New York Journal of Mathematics
New York J. Math. 24a(2018) 107–135.
Characterisations of the weak expectation property
Douglas Farenick, Ali S. Kavruk, Vern I. Paulsen and Ivan G. Todorov
Abstract. We use representations of operator systems as quotients to deduce various characterisations of the weak expectation property (WEP) for C∗-algebras. By Kirchberg’s work on WEP, these results give new formulations of Connes’ embedding problem.
Contents
Dedication 107
1. Introduction 108
2. Operator System Preliminaries 109
3. Characterisations of WEP via Group C∗-Algebras 112 4. Characterisations of WEP via Noncommutative n-Cubes 115
5. N C(n) as a quotient of C2n 123
6. The Riesz decomposition property 128
References 134
Dedication
Arveson introduced operator systems and was the first to fully appreciate and exploit the extent that many questions and results in the theory of C*-algebras could be reduced to the study of the matrix-order structure on these subspaces of C*-algebras. In this paper we exploit his viewpoint.
Bill’s kindness and humor will be surely missed, but his vision lives on.
He was a major founder of the “completely” revolution and he has influ- enced how many mathematicians think about certain problems and, more personally, how we behave professionally. We are a better field in many ways because of his influence.
Received October 25, 2013.
2010 Mathematics Subject Classification. Primary 46L06, 46L07; Secondary 46L05, 47L25, 47L90.
Key words and phrases. operator system, tensor product, discrete group, free group, free product, Kirchberg’s Conjecture, complete quotient maps, lifting property.
This work supported in part by NSERC (Canada), NSF (USA), and the Royal Society (United Kingdom).
ISSN 1076-9803/2018
107
D. FARENICK, A.S. KAVRUK, V.I. PAULSEN AND I.G. TODOROV
1. Introduction
The results of this paper were presented by the third author at the GPOTS2013 conference dedicated to the memory of W. B. Arveson.
In this paper we deduce various characterisations of Lance’s weak expec- tation property (WEP) for C*-algebras [21].
Using certain finite dimensional operator systems, their operator system duals, introduced by Choi-Effros, and a more recent construction of operator system quotients, we reduce questions about whether or not C*-algebras possess WEP to certain finite lifting problems. We show that there are many apparently different lifting problems that are all equivalent to the C*- algebra possessing WEP, and hence these lifting problems are all equivalent for C*-algebras.
Lance’s original definition of WEP requires thateveryfaithful representa- tion of the C*-algebra possesses a so-called weak expectation or, equivalently, that the universal representation, which is somewhat cumbersome, possesses a weak expectation. Given a unital C*-algebra Aand a faithful unital rep- resentationπ:A →B(H),then a weak expectation is a completely positive mapφ:B(H)→π(A)00 such thatφ(π(a)) =π(a) for everya∈ A.
One advantage of our results is that they give new characterisations of WEP in terms of a fixed given representation. Thus, one is free to choose a preferred faithful representation of the C*-algebra to attempt to determine if it has WEP. These results expand on earlier work of the first three authors [8] and of the second author [15] that also obtained such representation-free characterisations of WEP.
One major motivation for the desire to obtain such a plethora of charac- terisations of WEP are the results of Kirchberg, who proved that Connes’
embedding conjecture is equivalent to determining if certain C*-algebras have WEP. Thus, a wealth of characterisations of WEP could help to re- solve this conjecture.
Our technique is to first characterise WEP in terms of operator system tensor products with certain “universal” finite dimensional operator sys- tems. These universal operator systems have no completely order isomor- phic representations on finite dimensional spaces, but we then realise them as quotients of finite dimensional operator subsystems of matrix algebras.
This leads to characterisations of the C*-algebras possessing WEP as the C*-algebras for which these quotient maps remain quotient after tensoring with the algebra (see Theorem 4.3 for a precise formulation). Thus, WEP is realised as an “exactness” property for these operator system quotients or, equivalently, as a “lifting” property from a quotient. Because the liftings lie in finite dimensional matrix algebras, the question of the existence or non-existence of liftings can be reduced to a question about the existence of liftings satisfying elementary linear constraints.
As in the work of the second author, many of these characterisations of WEP reduce to interpolation or decomposition properties of the C*-algebra
of the type studied by F. Riesz in other contexts. In ordered function space theory, or in general, in ordered topological lattice theory, the vast use of Riesz interpolation and decomposition properties dates back to F. Riesz’s studies in the late ’30’s [29]. The reader may refer to [1] and the bibliography therein for broad applications of this concept. We also refer to [15] for a non- commutative Riesz interpolation property that characterises WEP. In this paper we give a characterisation of WEP in terms of a Riesz decomposition property.
2. Operator System Preliminaries
In this section, we introduce basic terminology and notation, and recall previous constructions and results that will be needed in the sequel. If V is a vector space, we let Mn,m(V) be the space of all n by mmatrices with entries in V. We set Mn(V) =Mn,n(V) and Mn=Mn(C). We let (Ei,j)i,j be the canonical matrix unit system inMn. For a mapφ:V →W between vector spaces, we let φ(n) : Mn(V) → Mn(W) be the nth ampliation of φ given by φ(n)((xi,j)i,j) = (φ(xi,j))i,j. For a Hilbert space H, we denote by B(H) the algebra of all bounded linear operators on H. An operator system is a subspace S of B(H) for some Hilbert space H which contains the identity operatorI and is closed under taking adjoints. The embedding ofMn(S) intoB(Hn) gives rise to the coneMn(S)+of all positive operators in Mn(S). The family (Mn(S)+)n∈N of cones is called the operator system structure of S. Every complex ∗-vector space equipped with a family of matricial cones and an order unit satisfying natural axioms can, by virtue of the Choi-Effros Theorem [5], be represented faithfully as an operator system acting on some Hilbert space. When a particular embedding is not specified, the order unit of an operator system will be denoted by 1. A mapφ:S → T between operator systems is called completely positive ifφ(n) positive, that is,φ(n)(Mn(S)+)⊆Mn(T)+, for everyn∈N. A linear bijection φ:S → T of operator systems S and T is a complete order isomorphism if both φ and φ−1 are completely positive. We refer the reader to [25] for further properties of operator systems and completely positive maps.
Anoperator system tensor product S ⊗τT of operator systemsSandT is an operator system structure on the algebraic tensor productS ⊗T satisfying a set of natural axioms. We refer the reader to [16], where a detailed study of such tensor products was undertaken. Suppose thatS1 ⊆ T1andS2 ⊆ T2are inclusions of operator systems. Letιj :Sj → Tj denote the inclusion maps ιj(xj) =xj forxj ∈ Sj,j = 1,2, so that the mapι1⊗ι2 :S1⊗S2→ T1⊗T2is a linear inclusion of vector spaces. Ifτ andσ are operator system structures on S1⊗ S2 and T1⊗ T2 respectively, then we use the notation
S1⊗τ S2 ⊆+ T1⊗σT2
D. FARENICK, A.S. KAVRUK, V.I. PAULSEN AND I.G. TODOROV
to express the fact thatι1⊗ι2 :S1⊗τS2 → T1⊗σT2 is a (unital) completely positive map. This notation is motivated by the fact that ι1⊗ι2 is a com- pletely positive map if and only if, for every n, the cone Mn(S1⊗τ S2)+ is contained in the cone Mn(T1⊗σ T2)+. If, in addition, ι1⊗ι2 is a complete order isomorphism onto its range, then we write
S1⊗τS2 ⊆coiT1⊗σ T2.
In particular, ifτ and σ are two operator system structures on S ⊗ T, then S ⊗τT = S ⊗σT means S ⊗τT ⊆coiS ⊗σT and S ⊗σT ⊆coiS ⊗τT . When S1⊗τ S2⊆+ S1⊗σS2,then we will also write τ ≥σ and say that τ majorises σ.
In the sequel, we will use extensively the following operator system tensor products introduced in [16]:
(a)The minimal tensor product min. IfS ⊆ B(H) andT ⊆ B(K), where H and K are Hilbert spaces, then S ⊗minT is the operator system arising from the natural inclusion ofS ⊗ T intoB(H ⊗ K).
(b)The maximal tensor productmax. For eachn∈N, letDn={A∗(P⊗ Q)A:A∈Mn,km(C), P ∈Mk(S)+, Q∈Mm(T)+}. The Archimedanisation [26] of the family (Dn)n∈Nof cones is an operator system structure onS ⊗T; the corresponding operator system is denoted by S ⊗maxT.
(c) The commuting tensor product c. By definition, an element X ∈ Mn(S ⊗ T) belongs to the positive cone Mn(S ⊗cT)+ if (φ·ψ)(n)(X) is a positive operator for all completely positive maps φ : S → B(H) and ψ:T → B(H) with commuting ranges. Here, the linear mapφ·ψ:S ⊗ T → B(H) is given byφ·ψ(x⊗y) =φ(x)ψ(y),x∈ S,y∈ T.
The tensor products min, c, and max are functorial in the sense that if τ denotes any of them, and φ : S1 → S2 and ψ :T1 → T2 are completely positive maps, then the tensor product mapφ⊗ψ:S1⊗τT1 → S2⊗τ T2 is completely positive. We will use repeatedly the following fact, extablished in [16]: If S is an operator system and A is a C*-algebra, then S ⊗cA = S ⊗maxA.
The three tensor products mentioned above satisfy the relations S ⊗maxT ⊆+ S ⊗cT ⊆+S ⊗minT
for all operator systems S and T.
For every operator system S, we denote by Sd the (normed space) dual of S. The space Mn(Sd) can be naturally identified with a subspace of the space L(S, Mn) of all linear maps from S into Mn. Taking the pre- image of the cone of all completely positive maps in L(S, Mn), we obtain a family (Mn(Sd)+)n∈N of matricial cones onSd. We have, in particular, that (Sd)+ consists of all positive functionals onS; the elementsφ∈(Sd)+ with φ(1) = 1 are called states of S. An important case arises when S is finite dimensional; in this case,Sd is an operator system when equipped with the
family of matricial cones just described and has an order unit given by any faithful state onS [5, Corollary 4.5].
We now move to the notion of quotients in the operator system category.
Definition 2.1. A linear subspace J ⊆ S of an operator system S is called a kernel if there is an operator system T and a completely positive linear map φ:S → T such that J = kerφ.
IfJ ⊆ S is kernel, then one may endow the ∗-vector spaceS/J with an operator system structure such that the canonical quotient map qJ :S → S/J is unital and completely positive [17]. An element (xi,j+J) is positive in Mn(S/J) if and only if for every > 0, there exist elements yi,j ∈ J such that (xi,j +yi,j) +1n ∈ Mn(S)+. Moreover, if J ⊆ kerφ for some completely positive mapφ:S → T, then there exists a completely positive map ˙φ : S/J → T such that φ = ˙φ◦qJ. A null subspace of S [14] is a subspace J which does not contain positive elements other than 0. It was shown in [14] that every null subspace is a kernel.
Definition 2.2. A unital completely positive map φ : S → T is called a complete quotient map if the natural quotient map φ˙ : S/kerφ → T is a complete order isomorphism.
Definition 2.3. Given an operator system T an element (ti,j) ∈ Mn(T) will be called strongly positive if there exists >0 such that (ti,j)−1n ∈ Mn(T)+.
Thus, an element of a C*-algebra is strongly positive if and only if it is positive and invertible and an element of an operator system is strongly positive if and only if its image under every unital completely positive map into a C*-algebra is positive and invertible.
We will write (ti,j) 0 to denote that (ti,j) is strongly positive. Given two self-adjoint elements (xi,j) and (yi,j) we will write (xi,j) (yi,j) or (yi,j)(xi,j) to indicate that (xi,j−yi,j) is strongly positive.
The concept of strongly positive element leads to the following useful characterisation of complete quotient maps [8, Proposition 3.2].
Proposition 2.4. Let S andT be operator systems and letφ:S → T be a unital completely positive surjection. Then φ is a complete quotient map if and only if, for every positive integer n, every strongly positive element of Mn(T)+ has a strongly positive pre-image.
We will frequently use the following result [10, Lemma 5.1].
Lemma 2.5. LetR,S,T andU be operator systems and assume that we are given linear maps ψ :R → S, θ:S → T, µ:R → U and ν :U → T, such that ν is a complete quotient map,µ is a complete order isomorphism, θ is a linear isomorphism, θ−1 is completely positive and θ◦ψ=ν◦µ. Then ψ is a complete quotient map if and only ifθis a complete order isomorphism.
D. FARENICK, A.S. KAVRUK, V.I. PAULSEN AND I.G. TODOROV
We recall the universal C*-algebra Cu∗(S) of an operator system S: it is the unique (up to a *-isomorphism) C*-algebra containingSand having the property that whenever ϕ:S → B(H) is a unital completely positive map, there exists a unique *-homomorphismπ :Cu∗(S)→ B(H) extending ϕ.
3. Characterisations of WEP via Group C∗-Algebras
If Gis a discrete group, we let C∗(G) denote, as is customary, the (full) group C*-algebra of G. Of particular interest are free groups with finitely many, say n, or countably many, generators, which we denote by Fn and F∞, respectively.
Kirchberg [19, Proposition 1.1(iii)] proved that a C*-algebraApossesses the weak expectation property (WEP) if and only if C∗(F∞) ⊗min A = C∗(F∞)⊗maxA. We will not use Lance’s original definition of WEP in this paper, only Kirchberg’s characterisation. In this sense our paper is re- ally about characterisations of C*-algebras that satisfy Kirchberg’s tensor formula and it is only and it is only because of Kirchberg’s theorem that these are characterisations of WEP.
Kirchberg’s Conjecture, on the other hand, asserts that the C*-algebra C∗(F∞) possesses WEP, i.e., that C∗(F∞)⊗min C∗(F∞) = C∗(F∞)⊗max C∗(F∞). Kirchberg proved that Connes’ Embedding Conjecture, which is a statement about type II1-factors, is equivalent to the statement thatC∗(F∞) possesses WEP. Consequently, many author’s refer to what we are calling Kirchberg’s Conjecture as Connes’ Embedding Problem. We prefer to dis- tinguish between the two to stress that we are using Kirchberg’s formulation.
A C∗-algebra A is said to have the quotient weak expectation property (QWEP) ifAis a quotient of a C∗-algebra Bthat has WEP. In many ways QWEP is a better behaved notion than WEP, as QWEP enjoys a number of permanence properties that are not necessarily shared by WEP: see, for example, [23, Proposition 4.1].
Lifting properties will play an important role in the sequel. If J is an ideal in a unital C∗-algebraBand ifqJ :B → B/J is the canonical quotient homomorphism, then a unital completely positive map φ : S → B/J of an operator system S into B/J is said to be liftable if there is a unital completely positive mapψ:S → Bsuch thatφ=qJ◦ψ. A unital C∗-algebra A has the lifting property (LP) if every unital completely positive map φ of A into B/J is liftable, for every unital C*-algebra B and every closed ideal J ⊆ B. A unital C∗-algebra A has the local lifting property (LLP) if for every unital completely positive map φ of A into B/J, the restriction of φto any finite dimensional operator subsystem S ⊆ A is liftable. In the operator system context, these lifting properties were studied in [17].
IfA1andA2are unital C*-algebras, we denote byA1∗A2the free product C*-algebra, amalgamated over the unit. The same notation is used for free products of groups. The following result, which combines results of Boca [2]
and Pisier [28, Theorem 1.11], will be useful for us in the sequel.
Theorem 3.1. Let A1, . . . ,An be unital C*-algebras and ϕi :Ai → B(H) be unital completely positive maps, i= 1, . . . , n. Then there exists a unital completely positive map ϕ : A1 ∗ · · · ∗ An → B(H) such that ϕ|Ai = ϕi. Furthermore, if eachAj is a separable C∗-algebra with LP, thenA1∗ · · · ∗ An has LP.
Example 3.2. The following group C∗-algebras have property LP:
(1) C∗(Fn), for alln∈N∪ {∞};
(2) C∗(SL2(Z));
(3) C∗(∗nj=1Z2), where ∗nj=1Z2 is the n-fold free product of n copies of Z2,n∈N.
Proof. The fact that C∗(Fn) andC∗(SL2(Z)) have the lifting property (LP) for all n∈N∪ {∞} was established by Kirchberg [19]. There are alternate proofs for the assertion that C∗(Fn) has LP: see [23, 28], for example.
Suppose that φ : C∗(Z2) → B/J is a unital completely positive map, where B is a unital C*-algebra and J ⊆ B is a closed ideal. Let b ∈ B be a selfadjoint contractive lifting of φ(h), where h is the generator ofC∗(Z2).
Then the linear map ˜φ : C∗(Z2) → B given by ˜φ(h) = b is unital and completely positive (see, e.g. Proposition 4.1 (2)), which is clearly a positive lifting of φ. To complete the proof observe that, because C∗(∗nj=1Z2) =
∗nj=1C∗(Z2), Theorem 3.1 implies that C∗(∗nj=1Z2) has LP.
We next record some observations that allow us to replace F∞ in the formulation of Kirchberg’s Conjecture by other discrete groups and, sub- sequently, to replace WEP by QWEP. Some of the results are certainly well-known, but we include their proofs for completeness.
Proposition 3.3. Let G1 and G2 be countable discrete groups that con- tain F2. If C∗(G1)⊗min C∗(G2) = C∗(G1)⊗maxC∗(G2), then Kirchberg’s Conjecture is true.
Proof. Since F2 contains F∞ as a subgroup, it follows by our assumption thatG1 andG2 do as well. By [28, Proposition 8.8], for i= 1,2 there exists a canonical compete order embedding ϕi :C∗(F∞)→ C∗(Gi) and a unital completely positive projection Pi : C∗(Gi) → C∗(F∞). Thus, there is a chain of completely positive maps
C∗(F∞)⊗minC∗(F∞)ϕ−→1⊗ϕ2 C∗(G1)⊗minC∗(G2) =
C∗(G1)⊗maxC∗(G2)P−→1⊗P2 C∗(F∞)⊗maxC∗(F∞),
and the result follows.
Definition 3.4. Consider the following two properties of a unital C∗-algebra A:
(1) Ahas WEP;
(2) C∗(G)⊗minA=C∗(G)⊗maxA.
D. FARENICK, A.S. KAVRUK, V.I. PAULSEN AND I.G. TODOROV
We say that a countable discrete group G detects WEP if (2) implies (1) and that G characterises WEP if (2) and (1) are equivalent.
Proposition 3.5. Every countable discrete group G that contains F2 as a subgroup detects WEP. If, in addition, C∗(G) has the local lifting property, thenG characterises WEP.
Proof. Suppose that G contains F2 and that A is a unital C∗-algebra for whichC∗(G)⊗minA=C∗(G)⊗maxA.SinceF2 containsF∞ as a subgroup, it follows by our assumption thatGdoes so as well. By [28, Proposition 8.8], there exists a canonical compete order embeddingϕ:C∗(F∞)→C∗(G) and a unital completely positive projection P :C∗(G)→ C∗(F∞). Thus, there is a chain of completely positive maps
C∗(F∞)⊗minAϕ⊗id→ C∗(G)⊗minA=C∗(G)⊗maxAP→⊗idC∗(F∞)⊗maxA, and so C∗(F∞)⊗minA=C∗(F∞)⊗maxA.
IfC∗(G) has the local lifting property andAhas WEP, thenC∗(G)⊗min A=C∗(G)⊗maxA, by [19, Proposition 1.1(i)].
Example 3.6. The following countable discrete groups characterise WEP:
(1) SL2(Z);
(2) ∗nj=1Z2, ifn≥3.
Proof. Recall that Example 3.2 shows that C∗(SL2(Z)) and C∗(∗nj=1Z2) have the lifting property, which is stronger than the local lifting property.
Moreover,SL2(Z) contains a copy ofF2(see,e.g., [19, p.486]) as does∗nj=1Z2
for n ≥ 3 (see, e.g., [12]). Thus, Proposition 3.5 applies to each of these
groups.
Remark. That the free product of n copies of Z2 detects WEP was also established by T. Fritz in [12] using a different method.
SinceSL3(Z) also containsF2this group detects WEP, but it is not known if C∗(SL3(Z)) has the lifting property. It would be interesting to know whether SL3(Z) characterises WEP. More generally, since every countable discrete group G that contains F2 as a subgroup and has the local lifting property characterises WEP (Proposition 3.5) it would be interesting to know if these two sufficient conditions are in fact necessary. A small step in this direction is the following proposition.
Proposition 3.7. If a finitely generated discrete subgroup G of GLn(C) detects WEP, then G⊇F2 and C∗(G) is a non-exact C∗-algebra.
Proof. If a finitely generated discrete subgroup GofGLn(C) does not con- tain F2 as a subgroup, then G contains a normal subgroupH such that H is solvable and G/H is finite (this is the “Tits Alternative”). As solvable and finite groups are amenable, Gis an extension of an amenable group by an amenable group and is, therefore, amenable. Hence, C∗(G) is nuclear
and therefore G cannot detect WEP. The only alternative, thus, is that G containsF2 as a subgroup.
Because G is a finitely generated linear group, every finitely generated subgroup ofGis maximally almost periodic (see,e.g., [19]). Thus, ifC∗(G) were exact, then it would in fact be nuclear [19, Theorem 7.5]; but then G
cannot detect WEP.
Observe that the argument of Proposition 3.7 applies to any countable discrete group G for which the Tits Alternative holds and it yields the inclusionG⊇F2.
Proposition 3.8. The following statements are equivalent for a countable discrete group G that contains F2 and such that C∗(G) has the local lifting property:
(1) Kirchberg’s Conjecture is true;
(2) C∗(G) has WEP;
(3) C∗(G) has QWEP.
Proof. (1) ⇒(2). If Kirchberg’s conjecture is true, thenC∗(F∞) has WEP and, hence by Proposition 3.5, C∗(G)⊗minC∗(F∞) =C∗(G)⊗maxC∗(F∞) and so by Kirchberg’s theorem, C∗(G) has WEP.
(2)⇒(3). Trivial
(3) ⇒(1). Assume C∗(G) has QWEP. Let C∗(G)⊆ C∗(G)∗∗ ⊆ B(Hu), where Hu is the Hilbert space of the universal representation of C∗(G).
Then C∗(G)∗∗ has QWEP also [19, Corollary 3.3(v)]. Since C∗(G) has the lifting property, C∗(G) is a unital C∗-subalgebra with LLP of a von Neumann subalgebra C∗(G)∗∗ of B(Hu) with QWEP. Therefore, by [19, Corollary 3.8(ii)], there is a unital completely positive map φ : B(Hu) → C∗(G)∗∗ such that φ(a) = afor every a∈C∗(G). Thus,C∗(G) has WEP.
Now by Kirchberg’s criterion for WEP [19, Proposition 1.1(iii)],C∗(G)⊗min C∗(F∞) = C∗(G) ⊗max C∗(F∞). Therefore, Propositon 3.3 implies that
Kirchberg’s Conjecture is true.
Corollary 3.9. Kirchberg’s conjecture is true if and only if C∗(F2) is a quotient of a C∗-algebra with WEP.
4. Characterisations of WEP via Noncommutative n-Cubes Let Fn be the free group on n generators and ∗nj=1Z2, be the n-fold free product of the group Z2 of two elements (n∈N). Following [17], we let
Sn= span{1, ui, u∗i : 1 = 1, . . . , n} ⊆C∗(Fn),
whereu1, . . . , un are the generators ofFn viewed as elements ofC∗(Fn) and u−1i =u∗i,i= 1, . . . , n. We also let [9] N C(n) be the operator system
N C(n) = span{1, hi:i= 1, . . . , n} ⊆C∗(∗nj=1Z2),
where h1, . . . , hn are the canonical generators of ∗nj=1Z2 (that is, hi is the non-trivial element of theith copy ofZ2 in∗nj=1Z2, viewed as an element of
D. FARENICK, A.S. KAVRUK, V.I. PAULSEN AND I.G. TODOROV
C∗(∗nj=1Z2)). The operator system N C(n) is called the operator system of the noncommutative n-cube.
We note that Sn and N C(n) are characterised by the following universal properties [9], [17]:
Proposition 4.1. Let S be operator system.
(1) Ifx1, . . . , xn∈ S are contractions then there exists a (unique) unital completely positive map ϕ : Sn → S such that ϕ(ui) = xi, i = 1, . . . , n.
(2) If x1, . . . , xn ∈ S are selfadjoint contractions then there exists a (unique) unital completely positive map ϕ0 :N C(n) → S such that ϕ0(hi) =xi, i= 1, . . . , n.
In fact, in [9], N C(n) is originally defined via this above universal prop- erty.
By [9, Proposition 5.7], the linear mapψ:Sn→N C(n) given byψ(1) = 1 and ψ(ui) = ψ(u∗i) = hi, i = 1, . . . , n, is a complete quotient map. Note that
kerψ=Sn0 def= ( n
X
i=−n
λiui:λ0 = 0, λi+λ−i = 0, i= 1, . . . , n )
. Let Tn+1 be the tridiagonal operator system in Mn+1, that is, Tn+1 = span{Ei,j : |i−j| ≤ 1}, where {Ei,j}i,j denote the standard matrix unit system. By [10, Theorem 4.2], the linear map φ : Tn+1 → Sn given by φ(Ei,j) = n+11 uj−i, is a complete quotient map. Note that
kerφ= (n+1
X
i=1
λiEi,i :
n+1
X
i=1
λi = 0 )
.
The symbolsφand ψwill be used to denote the maps introduced above.
Let Kn+1def=
(n+1 X
i=1
aiEi,i+
n
X
i=1
(biEi,i+1−biEi+1,i) :
n+1
X
i=1
ai = 0 )
⊆ Tn+1.
Proposition 4.2. The map ρ def= ψ◦φ : Tn+1 → N C(n) is a complete quotient map with kernelKn+1.
Proof. Sinceφandψare complete quotient maps, their composition is also a complete quotient (indeed, if X ∈ Mk(N C(n)) is strongly positive then by [8, Proposition 3.2] it has a strongly positive liftingY ∈Mk(Sn) and, by the same result, Y has a strongly positive lifting in Mk(Tn+1)). The image of an element u=Pn+1
i=1 aiEi,i+Pn
i=1(biEi,i+1+ciEi+1,i) underψ◦φis ψ(φ(u)) = 1
n+ 1
n+1
X
i=1
ai1 + 1 n+ 1
n
X
i=1
(bi+ci)hi;
thus, ψ(φ(u)) = 0 precisely when Pn+1
i=1 ai = 0 andbi+ci = 0, i= 1, . . . , n.
Theorem 4.3. The following statements are equivalent for a C*-algebra A:
(1) N C(n)⊗minA=N C(n)⊗maxA;
(2) the map ρ⊗min id : Tn+1 ⊗minA → N C(n)⊗minA is a complete quotient map.
Moreover, if n≥3, then these statements are also equivalent to:
(3) Apossesses WEP.
Proof. The mapρ⊗minid is completely positive by the functoriality of min.
On the other hand,ρ⊗maxid :Tn+1⊗maxA →N C(n)⊗maxAis a complete quotient map by [10, Proposition 1.6]. By [10, Proposition 4.1], and the fact thatTn+1⊗cA=Tn+1⊗maxA(see [16, Proposition 6.7]), the canonical map Tn+1⊗maxA → Tn+1⊗minA is a complete order isomorphism. It follows from Lemma 2.5 that (1) and (2) are equivalent.
Suppose n ≥3 and set B =C∗(∗nj=1Z2). Assume (3) holds. The group
∗nj=1Z2containsF2(see, for example, [12]) and hence, by Proposition 3.5 and Example 3.2 (3),B ⊗minA=B ⊗maxA. By the injectivity of min, we have N C(n)⊗minA ⊆coiB ⊗minA, and by [9, Lemma 6.2], N C(n)⊗maxA ⊆coi B ⊗maxA. It now follows that N C(n)⊗minA=N C(n)⊗maxA.
Finally, assume (1). By [9, Proposition 2.2], B = Ce∗(N C(n)). The natural inclusion of vector spacesN C(n)⊗minA → B ⊗maxAis completely positive as it is the composition of the completely positive mapsN C(n)⊗min A →N C(n)⊗maxAand N C(n)⊗maxA → B ⊗maxA. It follows from [17, Proposition 9.5] that the natural mapB ⊗minA → B ⊗maxA is completely positive and hence B ⊗minA=B ⊗maxA. Proposition 3.5 now shows that
A has WEP.
Corollary 4.4. The operator system N C(n) has the lifting property for every n∈ N and N C(n)⊗minB(H) = N C(n)⊗maxB(H) for every n ∈N and every Hilbert spaceH.
Proof. If ϕ:N C(n)→ B/J is a unital completely positive map, then the images of the generators ofN C(n) are hermitian contractions inB/J.But each hermitian contraction inB/J can be lifted to a hermitian contraction in B and these elements induce a unital completely positive lifting of φ by Proposition 4.1(2). The tensor equality follows from Theorem 4.3 and the
fact that B(H) possesses WEP.
Corollary 4.5. The map ρ⊗min id : T3 ⊗min A → N C(2)⊗min A is a complete quotient map for every unital C∗-algebraA. Hence, ifA0, A1, A2 ∈ Mk(A) are such that 1⊗A0 +h1 ⊗A1 +h2 ⊗A2 is strongly positive in N C(2)⊗minMk(A), then there exist elements A, B, C, X, Y ∈ Mk(A) with
D. FARENICK, A.S. KAVRUK, V.I. PAULSEN AND I.G. TODOROV
A+B+C=A0,X+X∗ =A1, Y +Y∗=A2 such that the matrix
A X 0
X∗ B Y 0 Y∗ C
is strongly positive in M3k(A).
Proof. By [9, Theorem 6.3],N C(2) is (min,c)-nuclear and now [16, Propo- sition 6.7] shows that N C(2)⊗minA = N C(2)⊗maxA. By Theorem 4.3, ρ⊗minid :T3⊗minA →N C(2)⊗minA is a complete quotient map. Hence, if u = 1⊗A0 +h1 ⊗A1 +h2 ⊗A2 is strongly positive in N C(2)⊗min Mk(A), then [8, Proposition 3.2] and Proposition 2.4 implies that there exist A, B, C, X, Y ∈Mk(A) such that
v=E1,1⊗A+E2,2⊗B+E3,3⊗C+E1,2⊗X+E2,1⊗X∗+E2,3⊗Y+E3,2⊗Y∗ is strongly positive inT3⊗minA and
u= (ρ⊗id)(k)(v) = 1
3(1⊗(A+B+C) +h1⊗(X+X∗) +h2⊗(Y +Y∗)).
It follows that 13(A+B +C) = A0, 13(X +X∗) = A1, 13(Y +Y∗) = A2. RescalingA, B, C, X andY by a factor of 13 shows the claim.
Corollary 4.6. The following statements are equivalent for a unital C∗- algebraA:
(1) Ahas WEP;
(2) whenever A0, A1, A2, A3 ∈Mk(A) are such that 1⊗A0+h1⊗A1+ h2⊗A2+h3⊗A3is strongly positive inN C(3)⊗minMk(A), then there exist elementsA, B, C, D, X, Y, Z ∈Mk(A)withA+B+C+D=A0, X+X∗ =A1, Y +Y∗ =A2 and Z+Z∗ =A3 such that the matrix
A X 0 0
X∗ B Y 0
0 Y∗ C Z
0 0 Z∗ D
is strongly positive inM4k(A).
Proof. As in the proof of Corollary 4.5, one can see that (2) is equivalent to the canonical mapρ⊗minid :T4⊗minA →N C(3)⊗minAbeing a complete quotient map. By Theorem 4.3, the latter condition is equivalent to A
having WEP.
We next include a characterisation of WEP in terms of liftings of strongly positive elements. We recall that the numerical radius w(x) of an element x of an operator systemS is given by w(x) = sup{|f(x)|:f a state ofS}.
Lemma 4.7. If S,T and R are operator systems and τ ∈ {min,c}, then ((S ⊕ T)⊗τR)+= ((S ⊗τR)⊕(T ⊗τ R))+.
Proof. It is clear that there is a linear identification ι : (S ⊕ T)⊗ R → (S ⊗ R)⊕(T ⊗ R). Suppose that u∈ ((S ⊗cR)⊕(T ⊗cR))+, and write u= (u1, u2), withu1∈(S ⊗cR)+andu2 ∈(T ⊗cR)+. Letf :S ⊕T → B(H) andg:R → B(H) be completely positive maps with commuting ranges. Let f1 =f|S and f2 =f|T. Then f1 and f2 are completely positive and hence f1·g(u1)≥0 andf2·g(u2)≥0. But thenf·g(ι−1(u)) =f1·g(u1)+f2·g(u2)≥ 0.
Conversely, assume that u ∈ ((S ⊕ T) ⊗cR)+. Write ι(u) = (u1, u2).
If f1 : S → B(H) and g : R → B(H) are completely positive maps with commuting ranges, then the map f :S ⊕ T → B(H) given by f((x, y)) = f1(x) is completely positive and hence f1 ·g(u1) = f ·g(u) ≥ 0. Thus, u1 ∈(S ⊗cR)+; similarly,u2∈(T ⊗cR)+.
The statement regarding min is immediate from the injectivity of this
tensor product.
Theorem 4.8. The following statements are equivalent for a unital C∗- algebraA:
(1) Ahas WEP;
(2) whenever
X = 1⊗A0+u1⊗A1+u∗1⊗A∗1+u2⊗A2+u∗2⊗A∗2
is a strongly positive element ofMk(S2⊗minA), where A0, A1, A2 ∈ Mk(A), there exist strongly positive elements B, C ∈ Mk(A) such that
A0= 1
2(B+C), w(B−12A1B−12)< 1
2 and w(C−12A2C−12)< 1 2. (3) whenever A1, A2 ∈Mk(A) satisfy w(A1, A2) <1/2 then there exist
positive invertible elements B, C∈Mk(A) such that 1
2(B+C) =I, w(B−12A1B−12)<1/2 andw(C−12A2C−12)<1/2.
Proof. We first prove the equivalence of (1) and (2). Let J = span{(1,−1)} ⊆ S1⊕ S1.
It follows from [15, Corollary 4.4] and [15, Proposition 4.7] that J is a kernel and S2 = (S1⊕ S1)/J. Let q :S1 ⊕ S1 → S2 be the corresponding (complete) quotient map. By [9, Proposition 3.3],S1is (min,c)-nuclear and henceS1⊗minA=S1⊗maxA. On the other hand, for every operator system Sand every unital C*-algebraB, we have thatMk(S ⊗minB) =S ⊗minMk(B) andMk(S ⊗maxB) =S ⊗maxMk(B),k∈N. It now follows from Lemma 4.7 that (S1⊕ S1)⊗minA= (S1⊕ S1)⊗maxA. In the diagram
(S1⊕ S1)⊗minA = (S1⊕ S1)⊗maxA
↓ ↓
S2⊗minA ← S2⊗maxA,
D. FARENICK, A.S. KAVRUK, V.I. PAULSEN AND I.G. TODOROV
the right arrow denotes a complete quotient map by [10, Proposition 1.6], while the left arrow denotes the completely positive map q⊗minid arising from the functoriality of min. By Lemma 2.5, S2⊗minA = S2 ⊗maxA if and only if q⊗minid is a complete quotient map. By [14, Theorem 5.9], it suffices to show that (2) is equivalent toq⊗minid being a complete quotient map.
To this end, suppose thatq⊗minid is a complete quotient and letXbe the strongly positive element ofMk(S2⊗minA) given in (2). By [8, Proposition 3.2], there exists a strongly positive element Y ∈ Mk((S1 ⊕ S1)⊗min A) with (q⊗min id)(k)(Y) = X. By virtue of Lemma 4.7, write Y = (Y1, Y2), where Y1 and Y2 are strongly positive elements of S1 ⊗min Mk(A). Write Y1= 1⊗B+ζ⊗B1+ ¯ζ⊗B2 and Y2 = 1⊗C+ζ⊗C1+ ¯ζ⊗C2, where we have denoted by ζ the generator of S1, viewed as the identity function on the unit circle T. It follows that
1⊗1
2(B+C) +u1⊗B1+u∗1⊗B2+u2⊗C1+u∗2⊗C2 =X, which shows that 12(B+C) =A0,B1=B2∗=A1,C1=C2∗=A2.
Suppose thatY1≥δ1. Then, for everyz∈Twe have thatB+zA1+¯zA∗1≥ δI inMk(A). Takingz=±1, we see thatB ≥δ1 and henceB is invertible.
Thus,I +zB−12A1B−12 + ¯zB−12A∗1B−12 ≥ kBkδ I inMk(A), for everyz ∈T. By [8, Theorem 1.1], this implies that w(B−12A1B−12)< 12. Similarly, C is invertible andw(C−12A2C−12)< 12.
Conversely, if (2) is satisfied then reversing the steps in the previous two paragraphs shows that the elementY = (Y1, Y2) is a strongly positive lifting of X. By [8, Proposition 3.2] and Proposition 2.4, q⊗minid is a complete quotient map. This proves the equivalence of (1) and (2).
We now show that (2) and (3) are equivalent.
Recall that w(A1, A2)<1/2 if and only if
I⊗I+A1⊗u1+A∗1⊗u∗1+A2⊗u2+A∗2⊗u∗2
is strictly positive. From this we see that (2) implies (3). Conversely, if (3) holds then (2) holds for the case thatA0=I. For the general case, use that fact that the strict positivity implies thatA0 is positive and invertible and conjugate by A−
1 2
0 .
Theorem 4.9. The following statements are equivalent forA=C∗(∗mj=1Z2):
(1) ρ⊗minid :Tn+1⊗minA →N C(n)⊗minAis a complete quotient map;
(2) N C(n)⊗minN C(m) =N C(n)⊗cN C(m).
Moreover, if n, m≥3, then these statements are equivalent to:
(3) Kirchberg’s Conjecture holds true.
Proof. (1) ⇒ (2). By Theorem 4.3, N C(n)⊗minA=N C(n)⊗maxA. On the other hand, N C(n)⊗minN C(m) ⊆coiN C(n)⊗minA by the injectivity
of min, whileN C(n)⊗cN C(m)⊆coiN C(n)⊗maxAby [9, Lemma 6.2]. It follows thatN C(n)⊗minN C(m) =N C(n)⊗cN C(m).
(2) ⇒ (1). Set B = C∗(∗nj=1Z2). By [9, Lemma 6.2], we have that N C(n)⊗cN C(m) ⊆coiB ⊗maxA. The assumption implies that the linear embedding β:N C(n)⊗minN C(m)→ B ⊗maxAis completely positive. On the other hand,N C(n)⊗minN C(m)⊆coiB⊗minAandβ(u⊗v) is unitary, for all canonical unitary generators u (resp. v) of the operator system N C(n) (resp. N C(m)). Sinceu⊗vfor suchuandvgenerateB ⊗minA, [17, Lemma 9.3] implies thatφ has an extension to a *-homomorphismπ :B ⊗minA → B⊗maxA. Thus, every positive element ofMk(N C(n)⊗minA) is sent viaπ(k) to a positive element ofMk(N C(n)⊗maxA) and, by Theorem 4.3,ρ⊗minid is a complete quotient map.
Suppose that n, m ≥ 3. Assuming (2), we have seen that B ⊗minA = B ⊗max A. By [12, Corollary C.4], C∗(F2) ⊗min C∗(F2) = C∗(F2) ⊗max C∗(F2), and hence Kirchberg’s Conjecture holds. Conversely, if Kirchberg’s Conjecture holds thenSn⊗minSm =Sn⊗maxSm. Denoting for a moment by ψn(resp. ψm) the canonical quotient map fromSn onto N C(n) introduced after Proposition 4.1, we have, by [10, Proposition 1.6], that ψn ⊗ψm : Sn⊗maxSm → N C(n)⊗maxN C(m) is a complete quotient map. Let γn : N C(n) → Sn be the linear map given by γn(hi) = hi+h2 ∗i, i= 1, . . . , n. By [9, Proposition 5.7], γn is a complete order isomorphism onto its range and a right inverse ofψn. Moreover, the map γn⊗γm :N C(n)⊗minN C(m)→ Sn⊗minSm sis completely positive. A standard diagram chase now shows that (2) holds: namely,N C(n)⊗minN C(m) =N C(n)⊗maxN C(m).
We conclude this section with another realisation ofN C(n) as a quotient of a matrix operator system, which leads to a different characterisation of WEP. Following [10], let
Wn= span{uiu∗j :i, j= 0,1, . . . , n} ⊆C∗(Fn),
where we have setu0 = 1. Letβ :Mn+1 → Wn be the linear map given by β(Ei,j) = n+11 u∗iuj,i, j= 0, . . . , n. It follows from [10] thatβ is a complete quotient map with kernel the space D0n+1 of all diagonal matrices of trace zero; thus, β : Mn+1/Dn+10 → Wn is a complete order isomorphism. (We note that the map sending Ei,j to uiu∗j was considered in [10] but since {u∗1, . . . , u∗n} is a set of universal unitaries whenever {u1, . . . , un} is such, the claims remain true with our definition as well.) Clearly, Sn⊆ Wn and
Rn+1 def= β−1(Sn) = span{E1,j, Ej,1, Ej,j :j= 1, . . . , n+ 1}.
Let γ denote the restriction of β to Rn+1. We claim that γ is a complete quotient map fromRn+1ontoSn. Indeed, ifXis a strongly positive element of Mk(Sn) then X is also strongly positive as an element of Mk(Wn). By [8, Proposition 3.2], there exists Y ∈ Mk(Mn+1) such that β(k)(Y) = X.
However, R∈Mk(Rn+1) by the definition ofRn+1.
D. FARENICK, A.S. KAVRUK, V.I. PAULSEN AND I.G. TODOROV
We note that the map γ is defined by the relations γ(Ei,i) = n+11 1, γ(E1,i) = n+11 u∗i,i= 1, . . . , n+ 1.
Proposition 4.10. The mapψ◦γ :Rn+1→N C(n) is a complete quotient map with kernel
Ln+1= span
n+1
X
i=1
aiEi,i+
n+1
X
j=2
bj(E1,j−Ej,1) :
n+1
X
i=1
ai = 0
.
Proof. Since both γ : Rn+1 → Sn and ψ : Sn → N C(n) are complete quotient maps, the mapψ◦γ is also a complete quotient. The identification
of its kernel is straightforward.
Since the graph underlying the operator systemRn+1 is chordal (in fact, it is a tree and hence does not have cycles), Rn+1⊗minA=Rn+1⊗maxA for any unital C*-algebra A (see [16, Proposition 6.7]). Thus, a version of Theorem 4.3 can be formulated with Rn+1 in the place of Tn+1. The methods in the proof of Corollary 4.6 can be used to obtain the following characterisation of WEP.
Corollary 4.11. The following statements are equivalent for a unital C*- algebraA:
(1) Ahas WEP;
(2) whenever A0, A1, A2, A3 ∈Mk(A) are such that 1⊗A0+h1⊗A1+ h2⊗A2+h3⊗A3 is strongly positive in N C(3)⊗minMk(A), there exist elementsA, B, C, D, X, Y, Z ∈Mk(A)withA+B+C+D=A0, X+X∗ =A1, Y +Y∗ =A2 and Z+Z∗ =A3 such that the matrix
A X Y Z
X∗ B 0 0 Y∗ 0 C 0
Z∗ 0 0 D
is strongly positive inM4k(A).
Although Corollary 4.6 and Corollary 4.11 both give characterisations of WEP in terms of “completions” of 4×4 matrices, there does not appear to be a direct connection between the two sets of conditions. In fact, even though both of these results arise from realising N C(3) as a quotient ofT4 and R4, respectively, we shall now show that these later operator systems are not completely order isomorphic.
Proposition 4.12. The operator systems Rn and Tn are not completely order isomorphic unless n∈ {1,2,3}.
Proof. For a graph G onnvertices, letSG be the “graph operator system”
(see [16])
SG= span{Ei,j, Ek,k :k= 1, . . . , n,(i, j)∈ G}.
We first claim that if G1 and G2 are connected graphs on n vertices and ϕ:SG1 → SG2 is a complete order isomorphism then there exists a unitary U ∈ Mn such that U∗SG1U = SG2. Indeed, since G1 and G2 are connected, the C*-algebrasC∗(SG1) andC∗(SG2) generated bySG1 andSG2, respectively, both coincide with Mn. Since Mn is simple, we have that the C*-envelopes Ce∗(SG1) and Ce∗(SG2) of SG1 and SG2, respectively, both coincide with Mn. The complete order isomorphismϕnow gives rise to an isomorphism between their C*-envelopes and hence there exists an isomorphsim ˜ϕ : Mn → Mn extendingϕ. LetU ∈Mnbe a unitary matrix with ˜ϕ(A) =U∗AU,A∈Mn; thenU∗SG1U =SG2.
Now note that Tn+1 and Rn+1 are both graph operator systems. Let Pk=U∗Ek,kU,k= 1, . . . , n+ 1, andC= span{Pk:k= 1, . . . , n+ 1}. Since Tn+1 is a bimodule over the algebra Dn+1 of all diagonal matrices, Rn+1 is a bimodule over C. Note that each Pk is a rank one operator. Assume that not all of P1, . . . , Pn+1 are equal to a diagonal matrix unit in Rn+1, suppose, for example, that P1 = (λiλj)n+1i,j=1 is not of the form Ek,k. Set Λ = {k : λk 6= 0}; then span{Ei,j : i, j ∈ Λ} ⊆ Rn+1. However, the only full matrix subalgebras of Rn+1 are of the form span{E1,1, E1,j, Ej,1, Ej,j}, for somej. Assume, without loss of generality, thatj= 1. But thenP1E1,3
hasλ2λ1 as its (2,3)-entry, contradicting the definition ofRn+1.
It follows that {Pk}n+1k=1 ={Ek,k}n+1k=1, so that there exists a permutation π of{1, . . . , n+ 1}withPk=Eπ(k),π(k),k= 1, . . . , n+ 1. If we letUπ denote the corresponding permutation unitary, then UπEk,kUπ∗ = Pk = U∗Ek,kU.
Hence, Uπ∗U∗Ek,kU Uπ = Ek,k for all k and consequently, U Uπ is diagonal.
Thus,UπTn+1Uπ∗ =U∗Tn+1U =Rn+1.This means thatπ defines an isomor- phism of the underlying graphs ofTn+1 andRn+1.This is a contradiction if n≥3 since the graph underlyingTn+1 has at least two vertices of degree 2, while the graph underlyingRn+1 has only one vertex of degree bigger than
1.
5. N C(n) as a quotient of C2n
In this section we representN C(n) as an operator system quotient of the abelian C*-algebraC2nin two different ways and include some consequences of these results. In the next section we will use these two representations to give two more characterisations of WEP. We first recall some basic facts about coproducts of operator systems. Coproducts in this category were used by D. Kerr and H. Li [18], where the authors described the amal- gamation process over a joint operator subsystem. T. Fritz demonstrated some applications of this concept in quantum information theory [11]. A categorical treatment and further results can be found in the thesis of the second author [14]. We next extend the results from [11] and [14] to deduce representations of the coproduct of (finitely) many operator systems.
LetS1, . . . ,Sn be operator systems. Then there exists a unique operator system U, along with the unital complete order embeddings im :Sm ,→ U,
