On the cohomology of joins of operator algebras

By analogy with the join in topology, the join A * B for operator algebras A and B acting on Hilbert spaces W and K, respectively, was defined by Gilfeather and Smith (Amer. J. Math. 116 (1994) 541-561). Assuming that K is finite dimensional, they calculated the Hochschild cohomology groups for A * B with coefficients in L(K circle plus H). We assume that U is a maximal abelian von Neumann algebra acting on H, A is a subalgebra of U (circle times) over barL(K) and B is an ultraweakly closed subalgebra of M-n (U) containing U circle times 1(n). We show that B may be decomposed into a finite sum of free modules. In this context, we redefine the join of A and B. generalize the calculations of Gilfeather and Smith, and calculate H-m (A * B, U (circle times) over bar L(C-n circle plus K)), for all m >= 0. (C) 2005 Elsevier Inc. All rights reserved.