$G$-amenability for Direct Sum, Tensor Product and Free Product of von-Neumann Algebras

Document Type : Research Paper

Author

Department of Mathematics, Khansar Campus, University of Isfahan, Isfahan, Iran

10.22108/msci.2025.144699.1734

Abstract

For a family of $W^*$-dynamical systems $(M_i, G, \alpha_i)_{i\in I}$, where $G$ is a locally compact group, we prove that if the direct sum $\bigoplus_{i \in I} M_i$ is $G$-amenable, then each $M_i$ is also $G$-amenable.
Conversely, if all $M_i$'s form a countable family of $G$-amenable von-Neumann algebras, then $\bigoplus_{i \in I} M_i$ is $G$-amenable as well. For two $W^*$-dynamical systems $(M, G, \alpha)$ and $(N, K, \beta)$, we show that the von-Neumann tensor product $M \bar{\otimes} N$ is $G \times K$-amenable if and only if $M$ is $G$-amenable and $N$ is $K$-amenable. We show that if the group $G$ is inner amenable, then the group von-Neumann algebra $VN(G)$ is also $G$-amenable. Furthermore, we prove that $VN(G) \bar{\otimes} M$ is $G$-amenable whenever the action $\alpha$ is inner amenable and $M$ is $G$-amenable. Finally, we show that von-Neumann algebras $M$ and $N$ are $G$-amenable if and only if their free product $M \bar{*} N$ is $G$-amenable. We also prove that the amenability of the group $G$ is equivalent to the $G$-amenability of $L^\infty(G) \bar{*} L^\infty(G)$.

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References

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History

  • Receive Date: 16 March 2025
  • Revise Date: 15 July 2025
  • Accept Date: 16 August 2025
  • Publish Date: 26 October 2025

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