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 H 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⊕H). We assume that A is a maximal abelian von Neumann algebra acting on H, A is a subalgebra of , and B is an ultraweakly closed subalgebra of Mn(A) containing A⊗1n. 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 , for all m⩾0.