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Fell's absorption principle states that the left regular representation of agroup absorbs any unitary representation of the group when tensored with it. Ina weakened form, this result carries over to the left regular representation ofa right LCM submonoid of a group and its Nica covariant isometricrepresentations but it fails if the semigroup does not satisfy independence. Inthis paper we explain how to extend Fell's absorption principle to an arbitrarysubmonoid $P$ of a group $G$ by using an enhanced version of the left regularrepresentation. Li's semigroup $C^*$-algebra $C^*_s(P)$ and its representationsappear naturally in our context. Using the enhanced left regularrepresentation, we not only provide a very concrete presentation for thereduced object for $C^*_s(P)$ but we also obtain very transparent proofs ofearlier results. We also address the non-selfadjoint theory and we show thatthe non-selfadjoint object attached to the enhanced left regular representationcoincides with that of the left regular representation. We obtain anon-selfadjoint version of Fell's absorption principle involving the tensoralgebra of a semigroup and we use it to improve recent results of Clouatre andDor-On on the residual finite dimensionality of certain $C^*$-algebrasassociated with such tensor algebras. We also give yet another proof for theexistence of a $C^*$-algebra which is co-universal for equivariant,Li-covariant representations of a submonoid $P$ of a group $G$.

Fell's absorption principle for semigroup operator algebras