GROUP-FREENESS AND CERTAIN AMALGAMATED FREENESS

Title & Authors
GROUP-FREENESS AND CERTAIN AMALGAMATED FREENESS
Cho, Il-Woo;

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 $\small{{\alpha}}$ : G$\small{{\rightarrow}}$ AutM be an action of G on M, where AutM is the group of all automorphisms on M. Then the crossed product $\small{\mathbb{M}=M{\times}{\alpha}}$ G of M and G with respect to $\small{{\alpha}}$ is a von Neumann algebra acting on $\small{H{\bigotimes}{\iota}^2(G)}$, generated by M and $\small{(u_g)_g{\in}G}$, where $\small{u_g}$ is the unitary representation of g on $\small{{\iota}^2(G)}$. We show that $\small{M{\times}{\alpha}(G_1\;*\;G_2)=(M\;{\times}{\alpha}\;G_1)\;*_M\;(M\;{\times}{\alpha}\;G_2)}$. We compute moments and cumulants of operators in $\small{\mathbb{M}}$. 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 $\small{F_N}$ is the free group with N-generators, then the crossed product algebra $\small{L_M(F_n){\equiv}M\;{\times}{\alpha}\;F_n}$ satisfies that $\small{L_M(F_n)=L_M(F_{{\kappa}1})\;*_M\;L_M(F_{{\kappa}2})}$, whenerver $\small{n={\kappa}_1+{\kappa}_2\;for\;n,\;{\kappa}_1,\;{\kappa}_2{\in}\mathbb{N}}$.
Keywords
crossed products of von Neumann algebras and groups;free product of algebras;moments and cumulants;
Language
English
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2.
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Applications of automata and graphs: Labeling operators in Hilbert space. II., Journal of Mathematical Physics, 2009, 50, 6, 063511
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ℂ-VALUED FREE PROBABILITY ON A GRAPH VON NEUMANN ALGEBRA, Journal of the Korean Mathematical Society, 2010, 47, 3, 601
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