TENSOR PRODUCTS OF MONOTONE COMPLETE C*-ALGEBRAS

Let A (circle times) over barL(K) be the Fubini product of a monotone complete C*-algebra A and L(K), where K is an arbitrary Hilbert space. Hamana defined a product 'circle' in A (circle times) over barL(K), which turns (A (circle times) over barL(K),circle) into a monotone complete C*-algebra. To help him to do this, he introduced a new concept of order convergence, which is subtly different from Kadison-Pedersen convergence, and, made use of the Choi-Effros product related to a completely positive idempotent map on the injective envelope of A, whose existence is a consequence of Zorn's lemma. When K is separable, Saio and Wright showed that the algebra (A (circle times) over barL(K), circle) is the quotient algebra of the Borel tensor product A (circle times) over barL(K) of the Pedersen-Baire envelope A(infinity) and L(K) by a s-ideal I. In this paper, when K is separable, it is shown, by making use of Kadison-Pedersen convergence, that there is a natural monotone sigma-closed two-sided ideal J of A(infinity)(circle times) over barL(K), such that the quotient algebra B = A(infinity) (circle times) over barL(K)/J is monotone s-complete, and the quotient map qJ restricts to a unital isometric bijection Phi from A (circle times) over barL(K) to B. Via the map Phi, we can transplant the multiplication on B into the multiplication 'center dot' on A (circle times) over barL(K) so that (A (circle times) over barL(K), center dot) is a monotone complete C*-algebra. It is also shown that the product 'center dot' coincides with 'circle' (and so I = J). So there is no need to appeal to Zorn's lemma here for our approach to defining the product 'circle'. Our construction sheds some fresh light on classification problems of cross product algebras associated with generic dynamics.