Regular dilation and Nica-covariant representation on right LCM semigroups. (English) Zbl 1456.43004
Generalizing the celebrated Sz. Nagy dilation of a single contraction, Brehmer studied regular dilations back in the early sixties. Since then the notion of regular dilation has been investigated by many researchers and has been generalized to product systems, lattice ordered semigroups, and recently to graph products of \(\mathbb{N}\). It was shown by the author of the present paper in [J. Funct. Anal. 273, No. 2, 799–835 (2017; Zbl 1459.47005)] that for such graph products, the existence of a \(*\)-regular dilation is equivalent to the existence of a minimal isometric Nica-covariant dilation.
In the paper under review, the author extends this result to right LCM semigroups, which are unital left cancellative semigroups \(P\) such that for any \(p, q\in P\), either \(pP\cap qP=\emptyset\) or \(pP\cap qP=rP\) for some \(r\in P\). Such an element \(r\) can be considered as a right least common multiple of \(p\) and \(q\) (hence the name right LCM). The author also shows the equivalence among a \(*\)-regular dilation, minimal isometric Nica-covariant dilation, and a Brehmer-type condition. This result unifies many previous results on regular dilations, including Brehmer’s theorem, Frazho-Bunce-Popescu’s dilation of noncommutative row contractions, regular dilations on lattice ordered semigroups and graph products of \(\mathbb{N}\). Applications to many important classes of right LCM semigroups are given. In the last section of the paper, the author obtains a characterization of \(*\)-regular representations of graph products of right LCM semigroups and an application to doubly commuting representations of direct sums of right LCM semigroups.
In the paper under review, the author extends this result to right LCM semigroups, which are unital left cancellative semigroups \(P\) such that for any \(p, q\in P\), either \(pP\cap qP=\emptyset\) or \(pP\cap qP=rP\) for some \(r\in P\). Such an element \(r\) can be considered as a right least common multiple of \(p\) and \(q\) (hence the name right LCM). The author also shows the equivalence among a \(*\)-regular dilation, minimal isometric Nica-covariant dilation, and a Brehmer-type condition. This result unifies many previous results on regular dilations, including Brehmer’s theorem, Frazho-Bunce-Popescu’s dilation of noncommutative row contractions, regular dilations on lattice ordered semigroups and graph products of \(\mathbb{N}\). Applications to many important classes of right LCM semigroups are given. In the last section of the paper, the author obtains a characterization of \(*\)-regular representations of graph products of right LCM semigroups and an application to doubly commuting representations of direct sums of right LCM semigroups.
Reviewer: Trieu Le (Toledo)
MSC:
| 43A65 | Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis) |
| 47A20 | Dilations, extensions, compressions of linear operators |
| 20F36 | Braid groups; Artin groups |
Citations:
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