Fell’s absorption principle for semigroup operator algebras

Abstract

Fell’s absorption principle states that the left regular representation of a group absorbs any unitary representation of the group when tensored with it. In a weakened form, this result carries over to the left regular representation of a right LCM submonoid of a group and its Nica-covariant isometric representations but it fails if the semigroup does not satisfy independence. In this paper, we explain how to extend Fell’s absorption principle to an arbitrary submonoid $P$ of a group $G$ by using an enhanced version of the left regular representation. Li’s semigroup ${\mathrm{C}}^{\ast }$-algebra ${\mathrm{C}}_s^{\ast }(P)$ and its representations appear naturally in our context. Using the enhanced left regular representation, we not only provide a very concrete presentation for the reduced object for ${\mathrm{C}}_s^{\ast }(P)$ but we also resolve open problems and obtain very transparent proofs of earlier results. In particular, we address the non-selfadjoint theory and we show that the non-selfadjoint object attached to the enhanced left regular representation coincides with that of the left regular representation. We obtain a non-selfadjoint version of Fell’s absorption principle involving the tensor algebra of a semigroup and we use it to improve recent results of Clouâtre and Dor-On on the residual finite dimensionality of certain ${\mathrm{C}}^{\ast }$-algebras associated with such tensor algebras. As another application, we give yet another proof for the existence of a ${\mathrm{C}}^{\ast }$-algebra which is co-universal for equivariant, Li-covariant representations of a submonoid $P$ of a group $G$.