Tensor Products of Convex Cones, Part II: Closed Cones in Finite-Dimensional Spaces
TTensor Products of Convex Cones, Part II:Closed Cones in Finite-Dimensional Spaces
Josse van Dobben de Bruyn24 September 2020
Abstract
In part I, we studied tensor products of convex cones in dual pairs of real vector spaces.This paper complements the results of the previous paper with an overview of the mostimportant additional properties in the finite-dimensional case. (i) We show that the projectivecone can be identified with the cone of positive linear operators that factor through a simplexcone. (ii) We prove that the projective tensor product of two closed convex cones is onceagain closed (Tam already proved this for proper cones). (iii) We study the tensor product ofa cone with its dual, leading to another proof (and slight extension) of a theorem of Barkerand Loewy. (iv) We provide a large class of examples where the projective and injective conesdiffer. As this paper was being written, this last result was superseded by a result of Aubrun,Lami, Palazuelos and Pl´avala, who independently showed that the projective cone E + ⊗ π F + is strictly contained in the injective cone E + ⊗ ε F + whenever E + and F + are closed, properand generating, with neither E + nor F + a simplex cone. Compared to their result, this paperonly proves a few special cases. Tensor products of ordered vector spaces have been studied extensively in functional analysis,mostly with a focus on Riesz spaces and Banach lattices (a large number of references can be foundin [Dob20]). Meanwhile, tensor products of closed convex cones in finite-dimensional spaces havereceived attention in various different fields, such as linear algebra ([Bar76, Tam92]), interpolationtheory ([Mul97]), polyhedral geometry ([BCG13]), operator systems ([PTT11, FNT17, HN18]),matrix convex sets ([PSS18]), and theoretical physics ([ALP19, ALPP19]). However, the majorityof results from functional analysis are not relevant in this setting, for the only closed latticecones in a finite-dimensional space are the ones isomorphic with the standard cone R n ≥ . As such,papers on tensor products of finite-dimensional cones deal with completely different problemsthan papers on tensor products of Riesz spaces and Banach lattices.Even if one is primarily interested in tensor products of infinite-dimensional ordered vectorspaces, it is good to be aware of the finite-dimensional theory, as various essential phenomenacan already be observed here. (For instance, the projective cone does not preserve subspaces andthe injective cone does not preserve quotients, even if the spaces are finite-dimensional and the Mathematics Subject Classification . Primary: 52A20. Secondary: 06F20, 46A40, 47L07.
Key words and phrases . Closed convex cone, partially ordered vector space, ordered tensor product. a r X i v : . [ m a t h . F A ] S e p ones are polyhedral; see [Dob20, Examples 3.7 and 4.20].) For this reason, this paper aims toprovide an overview of the most important additional properties in the finite-dimensional setting.The study of tensor products of closed, proper and generating convex cones in finite-dimensionalcan be traced back to the early 1970s. Important contributions were made by George PhillipBarker (e.g. [BL75, Bar76, Bar81]) and Bit-Shun Tam (e.g. [Tam77, Tam92]), among others. Thereferences in [Bar81] and [Tam92] provide an extensive overview of the literature from aroundthat time.More recently, tensor products of cones in finite-dimensional spaces have started to turn up inmany other fields. For example, Mulansky [Mul97] discusses the relevance of cone tensor productsto interpolation theory, and in polyhedral geometry the tensor product of two polyhedral cones isclosely related to the so-called Hom-polytope ([BCG13]). Furthermore, the tensor product of ageneral cone with a positive semidefinite cone plays a role in the theories of operator systems([PTT11, FNT17, HN18]) and matrix convex sets ([PSS18]), and recent results in theoreticalphysics show some of the applications of cone tensor products to so-called general probabilistictheories ([ALP19, ALPP19]).The aim of this paper is to complement the preceding paper [Dob20] with an overview of themost important additional properties in the finite-dimensional case. Contrary to the first paper,most of the results of this paper are not new, but merely slight extensions of existing results(Theorem B, Theorem C), or independent discovery of recent results (Theorem D, Theorem E).Nevertheless, all the proofs in this paper are original, and the general presentation is streamlined.
This paper studies tensor products of convex cones (otherwise known as wedges) in finite-dimensional real vector spaces. If E is a vector space and E + ⊆ E is a convex cone, we say E + is proper if E + ∩ − E + = { } and generating if E + − E + = E . In general, we do not assume conesto be proper, generating, or closed.If E and F are vector spaces and E + ⊆ E , F + ⊆ F are convex cones, then the projective cone E + ⊗ π F + ⊆ E ⊗ F and the injective cone E + ⊗ ε F + ⊆ E ⊗ F are given by E + ⊗ π F + := ( k X i =1 x i ⊗ y i : k ∈ N , x , . . . , x k ∈ E + , y , . . . , y k ∈ F + ) ; E + ⊗ ε F + := (cid:8) u ∈ E ⊗ F : h u, ϕ ⊗ ψ i ≥ ϕ ∈ E ∗ + , ψ ∈ F ∗ + (cid:9) ;where E ∗ + ⊆ E ∗ and F ∗ + ⊆ F ∗ denote the dual cones. For additional notation, see § Tensor products as spaces of positive linear maps If E + and F + are closed, then it is well-known that the natural isomorphism E ⊗ F ∼ = L( E ∗ , F )identifies the injective cone E + ⊗ ε F + with the cone of positive linear operators E ∗ → F ; that is,those linear maps T : E ∗ → F which satisfy T [ E ∗ + ] ⊆ F + . We give a similar classification of theprojective cone E + ⊗ π F + . Theorem A.
Let E and F be finite-dimensional real vector spaces, and let E + ⊆ E , F + ⊆ F beclosed convex cones. Then the natural isomorphism E ⊗ F ∼ = L( E ∗ , F ) identifies the projectivecone E + ⊗ π F + with the cone of positive linear operators E ∗ → F that factor ( positively ) throughsome R n with the standard cone R n ≥ . The proof of Theorem A will be given in § E + is a simplex cone (or Yudin cone ) if it is generated by a basis, or equivalently,if there is a linear isomorphism E ∼ = R n that identifies E + with the standard cone R n ≥ . Borrowingterminology from the theory of normed tensor products, Theorem A can be expressed as statingthat the projective cone is associated with the ideal of simplex-factorable operators in the semiringof positive operators , but we will not make extensive use of this terminology. The closure of the projective cone
In general, the projective tensor product of two Archimedean cones is not necessarily Archimedean(e.g. [PTT11, Remark 3.12]). Things are different in the finite-dimensional setting. Here a cone isArchimedean if and only if it is closed (cf. [AT07, Corollary 3.4]). Using a compactness argument,Tam [Tam77, Theorem 1] showed that the projective tensor product of two closed, proper andgenerating convex cones is once again closed. We give another (short) proof of this fact, andextend it to all closed convex cones. Furthermore, by starting with arbitrary cones and takingclosures, we prove the following slightly more general statement.
Theorem B.
Let E and F be finite-dimensional real vector spaces, and let E + ⊆ E , F + ⊆ F beconvex cones. Then the closure of the projective cone E + ⊗ π F + is the projective cone E + ⊗ π F + .In particular, the projective tensor product of closed convex cones is closed. The proof of Theorem B will be given in § E + ⊗ π F + ) ∗ = E ∗ + ⊗ ε F ∗ + (a similar statement holds in the infinite-dimensional case; see [Dob20, § E + ⊗ ε F + ) ∗ = E ∗ + ⊗ π F ∗ + whenever E and F are finite-dimensional; see Corollary 4.11(b). The tensor product of a closed convex cone with its dual cone
The simplest example where the projective and injective cones are different occurs when takingthe tensor product of a closed cone with its dual. For closed, proper, and generating cones, thiswas first observed by Barker and Loewy [BL75, Proposition 3.1], and later extended and simplifiedby Tam [Tam77, Theorem 3]. We reformulate their results in terms of tensor products, extendthem to all closed convex cones (not just proper ones), and give a slightly different proof.
Theorem C.
Let E be a finite-dimensional real vector space and let E + ⊆ E be a closed convexcone. Then the following are equivalent:(i) E + is a simplex cone;(ii) id E : E → E factors ( positively ) through some R n with the standard cone R n ≥ ;(iii) for every positive linear map T : E → E , one has tr( T ) ≥ ;(iv) for every finite-dimensional real vector space F and every closed convex cone F + ⊆ F , onehas E ∗ + ⊗ π F + = E ∗ + ⊗ ε F + ;(v) for every finite-dimensional real vector space F and every closed convex cone F + ⊆ F , onehas F + ⊗ π E + = F + ⊗ ε E + ;(vi) E ∗ + ⊗ π E + = E ∗ + ⊗ ε E + . The proof of Theorem C will be given in §
5. Our proof is much shorter than the original proofby Barker and Loewy, but comparable in size to the proof of Tam.3 any examples where the projective and injective cones differ
In the infinite-dimensional setting, the question of whether or not the projective cone E + ⊗ π F + is dense in the injective cone E + ⊗ ε F + has attracted some attention. For locally convex lattices E and F , Birnbaum [Bir76, Proposition 3] proved that E + ⊗ π F + is dense in E + ⊗ ε F + in theprojective topology (and therefore in every coarser topology), and followed this by an exampleshowing that this is not true for all ordered locally convex spaces (not necessarily lattice ordered).In general, however, this problem does not appear to be well understood.Inspired by recent results in the theory of operator systems ([FNT17, Theorem 4.7], [HN18]),this paper attempts to get a grip on this problem in the finite-dimensional setting. In light ofTheorem B, the projective cone E + ⊗ π F + is dense in the injective cone E + ⊗ ε F + if and onlyif E + ⊗ π F + = E + ⊗ ε F + , so the problem can be reduced to the case where E + and F + areclosed. We show that for most pairs of closed, proper, and generating convex cones, the projectiveand injective cones differ. Hence, in the infinite-dimensional setting, one should not expect theprojective cone to be dense in the injective cone.Recall that a closed, proper and generating convex cone is called strictly convex if everynon-zero boundary point is an extremal direction, and smooth if every non-zero boundary pointhas exactly one supporting hyperplane. It is well-known that E + is strictly convex if and onlyif E ∗ + is smooth, and vice versa. If the larger cone (in terms of dimension) is strictly convex orsmooth, then the projective and injective cones differ, unless the other cone is a simplex cone. Theorem D.
Let E , F be finite-dimensional real vector spaces, and let E + ⊆ E , F + ⊆ F beclosed, proper, and generating convex cones. If dim( E ) ≥ dim( F ) , and if E + is strictly convex orsmooth, then one has E + ⊗ π F + = E + ⊗ ε F + if and only if F + is a simplex cone. The proof of Theorem D will be given in § R n which are not smooth or not strictlyconvex form a meagre set in the Hausdorff metric. (This was later strengthened to σ -porosity byZamfirescu [Zam87].) As such, Theorem D shows that the projective and injective cones differ foralmost all pairs of closed, proper, and generating cones ( E + , F + ). Nevertheless, this result doesnot cover many standard cones, such as polyhedral cones and positive semidefinite cones. Wecomplement Theorem D with a similar result for combinations of standard cones. Theorem E.
Let E , F be finite-dimensional real vector spaces, and let E + ⊆ E , F + ⊆ F beclosed, proper, and generating convex cones. Assume that each of E + and F + is one of thefollowing ( all combinations allowed ) :(a) a polyhedral cone;(b) a second-order cone;(c) a ( real or complex ) positive semidefinite cone.Then one has E + ⊗ π F + = E + ⊗ ε F + if and only if at least one of E + and F + is a simplex cone. The proof of Theorem E will be given in §
8, using the concept of retracts studied in § Remark F.
As this series of papers was being written, the preceding results were superseded bya result of Aubrun, Lami, Palazuelos and Pl´avala [ALPP19]. Motivated by questions in theoreticalphysics, they proved that E + ⊗ π F + = E + ⊗ ε F + if and only if at least one of E + and F + isa simplex cone (provided that E + and F + are closed, proper, and generating; cf. Example 9.2below). Theorem D and Theorem E are merely special cases of this result. Furthermore, the resultsof Theorem E can already be deduced from an earlier paper of Aubrun, Lami and Palazuelos[ALP19]. We had not been aware of these results until shortly before publication, and our proofsdiffer from the proofs in [ALP19] and [ALPP19].4 pplications to operator systems In terms of operator systems (using notation from [FNT17]), our results prove the following.
Corollary G.
Let C ⊆ R d be a closed, proper, and generating convex cone. If d ≤ , or if C isstrictly convex, or smooth, or polyhedral, then the following are equivalent:(i) C is a simplex cone;(ii) the minimal and maximal operator systems C min and C max are equal;(iii) one has C min2 = C max2 . The proof of Corollary G will be given in § C . Throughout this article, all vector spaces are over the real numbers.Let E be a vector space. A ( convex ) cone is a non-empty subset K ⊆ E satisfying K + K ⊆ K and λ K ⊆ K for all λ ∈ R ≥ . If K is a convex cone, then lin( K ) := K ∩ −K is a linear subspace of E , called the lineality space of K . We say that K is proper if lin( K ) = { } , and generating if K − K = E . The ( algebraic ) dual cone K ∗ ⊆ E ∗ is the set of positive linear functionals: K ∗ := (cid:8) ϕ ∈ E ∗ : h x, ϕ i ≥ x ∈ K (cid:9) . If E is finite-dimensional, then K ∗ is closed, and the natural isomorphism E ∗∗ ∼ = E identifies thedouble dual cone K ∗∗ with the closure K .If K is a convex cone, then a base of K is a convex subset B ⊆ K \ { } such that each x ∈ K \ { } can be written uniquely as x = λb with λ > b ∈ B . If K is generating, then thebases of K are in bijective correspondence with the strictly K -positive linear functionals on E (cf. [AT07, Theorem 1.47]). Furthermore, every closed proper cone in a finite-dimensional spaceadmits a compact base (e.g. [AT07, Corollary 3.8]).We say that a convex cone E + ⊆ E is a simplex cone (or Yudin cone ) if it is generated by abasis of E , or equivalently, if every base of E + is a simplex. A simplex cone turns E into a Dedekindcomplete Riesz space (cf. [AT07, Theorem 3.17]). Furthermore, a cone in a finite-dimensionalspace is a simplex cone if and only if it is a closed lattice cone (cf. [AT07, Theorem 3.21]).We fix notation for a number of standard cones. For n ≥
1, we let L n ⊆ R n denote the n -dimensional second-order cone (or Lorentz cone , or ice cream cone ), L n := (cid:8) ( x , . . . , x n ) ∈ R n : q x + · · · + x n − ≤ x n (cid:9) . (By convention, L is just the standard cone R + ⊆ R .) Furthermore, let S n ⊆ R n × n denote thespace of real symmetric n × n matrices, and let H n ⊆ C n × n the space of complex hermitian n × n matrices. We denote the respective positive semidefinite cones by S n + and H n + : S n + := { n × n real positive semidefinite matrices } ; H n + := { n × n complex positive semidefinite matrices } . Some authors call this pointed or salient .
5e recall that S is isomorphic with the Lorentz cone L , for instance via the isomorphism S → R , (cid:18) a bb c (cid:19) ( a − c, b, a + c ) . (Use that A ∈ S is positive semidefinite if and only if tr( A ) ≥ A ) ≥ For the remainder of this paper, it will be convenient to reformulate questions regarding theprojective and injective cones in terms of positive linear operators.If E and F are preordered by convex cones E + ⊆ E , F + ⊆ F , then a positive linear operator T : E → F is called separable if it can be written as T = P ki =1 ϕ i ⊗ y i , where ϕ , . . . , ϕ k ∈ E ∗ + are positive linear functionals and y , . . . , y k ∈ F + are positive elements.If G is another finite-dimensional real vector space, preordered by a convex cone G + ⊆ G ,then we say that a positive linear map T : E → F factors through G is there exist positive linearmaps R : E → G and S : G → F such that T = S ◦ R . We say that T factors through a simplexcone if it factors through some R n , ordered by the standard cone R n ≥ . Proposition 3.1.
A positive linear map T : E → F is separable if and only if it factors througha simplex cone R n ≥ . If this is the case, one may take n ≤ dim( E ) × dim( F ) .Proof. “= ⇒ ”. Write T = P ki =1 ϕ i ⊗ y i with ϕ , . . . , ϕ k ∈ E ∗ + , y , . . . , y k ∈ F + . By Carath´eodory’stheorem (for cones), we may assume without loss of generality that k ≤ dim( E ∗ ⊗ F ) =dim( E ) × dim( F ). Now define R : E → R k and S : R k → F by R ( x ) := ( ϕ ( x ) , . . . , ϕ k ( x )); S ( λ , . . . , λ k ) := λ y + · · · + λ k y k . Then R and S are positive ( R k equipped with the standard cone R k ≥ ), and T = S ◦ R .“ ⇐ =”. Suppose that T factors as E R n F, R S with R and S positive ( R n equipped with the standard cone R n ≥ ). Let e , . . . , e n ∈ R n denote thestandard basis of R n , and e ∗ , . . . , e ∗ n the corresponding dual basis. Define ϕ , . . . , ϕ n ∈ E ∗ and y , . . . , y n ∈ F by setting ϕ i := e ∗ i ◦ R and y i := S ( e i ). Then ϕ , . . . , ϕ n ∈ E ∗ + , y , . . . , y n ∈ F + ,and T = P ni =1 ϕ i ⊗ y i . (cid:4) Corollary 3.2. If E + and F + are closed, then:(a) E ∗ + ⊗ ε F + is the set of all positive linear maps E → F ;(b) E ∗ + ⊗ π F + is the set of all positive linear maps E → F that factor through a simplex cone.Proof. It is well known that E ∗ + ⊗ ε F + can be identified with the cone of linear maps T : E → F satisfying T [ E + ] ⊆ F + (e.g. [Dob20, Remark 4.2]). Since E + and F + are closed, these are simplythe positive linear maps E → F .Under this identification, it is clear from the definition that E ∗ + ⊗ π F + is the subset of separablepositive linear maps E → F . By Proposition 3.1 these are precisely the positive linear maps E → F that factor through a simplex cone. (cid:4) The terminology was introduced by other authors in connection with quantum theory (e.g. [Hil08, ALP19]). The closure of the projective cone
In this section, we prove that the closure of the projective cone E + ⊗ π F + is equal to the projectivetensor product of E + and F + . In particular, the projective tensor product of closed convex conesis closed. In the case where E + and F + are closed, proper, and generating, this was alreadyestablished by Tam [Tam77].The proof is carried out in three steps. First we prove the result for closed proper cones,thereby giving another proof of the aforementioned result by Tam. Secondly, we extend this to allclosed convex cones by decomposing a closed convex cone as the sum of a closed proper cone anda subspace. After this, it will be relatively simple to deduce the general formula for the closure ofthe projective cone. Recall that a base (of E + ) is a convex subset B ⊆ E + with 0 / ∈ B such that every x ∈ E + \ { } can be written uniquely as x = λb with λ > b ∈ B . A key property is that a subset B ⊆ E + is a base if and only if there is a strictly positive linear functional f : E → R such that B = { x ∈ E + : f ( x ) = 1 } ; see e.g. [AT07, Theorem 1.47].Not every proper cone has a base. However, every closed proper cone in a finite-dimensionalspace has a compact base (e.g. [AT07, Corollary 3.8]), and conversely the convex cone generatedby a compact convex set S ⊆ R n \ { } is a closed proper cone (e.g. [AT07, Lemma 3.12]). Proposition 4.1.
Let
E, F be real vector spaces, and let E + ⊆ E , F + ⊆ F be convex coneshaving bases B E + ⊆ E + , B F + ⊆ F + . Then conv( B E + ⊗ B F + ) is a base of E + ⊗ π F + .Proof. Let f : E → R and g : F → R be strictly positive linear functionals such that B E + = { x ∈ E + : f ( x ) = 1 } and B F + = { y ∈ F + : g ( y ) = 1 } . Then f ⊗ g is a strictly positive linearfunctional on E ⊗ F (with respect to the projective cone), and we have B E + ⊗ B F + ⊆ { z ∈ E + ⊗ π F + : ( f ⊗ g )( z ) = 1 } . Since the right-hand side is convex, it follows thatconv( B E + ⊗ B F + ) ⊆ { z ∈ E + ⊗ π F + : ( f ⊗ g )( z ) = 1 } . (4.2)On the other hand, every non-zero element of E + ⊗ π F + can be written as a positive multiple ofan element in conv( B E + ⊗ B F + ), so we must have equality in (4.2). (cid:4) Corollary 4.3 ([Tam77]) . If E + ⊆ E , F + ⊆ F are closed proper cones, then E + ⊗ π F + is alsoa closed proper cone.Proof. Choose compact bases B E + ⊆ E + , B F + ⊆ F + . Then, by Proposition 4.1, conv( B E + ⊗ B F + )is a base for E + ⊗ π F + . In particular, 0 / ∈ conv( B E + ⊗ B F + ).The natural map E × F → E ⊗ F is continuous, so B E + ⊗ B F + is compact in E ⊗ F . Sincethe convex hull of a compact set in R n is also compact (e.g. [Rud91, Theorem 3.20(d)]), it followsthat conv( B E + ⊗ B F + ) is compact. Thus, E + ⊗ π F + is generated by a compact convex set notcontaining 0, so it follows that E + ⊗ π F + is a closed proper cone. (cid:4) Remark 4.4.
The proof of Corollary 4.3 shows directly that E + ⊗ π F + is proper whenever E + and F + are closed proper cones. As such, this provides yet another way to prove that theprojective tensor product of proper cones is always proper, in addition to the different waysdiscussed in [Dob20, Remark 3.12]. 7 .2 The projective tensor product of closed convex cones In order to extend Corollary 4.3 to all closed convex cones, we decompose each of E + and F + asthe sum of a closed proper cone and a subspace. The (straightforward) proof of the followingclassical result is omitted. Proposition 4.5.
Let E be a finite-dimensional, and let E + ⊆ E be a closed convex cone. Let lin( E + ) := E + ∩ − E + be the lineality space of E + , and let lin( E + ) ⊥ be any complementarysubspace of lin( E + ) . Then lin( E + ) ⊥ + := lin( E + ) ⊥ ∩ E + is a closed proper cone, and one has E + = lin( E + ) + lin( E + ) ⊥ + . Conversely, the sum of a closed proper cone and a subspace need not be closed. (Example: let E + be the cone generated by { ( x, y, ∈ R : ( x − + y ≤ } , and X := span { (0 , , } ⊆ R .)However, we have the following partial converse of Proposition 4.5. Proposition 4.6.
Let E be a finite-dimensional, let E + ⊆ E be a closed convex cone, and let X ⊆ E be a subspace. If span( E + ) ∩ X = { } , then E + + X is a closed convex cone.Proof. Extend span( E + ) to a complementary subspace X ⊥ of X . Let P : E → X ⊥ be theprojection x + x ⊥ x ⊥ . Then P is continuous, and E + + X = P − [ E + ], so E + + X is closed. (cid:4) The preceding propositions give us a way to decompose the cones and later put them backtogether. To see what happens when we lift the pieces separately, we use the following observation.
Proposition 4.7.
Let
E, F be real vector spaces, and let E + ⊆ E , F + ⊆ F be convex cones. Ifat least one of E + and F + is a subspace, then E + ⊗ π F + is a subspace as well.Proof. A convex cone G + is a subspace precisely when one has s ∈ G + if and only if − s ∈ G + .This property is preserved by the projective tensor product. (cid:4) We can now extend Corollary 4.3 to all closed convex cones.
Theorem 4.8.
Let
E, F be finite-dimensional, and let E + ⊆ E , F + ⊆ F be closed convex cones.Then E + ⊗ π F + is closed as well.Proof. Choose complementary subspaces lin( E + ) ⊥ ⊆ E and lin( F + ) ⊥ ⊆ F of lin( E + ) and lin( F + ).Then, by Proposition 4.5, we have E + = lin( E + ) + lin( E + ) ⊥ + and F + = lin( F + ) + lin( F + ) ⊥ + , withlin( E + ) ⊥ + and lin( F + ) ⊥ + closed proper cones. It follows that E + ⊗ π F + = subspace z }| {(cid:0) lin( E + ) ⊗ π lin( F + ) (cid:1) + subspace z }| {(cid:0) lin( E + ) ⊗ π lin( F + ) ⊥ + (cid:1) + subspace z }| {(cid:0) lin( E + ) ⊥ + ⊗ π lin( F + ) (cid:1) + (cid:0) lin( E + ) ⊥ + ⊗ π lin( F + ) ⊥ + (cid:1)| {z } closed proper cone . (The first three terms are subspaces by Proposition 4.7; the fourth is a closed proper cone byCorollary 4.3.) The three subspaces in the preceding formula are contained in the subspace(lin( E + ) ⊗ lin( F + )) + (lin( E + ) ⊗ lin( F + ) ⊥ ) + (lin( E + ) ⊥ ⊗ lin( F + )), whereas lin( E + ) ⊥ + ⊗ π lin( F + ) ⊥ + is a closed proper cone contained within lin( E + ) ⊥ ⊗ lin( F + ) ⊥ . These containing subspaces arecomplementary, so it follows from Proposition 4.6 that E + ⊗ π F + is closed. (cid:4) Remark 4.9.
We should point out that nothing like Theorem 4.8 is true in the infinite-dimensionalsetting. In fact, the projective tensor product of closed proper cones in Banach spaces might noteven be Archimedean (see e.g. [PTT11, Remark 3.12]).8 .3 The closure of the projective cone; duality
Using Theorem 4.8, it is now relatively easy to prove the following.
Theorem 4.10.
Let E and F be finite-dimensional real vector spaces, and let E + ⊆ E , F + ⊆ F beconvex cones. Then the closure of the projective cone E + ⊗ π F + is the projective cone E + ⊗ π F + .Proof. “ ⊇ ”. Given x ∈ E + , y ∈ F + , choose sequences { x n } ∞ n =1 and { y n } ∞ n =1 in E + and F + converging to x and y , respectively. Then x ⊗ y = lim n →∞ x n ⊗ y n ∈ E + ⊗ π F + .“ ⊆ ”. Evidently, E + ⊗ π F + ⊆ E + ⊗ π F + . By Theorem 4.8, E + ⊗ π F + is closed, so we also have E + ⊗ π F + ⊆ E + ⊗ π F + . (cid:4) Other consequences of Theorem 4.8 include the following.
Corollary 4.11.
Let
E, F be finite-dimensional, and let E + ⊆ E , F + ⊆ F be convex cones.Then:(a) E + ⊗ π F + is dense in E + ⊗ ε F + if and only if E + ⊗ π F + = E + ⊗ ε F + ;(b) ( E + ⊗ ε F + ) ∗ = E ∗ + ⊗ π F ∗ + .Proof. (a) It is clear from the definition that E + ⊗ ε F + = E + ⊗ ε F + , and that this cone is alwaysclosed. Thus, the conclusion follows immediately from Theorem 4.10.(b) We have ( E ∗ + ⊗ π F ∗ + ) ∗ = E ∗∗ + ⊗ ε F ∗∗ + = E + ⊗ ε F + . Using once again that E + ⊗ ε F + = E + ⊗ ε F + , we find that ( E ∗ + ⊗ π F ∗ + ) ∗ = E + ⊗ ε F + . By Theorem 4.8, E ∗ + ⊗ π F ∗ + is closed,so the result follows by duality. (cid:4) In other words, if the spaces are finite-dimensional and the cones are closed, then we have fullduality between the projective and injective cones.
The simplest case where the projective and injective cone are different occurs when considering thetensor product of a closed convex cone with its dual. The main result of this section, Theorem 5.1,is a slight extension of a well-known result of Barker and Loewy [BL75, Proposition 3.1] (see also[Tam77, Theorem 4]), who proved it for convex cones which are closed, proper and generating.The proof below is much simpler than the original proof in [BL75], since it was not realized atthe time that the projective tensor product of closed convex cones is automatically closed.If E + and F + are closed, then by Corollary 3.2 one has E ∗ + ⊗ π F + = E ∗ + ⊗ ε F + if and onlyif every positive linear map E → F factors through a simplex cone. This language makes itmuch easier to think about the difference between projective and injective cones. Note: if T or S factors through a simplex cone, then so does the composition S ◦ T . This shows that the separablepositive operators form an ideal in the semiring of positive operators. (Ideals of operators alsoplay an important role in the theory of normed tensor products; e.g. [DF93].)We proceed to examine the tensor product of E ∗ + and E + . Theorem 5.1 (cf. [BL75, Proposition 3.1], [Tam77, Theorem 4]) . Let E be finite-dimensionaland let E + ⊆ E be a closed convex cone. Then the following are equivalent:(i) E + is a simplex cone; ii) id E : E → E is separable ( i.e. factors through a simplex cone ) ;(iii) for every positive linear map T : E → E , one has tr( T ) ≥ ;(iv) for every finite-dimensional real vector space F and every closed convex cone F + ⊆ F , onehas E ∗ + ⊗ π F + = E ∗ + ⊗ ε F + ;(v) for every finite-dimensional real vector space F and every closed convex cone F + ⊆ F , onehas F + ⊗ π E + = F + ⊗ ε E + ;(vi) E ∗ + ⊗ π E + = E ∗ + ⊗ ε E + .Proof. (i) = ⇒ (ii) . Let x , . . . , x d be a basis of E which generates E + , and let x ∗ , . . . , x ∗ d be thecorresponding dual basis. Then x ∗ , . . . , x ∗ d ∈ E ∗ + , and id E = P di =1 x ∗ i ⊗ x i . (ii) ⇐⇒ (iii) . The trace tr ∈ L ( E, E ) ∗ = ( E ∗ ⊗ E ) ∗ = E ⊗ E ∗ is the transpose of the identityid E ∈ L ( E, E ) = E ∗ ⊗ E . Since ( E ∗ + ⊗ ε E + ) ∗ = E + ⊗ π E ∗ + , we see that the trace defines a positivelinear functional (i.e. tr ∈ ( E ∗ + ⊗ ε E + ) ∗ ) if and only if id E is separable (i.e. id E ∈ E ∗ + ⊗ π E + ). (ii) = ⇒ (iv) and (ii) = ⇒ (v) . Since id E : E → E factors through a simplex cone, so do allpositive linear maps to or from E : R m R m or F ∗ E E E E F. id E id E Therefore F + ⊗ π E + = F + ⊗ ε E + and E ∗ + ⊗ π F + = E ∗ + ⊗ ε F + . (iv) = ⇒ (vi) and (v) = ⇒ (vi) . Clear. (vi) = ⇒ (ii) . Note that id E : E → E is positive. (ii) = ⇒ (i) . Write id E = P ki =1 ϕ i ⊗ x i with ϕ , . . . , ϕ k ∈ E ∗ + and x , . . . , x k ∈ E + . Then,for arbitrary x ∈ E + we have x = id E ( x ) = P ki =1 ϕ i ( x ) x i with ϕ ( x ) , . . . , ϕ k ( x ) ≥
0, so wesee that E + is generated by x , . . . , x k . In particular, it follows that E + is a polyhedral cone.Furthermore, since we have E = ran(id E ) ⊆ span( x , . . . , x k ), it follows that x , . . . , x k mustspan E , so E + is generating. Dually, if x, − x ∈ E + , then ϕ ( x ) = . . . = ϕ k ( x ) = 0, hence x = id E ( x ) = P ki =1 ϕ i ( x ) x i = 0, which shows that E + is a proper cone.Since both E + and E ∗ + are proper polyhedral cones, each has a finite number of extremal raysgenerating the cone. Let { ψ i } ni =1 and { y j } mj =1 be (representatives of) the extremal directions of E ∗ + and E + , respectively. Writing every ϕ i and every x j as a positive combination of the extremalrays, we can expand our expression of id E toid E = n X i =1 m X j =1 λ ij ψ i ⊗ y j , with λ ij ≥ i and all j. For every j we have y j = id E ( y j ) = P ni =1 P mk =1 λ ik ψ i ( y j ) y k = P ni =1 λ ij ψ i ( y j ) y j , for by extremal-ity of y j the terms λ ik ψ i ( y j ) y k with k = j must be zero. It follows that P ni =1 λ ij ψ i ( y j ) = 1 forall j . Therefore:dim( E ) = tr(id E ) = tr n X i =1 m X j =1 λ ij ψ i ⊗ y j = m X j =1 n X i =1 λ ij ψ i ( y j ) = m X j =1 m. Since span( y , . . . , y m ) = span( E + ) = E , it follows that y , . . . , y m is a basis of E . This provesthat E + is a simplex cone. (cid:4) emark 5.2. Taking the tensor product of a space with its dual is also a common technique inthe theory of normed tensor products. For instance, Theorem 5.1 is very similar to a result aboutthe approximation property; see [DF93, Theorem 5.6].The following corollary is immediate.
Corollary 5.3.
Let E + ⊆ R n be a self-dual cone. Then E + ⊗ π E + = E + ⊗ ε E + if and only if E + is a simplex cone. In particular, it follows that S n + ⊗ π S n + = S n + ⊗ ε S n + and H n + ⊗ π H n + = H n + ⊗ ε H n + whenever n ≥
2. This has been known for a long time in relation to quantum theory and C ∗ -algebras, andis related to the difference between positive and completely positive operators. The interestedreader is encouraged to refer to the expository article by Ando [And04, § In this section, we prove Theorem D on the tensor product of a smooth or strictly convex cone E + with an arbitrary cone F + , provided that dim( E ) ≥ dim( F ). The argument is based on a(generalized) John’s decomposition of the identity.We recall some common terminology. A convex body is a compact convex set C ⊆ R n withnon-empty interior. An affine transformation is an invertible affine map R n → R n ; that is, a mapof the form x T x + y with T ∈ GL n ( R ) and y ∈ R n fixed.If C , C ⊆ R n are convex bodies, then a compactness argument shows that there is an affinetransformation T such that T [ C ] ⊆ C and vol( T [ C ]) is maximal among all affine transformations T for which T [ C ] ⊆ C . If the maximum is attained for T = I n (the identity transformation),then we say that C is in a maximum volume position inside C . Furthermore, we say that C is in John’s position inside C if C ⊆ C and there exist m ∈ N , x , . . . , x m ∈ ∂C ∩ ∂C , y , . . . , y m ∈ ∂C ◦ ∩ ∂C ◦ and λ , . . . , λ m > h x i , y i i = 1 for all i , and I n = m X i =1 λ i x i ⊗ y i and 0 = m X i =1 λ i x i = m X i =1 λ i y i . Gordon, Litvak, Meyer and Pajor [GLMP04] proved the following result, building on earlierextensions ([GPT01, BR02]) of Fritz John’s classical theorem ([Joh49]).
Theorem 6.1 ([GLMP04, Theorem 3.8]) . Let C , C ⊆ R n be convex bodies such that C is in amaximum volume position inside C . Then there exists z ∈ int( C ) such that C − z is in John’sposition inside C − z . For our purposes, we will only need the following (much weaker) corollary.
Corollary 6.2.
Let C , C ⊆ R n be convex bodies. Then there is an affine transformation T : R n → R n such that T [ C ] ⊆ C and ∂T [ C ] ∩ ∂C contains an affine basis of R n . (Equivalently, there is a T such that T [ C ] ⊆ C and the set of points where T [ C ] and C touch is not contained in an affine hyperplane.) Proof of Corollary 6.2.
Let T : R n → R n be an affine transformation such that T [ C ] is in amaximum volume position inside C . By Theorem 6.1, we may choose z ∈ int( T [ C ]), m ∈ N ,11 , . . . , x m ∈ ∂ ( T [ C ] − z ) ∩ ∂ ( C − z ), y , . . . , y m ∈ ∂ ( T [ C ] − z ) ◦ ∩ ∂ ( C − z ) ◦ and λ , . . . , λ m > h x i , y i i = 1 for all i , and I n = m X i =1 λ i x i ⊗ y i and 0 = m X i =1 λ i x i = m X i =1 λ i y i . ( ∗ )After an appropriate rescaling of the λ i , the second formula in ( ∗ ) shows that 0 ∈ aff( x , . . . , x m ), soit follows that aff( x , . . . , x m ) = span( x , . . . , x m ). Moreover, it follows immediately from the firstformula in ( ∗ ) that span( x , . . . , x m ) = R n , so we conclude that { x , . . . , x m } contains an affinebasis, say x , . . . , x n +1 . Consequently, x + z, . . . , x n +1 + z is an affine basis in ∂T [ C ] ∩ ∂C . (cid:4) Since every closed and proper convex cone has a compact base, the following homogenizationof Corollary 6.2 follows immediately.
Corollary 6.3.
Let E and F be finite-dimensional real vector spaces with dim( E ) = dim( F ) ,and let E + ⊆ E , F + ⊆ F be closed, proper, and generating convex cones. Then there exists apositive linear transformation T : E → F such that ∂T [ E + ] ∩ ∂F + contains a ( linear ) basis of F . Using the preceding results, we can prove the main theorem of this section.
Theorem 6.4.
Let E , F be finite-dimensional real vector spaces, and let E + ⊆ E , F + ⊆ F beclosed, proper, and generating convex cones. If dim( E ) ≥ dim( F ) , and if E + is strictly convex orsmooth, then one has E + ⊗ π F + = E + ⊗ ε F + if and only if F + is a simplex cone.Proof. First assume that E + ⊗ π F + = E + ⊗ ε F + , with E + strictly convex and dim( E ) ≥ dim( F ).Choose an interior point x ∈ E + and a linear subspace G ⊆ E of dimension dim( F ) through x . Then G + := G ∩ E + is closed, proper, and generating, so by Corollary 6.3 we may choose apositive linear isomorphism T : F ∗ → G such that ∂T [ F ∗ + ] ∩ ∂G + contains a basis { b , . . . , b m } of G . Note that every b i is also a boundary point of E + (this a basic property of topologicalboundaries), and therefore an extremal direction of E + (since E + is strictly convex). For all i ,write a i := T − ( b i ) ∈ ∂F ∗ + ; then { a , . . . , a m } is a basis of F ∗ .If ι : G , → E denotes the inclusion, then ι ◦ T : F ∗ → E is positive, so by assumption we maywrite ι ◦ T = P ki =1 y i ⊗ x i with x , . . . , x k ∈ E + and y , . . . , y k ∈ F + (where the y i act as linearfunctionals on F ∗ ). Since b i is extremal and b i = T ( a i ) = k X j =1 h y j , a i i x j , it follows that at least one of the x j is a positive multiple of b i , and h y j , a i i = 0 whenever x j isnot a positive multiple of b i . In particular, if x j is not a positive multiple of any one of the b i ,then h y j , a i i = 0 for all i , so y j = 0 (since { a , . . . , a m } is a basis of F ∗ ). Thus, after removingthe zero terms, every x j is a positive multiple of some b i , and so in particular belongs to T [ F ∗ + ].This shows that not only ι ◦ T , but also T is separable, and not only with respect to the cones F ∗ + and G + , but even with respect to the cones F ∗ + and T [ F ∗ + ]. Since id F ∗ = T − ◦ T , it follows fromthe ideal property of separable operators that id F ∗ is also separable. Hence, by Theorem 5.1, F ∗ + is a simplex cone. Since F + is closed, it follows that F + = F ∗∗ + is also a simplex cone.Now assume that E + ⊗ π F + = E + ⊗ ε F + with E + smooth and dim( E ) ≥ dim( F ). By duality(cf. Corollary 4.11(b)), it follows that E ∗ + ⊗ π F ∗ + = ( E + ⊗ ε F + ) ∗ = ( E + ⊗ π F + ) ∗ = E ∗ + ⊗ ε F ∗ + . Since E + is smooth, the dual cone E ∗ + is strictly convex, so it follows from the first part of theproof that F ∗ + must be a simplex cone. Since F + is closed, it follows that F + = F ∗∗ + is also asimplex cone. (cid:4) Retracts
The question of whether or not E + ⊗ π F + and E + ⊗ ε F + coincide can sometimes be reduced tolower dimensional spaces by using retracts.Let ( F, F + ) be a finite-dimensional preordered vector space. Then a subspace E ⊆ F is calledan order retract if there exists a positive projection F → E . More generally, another preorderedspace ( G, G + ) is isomorphically an order retract if there exist positive linear maps T : G → F and S : F → G such that S ◦ T = id G . Note that in this case T is automatically bipositive (i.e. apullback) and S is automatically a pushforward, and ran( T ) ⊆ F is a retract of F which is orderisomorphic to G .For simplicity, we shall omit the word order when talking about retracts, as there is minimalchance of confusion with other types of retracts (e.g. from topology).Although retracts do not appear to be a very common notion in the theory of ordered vectorspaces, some of the results from this section were discovered independently by Aubrun, Lami andPalazuelos [ALP19]. Remark 7.1.
Some basic properties of retracts:(a) if E is a retract of ( F, F + ) and F + is a proper cone, then E + := E ∩ F + is a proper cone(after all, E + is a subcone);(b) if E is a retract of ( F, F + ) and F + is generating, then E + is generating in E (after all, thereexists a surjective positive operator F → E );(c) if ( E, E + ) is isomorphically a retract of ( F, F + ), and if ( F, F + ) is isomorphically a retractof ( G, G + ), then ( E, E + ) is isomorphically a retract of ( G, G + ).(d) If ( E, E + ) is isomorphically a retract of ( F.F + ), then ( E ∗ , E ∗ + ) is isomorphically a retract of( F ∗ , F ∗ + ). After all, if T : E → F and S : F → E are positive linear maps with id E = S ◦ T ,then S ∗ : E ∗ → F ∗ and T ∗ : F ∗ → E ∗ are positive linear maps withid E ∗ = (id E ) ∗ = ( S ◦ T ) ∗ = T ∗ ◦ S ∗ . By Remark 7.1(b), if F + ⊆ F is generating, then a retract E ⊆ F is uniquely determined byits positive part E + := E ∩ F + , so instead of saying that E is a retract of ( F, F + ) we will simplysay that E + is a retract of F + . Example 7.2.
We present some examples of retracts.(a) If E is finite-dimensional and E + is a closed and proper convex cone, then every ray in E + is a retract. Indeed, let x ∈ E + \ { } be arbitrary, and let ϕ ∈ E ∗ + be a strictly positivelinear functional. Then ϕ ( x ) · ϕ ⊗ x defines a positive projection onto span( x ).(b) If n ≤ m , then R n ≥ is a retract of R m ≥ , for instance via the maps R n → R m and R m → R n given respectively by padding with zeroes and projecting onto the first n coordinates.Although (a) shows that these are not the only retracts, we will show in Corollary 7.6 thatevery retract of a simplex cone is once again a simplex cone.(c) In the same manner, if n ≤ m , then S n + is a retract of S m + , and H n + is a retract of H m + .(d) If n ≤ m , then L n is a retract of L m via the maps T : R n → R m , S : R m → R n given by T ( x , . . . , x n ) = ( x , . . . , x n − , , . . . , , x n ); S ( y , . . . , y m ) = ( y , . . . , y n − , y m ) . R n ≥ is a retract of S n + via the map T : R n → S n that maps x to the diagonal matrix whoseentries are specified by x , and the map S : S n → R n that maps A to the diagonal of A .(f) S n + is a retract of H n + , via the maps T : S n → H n , A A and S : H n → S n , A ( A + A ).(g) H n + is a retract of S n + , via the maps T : H n → S n and S : S n → H n given by T ( A + iB ) = (cid:18) A − BB A (cid:19) , S (cid:18) A A A A (cid:19) = 12 ( A + A ) + i A − A ) . A more advanced example occurs in polyhedral cones. If E + is a proper and generatingpolyhedral cone with extremal directions { x , . . . , x k } , then a vertex figure at x is a subcone ofthe form E + ∩ ker( ϕ ), where ϕ ∈ E ∗ is a linear form such that ϕ ( x ) < ϕ ( x i ) > i >
0. Vertex figures are combinatorially dual to facets (e.g. [Brø83, Theorem 11.5]).
Proposition 7.3.
Let E be finite-dimensional and let E + ⊆ E be a proper and generatingpolyhedral cone. Then every vertex figure and every facet of E + is a retract.Proof. Let x , . . . , x k be the extremal directions of E + , and let ϕ ∈ E ∗ be such that ϕ ( x ) = − ϕ ( x i ) > i >
0. We show that the vertex figure E + ∩ ker( ϕ ) is a retract.Define P ϕ : E → E by y y + ϕ ( y ) x . We show that P ϕ is a positive projection ontoker( ϕ ). For all y ∈ E we have ϕ ( P ϕ ( y )) = ϕ ( y ) + ϕ ( y ) ϕ ( x ) = ϕ ( y ) − ϕ ( y ) = 0 , which shows that ran( P ϕ ) ⊆ ker( ϕ ). Furthermore, if y ∈ ker( ϕ ), then P ϕ ( y ) = y + 0 = y , so P ϕ is a projection onto ker( ϕ ). To prove positivity, it suffices to show that P ϕ ( x i ) ∈ E + for all i . We distinguish two cases: • For i = 0, we have P ϕ ( x ) = x + ϕ ( x ) x = x − x = 0 ∈ E + . • For i >
0, we have ϕ ( x i ) >
0, hence P ϕ ( x i ) = x i + ϕ ( x i ) x ∈ E + .This shows that every vertex figure is a retract. Additionally, note that ker( P ϕ ) = span( x ); thiswill be used in the second part of the proof.Now let M ⊆ E + be a facet. Then M corresponds with an extremal direction ψ ∈ E ∗ + \ { } of the dual cone, in such a way that M ⊥ = span( ψ ). Choose a vertex figure N ⊆ E ∗ + at ψ . The preceding argument shows that there are positive linear maps T : span( N ) , → E ∗ and S : E ∗ (cid:16) span( N ) such that id span( N ) = ST . Furthermore, the construction gives us theadditional property that ker( S ) = span( ψ ). Dualizing the retract (cf. Remark 7.1(d)) showsthat span( N ) ∗ is isomorphically a retract of E , by means of the maps S ∗ : span( N ) ∗ , → E and T ∗ : E (cid:16) span( N ) ∗ . Since ran( S ∗ ) = ⊥ ker( S ) = ⊥ { ψ } = span( M ), this shows that M is aretract of E + . (cid:4) Remark 7.4.
In fact, retractions give a geometric duality between facets and vertex figures (inaddition to the well-known combinatorial duality). If M ⊆ E + is a facet corresponding to theextremal direction ψ ∈ E ∗ + \ { } of the dual cone, then one can show that: • every vertex figure at ψ admits a unique positive projection (namely, the one from theproof of Proposition 7.3); • there is a bijective correspondence between vertex figures at ψ and positive projections E → span( M ) that map every element of E + \ M in the relative interior of M .As we have no use for this, the proof is omitted.14etracts can be useful in the theory of ordered tensor products. For instance, in [Dob20] weproved that the projective cone does not preserve subspaces, and the injective cone does notpreserve quotients, but retracts are sufficiently rigid to be preserved by both. (A similar role isplayed by complemented subspaces in the theory of normed tensor products.) The following resultshows that retracts can also be useful for comparing the projective and injective cones. Proposition 7.5.
Let G and H be finite-dimensional real vector spaces, let G + ⊆ G , H + ⊆ H be closed convex cones, and let E ⊆ G and F ⊆ H be retracts. If E + ⊗ π F + = E + ⊗ ε F + , then G + ⊗ π H + = G + ⊗ ε H + .Proof. We prove the contrapositive: assuming that G + ⊗ π H + = G + ⊗ ε H + , we prove that E + ⊗ π F + = E + ⊗ ε F + . By Remark 7.1(d), we may identify E ∗ with a retract of G ∗ . Choosepositive projections π E ∗ : G ∗ (cid:16) E ∗ and π F : H (cid:16) F .Let T : E ∗ → F be a positive linear map. Since every positive operator G ∗ → H factorsthrough a simplex cone, we may choose positive operators S : G ∗ → R m and S : R m → H (where R m carries the standard cone R m ≥ ) so that the following diagram commutes: G ∗ R m HE ∗ E ∗ F F. π E ∗ S S π F id E ∗ T T id F Now T : E ∗ → F factors through a simplex cone, so T ∈ E + ⊗ π F + . (cid:4) Corollary 7.6.
Every retract of a ( finite-dimensional ) simplex cone is a simplex cone.Proof. Let F + be simplex cone and let E + be a retract of F + . By Theorem 5.1, we have E ∗ + ⊗ π F + = E ∗ + ⊗ ε F + , so it follows from Proposition 7.5 that E ∗ + ⊗ π E + = E ∗ + ⊗ ε E + . Anotherapplication of Theorem 5.1 shows that E + is a simplex cone. (cid:4) Remark 7.7.
In the supplementary material to [ALP19], the authors proved that, for n ≥ n -dimensional convex cones do not have an ( n − n − n − E + ⊗ π F + = E + ⊗ ε F + for a large class of cones.Combining Proposition 7.3 and Corollary 7.6, we get the following characterization of polyhe-dral cones in terms of their retracts. Corollary 7.8.
Let E be finite-dimensional and let E + ⊆ E be a proper and generating polyhedralcone. Then E + is a simplex cone if and only if every -dimensional retract of E + is a simplexcone.Proof. If dim( E ) ≤
2, then E + is automatically a simplex cone, and there are no 3-dimensionalretracts, so the statement is vacuously true.Assume dim( E ) ≥
3. If E + is a simplex cone, then every retract of E + is a simplex cone, byCorollary 7.6. Conversely, if E + is not a simplex cone, then one of the following must be true(use [Brø83, Theorem 12.19] and homogenization): • dim( E + ) = 3; 15 E + has a facet that is not a simplex cone; • E + has a vertex figure that is not a simplex cone.Since facets and vertex figures are retracts, it follows (by induction) that E + has a 3-dimensionalretract that is not a simplex cone. (cid:4) In this section, we prove Theorem E on tensor products of standard cones. The proof uses retracts(see §
7) to reduce the problem to the three-dimensional case. The case where E + and F + arepolyhedral is postponed until Theorem 8.3; we first put the pieces together. Standard conesTheorem 8.1.
Let G , H be finite-dimensional real vector spaces, and let G + ⊆ G , H + ⊆ H be closed, proper, and generating convex cones. Assume that each of G + and H + is one of thefollowing ( all combinations allowed ) :(a) a polyhedral cone;(b) a second-order cone L n ;(c) a ( real or complex ) positive semidefinite cone S n + or H n + .Then one has G + ⊗ π H + = G + ⊗ ε H + if and only if at least one of G + and H + is a simplexcone.Proof. Suppose that neither G + nor H + is a simplex cone. We claim that G + (resp. H + ) has athree-dimensional retract E + (resp. F + ) which is isomorphic with one of the following: • the three-dimensional Lorentz cone L (which is isomorphic to S ); • a proper and generating polyhedral cone P ⊆ R which is not a simplex cone.To prove the claim, we distinguish three cases: • If G + is polyhedral, then this follows from Corollary 7.8. • If G + = L n , then the assumption that G + is not a simplex cone forces n ≥
3. Hence itfollows from Example 7.2(d) that L is a retract of G + . • If G + = S n + or H n + , then the assumption that G + is not a simplex cone forces n ≥
2, so itfollows from Example 7.2(c) and Example 7.2(f) that S ( ∼ = L ) is a retract of G + .Next, we show that E + ⊗ π F + = E + ⊗ ε F + , again distinguishing three cases. • If E + = F + = L , then this follows from Corollary 5.3, since the Lorentz cone is self-dual. • If E + = L and F + is polyhedral (or vice versa), then this follows from Theorem 6.4, since L is strictly convex and dim( E ) = dim( F ). • The case where both E + and F + are polyhedral will be settled in Theorem 8.3 below.Since E + and F + are retracts of G + and H + satisfying E + ⊗ π F + = E + ⊗ ε F + , it follows fromProposition 7.5 that G + ⊗ π H + = G + ⊗ ε H + . (cid:4) perator systems Some of our results can be reformulated in terms of operator systems. Let C ⊆ R d be a closed,proper, and generating convex cone, and let n ∈ N be a positive integer. Following notationfrom [FNT17], we denote the projective and injective tensor products H n + ⊗ π C and H n + ⊗ ε C by C min n and C max n , respectively, and we write C min = { C min n } ∞ n =1 and C max = { C max n } ∞ n =1 . Corollary 8.2.
Let C ⊆ R d be a closed, proper, and generating convex cone. If d ≤ , or if C isstrictly convex, or smooth, or polyhedral, then the following are equivalent:(i) C is a simplex cone;(ii) the minimal and maximal operator systems C min and C max are equal;(iii) there exists n ≥ for which C min n = C max n ;(iv) one has C min2 = C max2 .Proof. (i) = ⇒ (ii) . This follows from Theorem 5.1. (ii) = ⇒ (iii) . Trivial. (iii) = ⇒ (iv) . If H n + ⊗ π C = H n + ⊗ ε C for some n ≥
2, then it follows from Proposition 7.5that H ⊗ π C = H ⊗ ε C , since H is a retract of H n + , by Example 7.2(c). (iv) = ⇒ (i) . First we prove that H is strictly convex. Indeed, the interior points of H arethe positive definite matrices, so the boundary points are the singular 2 × H must be a rank one positive semidefinitematrix, which is known to be extremal (it is a positive multiple of a rank one orthogonal projection).Now suppose that C is of one of the forms described in the theorem, but not a simplex cone.If d ≤ dim( H ) = 4, or if C is smooth or strictly convex, then it follows from Theorem 6.4that H ⊗ π C = H ⊗ ε C . If C is polyhedral (but not a simplex cone), then it follows fromTheorem 8.1 that H ⊗ π C = H ⊗ ε C . Either way, we have C min2 = C max2 . (cid:4) Tensor products of -dimensional polyhedral cones All that remains is to prove that the projective and injective tensor products of two 3-dimensionalpolyhedral cones are different, unless one of the two is a simplex cone. For this we use acombinatorial argument, based on the results of the previous paper [Dob20].The proof essentially boils down to finding a combinatorial obstruction. If E + ⊗ π F + = E + ⊗ ε F + , then ( E + ⊗ π F + ) ∗ = E ∗ + ⊗ π F ∗ + , which gives us enough information to determine theface lattice of E + ⊗ π F + . If both E + and F + have at least 4 extremal rays, then it turns out thatthe lattice thus obtained is not graded (in other words, it contains maximal chains of differentlengths), which contradicts a well-known property of polyhedral cones.(Note: the previous paragraph serves to outline the high-level ideas behind the proof. Theproof below uses slightly different terminology and does not proceed by contradiction.) Theorem 8.3.
Let E + , F + ⊆ R be proper and generating polyhedral cones. Then E + ⊗ π F + = E + ⊗ ε F + if and only if at least one of E + and F + is a simplex cone.Proof. A proper and generating polyhedral cone in R is the homogenization of a polygon. Let v , . . . , v m ∈ R be (representatives of) the extremal directions of E + in such a way that theneighbours of v i are v i − and v i +1 (modulo m ). In the same way, let w , . . . , w n ∈ F + be theextremal directions of F + (in cyclic order). Furthermore, let ϕ , . . . , ϕ m ∈ E ∗ + and ψ , . . . , ψ n ∈ F ∗ + be the extremal directions of E ∗ + and F ∗ + , in such a way that ϕ i (resp. ψ j ) represents the facet of E + (resp. F + ) that contains v i and v i +1 (resp. w j and w j +1 ).17f E + or F + is a simplex cone, then E + ⊗ π F + = E + ⊗ ε F + by Theorem 5.1. So assume thatneither E + nor F + is a simplex cone, i.e. m, n ≥
4. We show by a combinatorial argument that( E + ⊗ π F + ) ∗ must be larger than E ∗ + ⊗ π F ∗ + = ( E + ⊗ ε F + ) ∗ .By [Dob20, Theorem 3.20], the extremal directions of the projective cone E + ⊗ π F + aregiven by { v i ⊗ w j : i ∈ [ m ] , j ∈ [ n ] } , and the extremal directions of E ∗ + ⊗ π F ∗ + are given by { ϕ i ⊗ ψ j : i ∈ [ m ] , j ∈ [ n ] } . Furthermore, by [Dob20, Corollary 5.4(b)], the extremal directionsof E ∗ + ⊗ π F ∗ + are also extremal for the (larger) cone ( E + ⊗ π F + ) ∗ = E ∗ + ⊗ ε F ∗ + . To complete theproof, we show that this larger cone must have extremal directions which are not of the form ϕ i ⊗ ψ j . (By [Dob20, Corollary 5.4(b)], these must have rank ≥ E + ⊗ π F + must have facets which cannot be written as the tensor product of a facet in E + and a facet in F + .Given k ∈ [ m ] and ‘ ∈ [ n ], let F k,‘ denote the facet of E + ⊗ π F + corresponding to theextremal direction ϕ k ⊗ ψ ‘ ∈ ( E + ⊗ π F + ) ∗ = E ∗ + ⊗ ε F ∗ + . The extremal directions in F k,‘ arethose v i ⊗ w j with i ∈ { k, k + 1 } or j ∈ { ‘, ‘ + 1 } , for one has v i ⊗ w j ∈ F k,‘ if and only if0 = h v i ⊗ w j , ϕ k ⊗ ψ ‘ i = h v i , ϕ k ih w j , ψ ‘ i . In particular, F k,‘ contains 2 m + 2 n − F k,‘ , namely C := F , ∩ F , ∩ F , ∩ F , . Clearly C is a face of E + ⊗ π F + . We claim that C contains only 4 extremal rays. To that end, notethat F , ∩ F , contains exactly 8 extremal rays, namely v i ⊗ w j with ( i, j ) ∈ ( { , } × { , } ) ∪ ( { , } × { , } ). This is illustrated in the figure below (with m = 6 and n = 8). m n F , F , F , ∩ F , ∩ =Similarly, F , ∩ F , has the same pattern, but shifted one step in the second coordinate, so wesee that the intersection of F , ∩ F , and F , ∩ F , contains 4 extremal rays: m n F , ∩ F , F , ∩ F , C ∩ =(We have to be aware of a subtlety here: if n = 4, then the pattern of F , ∩ F , is “wrappedaround” from right to left, but this does not affect the conclusion.)Since C contains 4 extremal rays, we have dim( C ) ≤
4. However, note that the only F k,‘ containing C are the four facets defining C . Since we know from classical polyhedral geometrythat C must be contained in at least 9 − dim( C ) ≥ E + ⊗ π F + hasfacets which are not of the form F k,‘ . Equivalently, the dual cone ( E + ⊗ π F + ) ∗ = E ∗ + ⊗ ε F ∗ + hasextremal directions which are not of the form ϕ i ⊗ ψ j , so we have E ∗ + ⊗ ε F ∗ + = E ∗ + ⊗ π F ∗ + . Byduality, it follows that E + ⊗ π F + = E + ⊗ ε F + . (cid:4) Closing remarks
As mentioned before, Aubrun, Lami, Palazuelos and Pl´avala [ALPP19] independently proved thefollowing even stronger result.
Theorem 9.1 ([ALPP19, Theorem A]) . Let E , F be finite-dimensional real vector spaces, andlet E + ⊆ E , F + ⊆ F be closed, proper, and generating convex cones. Then one has E + ⊗ π F + = E + ⊗ ε F + if and only if at least one of E + and F + is a simplex cone. The following example shows that this is no longer true if we omit the requirement that E + or F + is proper or generating. Example 9.2.
Let F + ⊆ F be a “partial simplex cone”; that is, a cone generated by m < dim( F )linearly independent vectors x , . . . , x m ∈ F . Furthermore, let E be another finite-dimensionalspace, and let E + ⊆ E be an arbitrary closed, proper, and generating cone.Since E + is generating, every positive linear map T : E → F has its range contained inspan( F + ). Since span( F + ) is ordered by a simplex cone, this shows that every positive linear map E → F is simplex-factorable, hence E ∗ + ⊗ π F + = E ∗ + ⊗ ε F + . Note: in the preceding example, E ∗ + and F + are closed and proper (and E ∗ + is generating),so the requirement that F + is generating cannot be omitted from Theorem 9.1. Furthermore,by duality, we also have E + ⊗ π F ∗ + = E + ⊗ ε F ∗ + , which shows that the requirement that F + isproper cannot be omitted either.In a sense, a partial simplex cone (or its dual) is almost a simplex cone. In fact, we can extendTheorem 9.1 to show that all examples must be of this form. If E + is a closed convex cone,then we define the proper reduction prop( E + ) of E + as the positive cone of span( E + ) / lin( E + ).Equivalently, choose subspaces E , E , E ⊆ E such that E = lin( E + ), E ⊕ E = span( E + ),and E ⊕ E ⊕ E = E ; then the proper reduction of E + is the positive cone ( E ) + := E ∩ E + of E , viewed as a closed, proper, and generating cone in E . It is readily verified that the projection E → E , ( e , e , e ) e is positive (every projection onto span( E + ) is positive, and adding orsubtracting elements of the lineality space does not affect positivity), so prop( E + ) is a retract of E + . Therefore Theorem 9.1 has the following extension. Corollary 9.3.
Let E , F be finite-dimensional real vector spaces, and let E + ⊆ E , F + ⊆ F beclosed convex cones. If E + ⊗ π F + = E + ⊗ ε F + , then at least one of prop( E + ) and prop( F + ) is asimplex cone.Proof. Since prop( E + ) and prop( F + ) are retracts of E + and F + , it follows from Proposition 7.5that prop( E + ) ⊗ π prop( F + ) = prop( E + ) ⊗ ε prop( F + ). But prop( E + ) and prop( F + ) are closed,proper, and generating, so it follows from Theorem 9.1 that at least one of prop( E + ) and prop( F + )must be a simplex cone. (cid:4) The converse is not true; it can happen that prop( E + ) and prop( F + ) are simplex cones but E + ⊗ π F + = E + ⊗ ε F + . This is because prop( E + ) ⊗ π prop( F + ) = prop( E + ) ⊗ ε prop( F + ) doesnot necessarily imply E + ⊗ π F + = E + ⊗ ε F + ; the implication of Proposition 7.5 only runs in theother direction. As an extreme example, consider the case where E + = { } and F + = F ; thenone has E + ⊗ π F + = { } but E + ⊗ ε F + = E ⊗ F . Acknowledgements
I am grateful to Dion Gijswijt, Onno van Gaans, and Marcel de Jeu for helpful discussions andcomments. Part of this work was carried out while the author was partially supported by theDutch Research Council (NWO), project number 613.009.127.19 eferences [AT07] C.D. Aliprantis, R. Tourky,
Cones and Duality (2007), Graduate Studies in Mathematics 84,American Mathematical Society.[And04] T. Ando,
Cones and norms in the tensor product of matrix spaces , Linear Algebra and itsApplications, vol. 379 (2004), pp. 3–41.[ALP19] G. Aubrun, L. Lami, C. Palazuelos,
Universal entangleability of non-classical theories , preprint(2019), https://arxiv.org/abs/1910.04745v1 .[ALPP19] G. Aubrun, L. Lami, C. Palazuelos, M. Pl´avala,
Entangleability of cones , preprint (2019), https://arxiv.org/abs/1911.09663v2 .[BL75] G.P. Barker, R. Loewy,
The structure of cones of matrices , Linear Algebra and its Applications,vol. 12 (1975), issue 1, pp. 87–94.[Bar76] G.P. Barker,
Monotone norms and tensor products , Linear and Multilinear Algebra, vol. 4(1976), issue 3, pp. 191–199.[Bar81] G.P. Barker,
Theory of Cones , Linear Algebra and its Applications, vol. 39 (1981), pp. 263–291.[BR02] J. Bastero, M. Romance,
John’s decomposition of the identity in the non-convex case , Positivity,vol. 6 (2002), issue 1, pp. 1–16.[Bir76] D.A. Birnbaum,
Cones in the tensor product of locally convex lattices , American Journal ofMathematics, vol. 98 (1976), issue 4, pp. 1049–1058.[BCG13] T. Bogart, M. Contois, J. Gubeladze,
Hom-polytopes , Mathematische Zeitschrift, vol. 273(2013), issue 3–4, pp. 1267–1296.[Brø83] A. Brøndsted,
An Introduction to Convex Polytopes (1983), Graduate Texts in Mathematics90, Springer.[DF93] A. Defant, K. Floret,
Tensor Norms and Operator Ideals (1993), Mathematics Studies 176,North-Holland.[Dob20] J. van Dobben de Bruyn,
Tensor products of convex cones, part I: mapping properties, faces,and semisimplicity (2020), arXiv preprints.[FNT17] T. Fritz, T. Netzer, A. Thom,
Spectrahedral Containment and Operator Systems with Finite-Dimensional Realization , SIAM Journal on Applied Algebra and Geometry, vol. 1 (2017),issue 1, pp. 556–574.[GPT01] A. Giannopoulos, I. Perissinaki, A. Tsolomitis,
John’s theorem for an arbitrary pair of convexbodies , Geometriae Dedicata, vol. 84 (2001), pp. 63–79.[GLMP04] Y. Gordon, A.E. Litvak, M. Meyer, A. Pajor,
John’s decomposition in the general case andapplications , Journal of Differential Geometry, vol. 68 (2004), issue 1, pp. 99–119.[Hil08] R. Hildebrand,
Semidefinite descriptions of low-dimensional separable matrix cones , LinearAlgebra and its Applications, vol. 429 (2008), issue 4, pp. 901–932.[HN18] B. Huber, T. Netzer,
A note on non-commutative polytopes and polyhedra , preprint (2018), https://arxiv.org/abs/1809.00476v3 .[Joh49] F. John,
Extremum problems with inequalities as subsidiary conditions . In:
Studies and Essayspresented to R. Courant on his 60th Birthday (1948), pp. 187–204, Interscience.[Kle59b] V. Klee,
Some new results on smoothness and rotundity in normed linear space , MathematischeAnnalen, vol. 139 (1959), issue 1, pp. 51–63.[Mul97] B. Mulansky,
Tensor Products of Convex Cones . In: N¨urnberger, Schmidt, Walz (editors),
Multivariate Approximation and Splines , International Series on Numerical Mathematics 125,Springer.[PSS18] B. Passer, O.M. Shalit, B. Solel,
Minimal and maximal matrix convex sets , Journal ofFunctional Analysis, vol. 274 (2018), issue 11, pp. 3197–3253. PTT11] V.I. Paulsen, I.G. Todorov, M. Tomforde,
Operator system structures on ordered spaces ,Proceedings of the London Mathematical Society, vol. 102 (2011), issue 1, pp. 25–49.[Rud91] W. Rudin,
Functional Analysis , Second Edition (1991), McGraw–Hill.[Tam77] B.-S. Tam,
Some results of polyhedral cones and simplicial cones , Linear and MultilinearAlgebra, vol. 4 (1977), issue 4, pp. 281–284.[Tam92] B.-S. Tam,
On the structure of the cone of positive operators , Linear Algebra and its Applica-tions, vol. 167 (1992), pp. 65–85.[Zam87] T. Zamfirescu,
Nearly all convex bodies are smooth and strictly convex , Monatshefte f¨urMathematik, vol. 103 (1987), issue 1, pp. 57–62.
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