## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Operator Algebras and Mathematical Physics, M. Izumi, Y. Kawahigashi, M. Kotani, H. Matui and N. Ozawa, eds. (Tokyo: Mathematical Society of Japan, 2019), 189 - 210

### Actions of amenable groups and crossed products of $\mathcal{Z}$-absorbing $\mathrm{C}^*$-algebras

#### Abstract

We study actions of countable discrete amenable groups on unital separable simple nuclear $\mathcal{Z}$-absorbing $\mathrm{C}^*$-algebras. Under a certain assumption on tracial states, which is automatically satisfied in the case of a unique tracial state, the crossed product is shown to absorb the Jiang-Su algebra $\mathcal{Z}$ tensorially.

#### Article information

**Dates**

Received: 27 December 2016

Revised: 28 February 2017

First available in Project Euclid:
21 August 2019

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1566404316

**Digital Object Identifier**

doi:10.2969/aspm/08010189

**Mathematical Reviews number (MathSciNet)**

MR3966590

**Zentralblatt MATH identifier**

07116429

**Subjects**

Primary: 46L06: Tensor products of $C^*$-algebras 46L35: Classifications of $C^*$-algebras 46L40: Automorphisms 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]

#### Citation

Sato, Yasuhiko. Actions of amenable groups and crossed products of $\mathcal{Z}$-absorbing $\mathrm{C}^*$-algebras. Operator Algebras and Mathematical Physics, 189--210, Mathematical Society of Japan, Tokyo, Japan, 2019. doi:10.2969/aspm/08010189. https://projecteuclid.org/euclid.aspm/1566404316