Journal of the Iranian Mathematical Society

Journal of the Iranian Mathematical Society

Inner amenability of certain Lau algebras associated to discrete crossed products

Document Type : Research Article

Author
M. R. Ghanei
Department of Mathematics, Khansar Campus, University of Isfahan, Isfahan, Iran.
10.30504/jims.2024.470441.1197
Abstract
For a discrete group $\Gamma$, a Hopf von Neumann algebra $(\mathfrak{M},\Delta)$ and a $W^*$-dynamical system $(\mathfrak{M},\Gamma,\alpha)$ such that $(\alpha_s\otimes\alpha_s)\circ\Delta=\Delta\circ\alpha_s$, we show that the crossed product $\mathfrak{M}\rtimes_\alpha\Gamma$ with a co-multiplication is a Hopf von Neumann algebra.
Furthermore, we prove that the inner amenability of the predual $\mathfrak{M}_*$ is equivalent to the inner amenability of $(\mathfrak{M}\rtimes_\alpha\Gamma)_*$. Finally, we conclude that if the action $\alpha:\Gamma\rightarrow\mathrm{Aut}(\ell^\infty(\Gamma))$ is defined by $\alpha_s(f)(t)=f(s^{-1}ts)$, then the inner amenability of discrete group $\Gamma$ is equivalent to the inner amenability of $(\ell^\infty(\Gamma)\rtimes_\alpha\Gamma)_*$.
Keywords
Discrete Crossed product
W$^*$-Dynamical systems
Hopf von Neumann algebras
Inner amenability of Lau algebras
Subjects
[1] H. R. Ebrahimi Vishki and A. R. Khoddami, Character inner amenability of certain Banach algebras, Colloq. Math. 122 (2011), no. 2, 225--232.
[2] M. Fujita, Banach algebra structure in Fourier spaces and generalization of harmonic analysis on locally compact groups, J. Math. Soc. Japan 31 (1979), no. 1, 53--67.
[3] M. R. Ghanei and M. Nemati, On the amenability of a class of Banach algebras with application to measure algebra, Math. Slovaca 69 (2019), no. 5, 1177--1184.
[4] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis I, Springer-Verlag, Berlin, 1970.
[5] A. T. -M. Lau, Analysis on a class of Banach algebras with application to harmonic analysis on locally compact groups and semigroups, Fund. Math. 118 (1983), no. 3, 161--175.
[6] A. T. -M. Lau, Uniformly continuous functionals on Banach algebras, Colloq. Math. 51 (1987), no. 1, 195--205.
[7] A. McKee, Weak amenability for dynamical systems, Studia Math. 258 (2021), no. 1, 53--70.
[8] R. Nasr-Isfahani, Inner amenability of Lau algebras, Arch. Math. (Brno) 37 (2001), no. 1, 45--55.
[9] J. P. Pier, Amenable Banach algebras, Pitman research notes in mathematical series, 172, Longeman, Essex, 1988.
[10] M. Ramezanpour and H. R. Ebrahimi Vishki, Reiter’s properties for the actions of locally compact quantum groups on von Neumann algebras, Bull. Iranian Math. Soc. 36 (2010), no. 2, 1--17.
[11] Z. Ruan, Amenability of Hopf von Neumann algebras and Kac algebras, J. Funct. Anal. 139 (1996), no. 2, 466--499.
Journal of the Iranian Mathematical Society
Volume 5, Issue 2
March 2024
Pages 253-260