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We prove a number of results having to do with equipping type-I\mathrm{C}^*-algebras with compact quantum group structures, the two main ones being that such a compact quantum group is necessarily co-amenable, and that if the\mathrm{C}^*-algebra in question is an extension of a non-zero finite direct sum of elementary\mathrm{C}^*-algebras by a commutative unital\mathrm{C}^*-algebra then it must be finite-dimensional.
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This content will become publicly available on April 3, 2026
Fell’s absorption principle for semigroup operator algebras
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 submonoidPof a groupGby using an enhanced version of the left regular representation. Li’s semigroup\mathrm{C}^{*}-algebra\mathrm{C}^{*}_{s}(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}(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}^{*}-algebras associated with such tensor algebras. As another application, we give yet another proof for the existence of a\mathrm{C}^{*}-algebra which is co-universal for equivariant, Li-covariant representations of a submonoidPof a groupG.
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- Award ID(s):
- 2054781
- PAR ID:
- 10627847
- Editor(s):
- Cuntz, J
- Publisher / Repository:
- EMS
- Date Published:
- Journal Name:
- Journal of Noncommutative Geometry
- Volume:
- 19
- Issue:
- 2
- ISSN:
- 1661-6952
- Page Range / eLocation ID:
- 401 to 428
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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