University of Isfahan Mathematics and Society 2345-6493 11 2 2025 10 26 $G$-amenability for Direct Sum, Tensor Product and Free Product of von-Neumann Algebras $G$-amenability for Direct Sum, Tensor Product and Free Product of von-Neumann Algebras 65 80 29750 10.22108/msci.2025.144699.1734 FA Mohammad Reza Ghanei Department of Mathematics, Khansar Campus, University of Isfahan, Isfahan, Iran Journal Article 2025 03 16 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.<br />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 $L^\infty(G) \bar{*} L^\infty(G)$. 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.<br />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 $L^\infty(G) \bar{*} L^\infty(G)$. https://math-sci.ui.ac.ir/article_29750_b93588e4753618c2d8319df3c1b042bf.pdf