Nonself-adjoint Semicrossed Products by Abelian Semigroups

Fuller, Adam H.

doi:10.4153/cjm-2012-051-8

Cited by 13 publications

(23 citation statements)

References 31 publications

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“…They posed a question [6, Question 2.5.11] of whether regularity is automatic for Nica-covariant representations. Fuller [8] established this for certain abelian semigroups.…”

Section: Introductionmentioning

confidence: 89%

“…Similarly, V (s)π(α s (a))h 2 , V (t)h = π(α t (a))T (t − t ∧ s) * T (s − t ∧ s)h 2 , h , This lifting of contractive Nica-covariant pairs to isometric Nica-covariant pairs has significant implication in its associated semi-crossed product. A family of covariant pairs gives rise to a semi-crossed product algebra in the following way [8,6]. For a C * -dynamical system (A, α, P ), denote P(A, P ) be the algebra of all formal polynomials q of the form q = n i=1 e p i a p i , where p i ∈ P and a p i ∈ A.…”

Section: Now Notice Thatmentioning

confidence: 99%

“…This field has also been explored in [16]. Davidson, Fuller, and Kakariadis [8,6] defined and studied contractive Nica-covariant representation on lattice ordered semigroups. The regularity of such representations was seen as a critical property in describing the C * -envelope of semicrossed products.…”

Section: Introductionmentioning

confidence: 99%

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Regular representations of lattice ordered semigroups

Li¹

2016

J. Operator Theory

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Abstract. We establish a necessary and sufficient condition for a representation of a lattice ordered semigroup to be regular, in the sense that certain extensions are completely positive definite. This result generalizes a theorem due to Brehmer where the lattice ordered group was taken to be Z Ω + . As an immediate consequence, we prove that contractive Nica-covariant representations are regular. We also introduce an analog of commuting row contractions on lattice ordered group and show that such a representation is regular.

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“…They posed a question [6, Question 2.5.11] of whether regularity is automatic for Nica-covariant representations. Fuller [8] established this for certain abelian semigroups.…”

Section: Introductionmentioning

confidence: 89%

Section: Now Notice Thatmentioning

confidence: 99%

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Regular representations of lattice ordered semigroups

Li¹

2016

J. Operator Theory

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“…A doubly commuting representation of N k is always regular ( [3], see also [25] for an alternative proof using C * -algebra and completely positive maps). Fuller [10,Theorem 2.4] proved that a doubly commuting representation of ⊕S i is always regular, where S i is a countable additive subgroup of R + . We are now going to extend all these results to direct sums of right LCM semigroups.…”

Section: The Graph Product Of Right Lcm Semigroupsmentioning

confidence: 99%

Regular Dilation and Nica-covariant Representation on Right LCM Semigroups

2019

Integr. Equ. Oper. Theory

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Regular dilation has recently been extended to graph product of N, where having a * -regular dilation is equivalent to having a minimal isometric Nica-covariant dilation. In this paper, we first extend the result to right LCM semigroups, and establish a similar equivalence among * -regular dilation, minimal isometric Nica-covariant dilation, and a Brehmer-type condition. This result can be applied to various semigroups to establish conditions for * -regular dilation.

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“…This obstruction spurred on dilation theories in other contexts [2,8,11,14,23] and many other generalizations. One recent usage of dilations of doubly commuting contractions is the dilation of Nica covariant representations of lattice-ordered semigroups [15,17].…”

Section: Introductionmentioning

confidence: 99%

Unitary dilation of freely independent contractions

Atkinson

Ramsey

2016

Proc. Amer. Math. Soc.

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Abstract. Inspired by the Sz.-Nagy-Foias dilation theorem we show that n freely independent contractions dilate to n freely independent unitaries. IntroductionThe Sz.-Nagy-Foias dilation theorem is a celebrated result in classical dilation theory. It says that n doubly commuting contractions can be simultaneously dilated to n doubly commuting unitaries. This was the original multivariable dilation theory context proven by Brehmer and Sz.-Nagy [7,25,26] until Andô [1] proved that one can do this for just commuting and not doubly commuting contractions when n = 2. However, it was subsequently shown in [20] and [27] that there are three commuting contractions which do not dilate to three commuting unitaries. This obstruction spurred on dilation theories in other contexts [2,8,11,14,23] and many other generalizations. One recent usage of dilations of doubly commuting contractions is the dilation of Nica covariant representations of lattice-ordered semigroups [15,17].Doubly commuting is one of two ingredients in the notion of tensor independence (or classical independence). It is natural then to ask whether n tensor independent contractions can be dilated to n tensor independent unitaries. The answer is yes (Theorem 2.2) and begs the question whether this can be done with other notions of non-commutative probability, namely free probability.Stemming from the notion of reduced free product [3,28] Voiculescu developed the theory of free probability in the 1980's with the goal of solving the free group factor problem. While this still remains unsolved, free probability has become a very important field of mathematical research. For further reading see [16,19]. This paper culminates in Theorem 3.2, that n freely independent contractions do indeed dilate to n freely independent unitaries. In a dilation theory context this has been done by Boca in [5] where he gives the most general unitary dilation of n contractions. The only free probability dilation result we know of is the unitary dilation of L-free sets of contractions of Popa and Vaes [22].Acknowledgements. We would like to thank David Sherman and Stuart White for some very helpful discussions during White's visit to the University of Virginia sponsored by the Institute of Mathematical Science.2010 Mathematics Subject Classification. 47A20, 46L54, 46L09.

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Nonself-adjoint Semicrossed Products by Abelian Semigroups

Cited by 13 publications

References 31 publications

Regular representations of lattice ordered semigroups

Regular representations of lattice ordered semigroups

Regular Dilation and Nica-covariant Representation on Right LCM Semigroups

Unitary dilation of freely independent contractions

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