## Rocky Mountain Journal of Mathematics

### Invariant means and property $T$ of crossed products

#### Abstract

Let $\Gamma$ be a discrete group that acts on a semi-finite measure space $(\Omega , \mu )$ such that there is no $\Gamma$-invariant function in $L^1(\Omega , \mu )$. We show that the absence of the $\Gamma$-invariant mean on $L^\infty (\Omega ,\mu )$ is equivalent to the property $T$ of the reduced $C^*$-crossed product of $L^\infty (\Omega ,\mu )$ by $\Gamma$. In particular, if $\Lambda$ is a countable group acting ergodically on an infinite $\sigma$-finite measure space $(\Omega , \mu )$, then there exists a $\Lambda$-invariant mean on $L^\infty (\Omega , \mu )$ if and only if the corresponding crossed product does not have property $T$. Moreover, if $\Gamma$ is an ICC group, then $\Gamma$ is inner amenable if and only if $\ell ^\infty (\Gamma \setminus \{e\})\rtimes _{\mathbf {i},r} \Gamma$ does not have property $T$, where $\mathbf {i}$ is the conjugate action. On the other hand, a non-compact locally compact group $G$ is amenable if and only if $L^\infty (G)\rtimes _{\mathbf {lt}, r} G_\mathrm {d}$ does not have property $T$, where $G_\mathrm {d}$ is the group $G$ equipped with the discrete topology and $\mathbf {lt}$ is the left translation.

#### Article information

Source
Rocky Mountain J. Math., Volume 48, Number 3 (2018), 905-912.

Dates
First available in Project Euclid: 2 August 2018

https://projecteuclid.org/euclid.rmjm/1533230831

Digital Object Identifier
doi:10.1216/RMJ-2018-48-3-905

Mathematical Reviews number (MathSciNet)
MR3835578

Zentralblatt MATH identifier
06917353

#### Citation

Meng, Qing; Ng, Chi-Keung. Invariant means and property $T$ of crossed products. Rocky Mountain J. Math. 48 (2018), no. 3, 905--912. doi:10.1216/RMJ-2018-48-3-905. https://projecteuclid.org/euclid.rmjm/1533230831

#### References

• M.B. Bekka, Property (T) for $C^*$-algebras, Bull. Lond. Math. Soc. 38 (2006), 857–867.
• F. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand, Princeton, 1969.
• U. Haagerup, The standard form of von Neumann algebras, Math. Scand. 37 (1975), 271–283.
• ––––, On the dual weights for crossed products of von Neumann algebras, I, Removing separability conditions, Math. Scand. 43 (1978), 99–118.
• E. Kaniuth and A. Markfort, The conjugation representation and inner amenability of discrete groups, J. reine angew. Math. 432 (1992), 23–37.
• C.W. Leung and C.K. Ng, Property (T) and strong property (T) for unital $C^*$-algebras, J. Funct. Anal. 256 (2009), 3055–3070.
• H. Li and C.K. Ng, Spectral gap actions and invariant states, Int. Math. Res. Not. 18 (2014), 4917–4931.
• Q. Meng and C.K. Ng, A full description of property (T) of unital $C^*$-crossed products, preprint.
• D.P. Williams, Crossed products of $C^*$-algebras, Math. Surv. Mono. 134 (2007).