AN n×n MATRIX OF LINEAR FUNCTIONALS OF C∗-ALGEBRAS

We show that any bounded matrix of linear functionals [fij]:Mn(A)→Mn(ℂ) has a representation fij(a)=〈Tπ(a)xj,xi〉, a∈A, i,j=1,2,…,n, for some representation π on a Hilbert space K and an n vectors x1,x2,…,xn in K.


Introduction.
Let M n be the C * -algebras of complex n × n matrices generated as a linear space by the matrix units E ij (i, j = 1, 2,...,n) and let B(H) denotes the algebra of all bounded linear operators on a Hilbert space H. Let A and B denote C * -algebras and L : A → B be a bounded linear map.The map L is positive provided L(a) is positive whenever a is positive.The map L is said to be completely positive if The map L is said to be completely bounded if sup n L ⊗ I n is finite.We set L cb = sup n L ⊗ I n , L * (a) = L(a) * .Given S ⊆ B(H), and let S denote its commutant.An n × n matrix [f ij ] of linear functionals on a C * -algebra A is positive if [f ij (a ij )] is positive whenever [a ij ] is positive in A ⊗ M n .

A positive matrix of linear functionals.
The following result [7, Corollary 2.3] is well known.Theorem 2.1.Let F be a linear map from a C * -algebra A to M n and let the functional f : Depending on the previous result, Suen [8] proved the following theorem.
In what follows we give a new proof to this result.

Proof. Define
and a complete positive map δ : as As F , δ are positive maps, then Φ is positive.Since C is commutative, then by [2] Φ is completely positive.The complete positivity of Φ and δ insures the complete positivity of F .
Choi [2] showed that any n-positive map from a C * -algebra A to M n is completely positive.The following is a generalization of a special case.

Theorem 2.3. Via the linear functionals
Therefore, F is completely positive by Theorem 2.2, which in return gives that Ψ is completely positive.

S, T ∈ B(H) with T being positive and invertible. Then
Proof.(a) This follows from the identity and if (b) Follows from the following two identities: (2.12) Proof.Without loss of generality, assume that F ≤1. Therefore, by Lemma 2.4(b), this also follows from Lemma 2.4(a) by noticing that which implies that is completely positive and hence completely bounded.Therefore F is completely bounded.
Theorem 2.6.Let G : A → M n (C) be a bounded map defined by G(a) = [g ij (a)] ij .Then there is a representation π of A, a Hilbert space K, an isometry V : H → K, and an operator Proof.Since G is bounded, then by [5,Lemma 6] G is completely bounded.By [6, Theorem 2.5] there exist completely positive maps φ The map is completely positive, as it is identified with the map M ij Ψ M ij , which is completely positive as is completely positive.By setting Φ ij = (φ ii + ϕ jj )/2, we have for any λ for which and the maps Φ ± Re(f ij ) and Φ ± Im(f ij ) are completely positive.Let (π,V ,K) be the minimal Stinespring representation of Φ, that is, K is a Hilbert space, V : The following theorem generalizes [4,Proposition 2.4].
Then there is a representation π of A on a Hilbert space K and n vectors x 1 ,x 2 ,...,x n in K, an operator is completely positive.For |λ| = 1, the map the maps Φ ± Re(F ) and Φ ± Im(F ) are completely positive.Since Φ ≥ (Φ + Re(F ))/2, then by [4, Theorem 2.1] let π be the representation engendered by Φ on a Hilbert space K such that Φ ij (a) = π(a)x j ,x i , for some generating set of vectors x 1 ,x 2 ,...,x n for π(A).By [4, Proposition 2.4], there is a positive operator H in the unit ball of π(A) such that The following is a generalization of [8,Proposition 2.7].
, is completely bounded, then there is a representation π of A on a Hilbert space K, an isometry V : H → K, and an operator Proof.The proof it follows by the same technique used in the proof of Theorem 2.6.
The following generalizes [7,Proposition 4.2] for a special case.

Proof. By the following diagram
(2.26) The positivity of φ, F , and γ implies the positivity of Ψ .By Theorem 2.3, Ψ is completely positive.The complete positivity of Ψ , F , and γ insures the complete positivity of φ.
Theorem 2.10.There is a one-to-one correspondence between the set of all bounded linear functionals f = [f ij ] of a C * -algebra A and the set of all bounded maps F : Proof.The map f is completely bounded, by [8,Theorem 2.2].By [6, Theorem 2.5], there exist completely positive maps φ, ϕ :