Completely Bounded Maps and Operator Algebras: Contents

By Vern I. Paulsen. 2003

1,634 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/58ce81dfb3e43291a2f23132e40c5cb8b39f6cf7

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Citations in the corpus

Representations of product systems over semigroups and dilations of commuting CP maps (2005, background): “Since it is known that, in general, one cannot dilate simultanuously a commuting triple of contractions to a commuting triple of isometries (see [ 27, Chapter 5 ]), one cannot hope to have a general isometric dilation result for representations of product systems over Nk for k > 2.”
The Tracial Hahn-Banach Theorem, Polar Duals, Matrix Convex Sets, and Projections of Free Spectrahedra (2014, background): “s. 3.1. Completely Positive Maps over the Reals. This subsection collects basic facts about completely positive (cp) maps ˚: S!M d, where Sis a subspace of M nclosed under transpose (see for instance [Pau02]) containing a positive denite matrix. We call such an S a operator system. Because the resul”
Characterisations of the weak expectation property (2013, background): “We refer the reader to [25] for further properties of operator systems and completely positive maps.”
Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance (2016, background): “Many readers will have this familiarity, but for background on C∗ and von Neumann algebras, we refer to Sakai [63], and for completely positive maps to Paulsen [58].”
Entanglement of assistance and multipartite state distillation (2005, background): “The appropriate distance measure for quantum channels T and T ′ to be used here is the completely bounded norm (cb-norm) [26].”
About the Connes embedding conjecture (2013, background): “1 in [Pa]).”
Interpolation and transfer-function realization for the noncommutative Schur-Agler class (2016, background): “Recall (see [68, 40]) that a normed linear space W is said to be an operator space if it is equipped with a system of norms ‖ · ‖n on n × n matrices over W so that there is a map φ : W → L(X ) (where X is some Hilbert space) so that, for each n = 1, 2, .”
The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement (2012, methodology): “For its proof we refer to the original work [133] and Paulsen’s book [112].”
Minimal and maximal matrix convex sets (2017, neutral): “ce His denoted by B(H), B(H)d denotes d-tuples of operators, and B(H)d sa denotes d-tuples of self-adjoint operators. We will use basic results on C*-algebras and operator systems freely (see [5] and [20], respectively), but let us recall a few denitions. If Aand Bare C*-algebras, a linear map ˚: A!”
Noncommutative Choquet Theory (2019, neutral): “For a detailed reference on operator systems, we direct the reader to the book of Paulsen [42].”
Noncommutative reproducing kernel Hilbert spaces (2016, methodology): “Such positive kernels can be seen as more elaborate versions of the notion of completely positive map as occurs in the operator algebra literature (see [40]) as well as of the notion of completely positive kernel as introduced and developed by Barreto-Bhat-Liebscher-Skeida [16] (see Section 3.”
Connes' embedding problem and Tsirelson's problem (2010, background): “It is not difficult to see that min is the minimal norm in the category of operator systems (see [21]).”
The structure of degradable quantum channels (2008, methodology): “We explain this following the notation in Appendix A of [23], where it was observed that the ancilla representation (1) is a special case of Stinespring’s fundamental representation theorem [1, 30, 37].”