| Let U be an abelian von Neumann algebra acting on a Hilbert space H . Then Mn( U ) is a Hilbert C*-module over U⊗C1n . C*-modules were originally defined and studied by Kaplansky and we outline the foundations of the theory and particular properties of Mn( U ). Furthermore, we prove a structure theorem for ultraweakly closed submodules of Mn( U ), using techniques from the theory of type I finite von Neumann algebras.;By analogy with the classical 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. Assuming that K is finite dimensional, Gilfeather and Smith calculated the Hochschild cohomology groups for A*B with coefficients in LCn⊕ K .;We assume that U is a maximal abelian von Neumann algebra acting on H , A is a subalgebra of U⊗L K , and B is an ultraweakly closed subalgebra of Mn( U ) containing U⊗C1n . In this new context, we redefine the join A*B and generalize the calculations of Gilfeather and Smith to multilinear maps on A*B with values in U⊗L Cn⊕K . We then calculate Hm( A*B,U⊗ LCn⊕ K ), for all m ≥ 0. |
