Completely Positive Maps Paul Skoufranis August 21, 2014 Abstract These notes are based on my knowledge of completely positive maps that I have gained throughout my studies. Most of the results contained in these notes may be found in Vern Paulsen's Completely Bounded Maps and Operators Algebras which is a very complete resource on these topics. However, my notes will focus on those ideas that are essential to be able to study C∗-algebras and will take a slightly different point of view towards the subject matter. These notes will assume that the reader has a basic knowledge of C∗-algebras including knowledge of the Continuous Functional Calculus for Normal Operators, a basic knowledge about positive operators, and knowledge of C∗-bounded approximate identities for C∗-algebras. All C∗-algebras will be non-unital unless otherwise specified and all inner products will be linear in the first variable. This document is for educational purposes and should not be referenced. Please contact the author of this document if you need aid in finding the correct reference. Comments, corrections, and recom- mendations on these notes are always appreciated and may be e-mailed to the author (see his website for contact info). Contents 1 Introduction 2 2 Positive Maps 15 3 Completely Positive and Completely Bounded Maps 23 4 Arveson's Extension Theorem and Stinespring's Theorem 34 5 Applications of Completely Positive Maps 49 6 Liftings of Completely Positive Maps 59 7 Wittstock's Theorem 62 1 Introduction Completely positive maps are an important collection of morphisms between C∗-algebras. These maps have many of the same properties as ∗-homomorphism even though they are generally not multiplicative. Various interesting properties of C∗-algebra can be developed by considering how completely positive maps behave on these C∗-algebras. To begin these notes, we will review how positive linear functionals on C∗-algebra behave. Definition 1.1. Let A be a C∗-algebra. A linear functional ' : A ! C is said to be positive if '(A) ≥ 0 whenever A 2 A and A ≥ 0. A linear functional ' : A ! C is said to be a state if ' is positive and k'k = 1. States are the building blocks of the proof of the GNS theorem; the fact that for every C∗-algebra A there exists a Hilbert space H such that A may be viewed as a C∗-subalgebra of B(H). Before we give some examples of positive linear functionals, we first note the following simple yet important observation. Lemma 1.2. Let A be a C∗-algebra and let ' : A ! C be a positive linear functional. Then '(A) 2 R whenever A is a self-adjoint element of A. Moreover '(A∗) = '(A) for all A 2 A. Proof. Suppose A 2 A be self-adjoint. By the Continuous Functional Calculus for Normal Operators, A = A+ − A− where A+ and A− are positive operators. Since ' is a positive linear functional, '(A+) and '(A−) are positive scalars and thus '(A) = '(A+) − '(A−) 2 R. Next suppose that A 2 A is an arbitrary operator. Then '(Re(A));'(Im(A)) 2 R so '(A∗) = '((Re(A) + iIm(A))∗) = '(Re(A) − iIm(A)) = '(Re(A)) − i'(Im(A)) = '(Re(A)) + i'(Im(A)) = '(A) as desired. Examples of positive linear functionals are abundant in mathematics. Example 1.3. Let X be a compact Hausdorff space and let µ be a probability measure on X. If ' : C(X) ! is defined by C Z '(f) = f(x)dµ(x) X for each f 2 C(X), then ' is a state on C(X). To see this we first notice that ' is clearly linear by integration theory. Next if f 2 C(X) is positive then f = g∗g for some g 2 C(X). Whence for all x 2 X f(x) = g∗(x)g(x) = g(x)g(x) ≥ 0. Therefore '(f) is the integral of a continuous function that is positive everywhere and thus '(f) ≥ 0. Moreover Z Z j'(f)j ≤ jf(x)jdµ(x) ≤ kfk1 dµ(x) ≤ kfk1 µ(X) = kfk1 X X for all f 2 C(X). Thus k'k ≤ 1. Since '(IC(X)) = µ(X) = 1, k'k = 1. Whence ' is a state on C(X). In particular the map f 7! f(x) is defines a state on C(X) for all x 2 X. Example 1.4. Let A be a C∗-algebra and let π : A !B(H) be a ∗-homomorphism. For each ξ 2 H define 'ξ : A ! C by 'ξ(A) = hπ(A)ξ; ξi for all A 2 A. It is clear that 'ξ is well-defined and linear. In addition if A 2 A is positive then A = B∗B for some B 2 A and thus ∗ 'ξ(A) = hπ(B B)ξ; ξi = hπ(B)ξ; π(B)ξi = kπ(B)ξk ≥ 0: Whence each 'ξ is a positive linear functional. Note that for all A 2 A 2 j'ξ(A)j ≤ kπ(A)ξk kξk ≤ kAk kξk 2 2 2 so k'ξk ≤ kξk . Moreover, if A is unital and π(IA) = IH then k'ξk = kξk as 'ξ(IA) = kξk . If 'ξ is a state, we call 'ξ a vector state on A. 2 In fact it will be shown later that every state on a C∗-algebra is a vector state on A. 1 Pn Example 1.5. Let A = Mn(C) and define tr : A ! C by tr([ai;j]) = n j=1 aj;j for all [ai;j] 2 Mn(C). It is clear that tr is a linear functional. To see that tr is positive, we notice that for all A = [ai;j] 2 A " n #! ∗ X tr(A A) = tr ak;iak;j k=1 n n 1 X X = a a n k;j k;j j=1 k=1 n n 1 X X = ja j2 ≥ 0: n k;j j=1 k=1 Whence tr is a positive linear functional. We claim that tr is a state on A. To see this, we notice that tr(In) = 1 so ktrk ≥ 1. To prove the other inequality, let A = [ai;j] 2 A be arbitrary and let Ei;j be the canonical matrix units of Mn(C). Since Ei;i is a projection, kEi;ik = 1. Then jai;ij = kai;iEi;ik = kEi;iAEi;ik ≤ kEi;ik kAk kEi;ik = kAk : Thus n n 1 X 1 X jtr(A)j ≤ ja j ≤ kAk = kAk : n j;j n j=1 j=1 Hence ktrk = 1 and tr is a state on A. Notice that all of the positive linear functionals given in the above examples were continuous even though continuity was not required in Definition 1.1. It turns out that if a linear functional is positive then it is automatically continuous. To prove this, we begin with a lemma about convergence of positive elements in a C∗-algebra that the reader may not be familiar with. ∗ Lemma 1.6. Let A be a C -algebra. Suppose (An)n≥1 2 A is a sequence such that limn!1 An = A 2 A and An ≥ 0 for all n 2 N. Then A is positive. Proof. By considering the unitization A~ of A, we may assume that A is unital. By the continuity of the ∗ ∗ adjoint A = limn!1 An = limn!1 An = A as each An is self-adjoint. Let c := supn≥1 kAnk < 1. Thus kAk ≤ c. Since 0 ≤ An ≤ cIA for all n, 0 ≤ 2An ≤ 2cIA for all n and thus −cIA ≤ 2An − cIA ≤ cIA for all n. Thus (by the Continuous Functional Calculus) k2An − cIAk ≤ c for all n 2 N. As limn!1 An = A, limn!1 2An − cIA = 2A − cIA so k2A − cIAk ≤ c. Whence −cIA ≤ 2A − cIA ≤ cIA and thus 0 ≤ A ≤ cIA as desired. Proposition 1.7. Let A be a C∗-algebra and let ' : A ! C be a positive linear functional. Then ' is continuous. Proof. First we claim that f'(A) j A 2 A;A ≥ 0; kAk ≤ 1g is a bounded subset of R≥0 (that is, we claim ' is bounded on the positive elements of norm at most one). To see this, suppose otherwise. Then 2 for each n 2 N there would exists an An 2 A such that An ≥ 0, kAnk ≤ 1, and '(An) ≥ n . Consider P1 1 1 B := n≥1 n2 An. Notice that B is a well-defined operator in A as ( n2 An)n≥1 is an absolutely summable 1 1 sequence since n2 An ≤ n2 . For each m 2 N m 1 X 1 X 1 B − A = A ≥ 0 n2 n n2 n n=1 n≥m+1 3 by Lemma 1.6. Whence m m ! X 1 X 1 ' (B) − ' A = ' B − A ≥ 0 n2 n n2 n n=1 n=1 so m m X 1 X '(B) ≥ '(A ) ≥ 1 = m n2 n n=1 n=1 for every m 2 N. As this is an impossibility, we must have that f'(A) j A 2 A;A ≥ 0; kAk ≤ 1g is bounded in R≥0. Let M := supf'(A) j A 2 A;A ≥ 0; kAk ≤ 1g and let A 2 A be arbitrary. Write A = Re(A) + iIm(A) and recall kRe(A)k ; kIm(A)k ≤ kAk. By the Continuous Functional Calculus, both Re(A) and Im(A) can be written as the difference of two positive elements each with norm at most kRe(A)k and kIm(A)k respectively. Whence A = P1 − P2 + iP3 − iP4 where Pj 2 A are positive elements with kPjk ≤ kAk.