Universal $C^∗$-algebras with the local lifting property

2021 ◽  
Vol 127 (2) ◽  
pp. 361-381
Author(s):  
Kristin E. Courtney
Keyword(s):  
Completely Positive Maps ◽  
Completely Positive ◽  
C Algebras ◽  
Positive Maps

The Local Lifting Property (LLP) is a localized version of projectivity for completely positive maps between $\mathrm{C}^*$-algebras. Outside of the nuclear case, very few $\mathrm{C}^*$-algebras are known to have the LLP\@. In this article, we show that the LLP holds for the algebraic contraction $\mathrm{C}^*$-algebras introduced by Hadwin and further studied by Loring and Shulman. We also show that the universal Pythagorean $\mathrm{C}^*$-algebras introduced by Brothier and Jones have the Lifting Property.

MODULES OVER MONOTONE COMPLETE C*-ALGEBRAS

1992 ◽  
Vol 03 (02) ◽  
pp. 185-204 ◽  
Author(s):  
MASAMICHI HAMANA
Keyword(s):  
Completely Positive Maps ◽  
Completely Positive ◽  
C Algebra ◽  
C Algebras ◽  
Positive Maps

The main result asserts that given two monotone complete C*-algebras A and B, B is faithfully represented as a monotone closed C*-subalgebra of the monotone complete C*-algebra End A(X) consisting of all bounded module endomorphisms of some self-dual Hilbert A-module X if and only if there are sufficiently many normal completely positive maps of B into A. The key to the proof is the fact that each pre-Hilbert A-module can be completed uniquely to a self-dual Hilbert A-module.

scholarly journals BURES DISTANCE FOR COMPLETELY POSITIVE MAPS

2013 ◽  
Vol 16 (04) ◽  
pp. 1350031 ◽  
Author(s):  
B. V. RAJARAMA BHAT ◽  
K. SUMESH
Keyword(s):  
Von Neumann Algebra ◽  
Rigidity Theorem ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Von Neumann ◽  
C Algebras ◽  
Positive Maps ◽  
Normal States ◽  
Bures Distance

Bures had defined a metric on the set of normal states on a von Neumann algebra using GNS representations of states. This notion has been extended to completely positive maps between C*-algebras by Kretschmann, Schlingemann and Werner. We present a Hilbert C*-module version of this theory. We show that we do get a metric when the completely positive maps under consideration map to a von Neumann algebra. Further, we include several examples and counter examples. We also prove a rigidity theorem, showing that representation modules of completely positive maps which are close to the identity map contain a copy of the original algebra.

scholarly journals Topological entropy for the canonical completely positive maps on graph C*-Algebras

2004 ◽  
Vol 70 (1) ◽  
pp. 101-116 ◽  
Author(s):  
Ja A. Jeong ◽  
Gi Hyun Park
Keyword(s):  
Topological Entropy ◽  
Finite Graph ◽  
Infinite Graph ◽  
Completely Positive Map ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Locally Finite ◽  
Finite Graphs ◽  
C Algebras ◽  
Positive Maps

Let C*(E) = C*(se, pv) be the graph C*-algebra of a directed graph E = (E0, E1) with the vertices E0 and the edges E1. We prove that if E is a finite graph (possibly with sinks) and φE: C*(E) → C*(E) is the canonical completely positive map defined by then Voiculescu's topological entropy ht(φE) of φE is log r(AE), where r(AE) is the spectral radius of the edge matrix AE of E. This extends the same result known for finite graphs with no sinks. We also consider the map φE when E is a locally finite irreducible infinite graph and prove that , where the supremum is taken over the set of all finite subgraphs of E.

On Approximately Finite-Dimensional Von Neuman Algebras, II

1978 ◽  
Vol 21 (4) ◽  
pp. 415-418 ◽  
Author(s):  
George A. Elliott
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Von Neumann ◽  
Intrinsic Characterization ◽  
C Algebras ◽  
Finite Dimensional ◽  
Positive Maps ◽  
Injective Von Neumann Algebra

AbstractAn intrinsic characterization is given of those von Neumann algebras which are injective objects in the category of C*-algebras with completely positive maps. For countably generated von Neumann algebras several such characterizations have been given, so it is in fact enough to observe that an injective von Neumann algebra is generated by an upward directed collection of injective countably generated sub von Neumann algebras. The present work also shows that three of the intrinsic characterizations known in the countably generated case hold in general.

scholarly journals Asymmetric invariant sets for completely positive maps on C*-algebras

1986 ◽  
Vol 33 (3) ◽  
pp. 471-473 ◽  
Author(s):  
A. Guyan Robertson
Keyword(s):  
Invariant Sets ◽  
Completely Positive Map ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Positive Map ◽  
C Algebras ◽  
Positive Maps

Let A be a noncommutative C*-algebra other than M2(I). We show that there exists a completely positive map φ of norm one on A and an element a ɛ A such that φ(a) = a, φ(a*a) = a*a, but φ(aa*) ≠ aa*.

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