GROUP-FREENESS AND CERTAIN AMALGAMATED FREENESS

Abstract

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 Hand G, a group and let <TEX>${\alpha}$</TEX> : G<TEX>${\rightarrow}$</TEX> AutM be an action of G on M, where AutM is the group of all automorphisms on M. Then the crossed product <TEX>$\mathbb{M}=M{\times}{\alpha}$</TEX> G of M and G with respect to <TEX>${\alpha}$</TEX> is a von Neumann algebra acting on <TEX>$H{\bigotimes}{\iota}^2(G)$</TEX>, generated by M and <TEX>$(u_g)_g{\in}G$</TEX>, where <TEX>$u_g$</TEX> is the unitary representation of g on <TEX>${\iota}^2(G)$</TEX>. We show that <TEX>$M{\times}{\alpha}(G_1\;*\;G_2)=(M\;{\times}{\alpha}\;G_1)\;*_M\;(M\;{\times}{\alpha}\;G_2)$</TEX>. We compute moments and cumulants of operators in <TEX>$\mathbb{M}$</TEX>. 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 <TEX>$F_N$</TEX> is the free group with N-generators, then the crossed product algebra <TEX>$L_M(F_n){\equiv}M\;{\times}{\alpha}\;F_n$</TEX> satisfies that <TEX>$$L_M(F_n)=L_M(F_{{\kappa}1})\;*_M\;L_M(F_{{\kappa}2})$$</TEX>, whenerver <TEX>$n={\kappa}_1+{\kappa}_2\;for\;n,\;{\kappa}_1,\;{\kappa}_2{\in}\mathbb{N}$</TEX>.