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Register a new userA Note on Decomposable Maps on Operator Systems
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Authors
Sriram Balasubramanian
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This article contains a characterization of operator systems $cS$ with the property that every positive map $phi:cS rightarrow M_n$ is decomposable, as well as an alternate and a more direct proof of a characterization of decomposable maps due to E. Sto rmer.
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We define a strong Morita-type equivalence $sim _{sigma Delta }$ for operator algebras. We prove that $Asim _{sigma Delta }B$ if and only if $A$ and $B$ are stably isomorphic. We also define a relation $subset _{sigma Delta }$ for operator algebras. We prove that if $A$ and $B$ are $C^*$-algebras, then $Asubset _{sigma Delta } B$ if and only if there exists an onto $*$-homomorphism $theta :Botimes mathcal K rightarrow Aotimes mathcal K,$ where $mathcal K$ is the set of compact operators acting on an infinite dimensional separable Hilbert space. Furthermore, we prove that if $A$ and $B$ are $C^*$-algebras such that $Asubset _{sigma Delta } B$ and $Bsubset _{sigma Delta } A $, then there exist projections $r, hat r$ in the centers of $A^{**}$ and $B^{**}$, respectively, such that $Arsim _{sigma Delta }Bhat r$ and $A (id_{A^{**}}-r) sim _{sigma Delta }B(id_{B^{**}}-hat r). $
We show that the definition of parabolic-like map can be slightly modified, by asking $partial Delta$ to be a quasiarc out of the parabolic fixed point, instead of the dividing arcs to be $C^1$ on $[-1,0]$ and $[0,1]$.
We study the structure of C*-algebras associated with compactly aligned product systems over group embeddable right LCM-semigroups. Towards this end we employ controlled maps and a controlled elimination method that associates the original cores to those of the controlling pair, and we combine with applications of the C*-envelope theory for cosystems of nonselfadjoint operator algebras recently produced. We derive several applications of these methods that generalize results on single C*-correspondences. First we show that if the controlling group is exact then the co-universal C*-algebra of the product system coincides with the quotient of the Fock C*-algebra by the ideal of strong covariance relations. We show that if the controlling group is amenable then the product system is amenable. In particular if the controlling group is abelian then the co-universal C*-algebra is the C*-envelope of the tensor algebra. Secondly we give necessary and sufficient conditions for the Fock C*-algebra to be nuclear and exact. When the controlling group is amenable we completely characterize nuclearity and exactness of any equivariant injective Nica-covariant representation of the product system. Thirdly we consider controlled maps that enjoy a saturation property. In this case we induce a compactly aligned product system over the controlling pair that shares the same Fock representation, and preserves injectivity. By using co-universality, we show that they share the same reduced covariance algebras. If in addition the controlling pair is a total order then the fixed point algebra of the controlling group induces a super-product system that has the same reduced covariance algebra and is moreover reversible.
Given an Archimedean order unit space (V,V^+,e), we construct a minimal operator system OMIN(V) and a maximal operator system OMAX(V), which are the analogues of the minimal and maximal operator spaces of a normed space. We develop some of the key properties of these operator systems and make some progress on characterizing when an operator system S is completely boundedly isomorphic to either OMIN(S) or to OMAX(S). We then apply these concepts to the study of entanglement breaking maps. We prove that for matrix algebras a linear map is completely positive from OMIN(M_n) to OMAX(M_m) if and only if it is entanglement breaking.
Frames on Hilbert C*-modules have been defined for unital C*-algebras by Frank and Larson and operator valued frames on a Hilbert space have been studied in arXiv.0707.3272v1.[math.FA]. Goal of the present paper is to introduce operator valued frames on a Hilbert C*-module for a sigma-unital C*-algebra. Theorem 1.4 reformulates the definition given by Frank and Larson in terms of a series of rank-one operators converging in the strict topology. Theorem 2.2. shows that the frame transform and the frame projection of an operator valued frame are limits in the strict topology of a series of elements in the multiplier algebra and hence belong to it. Theorem 3.3 shows that two operator valued frames are right similar if and only if they share the same frame projection. Theorem 3.4 establishes a one to one correspondence between Murray-von Neumann equivalence classes of projections in the multiplier algebra and right similarity equivalence classes of operator valued frames and provides a parametrization of all Parseval operator-valued frames on a given Hilbert C*-module. Left similarity is then defined and Proposition 3.9 establishes when two left unitarily equivalent frames are also right unitarily equivalent.
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