# An ultrapower construction of the multiplier algebra of a $C^{\ast}$-algebra and an application to boundary amenability of groups

## TL;DR

This paper introduces a novel ultrapower-based method for constructing the multiplier algebra of any $C^{st}$-algebra, extending previous work to more general cases and applying it to prove boundary amenability of certain groups.

## Contribution

It provides a new ultrapower construction of the multiplier algebra applicable to noncommutative and nonseparable $C^{st}$-algebras, and offers a new proof of boundary amenability for specific group actions.

## Key findings

- Extended the construction of multiplier algebras to noncommutative, nonseparable cases.
- Provided a new proof of boundary amenability for groups acting on trees.
- Generalized previous results to broader classes of $C^{st}$-algebras.

## Abstract

Using ultrapowers of $C^{\ast}$-algebras we provide a new construction of the multiplier algebra of a $C^{\ast}$-algebra. This extends the work of Avsec and Goldbring [Houston J. Math., to appear, arXiv:1610.09276.] to the setting of noncommutative and nonseparable $C^{\ast}$-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.