Characterizations of Ordered Self-adjoint Operator Spaces

Russell, Travis

doi:10.48550/arxiv.1508.06272

Cited by 5 publications

(14 citation statements)

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“…Following Connes and van Suijlekom [4], we will make use of Werner's [24] characterization of matrix ordered spaces with this property in terms of partial unitizations (see below), although other characterizations are also known (see e.g. [21]).…”

Section: Preliminaries On Operator Systemsmentioning

confidence: 99%

Nonunital operator systems and noncommutative convexity

Kennedy¹,

Kim²,

Manor³

2021

Preprint

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We establish the dual equivalence of the category of (potentially non-unital) operator systems and the category of pointed compact nc (noncommutative) convex sets, extending a result of Davidson and the first author. We then apply this dual equivalence to establish a number of results about operator systems, some of which are new even in the unital setting.For example, we show that the maximal and minimal C*-covers of an operator system can be realized in terms of the C*-algebra of continuous nc functions on its nc quasistate space, clarifying recent results of Connes and van Suijlekom. We also characterize "C*-simple" operator systems, i.e. operator systems with simple minimal C*-cover, in terms of their nc quasistate spaces.We develop a theory of quotients of operator systems that extends the theory of quotients of unital operator algebras. In addition, we extend results of the first author and Shamovich relating to nc Choquet simplices. We show that an operator system is a C*-algebra if and only if its nc quasistate space is an nc Bauer simplex with zero as an extreme point, and we show that a second countable locally compact group has Kazhdan's property (T) if and only if for every action of the group on a C*-algebra, the set of invariant quasistates is the quasistate space of a C*-algebra. ContentsM. KENNEDY, S.J. KIM, AND N. MANOR 5. Pointed noncommutative functions 23 6. Minimal and maximal C*-covers 24 7. Characterization of unital operator systems 29 8. Quotients of operator systems 31 9. Noncommutative faces 35 10. C*-simplicity 37 11. Characterization of C*-algebras 39 12. Stable equivalence 42 13. Dynamics and Kazhdan's property (T) 43 References 47

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Section: Preliminaries On Operator Systemsmentioning

confidence: 99%

Nonunital operator systems and noncommutative convexity

Kennedy¹,

Kim²,

Manor³

2021

Preprint

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“…The ordered selfadjoint operator space language. An ordered selfadjoint operator space, as defined in [8,62], is a matricially normed and matricially ordered * -vector space that admits a selfadjoint completely isometric complete order embedding into a C*-algebra. Concretely, one can defined an ordered selfadjoint operator space as a selfadjoint closed subspace of B(H) with the inherited matricial norms, matricial positive cones, and involution.…”

Section: Languages For C*-algebrasmentioning

confidence: 99%

“…Concretely, one can defined an ordered selfadjoint operator space as a selfadjoint closed subspace of B(H) with the inherited matricial norms, matricial positive cones, and involution. Ordered selfadjoint operator spaces have been abstractly characterized in [62,63,68], and further studied in [8,9,44,45,56,69]. For ordered operator spaces X and Y , we denote by CPC(X, Y ) the set of all selfadjoint completely positive completely contractive linear maps X → Y .…”

Section: Languages For C*-algebrasmentioning

confidence: 99%

Applications of model theory to C*-dynamics

Gardella

Lupini

2018

Journal of Functional Analysis

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We initiate the study of compact group actions on C*-algebras from the perspective of model theory, and present several applications to C*-dynamics. Firstly, we prove that the continuous part of the central sequence algebra of a strongly self-absorbing action is indistinguishable from the continuous part of the sequence algebra, and in fact equivariantly isomorphic under the Continuum Hypothesis. As another application, we present a unified approach to several dimensional inequalities in C*-algebras, which is done through the notion of order zero dimension for an (equivariant) *-homomorphism. Finiteness of the order zero dimension implies that the dimension of the target algebra can be bounded by the dimension of the domain. The dimension can be, among others, decomposition rank, nuclear dimension, or Rokhlin dimension. As a consequence, we obtain new inequalities for these quantities.As a third application we obtain the following result: if a C*-algebra A absorbs a strongly self-absorbing C*algebra D, and α is an action of a compact group G on A with finite Rokhlin dimension with commuting towers, then α absorbs any strongly self-absorbing action of G on D. This has a number of interesting consequences, already in the case of the trivial action on D. For example, we deduce that D-stability passes from A to the crossed product. Additionally, in many cases of interest, our result restricts the possible values of the Rokhlin dimension to 0, 1 and ∞, showing a striking parallel to the behavior of the nuclear dimension for simple C*-algebras. We also show that an action of a finite group with finite Rokhlin dimension with commuting towers automatically has the Rokhlin property if the algebra is UHF-absorbing.2000 Mathematics Subject Classification. Primary 03C98, 46L55; Secondary 28D05, 46L40, 46M07. 1 2 EUSEBIO GARDELLA AND MARTINO LUPINIproperty. In the context of C*-algebras, relative commutants were used in connection with the Rokhlin property for group actions in the work of Herman-Jones [37], Kishimoto [51], Izumi [42], , and the first-named author [28]. The study of Rokhlin dimension has also made extensive use of these tools, for example in [28] and [38], as well as the more recent work on strongly self-absorbing actions [64]. As is clear from these works, the use of sequence algebras in the equivariant setting becomes even more delicate when the acting group is not discrete, since a continuous action on an operator algebra may induce a discontinuous action on its relative commutant. As such, equivariant (central) sequence algebras are interesting objects whose systematic study is justified by their wide application in the literature.The present work takes up this task. For a given compact second countable group G, we consider actions of G on C*-algebras (G-C*-algebras) as structures in the framework of continuous model theory. When the group G is finite, one can regard a G-action as a usual metric structure by adding a function symbol for every element of the group. This does not work for a general compact g...

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“…We plan to study these concepts in a future paper. We refer the reader to an earlier paper [19] for a few details concerning quotients, as well as "commutative" versions of several results presented in this paper.…”

Section: Final Remarks and Acknowledgmentsmentioning

confidence: 99%

“…The author would like to thank the following individuals for helpful comments and discussions related to this paper and its predecessor [19]: David Blecher, Allan Donsig, Douglas Farenick, Rupert Levene, Vern Paulsen, David Pitts, and Mark Rieffel, as well as Mark Tomforde for making several fruitful conversations possible. The author also extends his gratitude to the Associate Editor and referees for their careful reading of this manuscript and their valuable suggestions.…”

Section: Final Remarks and Acknowledgmentsmentioning

confidence: 99%

Characterizations of ordered operator spaces

Russell

2017

Journal of Mathematical Analysis and Applications

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We demonstrate new abstract characterizations for unital and non-unital operator spaces. We characterize unital operator spaces in terms of the cone of accretive operators (operators whose real part is positive). We show that matrix norms and accretive cones are induced by gauges, although inducing gauges are not unique in general. Finally, we show that completely positive completely contractive linear maps on non-unital operator spaces extend to any containing operator system if and only if the operator space is induced by a unique gauge.

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Characterizations of Ordered Self-adjoint Operator Spaces

Cited by 5 publications

References 0 publications

Nonunital operator systems and noncommutative convexity

Nonunital operator systems and noncommutative convexity

Applications of model theory to C*-dynamics

Characterizations of ordered operator spaces

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