A new look at the crossed-product of a C *-algebra by an endomorphism

Exel, Ruy

doi:10.1017/s0143385702001797

Cited by 122 publications

(227 citation statements)

References 19 publications

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“…In [9], Exel constructs a crossed product A × α,L N using a correspondence M L over A. To construct M L , he endows the vector space A L := A with the right action of A given by m · a := mα(a) and the pre-inner product…”

Section: Multi-resolution Analyses From Transfer Operatorsmentioning

confidence: 99%

“…For modules such as X = , we usually expect the dilation to be an operator from X to itself, and to achieve this we need to choose appropriate correspondences M. We use correspondences M L which were introduced by Exel in his study of irreversible dynamics [9], [10], and which are associated to an endomorphism α of a C * -algebra A and a transfer operator L for α. When A is C(T) or C(T n ), there are natural transfer operators such that the filters in M L are the filters arising in wavelet theory and signal processing, and for these filters we can identify the underlying vector spaces of M L with A and X ⊗ A M L with X.…”

Section: Introductionmentioning

confidence: 99%

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Projective multi-resolution analyses arising from direct limits of Hilbert modules

Larsen¹,

Raeburn²

2007

MATH. SCAND.

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The authors have recently shown how direct limits of Hilbert spaces can be used to construct multi-resolution analyses and wavelets in L 2 (R). Here they investigate similar constructions in the context of Hilbert modules over C * -algebras. For modules over C(T n ), the results shed light on work of Packer and Rieffel on projective multi-resolution analyses for specific Hilbert C(T n )-modules of functions on R n . There are also new applications to modules over C(C) when C is the infinite path space of a directed graph.

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Section: Multi-resolution Analyses From Transfer Operatorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Projective multi-resolution analyses arising from direct limits of Hilbert modules

Larsen¹,

Raeburn²

2007

MATH. SCAND.

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“…В действи-тельности это исследование было инспирировано пионерскими работами Арзу-маняна и Вершика [22][23][24][25], в которых были введены и изучены соответствующие объекты в лебеговых пространствах. В последнее время данная тематика полу-чила новое развитие в работах Экселя [26,27] и Экселя и Вершика [28].…”

Section: скрещенные произведения и приведенные скрещенные произведенияunclassified

Топологически Свободные Частичные Действия И Точные Представления Скрещенных Произведений

Лебедев¹

2005

Функц. анализ и его прил.

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“…There has been a continuous interest for a generalised C*-crossed product construction of possibly noninvertible semigroup actions. Notable examples from the one-variable case are the works of Paschke [30], Peters [31], Stacey [34], Murphy [28], and Exel [12] (to mention only a few). The key idea is to consider the quotient of a Fock algebra by an ideal of redundancies.…”

Section: Introductionmentioning

confidence: 99%

On Nica-Pimsner Algebras of C*-Dynamical Systems Over $\mathbb{Z}_+^n$

Kakariadis

2016

Int Math Res Notices

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Abstract. We examine Nica-Pimsner algebras associated with semigroup actions of Z n + on a C*-algebra A by * -endomorphisms. We give necessary and sufficient conditions on the dynamics for exactness and nuclearity of the Nica-Pimsner algebras. Furthermore we parameterize the KMS states at finite temperature by the tracial states on A. A parametrization is also shown for KMS states at zero temperature (resp. ground states) by the tracial states on A (resp. states on A). Finally we give a formula for obtaining tracial states on the Nica-Pimsner algebras. IntroductionThis project lies in the intersection of three ongoing programs on analysing C*-algebras associated with C*-dynamics. There has been a continuous interest for a generalised C*-crossed product construction of possibly noninvertible semigroup actions. (to mention only a few). The key idea is to consider the quotient of a Fock algebra by an ideal of redundancies. Our standing point of view follows Arveson's program on the C*-envelope [19], and suggests that this ideal is theŠilov ideal of a nonselfadjoint operator algebra in the sense of Arveson [2]. In contrast to the case of the usual C*-crossed products, these redundancies may be very complicated to allow an in-depth exploration. An alternate route is suggested in the joint work of the author with Davidson and Fuller [9, Introduction]: dilate a possibly non-invertible semigroup action α : P → End(A) to a group action β : G → Aut( B) such that the distinguished quotient related to α : Z n + → End(A) is a full corner of the usual C*-crossed product related to β : G → Aut( B). Consequently the examination of the distinguished quotient can be induced via the more exploit-able and more exploit-ed theory of usual C*-crossed products. This pathway was met with success for several semigroup actions including Ore semigroups and spanning cones in [9], and in earlier works of the author for P = Z + [16], and of the author with Katsoulis for P = F n + [18].2010 Mathematics Subject Classification. 46L05, 46L55, 46L30, 58B34.

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A new look at the crossed-product of a C *-algebra by an endomorphism

Cited by 122 publications

References 19 publications

Projective multi-resolution analyses arising from direct limits of Hilbert modules

Projective multi-resolution analyses arising from direct limits of Hilbert modules

Топологически Свободные Частичные Действия И Точные Представления Скрещенных Произведений

On Nica-Pimsner Algebras of C*-Dynamical Systems Over $\mathbb{Z}_+^n$

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