GROUP-FREENESS AND CERTAIN AMALGAMATED FREENESS

In this paper, we will consider certain amalgamated free product structure in crossed product algebras. Let M be a von Neumann algebra acting on a Hilbert space H and G, a group and let α : G → AutM be an action of G on M, where AutM is the group of all automorphisms on M: Then the crossed product M = M ×α G of M and G with respect to ® is a von Neumann algebra acting on H ­ l²(G), generated by M and {ug}g∈G, where ug is the unitary representation of g on l²(G), We show that M ×α(G₁* G₂) = (M ×α G₁) *M (M ×α G₂). We compute moments and cumulants of operators in M. By doing that, we can verify that there is a close relation between Group Freeness and Amalgamated Freeness under the crossed product. As an application, we can show that if FN is the free group with N-generators, then the crossed product algebra LM(Fn) ≡ M ×α Fn satisfies that LM(Fn) = LM(Fk₁ ) *M LM(Fk₂ ), whenever n = k₁ + k₂ for n, k₁, k₂∈ N. KCI Citation Count: 5