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arXiv:0904.0103

Decomposable and atomic projection maps

Erling Størmer

Published Apr 1, 2009• 0 citations
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Abstract

It is shown that a trace invariant projection map, i.e. a positive unital idempotent map, of a finite dimensional C*-algebra into itself is non-decomposable if and only if it is atomic, or equivalently not the sum of a 2-positive and a 2-copositive map. In particular projections onto spin factors of dimension greater than 6 are atomic.
DOI
10.48550/arxiv.0904.0103

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A factorization property of positive maps on C*-algebras

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The purpose of this short paper is to clarify and present a general version of an interesting observation by [Piani and Mora, Phys. Rev. A 75 (2007) 012305], linking complete positivity of linear maps on matrix algebras to decomposability of their ampliations. Let [Formula: see text], [Formula: see text] be unital C*-algebras and let [Formula: see text] be positive linear maps from [Formula: see text] to [Formula: see text] [Formula: see text]. We obtain conditions under which any positive map [Formula: see text] from the minimal C*-tensor product [Formula: see text] to [Formula: see text], such that [Formula: see text], factorizes as [Formula: see text] for some positive map [Formula: see text]. In particular, we show that when [Formula: see text] are completely positive (CP) maps for some Hilbert spaces [Formula: see text] [Formula: see text], and [Formula: see text] is a pure CP map and [Formula: see text] is a CP map so that [Formula: see text] is also CP, then [Formula: see text] for some CP map [Formula: see text]. We show that a similar result holds in the context of positive linear maps when [Formula: see text] and [Formula: see text]. As an application, we extend IX Theorem of Ref. 4 (revisited recently by [Huber et al., Phys. Rev. Lett. 121 (2018) 200503]) to show that for any linear map [Formula: see text] from a unital C*-algebra [Formula: see text] to a C*-algebra [Formula: see text], if [Formula: see text] is decomposable for some [Formula: see text], where [Formula: see text] is the identity map on the algebra [Formula: see text] of [Formula: see text] matrices, then [Formula: see text] is CP.

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A factorization property of positive maps on C∗C^*C∗-algebras
The purpose of this short note is to clarify and present a general version of an interesting observation by Piani and Mora (Physic. Rev. A 75, 012305 (2007)), linking complete positivity of linear maps on matrix algebras to decomposability of their ampliations. Let AiA_iAi​, CiC_iCi​ be unital C*-algebras and let αiα_iαi​ be positive linear maps from AiA_iAi​ to Ci,C_i,Ci​, i=1,2i=1,2i=1,2. We obtain conditions under which any positive map βββ from the minimal C*-tensor product A1⊗minA2A_1 \otimes_{min} A_2A1​⊗min​A2​ to C1⊗minC2C_1 \otimes_{min} C_2C1​⊗min​C2​, such that $ α_1 \otimes α_2 \geq β$, factorizes as β=γ⊗α2β= γ\otimes α_2β=γ⊗α2​ for some positive map γγγ. In particular we show that when αi ⁣:Ai→B(Hi)α_i \colon A_i \rightarrow B(\mathcal H_i)αi​:Ai​→B(Hi​) are completely positive (CP) maps for some Hilbert spaces Hi\mathcal H_iHi​ (i=1,2)(i=1,2)(i=1,2), and α2α_2α2​ is a pure CP map and βββ is a CP map so that α1⊗α2−βα_1 \otimes α_2 - βα1​⊗α2​−β is also CP, then β=γ⊗α2β= γ\otimes α_2β=γ⊗α2​ for some CP map γγγ. We show that a similar result holds in the context of positive linear maps when A2=C2=B(H)A_2 = C_2 = B(\mathcal H)A2​=C2​=B(H) and α2=idα_2 = idα2​=id. As an application we extend \cite[IX Theorem]{PM}( revisited recently by Huber et al in \cite{HLLM}) to show that for any linear map τττ from a unital C*-algebra AAA to a C*-algebra CCC, if τ⊗idkτ\otimes id_kτ⊗idk​ is decomposable for some k≥2k \geq 2k≥2, where idkid_kidk​ is the identity map on the algebra Mk(C)M_k(\mathbb {C} )Mk​(C) of k×kk\times kk×k matrices, then τττ is completely positive.
The purpose of this short paper is to clarify and present a general version of an interesting observation by [Piani and Mora, Phys. Rev. A 75 (2007) 012305], linking complete positivity of linear maps on matrix algebras to decomposability of their ampliations. Let [Formula: see text], [Formula: see text] be unital C*-algebras and let [Formula: see text] be positive linear maps from [Formula: see text] to [Formula: see text] [Formula: see text]. We obtain conditions under which any positive map [Formula: see text] from the minimal C*-tensor product [Formula: see text] to [Formula: see text], such that [Formula: see text], factorizes as [Formula: see text] for some positive map [Formula: see text]. In particular, we show that when [Formula: see text] are completely positive (CP) maps for some Hilbert spaces [Formula: see text] [Formula: see text], and [Formula: see text] is a pure CP map and [Formula: see text] is a CP map so that [Formula: see text] is also CP, then [Formula: see text] for some CP map [Formula: see text]. We show that a similar result holds in the context of positive linear maps when [Formula: see text] and [Formula: see text]. As an application, we extend IX Theorem of Ref. 4 (revisited recently by [Huber et al., Phys. Rev. Lett. 121 (2018) 200503]) to show that for any linear map [Formula: see text] from a unital C*-algebra [Formula: see text] to a C*-algebra [Formula: see text], if [Formula: see text] is decomposable for some [Formula: see text], where [Formula: see text] is the identity map on the algebra [Formula: see text] of [Formula: see text] matrices, then [Formula: see text] is CP.