Fell’s absorption principle for semigroup operator algebras

  • Elias G. Katsoulis

    East Carolina University, Greenville, USA
Fell’s absorption principle for semigroup operator algebras cover
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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 of a group by using an enhanced version of the left regular representation. Li’s semigroup -algebra 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 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 -algebras associated with such tensor algebras. As another application, we give yet another proof for the existence of a -algebra which is co-universal for equivariant, Li-covariant representations of a submonoid of a group .

Cite this article

Elias G. Katsoulis, Fell’s absorption principle for semigroup operator algebras. J. Noncommut. Geom. 19 (2025), no. 2, pp. 401–428

DOI 10.4171/JNCG/566