${\mathrm{C}}^{\ast }$-selflessness of groups of homeomorphisms

Abstract

We prove that ${\mathrm{C}}^{\ast }$-simple groups which admit a faithful extreme boundary are ${\mathrm{C}}^{\ast }$-selfless. More precisely, we establish complete selflessness of the canonical inclusion of the reduced group ${\mathrm{C}}^{\ast }$-algebra into the reduced crossed product. In particular, the Higman--Thompson groups and the Brin--Thompson groups are ${\mathrm{C}}^{\ast }$-selfless.

arXiv comment10 pages(v1, v2); title changed, main theorem generalized and proof revised