An ultrapower construction of the multiplier algebra of a $C^*$-algebra‎ ‎and an application to boundary amenability of groups

Abstract

‎‎‎‎Using ultrapowers of $C^*$-algebras we provide a new construction of the multiplier algebra of a $C^*$-algebra‎. ‎This extends the work of Avsec and Goldbring [Houston J‎. ‎Math.‎, ‎to appear‎, ‎arXiv:1610.09276]‎ ‎to the setting of noncommutative and non separable $C^*$-algebras‎. ‎We also extend their work‎ ‎to give a new proof of the fact that groups acting transitively on locally finite trees with boundary amenable stabilizers are‎ ‎boundary amenable‎.