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Lecture Notes on “Completely Positive Semigroups and Applications to II1

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Lecture Notes on “Completely Positive Semigroups and Applications to II1 Lecture notes on \Completely positive semigroups and applications ∗ to II1 factors" Jesse Peterson May 24, 2013 Contents 1 Completely positive maps 1 1.1 Stinespring's Dilation Theorem . 2 1.2 Bhat's Dilation Theorem . 6 1.3 Poisson boundaries . 9 2 The Poisson boundary of a finite von Neumann algebra 12 3 Derivations 15 3.1 Examples . 21 3.1.1 Inner derivations . 21 3.1.2 Approximately inner derivations . 21 3.1.3 The difference quotient . 22 3.1.4 Group algebras and group-measure space constructions . 22 3.1.5 Equivalence relations . 23 3.1.6 Free products . 24 3.1.7 Free Brownian motion . 24 3.1.8 Tensor products . 24 3.1.9 q-Gaussians . 25 3.1.10 Conditionally negative definite kernels . 26 4 Approximate bimodularity 26 1 Completely positive maps An operator system E is a closed self adjoint subspace of a unital C∗-algebra A such that ∗ 1 2 E. We denote by Mn(E) the space of n × n matrices over E. If A is a C -algebra, then ∗From the minicourse given at the workshop on Operator Algebras and Harmonic Analysis, at Instituto de Ciencias Mathem´aticas 1 ∼ ∗ Mn(A) = A ⊗ Mn(C) has a unique norm for which it is again a C -algebra, where the adjoint given ∗ ∗ ∗ by [ai;j] = [aj;i]. This can be seen easily for C -subalgebras of B(H), and the general case then follows since every C∗-algebra is isomorphic to a C∗-subalgebra of B(H) by the GNS-construction. In particular, if E ⊂ A is an operator system then Mn(E) is again an operator system when viewed ∗ as a subspace of the C -algebra Mn(A). (n) If φ : E ! F is a linear map between operator systems, then we denote by φ : Mn(E) ! (n) Mn(F ) the map defined by φ ([ai;j]) = [φ(ai;j)]. We say that φ is positive if φ(a) ≥ 0, whenever (n) a ≥ 0. If φ is positive then we say that φ is n-positive and if φ is n-positive for every n 2 N then we say that φ is completely positive. We'll say that φ is unital if φ(1) = 1. Throughout these notes we will use the abbreviation u.c.p. for unital completely positive. If φ : E ! F is positive then it follows easily that φ is Hermitian, i.e., φ(x∗) = φ(x)∗, for all x 2 E. Also note that positive maps are continuous. Indeed, if φ : E ! F is positive and fxngn is a sequence which converges to 0 in E, such that limn!1 φ(xn) = y, then since ! ◦ φ is positive (and hence continuous by the Cauchy-Schwarz inequality) for any state ! 2 S(B), we then have !(y) = 0. Since ! was arbitrary it then follows that y = 0 and hence φ is bounded by the closed graph theorem. We also remark that it follows form the Hahn-Banach theorem that any positive linear functional on E extends to a positive linear functional on A which has the same norm. Lemma 1.1. If A and B are unital C∗-algebras and φ : A ! B is a unital contraction then φ is positive. Proof. We first show that φ is Hermitian. Suppose x = x∗ 2 A such that φ(x) = a + ib where a; b 2 B are self-adjoint. Assume kxk ≤ 1. If λ 2 σ(b) then for all t 2 R we have (λ + t)2 ≤ kb + tk2 ≤ kφ(x + it)k2 ≤ kx + itk2 ≤ 1 + t2: Hence λ2 + tλ ≤ 1, and as this is true for all t we must then have λ = 0, and hence b = 0. Thus, ! ◦ φ is a state, for any state ! 2 S(B). Hence if x ≥ 0 then for any state ! we have !(a) = ! ◦ φ(x) ≥ 0. We must then have a ≥ 0, and hence φ is positive. 1.1 Stinespring's Dilation Theorem If π : A !B(K) is a representation of a C∗-algebra A and V 2 B(H; K), then the map φ : A !B(H) given by φ(x) = V ∗π(x)V is completely positive. Indeed, if we consider the operator (n) ⊕n ⊕n (n) V 2 B(H ; K ) given by V ((ξi)i) = (V ξi)i then for all x 2 Mn(A) we have ∗ φ(n)(x∗x) = V (n) π(n)(x∗x)V (n) = (π(n)(x)V (n))∗(π(n)(x)V (n)) ≥ 0: Generalizing the GNS construction, Stinespring showed that every completely positive map from A to B(H) arises in this way. Theorem 1.2 (Stinespring [Sti55]). Let A be a unital C∗-algebra, and suppose φ : A !B(H), then φ is completely positive if and only if there exists a representation π : A !B(K) and a bounded operator V 2 B(H; K) such that φ(x) = V ∗π(x)V . We also have kφk = kV k2, and if φ is unital then V is an isometry. Moreover, if A is a von Neumann algebra and φ is a normal completely positive map, then π is a normal representation. 2 ∗ Proof. Consider the sesquilinear form on A ⊗ H given by ha ⊗ ξ; b ⊗ ηiφ = hφ(b a)ξ; ηi, for a; b 2 A, ⊕n ⊕n ξ; η 2 H. If (ai)i 2 A , and (ξi)i 2 H , then we have X X X ∗ h ai ⊗ ξi; aj ⊗ ξjiφ = hφ(aj ai)ξi; ξji i j i;j ∗ = hφ((ai)i (ai)i)(ξi)i; (ξi)ii ≥ 0: Thus, this form is non-negative definite and we can consider Nφ the kernel of this form so that h·; ·iφ is positive definite on K0 = (A ⊗ H)=Nφ. Hence, we can take the Hilbert space completion K = K0. As in the case of the GNS construction, we define a representation π : A !B(K) by first setting ∗ ∗ π0(x)(a ⊗ ξ) = (xa) ⊗ ξ for a ⊗ ξ 2 A ⊗ H. Note that since φ is positive we have φ(a x xa) ≤ 2 ∗ (n) P 2 2 P 2 kxk φ(a a), applying this to φ we see that kπ0(x) i ai ⊗ ξikφ ≤ kxk k i ai ⊗ ξikφ. Thus, π0(x) descends to a well defined bounded map on K0 and then extends to a bounded operator π(x) 2 B(K). If we define V0 : H!K0 by V0(ξ) = 1 ⊗ ξ, then we see that V0 is bounded by kφ(1)k and hence extends to a bounded operator V 2 B(H; K). For any x 2 A, ξ; η 2 H we then check that ∗ hV π(x)V ξ; ηi = hπ(x)(1 ⊗ ξ); 1 ⊗ ηiφ = hx ⊗ ξ; 1 ⊗ ηiφ = hφ(x)ξ; ηi: Thus, φ(x) = V ∗π(x)V as claimed. Corollary 1.3 (Kadison [Kad52]). If A and B are unital C∗-algebras, and φ : A ! B is u.c.p. then for all x 2 A we have φ(x)∗φ(x) ≤ φ(x∗x) Proof. We may assume that B ⊂ B(H). If we consider the Stinespring dilation φ(x) = V ∗π(x)V , then since φ is unital we have that V is an isometry. Hence 1 − VV ∗ ≥ 0 and so we have φ(x∗x) − φ(x)∗φ(x) = V ∗π(x∗x)V − V ∗π(x)∗VV ∗π(x)V = V ∗π(x∗)(1 − VV ∗)π(x)V ≥ 0: Lemma 1.4. A matrix a = [ai;j] 2 Mn(A) is positive if and only if n X ∗ xi ai;jxj ≥ 0; i;j=1 for all x1; : : : ; xn 2 A. Pn ∗ Proof. For all x1; : : : ; xn 2 A we have that i;j=1 xi ai;jxj is the conjugation of a by the 1 × n Pn ∗ column matrix with entries x1; : : : ; xn, hence if a is positive then so is i;j=1 xi ai;jxj. 3 Pn ∗ Conversely, if i;j=1 xi ai;jxj ≥ 0, for all x1; : : : ; xn then for any representation π : A !B(H), and ξ 2 H we have 0 1 0 1 * π(x1)ξ π(x1)ξ + n B . C B . C X (id ⊗π)(a) @ . A ; @ . A = hπ(ai;j)π(xj)ξ; π(xi)ξi i;j=1 π(xn)ξ π(xn)ξ * 0 n 1 + X ∗ = π @ xi ai;jxjA ξ; ξ ≥ 0: i;j=1 Thus, if H has a cyclic vector, then (id ⊗π)(a) ≥ 0. But since every representation is decomposed into a direct sum of cyclic representations it then follows that (id ⊗π)(a) ≥ 0 for any representation, and hence a ≥ 0 by considering a faithful representation. Proposition 1.5. Let E be an operator system, and let B be an abelian C∗-algebras. If φ : E ! B is positive, then φ is completely positive. Proof. Since B is commutative we may assume B = C0(X) for some locally compact Hausdorff space X. If a = [ai;j] 2 Mn(E) such that a ≥ 0, then for all x1; : : : ; xn 2 B, and ! 2 X we have 0 1 0 1 X ∗ X @ xi φ(ai;j)xjA (!) = @ φ(xi(!)xj(!)ai;j)A (!) i;j i;j 00 1∗ 0 11 x1(!) x1(!) BB . C B . CC = φ @@ . A a @ . AA (!) ≥ 0: xn(!) xn(!) By Lemma 1.4, and since n was arbitrary, we then have that φ is completely positive. Proposition 1.6. Let A and B be unital C∗-algebras such that A is abelian. If φ : A ! B is positive, then φ is completely positive.
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