Abstract :
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 A⊗L(K), 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 Hm(A∗ B,A⊗L(Cn ⊕K)),
for all m 0.
© 2005 Elsevier Inc. All rights reserved.
