx)=x*} satisfies R*(A) n iR*(A) = 0, C*(A) = R*(A) Ea iR*(A) and inherits from C*(A) the following universal property:- each Jordan homomorphism lr: A~ ~ .a' where ~ is a real C*- algebra, extends uniquely to a real C*-algebra homomorphism ft: R*(A)~ . For this reason R*(A) is called the enveloping real C*-algebra of A. If A c ce s . a , where ce is a complex ce*-algebra, we let R(A) and, respectively [A], denote the real and complex C*-algebra in ce generated by A. If I is a norm closed ideal of A then [I] is an ideal of ce and I = [I] n A [2, Theorem 2.]. This, together with a simple complexification argument, shows that R(I) is an ideal of R(A) and that 1= R(I) cA. In the special case of the canonical inclusion A c R*(A) c C*(A) we have [4, Theorem 4.1] that C*(I) is identified with [I], and C*(AjI) with C*(A)jC*(I). Similarly R*(I) is identified with R(I) and R*(AjI) with R*(A)jR*(I). § 1. The definition and representations of tensor products Let A, B be any pair of fC-algebras (not necessarily unital). We may suppose that A and B are canonically embedded in their respective universal enveloping C*-algebras C*(A), C*(B) so that A ® Be (C*(A) ® C*(B»s.a c (C*(A) ~ C*(B) )s.a, where A is any C*-norm on C*(A) ® C*(B). Quart. J. Math. Oxford (2), 4S (1994), 77-90 © 1994 Oxford University Press