## Annals of Functional Analysis

### Cone isomorphisms and expressions of some completely positive maps

#### Abstract

Let $B{\mathcal{(H)}}$, $K{\mathcal{(H)}}$ and $T{\mathcal{(H)}}$ be the set of all bounded linear operators, compact operators, and trace-class operators on the Hilbert space $\mathcal{H}$. The cone of all completely positive maps from $K{\mathcal{(H)}}$ into $T{\mathcal{(K)}}$ and all normal completely positive maps from $B{\mathcal{(K)}}$ into $T{\mathcal{(H)}}$ is denoted by $\mathit{CP}(K{\mathcal{(H)}},T{\mathcal{(K)}})$ and $\mathit{NCP}(B{\mathcal{(K)}},T{\mathcal{(H)}})$, respectively. In this note, the order structures of the positive cones $\mathit{CP}(K{\mathcal{(H)}},T{\mathcal{(K)}})$ and $\mathit{NCP}(B{\mathcal{(K)}},T{\mathcal{(H)}})$ are investigated. First, we show that $\mathit{CP}(K{\mathcal{(H)}},T{\mathcal{(K)}})$, $\mathit{NCP}(B{\mathcal{(K)}},T{\mathcal{(H)}})$, and $T{\mathcal{(K\otimesH)}}^{+}$ are cone-isomorphic. Then we give the operator sum representation for the map $\Phi\in\mathit{CP}(K{\mathcal{(H)}},T{\mathcal{(K)}})$.

#### Article information

Source
Ann. Funct. Anal., Volume 8, Number 1 (2017), 27-37.

Dates
Accepted: 21 May 2016
First available in Project Euclid: 14 October 2016

https://projecteuclid.org/euclid.afa/1476450345

Digital Object Identifier
doi:10.1215/20088752-3720566

Mathematical Reviews number (MathSciNet)
MR3558302

Zentralblatt MATH identifier
1356.47059

Subjects
Secondary: 47L50: Dual spaces of operator algebras

#### Citation

Sun, Xiuhong; Li, Yuan. Cone isomorphisms and expressions of some completely positive maps. Ann. Funct. Anal. 8 (2017), no. 1, 27--37. doi:10.1215/20088752-3720566. https://projecteuclid.org/euclid.afa/1476450345

#### References

• [1] M.-D. Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl. 10 (1975), 285–290.
• [2] J. B. Conway, A Course in Operator Theory, Grad. Stud. Math. 21, Amer. Math. Soc., Providence, 2000.
• [3] R. G. Douglas, On majorization, factorization and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413–415.
• [4] E. G. Effros and Z.-J. Ruan, Operator Spaces, London Math. Soc. Monogr. Ser. 23, Oxford University Press, New York, 2000.
• [5] M. H. Hsua, D. W. Kuo, and M. C. Tsai, Completely positive interpolations of compact, trace-class and Schatten-p class operators, J. Funct. Anal. 267 (2014), no. 4, 1205–1240.
• [6] K. Kraus, General state changes in quantum theory, Ann. Physics 64 (1971), 311–335.
• [7] C.-K. Li and Y.-T. Poon, Interpolation by completely positive maps, Linear Multilinear Algebra 59 (2011), no. 10, 1159–1170.
• [8] Y. Li, Fixed points of dual quantum operations, J. Math. Anal. Appl. 382 (2011), no. 1, 172–179.
• [9] Y. Li and H.-K. Du, Interpolations of entanglement breaking channels and equivalent conditions for completely positive maps, J. Funct. Anal. 268 (2015), no. 11, 3566–3599.
• [10] B. Magajna, Fixed points of normal completely positive maps on $\mathrm{B}(\mathcal{H})$, J. Math. Anal. Appl. 389 (2012), no. 2, 1291–1302.
• [11] V. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge Stud. Adv. Math. 78, Cambridge University Press, Cambridge, 2002.
• [12] V. Paulsen and F. Shultz, Complete positivity of the map from a basis to its dual basis, J. Math. Phys. 54 (2013), no. 7, art. ID 072201.
• [13] V. Paulsen, I. Todorov, and M. Tomforde, Operator system structures on ordered spaces, Proc. Lond. Math. Soc. (3) 102 (2011), no. 1, 25–49.
• [14] E. Størmer, Extension of positive maps into $B{\mathcal{(H)}}$, J. Funct. Anal. 66 (1986), no. 2, 235–254.
• [15] E. Størmer, Positive Linear Maps of Operator Algebras, Springer Monogr. Math., Springer, Heidelberg, 2013.
• [16] X.-H. Sun and Y. Li, The range of generalized quantum operations, Linear Algebra Appl. 452 (2014), 120–129.