Decomposable Positive Maps on C*-Algebras 403

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 86, Number 3, November 1982

DECOMPOSABLEPOSITIVE MAPS ON C*-ALGEBRAS

ERLING ST0RMER

Abstract. It is shown that a positive linear map of a C*-algebra A into B(H) is decomposable if and only if for all n G N whenever (x,j) and (*,,) belong to M„(A)+ then (<¡>(*,7))belongs to M„(B(H))+ .

A positive linear map (x)= v*ir(x)v for all x G A. Such maps have been studied in [2, 3, 5, 7, 8, 9], and are the natural symmetrization of the completely positive ones, defined as those : A -> B(H) is completely positive if and only if for all n G N whenever (x¡¡) G Mn(A)+ then («H*,,)) G Mn(B(H))+. It is the purpose of the present note to provide an analogous characterization of decomposable maps.

Theorem. Let A be a C*-algebra and is decomposable if and only if for all n G N whenever (x,¡) and (Xj¡) belong to Mn(A)+ then ((XlJ)) G Mn(B(H))+ . Proof. Suppose is decomposable, so of the form v*rrv. If n is a homomorphism (resp. antihomomorphism) and (xi}) (resp. (xj¡)) belongs to Mn(A)+ then (transpose map on B(L) with respect to some orthonormal basis. Let

Then Kis a self adjoint subspace of M2(B(L)) containing the identity. Define 0n on Mn(B(L)) by 8n((Xjj)) = (xj,). Then 6 is an antiautomorphism of order 2. Hence if

Received by the editors September 25, 1981 and, in revised form, December 4, 1981. 1980 Mathematics Subject Classification. Primary 46L05; Secondary 46L50. Key words and phrases. Positive maps.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use DECOMPOSABLE POSITIVE MAPS ON C*-ALGEBRAS 403

(x,,) G M„(A) then (*,,) G M„(A)+ if and only if (*?,) = 6n((Xj,)) Mn(B(L))+. Therefore both (jc¿ ) and (Xß) belong to Mn(A)+ if and only if

0 M GMn(V)+. Let :V -> B(H) be defined by

0 = *(*)■

Then is completely positive in the sense of [1] by our hypothesis on

x 0 ir2(x) A. 0 x' Then w = vt, o 7t2 is a Jordan homomorphism of A into #(/0 such that (x) = v*tt(x)v for all x G A, hence # is decomposable. The proof is complete. The first example of a nondecomposable positive map was exhibited by Choi [2]. An extension of his example was reproduced in [3] together with a complete proof based on nontrivial results on biquadratic forms. We conclude by giving a short proof of his result. The example is :Af3(C) -» M3(C) defined by

-M2 lt?3 0 \ '21 i22 *21 *22 '23 + P 0 «31 l32 «33/ 0 zr \ (22/ where p > 1. It was shown by Choi that d>is positive. We show <¡>is not decomposable. Let (x¡j) G M3(M3(C)) be the matrix:

2p 0 0 2/1 0 2p 0 4p2 0 0 0 0 0 0 0 0 0 0

(*,/) =

0 0 0 0 v 0 0 0 0 0 0 0 1 0 0 0 2jj. 0 0 2p

Then both (x,7) and (xJt) belong to Af3(Af3(C))+ while it is easily seen that the matrix (is not decomposable by the theorem.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 404 ERLING ST0RMER

References 1. W. Arveson,Subalgebras of C*-algebras,Acta Math. 123 (1969), 141-224. 2. M. D. Choi, Positive semidefinite biquadratic forms. Linear Algebra and Appl. 12(1975), 95-100. 3. _, Some assorted inequalities for positive linear maps on C*-algebras, J. Operator Theory 4 (1980), 271-285. 4. W. F. Stinespring, Positive functions on C*-algebras, Proc. Amer. Math. Soc. 6 (1955), 211-216. 5. E. Stornier, Positive linear maps of operator algebras, Acta Math. 110 (1963), 233-278. 6. _, On the Jordan structure of C*-algebras, Trans. Amer. Math. Soc. 120 (1965), 438-447. 7. _, Decomposition of positive projections on C-algebras, Math. Ann. 27 (1980), 21-41. 8. S. L. Woronowicz, Nonextendiblepositive maps, Comm. Math. Phys. 51 (1976), 243-282. 9. _, Positive maps of low dimensional matrix algebras, Rep. Math. Phys. 10 (1976), 165-183. Department of Mathematics, University of Oslo, Oslo, Norway

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