Crossed products by semigroups of endomorphisms and the Toeplitz algebras of ordered groups

Adji, Sriwulan; Laca, Marcelo; Nilsen, May; Raeburn, Iain

doi:10.1090/s0002-9939-1994-1215024-1

Cited by 60 publications

(58 citation statements)

References 11 publications

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“…Suppose ψ is a (nondegenerate) Nica-covariant Toeplitz representation of X. Proposition 6.1 (2) gives us a covariant representation (L ψ , ψ) of (B P , P, τ, X), and hence a representation L ψ × ψ of B P ⋊ τ,X P such that (L ψ × ψ) • i X = ψ. Restricting L ψ × ψ to T cov (X) gives the required representation ψ * .…”

Section: If the Left Action Of A On Each Fiber X S Is By Compact Opermentioning

confidence: 99%

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Discrete product systems of Hilbert bimodules

Fowler¹

2002

Pacific J. Math.

122

335

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A Hilbert bimodule is a right Hilbert module X over a C * -algebra A together with a left action of A as adjointable operators on X. We consider families X = {X s : s ∈ P } of Hilbert bimodules, indexed by a semigroup P , which are endowed with a multiplication which implements isomorphisms X s ⊗ A X t → X st ; such a family is a called a product system. We define a generalized Cuntz-Pimsner algebra O X , and we show that every twisted crossed product of A by P can be realized as O X for a suitable product system X. Assuming P is quasi-lattice ordered in the sense of Nica, we analyze a certain Toeplitz extension T cv (X) of O X by embedding it in a crossed product B P τ,X P which has been "twisted" by X; our main Theorem is a characterization of the faithful representations of B P τ,X P .Introduction.

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Section: If the Left Action Of A On Each Fiber X S Is By Compact Opermentioning

confidence: 99%

“…By [2,Lemma 1.3], to prove that L Ψ × Ψ is faithful on (B P ⋊ τ,X P ) δ θ it is enough to prove it is faithful on each of the subalgebras U F . We shall accomplish this by inducting on |F |.…”

Section: Amenabilitymentioning

confidence: 99%

Discrete product systems of Hilbert bimodules

Fowler¹

2002

Pacific J. Math.

122

335

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“…Crossed products of C * -algebras by semigroups of endomorphisms have been profitably used to model Toeplitz algebras [2,1,13] and the Hecke algebras arising in the Bost-Connes analysis of phase transitions in number theory [14,3,11]. There are two main ways of studying such a crossed product.…”

mentioning

confidence: 99%

“…The algebra on which the group acts is typically a direct limit, and the success of this approach depends on being able to recognise the direct limit and the action on it [7,23,17]. Or, second, one can use the techniques developed in [5,2,13] to deal directly with the semigroup crossed product and its representation theory. Here the goal is a characterisation of the faithful representations of the crossed product, and such characterisations have given important information about a wide range of semigroup crossed products [2,13,14,3].…”

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confidence: 99%

Discrete Product Systems and Twisted Crossed Products by Semigroups

Fowler¹,

Raeburn²

1998

Journal of Functional Analysis

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A product system E over a semigroup P is a family of Hilbert spaces [E s : s # P] together with multiplications E s _E t Ä E st . We view E as a unitary-valued cocycle on P, and consider twisted crossed products A < ;, E P involving E and an action ; of P by endomorphisms of a C*-algebra A. When P is quasi-lattice ordered in the sense of Nica, we isolate a class of covariant representations of E, and consider a twisted crossed product B P < {, E P which is universal for covariant representations of E when E has finite-dimensional fibres, and in general is slightly larger. In particular, when P=N and dim E 1 = , our algebra B N < {, E N is a new infinite analogue of the Toeplitz-Cuntz algebras TO n . Our main theorem is a characterisation of the faithful representations of B P < {, E P. Academic PressCrossed products of C*-algebras by semigroups of endomorphisms have been profitably used to model Toeplitz algebras [2,1,13], and the Hecke algebras arising in the Bost Connes analysis of phase transitions in number theory [3,11,14]. There are two main ways of studying such a crossed product. First, one can try to embed it as a corner in a crossed product by an automorphic action of an enveloping group, and then apply the established theory. The algebra on which the group acts is typically a direct limit, and the success of this approach depends on being able to recognise the direct limit and the action on it [7,17,23]. Or, second, one can use the techniques developed in [2,5,13] to deal directly with the semigroup crossed product and its representation theory. Here the goal is a characterisation of the faithful representations of the crossed product, and such characterisations have given important information about a wide range of semigroup crossed products [2,3,13,14].For ordinary crossed products A < : G (those involving an action : of G by automorphisms of A ), an important adjunct are the twisted crossed article no. FU973227

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“…2, 3], где изложены полные доказательства и представлены разнооб-разные приложения) для произвольной C * -алгебры и аменабельной дискрет-ной группы. Связь соответствующего свойства с точными представлениями скрещенных произведений на эндоморфизмы, порожденные изометриями, бы-ла исследована в [16], [17]. Роль свойств типа ( * ) в теории алгебр расслоений Фелла изучалась в [18].…”

Section: теорема 22 (см [11; теорема 28])unclassified