An ultrapower construction of the multiplier algebra of a \(C^*\)-algebra and an application to boundary amenability of groups. (English) Zbl 1458.46045

Poggi, Facundo; Sasyk, Roman

Adv. Oper. Theory 4, No. 4, 852-864 (2019).

Summary: Using ultrapowers of \(C^*\)-algebras we provide a new construction of the multiplier algebra of a \(C^*\)-algebra. This extends the work of S.Avsec and I.Goldbring [Houston J.Math.45, No.3, 731–741 (2019; Zbl 1447.46043)] to the setting of noncommutative and nonseparable \(C^*\)-algebras. We also extend their work to give a new proof of the fact that groups that act transitively on locally finite trees with boundary amenable stabilizers are boundary amenable.

MSC:

46L05	General theory of \(C^*\)-algebras
46M07	Ultraproducts in functional analysis
03C20	Ultraproducts and related constructions
20F65	Geometric group theory

Keywords:

multiplier algebra; ultraproducts of \(C^*\)-algebra; boundary amenable group

Citations:

Zbl 1447.46043