Tensor Products of Convex Cones, Part I: Mapping Properties, Faces, and Semisimplicity
TTensor Products of Convex Cones, Part I:Mapping Properties, Faces, and Semisimplicity
Josse van Dobben de Bruyn24 September 2020
Abstract
The tensor product of two ordered vector spaces can be ordered in more than one way,just as the tensor product of normed spaces can be normed in multiple ways. Two naturalorderings have received considerable attention in the past, namely the ones given by the projective and injective (or biprojective ) cones. This paper aims to show that these two conesbehave similarly to their normed counterparts, and furthermore extends the study of thesetwo cones from the algebraic tensor product to completed locally convex tensor products.The main results in this paper are the following: (i) drawing parallels with the normed theory,we show that the projective/injective cone has mapping properties analogous to those ofthe projective/injective norm; (ii) we establish direct formulas for the lineality space of theprojective/injective cone, in particular providing necessary and sufficient conditions for thecone to be proper; (iii) we show how to construct faces of the projective/injective cone fromfaces of the base cones, in particular providing a complete characterization of the extremalrays of the projective cone; (iv) we prove that the projective/injective tensor product of two closed proper cones is contained in a closed proper cone (at least in the algebraic tensorproduct).
Tensor products of ordered (topological) vector spaces have been receiving attention for morethan 50 years ([Mer64], [HF68], [Pop68], [PS69], [Pop69], [DS70], [vGK10], [Wor19]), but thefocus has mostly been on Riesz spaces ([Sch72], [Fre72], [Fre74], [Wit74], [Sch74, § IV.7], [Bir76],[FT79], [Nie82], [GL88], [Nie88], [Bla16]) or on finite-dimensional spaces ([BL75], [Bar76], [Bar78a],[Bar78b], [Bar81], [BLP87], [ST90], [Tam92], [Mul97], [Hil08], [HN18], [ALPP19]). In the generalsetting, some of the basic questions are left unanswered (or, on one occasion, escaped fromcollective memory, as we point out below). Meanwhile, the projective and injective cones arestarting to turn up in other areas of mathematics, including interpolation theory ([Mul97]),operator systems and matrix convex sets ([PTT11], [FNT17], [HN18], [PSS18]), and theoreticalphysics ([ALP19], [ALPP19]).One of the cornerstones of the theory of ordered tensor products is the article of Peressiniand Sherbert [PS69]. It contains an in-depth study of the properties of two specific cones in thetensor product of two ordered vector spaces, namely the projective and injective (or biprojective ) Mathematics Subject Classification . 46A40, 06F20, 46A32, 52A05.
Key words and phrases . Convex cone, partially ordered vector space, ordered tensor product, face, order ideal. a r X i v : . [ m a t h . F A ] S e p ones. Peressini and Sherbert answered various topological and order-theoretic question aboutthese cones, for instance relating to normality and order units. Furthermore, they found varioussufficient conditions for each of these to be proper, but a general answer was not found.Conditions for the projective cone to be proper were quickly provided by Dermenjian andSaint-Raymond [DS70], but their result seems to have been unknown to later generations ofmathematicians. Only recently was this question answered (again) by Wortel [Wor19]; until thenonly special cases were assumed to be known in the literature. We shall give yet another proofthat we believe to be simpler.The injective cone seems to have received even less attention in the literature, although manyresults about spaces of positive operators can be translated to analogous statements for theinjective cone. In the finite-dimensional case (with closed, proper, and generating cones), the basicproperties of the injective cone are outlined by Mulansky [Mul97], and more advanced resultshave been obtained in the study of cones of positive operators (e.g. [Bar81], [Tam92]). Beyondthe finite-dimensional setting, we are not aware of precise necessary and sufficient criteria for theinjective cone to be proper. In this paper, we fully answer this question, and we obtain directformulas for the lineality spaces of both the projective and the injective cone.The lack of answers to even these basic questions is in sharp contrast with the breadth anddepth of the theory of normed tensor products. The latter has flowered after the groundbreakingwork by Grothendieck in the 1950s, and has since come to play a central role in the theory ofBanach spaces. This motivates us to take some of the ideas from the normed theory into theordered theory. In particular, we show that the mapping properties of the projective and injectivenorms have direct analogues in the ordered setting.To illustrate the usefulness of these mapping properties, we use them to construct faces of theprojective cone from faces of the base cones. Not all faces of the projective cone are necessarily ofthis form, but we do obtain a complete description of the extremal rays.Finally, we match the construction of faces of the projective cone with a dual constructionin the injective cone. (Some special cases of this construction have already been studied inthe finite-dimensional case; e.g. [Bar78b, §
4] and [Tam92, § This paper studies the projective and injective tensor products of two convex cones (otherwiseknown as wedges) in real vector spaces. The lineality space of a convex cone E + is the linearsubspace lin( E + ) := E + ∩ − E + , and we say that a convex cone E + is proper if lin( E + ) = { } and semisimple if its weak closure E + w is proper. For additional notation, see § Cones in the completed tensor product
In the context of Banach lattices, many attempts have been made to define a natural cone in asuitable completion of the tensor product E ⊗ F (e.g. [Sch72], [Fre74], [Wit74]), but for generalcones the projective/injective cone has been studied only in the algebraic tensor product. Sincethe algebraic tensor product is often of limited use in functional analysis, this paper attempts toinitiate a study of these cones in completed locally convex tensor products. Most authors have called this the biprojective cone . We aim to show that it is in many ways analogous to theinjective norm, and as such deserves the name injective cone . This term has occasionally been used before, forinstance by Wittstock [Wit74] and Mulansky [Mul97]. topological tensor products, one has to define the topology before takingthe completion, for obviously the completion depends on the chosen topology. On the other hand,the cone is unrelated to the topology, and can therefore be defined directly on the completion.This gives rise to a natural extension of the injective cone to the completed tensor product, whichwe will also study in this paper. On the other hand, the projective cone in the completed tensorproduct E ˜ ⊗ α F is merely the same cone embedded in a larger ambient space , so there is littlereason to study this cone separately.An overview of the cones under consideration, their notation, and their domains of definition,is given in Table 1.1. (In all cases, E + ⊆ E and F + ⊆ F are convex cones in the primal spaces.)Table 1.1: The domain of definition of the projective and injective cones.Cone Ambient space Notation Domain of definitionProjective E ⊗ F E + ⊗ π F + E and F vector spacesInjective E ⊗ F E + ⊗ ε F + h E, E i and h F, F i dual pairsInjective E ˜ ⊗ α F E + ˜ ⊗ εα F + E and F complete locally convex spaces, α a compatible locally convex topology on E ⊗ F In the following subsections, we state the main properties of the projective and injective conesin the algebraic tensor product. Similar results hold for the injective cone in the completed locallyconvex tensor product, but these are harder to state, as they often require additional (topological)assumptions. Precise statements can be found in § E ⊗ F and E ˜ ⊗ α F from properties of the positive cone in a larger space (of bilinear forms). Mapping properties
In the normed theory, it is well-known that the projective norm preserves metric surjections(quotients) and the injective norm preserves metric injections (isometries). By looking at thecorresponding types of positive linear maps (see § Theorem A.
The projective cone preserves positive linear maps, ( approximate ) pushforwards,and order retracts, but not ( approximate ) pullbacks.The injective cone preserves weakly continuous positive linear maps, approximate pullbacks,and topological order retracts, but not pullbacks or ( approximate ) pushforwards. The proof of Theorem A will be given in § § approximate pullbacks. It is not so strange that itdoes not preserve all pullbacks: the injective cone does not see the difference between E + and E + w , and a pullback for E + is not necessarily a pullback for E + w . (For details, see § E + w and F + w rather than E + and F + . (The projective cone, on the other hand, does see the difference between E + and E + w .)An overview of the mapping properties of the projective/injective cone is given in Table 1.2. We define the projective cone algebraically, without taking its closure. This is the prevalent definition in theliterature, but might not be appropriate for all applications. A result about its closure will be given in Theorem G. Other authors have denoted the projective and injective cones by a variety of different notations; for instance, K p and K b (e.g. [PS69]), or E + (cid:12) F + and E + (cid:126) F + (e.g. [ALPP19]), among others. (cid:88) (cid:88) Pushforward (cid:88)
Approximate pushforward (cid:88)
PullbackApproximate pullback (cid:88)
Retract (positive projection) (cid:88) (cid:88)
Criteria for properness; lineality space
Criteria for the projective cone to be proper were discovered by Dermenjian and Saint-Raymond[DS70], and rediscovered in recent years by Wortel [Wor19]. We give yet another proof that webelieve to be simpler, and we also provide criteria for the injective cone to be proper.
Theorem B.
The projective cone E + ⊗ π F + is proper if and only if E + = { } , or F + = { } , orboth E + and F + are proper cones. (cf. [DS70, Th´eor`eme 2])The injective cone E + ⊗ ε F + is proper if and only if E = { } , or F = { } , or both E + w and F + w are proper cones. The proof of Theorem B will be given in § § cones is trivial, whereas the corner case for the injectivecone is when one of the spaces is trivial. This discrepancy can be explained by looking at thedirect formulas for the lineality spaces. Theorem C.
The lineality space of the projective/injective cone is lin( E + ⊗ π F + ) = (lin( E + ) ⊗ span( F + )) + (span( E + ) ⊗ lin( F + ));lin( E + ⊗ ε F + ) = (lin( E + w ) ⊗ F ) + ( E ⊗ lin( F + w )) . The proof of Theorem C will be given in Corollary 3.17 (projective cone) and Corollary 4.37(c)(injective cone). Note: although Theorem C contains Theorem B as a special case, we prove thesimpler Theorem B before proving Theorem C.
Faces and extremal rays
By combining the mapping properties with the properness criteria, we can show that the projectivecone preserves faces.
Theorem D. If M ⊆ E + and N ⊆ F + are faces, then ( M ⊗ π F + )+( E + ⊗ π N ) and ( M ⊗ π N )+lin( E + ⊗ π F + ) are faces of the projective cone E + ⊗ π F + .In particular, if E + and F + are proper cones, then M ⊗ π N is a face of E + ⊗ π F + . The proof of Theorem D will be given in § The result of Dermenjian and Saint-Raymond seems to have been unknown to later generations of mathemati-cians, and until recently only special cases were assumed to be known in the literature.
4s an application, we prove in § ideals (in the sense of Kadison [Kad51] and Bonsall [Bon54]; see also § Theorem E. If I ⊆ E and J ⊆ F are ideals with respect to E + w and F + w , then ( I ⊗ J ) +lin( E + ⊗ ε F + ) is an ideal with respect to the injective cone E + ⊗ ε F + .Additionally, if I is weakly closed and ( E/I ) + is semisimple, or if J is weakly closed and ( F/J ) + is semisimple, then ( I ⊗ F ) + ( E ⊗ J ) is also an ideal with respect to the injective cone. The proof of Theorem E will be given in § E/I ) + or ( F/J ) + is semisimple cannot be omitted (cf. Example 4.35).As a special case of Theorem D and Theorem E, we show that the projective and injectivecones preserve extremal rays. Theorem F.
A vector u ∈ E ⊗ F is an extremal direction of the projective cone E + ⊗ π F + ifand only if u can be written as u = x ⊗ y , where x and y are extremal directions of E + and F + .If x and y are extremal directions of E + w and F + w , then x ⊗ y is an extremal direction ofthe injective cone E + ⊗ ε F + . All extremal directions of ( tensor ) rank one are of this form, butthere may also be extremal directions of larger rank. The proof of Theorem F will be given in § § § Semisimplicity
In [vGK10], Van Gaans and Kalauch showed that the projective tensor product of two Archimedeanproper cones is contained in an Archimedean proper cone. We prove a parallel result: if E + and F + are weakly closed proper cones (this is a stronger assumption), then their projective/injectivetensor product is contained in a weakly closed proper cone (i.e. it is semisimple). Theorem G. If E + , F + = { } , then the projective cone E + ⊗ π F + is semisimple if and only if E + and F + are semisimple.If E, F = { } , then the injective cone E + ⊗ ε F + is semisimple if and only if E + and F + aresemisimple. The proof of Theorem G will be given in § E + , F + = { } or E, F = { } ) correspond with those of Theorem B.The injective cone remains semisimple in the completed injective tensor product, and moregenerally, in every completion E ˜ ⊗ α F for which the natural map E ˜ ⊗ α F → E ˜ ⊗ ε F is injective(cf. Theorem 5.14). However, we do not know whether the projective cone remains semisimple inthe completed projective tensor product E ˜ ⊗ π F ; see Question 5.16. As we shall see in § Appendix: faces and ideals
The main body of this paper is complemented by an appendix on faces and ideals of convex conesin infinite-dimensional spaces. This material is not directly related to tensor products, but will beused extensively in the proofs. Faces and ideals are closely related. We will prove in Appendix A.1 that the map I I + defines a surjectivemany-to-one correspondence between the order ideals of the preordered vector space ( E, E + ) and the faces of E + . Van Gaans and Kalauch only prove this for generating
Archimedean cones, but this easily implies the generalresult stated above. I I + defines a surjective many-to-one correspondence between ideals and faces (AppendixA.1). Going back and forth between faces and ideals is crucial in our study of the faces of theprojective/injective cone.In Appendix A.2, we outline to which extent the homomorphism and isomorphism theoremshold for ideals in ordered vector spaces. We use these to show that the maximal order ideals areprecisely the supporting hyperplanes of the positive cone. This shows that the order ideals can bethought of as being the “supporting subspaces” of E + .In Appendix A.3, we extend the notion of exposed faces to the infinite-dimensional setting.We show that we have to distinguish between dual and exposed faces, although the two notionscoincide if the ambient space is a separable normed space. A Ideals, faces, and duality 51
A.1 Faces and ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51A.2 The homomorphism and isomorphism theorems . . . . . . . . . . . . . . . . . . . 53A.3 Dual and exposed faces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
References 59 Notation and prerequisites
Throughout this article, all vector spaces are over R . Dual spaces and dual pairs
We denote algebraic duals as E ∗ and topological duals as E .In situations where the topology comes into play, we shall mainly be interested in propertieswhich depend not on the topology of E , but only on the dual pair h E, E i . When we say Let h E, E i be a dual pair , (2.1)we shall mean that E is a topological vector space whose (topological) dual E separates points.(Alternatively, E could be a vector space with no topology, and E := E ∗ its algebraic dual.)Statements following assumptions like (2.1) will not depend on the topology of E , but only onthe dual pair.If h E, E i is a dual pair, then the σ ( E, E )-topology on E will be called the weak topology,and the σ ( E , E )-topology on E will be called the weak- ∗ topology. The resulting topologicalvector spaces are denoted by E w and E w ∗ . Likewise, the weak closure of a subset M ⊆ E isdenoted by M w , and the weak- ∗ closure of a subset N ⊆ E is denoted by N w ∗ . Linear maps If E and F are vector spaces, then the space of linear maps E → F is denoted by L( E, F ). If E and F are topological vector spaces, then the space of continuous linear maps E → F is denotedby L ( E, F ). If the (topological) duals separate points, then every continuous map E → F is alsoweakly continuous (see e.g. [K¨ot83, § L ( E, F ) ⊆ L ( E w , F w ) ⊆ L( E, F ) . If E and F are vector spaces without topologies, then every linear map E → F is σ ( E, E ∗ )- σ ( F, F ∗ )-continuous (since ψ ◦ T is σ ( E, E ∗ )-continuous for every ψ ∈ F ∗ ), so we have L ( E w , F w ) = L( E, F ) . (if E = E ∗ , F = F ∗ )The adjoint of a (continuous) linear map T : E → F is denoted T ∗ : F ∗ → E ∗ (algebraic adjoint)or T : F → E (topological adjoint). Bilinear maps
For topological vector spaces
E, F , let B i ‘ ( E × F ) ⊆ Bil ( E × F ) ⊆ Bil( E × F )denote (from left to right) the spaces of continuous, separately continuous, and all bilinear forms E × F → R . Some authors treat dual pairs symmetrically, and refer to both topologies as weak . We must not do so: ourdual pairs consist of a topological vector space and its dual, and as such are not symmetric. Concretely, if E is aBanach space, then it can be important to differentiate between the weak and weak- ∗ topologies on E . Note: with this notation it is possible to confuse B i ‘ ( E × F ) with Bil ( E × F ), but notation like this appearsto be at least moderately common (e.g. [Sch99, p. 91], [K¨ot79, p. 154]). Bil ( E × F ), the space of separately continuousbilinear forms. It will be used extensively in the study of the injective cone in § b ∈ Bil( E × F ) and x ∈ E , y ∈ F , let b ( x , · ) ∈ F ∗ and b ( · , y ) ∈ E ∗ denote the linear functionals b ( x , · ) := (cid:0) y b ( x , y ) (cid:1) ; b ( · , y ) := (cid:0) x b ( x, y ) (cid:1) . Using this notation, we see that b is separately continuous if and only if one has b ( x , · ) ∈ F for all x ∈ E and b ( · , y ) ∈ E for all y ∈ F . In particular, it follows that Bil ( E × F ) does notdepend on the topologies of E and F , but only on the dual pairs h E, E i , h F, F i . Likewise, itfollows that Bil ( E × F ) = Bil( E × F ) whenever E = E ∗ and F = F ∗ .It follows from [K¨ot79, § b ( x b ( x, · ))and b ( y b ( · , y )) define linear isomorphisms Bil ( E × F ) = Bil ( E w × F w ) ∼ = L ( E w , F w ∗ ) ∼ = L ( F w , E w ∗ ) . (The isomorphism L ( E w , F w ∗ ) ∼ = L ( F w , E w ∗ ) is simply T T .) Tensor products
We assume the reader to be familiar with the basics of the (algebraic) theory of tensor products.We will need very little on the side of topological tensor products (but many results in this paperare inspired by the theory of normed tensor products).For clarity, we shall occasionally use the following notation: if M ⊆ E and N ⊆ F are subsets,then we define the “set-wise” tensor product M ⊗ s N := { x ⊗ y : x ∈ M, y ∈ N } ⊆ E ⊗ F. Many of the properties of a convex cone E + in topological vector space E depend only on thegeometry of E + and on the dual pair h E, E i . In light of this, the (notoriously complicated) theoryof locally convex tensor products is largely irrelevant to the present paper. We only need to knowto which dual pair the topological tensor product E ⊗ F belongs, and our main results can bestated for a wide range of reasonable duals of E ⊗ F (see below).Throughout this paper, we encode the “input spaces” E and F and the “output space” E ⊗ F by the dual pairs to which they belong. Here we briefly discuss how to handle subspaces, quotients,and tensor products of dual pairs.Questions about the projective/injective cone that depend not only on the dual pair, but alsoon a specific topology on E ⊗ F , will not be treated in this paper. In particular, for questionsabout normality of the projective/injective cone, we refer the reader to [PS69]. Remark 2.2.
Because we formulate all results in terms of dual pairs h E, E i , we will often referto the weak closure of a convex cone, or a weakly closed subspace. We should point out that theadjective “weak” can be omitted if E is a locally convex space (and E is its topological dual), forin this setting the weak and original closure of a convex set (in particular, a convex cone or asubspace) coincide, by [Rud91, Theorem 3.12].If E is a topological vector space which is not locally convex, then the adjective “weak” cannotbe omitted. 8 ubspaces If h E, E i is a dual pair and if I ⊆ E is a subspace, then we will understand I to belong to thedual pair h I, E /I ⊥ i . We point out that this is usually, but not always, the natural dual pair for I .Assume that E a topological vector space, E is its (topological) dual, and I carries thesubspace topology. Let T : I , → E denote the inclusion and T : E → I its adjoint.If E is locally convex, then every continuous linear functional on I can be extended to E ,so T is surjective. Clearly ker( T ) = I ⊥ , so T restricts to a linear isomorphism E /I ⊥ → I .Furthermore, the relative σ ( E, E )-topology on I coincides with the σ ( I, E /I ⊥ )-topology (even if I is not closed), so we may unambiguously refer to this as the weak topology on I . On the otherhand, the σ ( E /I ⊥ , I )-topology on E /I ⊥ = I coincides with the quotient topology E w ∗ /I ⊥ ifand only if I is closed (see e.g. [Sch99, § IV.4.1, Corollary 1]).If E is not locally convex, then I may have continuous linear functionals that cannot beextended. In this case one still has ker( T ) = I ⊥ , but T is not surjective, so I = E /I ⊥ .Nevertheless, E /I ⊥ is the dual of I with respect to the relative σ ( E, E )-topology on I . Quotients If E is a topological vector space and if I ⊆ E is a closed subspace, then E/I is a Hausdorfftopological vector space. Every continuous linear functional
E/I → R extends to a continuouslinear functional E → R that vanishes on I . Conversely, if ϕ : E → R is a continuous linearfunctional that vanishes on I , then ϕ factors through E/I , by the universal property of quotients.Therefore: (
E/I ) ∼ = I ⊥ as vector spaces.Thus, if h E, E i is a dual pair and if I ⊆ E is a weakly closed subspace, then we can understand E/I to belong to the dual pair h E/I, I ⊥ i . The quotient topology on E w /I coincides with the σ ( E/I, I ⊥ )-topology, and the subspace topology on I ⊥ ⊆ E w ∗ coincides with the σ ( I ⊥ , E/I )-topology (see e.g. [Sch99, § IV.4.1, Corollary 1]), so we may unambiguously refer to these as theweak topology on
E/I and the weak- ∗ topology on I ⊥ , respectively.The only caveat with this approach is that we cannot “see” all quotients of E . If E is locallyconvex, then every closed subspace is also weakly closed, but this may fail for general topologicalvector spaces. However, if I is closed but not weakly closed, then the quotient E/I is Hausdorff,but its topological dual (
E/I ) = I ⊥ does not separate points. Throughout this paper, we assumethat all duals separate points, so we only consider quotients E/I where I is weakly closed. Tensor products
Recall that the algebraic dual of E ⊗ F is the space Bil( E × F ) of bilinear forms E × F → R . If h E, E i , h F, F i are dual pairs, we say that a subspace G ⊆ Bil( E × F ) is a reasonable dual of E ⊗ F if E ⊗ F ⊆ G ⊆ Bil ( E × F ) . (See § Bil ( E × F ) of separately continuous bilinear forms.)Using this definition, we may treat topological tensor products without having to deal withthe specifics of their topologies. We show that the definition covers all important cases.First, if E, F are locally convex and E ⊗ F carries a compatible topology α (in the sense ofGrothendieck [Gro55, p. 89]; see also [K¨ot79, § E ⊗ α F ) is a reasonable dual of E ⊗ F . On the one hand, one of the requirements for α to becompatible is E ⊗ F ⊆ ( E ⊗ α F ) . On the other hand, every compatible topology is coarserthan the inductive topology, whose dual is Bil ( E × F ) (see e.g. [K¨ot79, § E ⊗ α F ) ⊆ Bil ( E × F ). This shows that ( E ⊗ α F ) is a reasonable dual.9econd, if E and F originate from spaces without topologies, then we understand these tobelong to the dual pairs h E, E ∗ i , h F, F ∗ i . In this case we have Bil ( E × F ) = Bil( E × F ) (see § E ⊗ F ) ∗ = Bil( E × F ) is a reasonable dual of E ⊗ F . This is useful whenapplying topological results in the non-topological setting (e.g. Corollary 3.4). Convex cones and their duals
Let E be a (real) 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 linearsubspace of E , called the lineality space of K . We say that K is proper if lin( K ) = { } , and generating if K − K = E .If K ⊆ E is a convex cone, then its algebraic dual cone K ∗ is the set of all positive linearfunctionals: K ∗ := (cid:8) ϕ ∈ E ∗ : ϕ ( x ) ≥ x ∈ K (cid:9) . If h E, E i is a dual pair, then we define K := K ∗ ∩ E (the dual cone for the dual pair h E, E i ).The dual cone of K ⊆ E with respect to the dual pair h E , E i is the bipolar cone K := (cid:8) x ∈ E : h x, ϕ i ≥ ϕ ∈ K (cid:9) = ( K ) . Using the (one-sided) bipolar theorem, one easily shows that K = K w . It follows that ⊥ ( K ) = K ∩ −K = lin( K w ). In particular, K w is a proper cone if and only if K separates the pointsof E . If this is the case, then we say that K is semisimple . (For more equivalent definitions ofsemisimplicity, see [Dob20a].) Positive linear maps
Let E and F be vector spaces, and let E + ⊆ E and F + ⊆ F be convex cones. We say that a linearmap T ∈ L( E, F ) is positive if T [ E + ] ⊆ F + , a pullback (or bipositive operator ) if E + = T − [ F + ],and a pushforward if T [ E + ] = F + .Likewise, if h E, E i and h F, F i are dual pairs, then we say that an operator T ∈ L ( E w , F w ) is approximately positive if T [ E + w ] ⊆ F + w , an approximate pullback (or approximately bipositive )if E + w = T − [ F + w ], and an approximate pushforward if T [ E + ] w = F + w . A continuous positivemap (resp. pushforward) is also approximately positive (resp. an approximate pushforward),but a pullback is not necessarily an approximate pullback. (Concrete example: let F = R with F + = { ( x, y ) : x > } ∪ { (0 , } , let E := span { (0 , } ⊆ F with F + := F + ∩ E , and let T be theinclusion E , → F .)These approximate type operators are not particularly natural from the perspective of orderedvector spaces, but they come into play as soon as one starts to make use of duality. Given T ∈ L ( E w , F w ), it is not hard to show that the adjoint T ∈ L ( F w ∗ , E w ∗ ) is positive if and onlyif T is approximately positive. In addition, it is shown in [Dob20c] that T is an approximatepullback if and only if T is a weak- ∗ approximate pushforward, and vice versa. This is no longertrue if the adjective “approximate” is omitted, even if the spaces are finite-dimensional and thecones are closed (see [Dob20c, § A note about terminology: some authors call this a wedge , and reserve the term cone for what we call a propercone (e.g. [Day62], [Per67], [AT07]). Some authors call this pointed or salient .
10t will be helpful to pass to the corresponding approximate versions. (In particular, we show thatthe injective cone preserves approximate pullbacks, but not all pullbacks.)Note that every linear map E → F can be made a pullback/pushforward by choosingappropriate cones. In particular, a pullback is not necessarily injective, and a pushforward isnot necessarily surjective. However, if E + is a proper cone, then every pullback T : E → F isinjective (since ker( T ) ⊆ T − [ F + ] = E + ), and if F + is generating then every pushforward E → F is surjective.Table 2.3: Ordered analogues of common concepts in the normed theory.Normed theory Ordered theoryContinuous operator Positive operatorMetric injection (isometry) Pullback (bipositive operator)Metric surjection (quotient) Pushforward (quotient)Projection (complemented subspace) Positive projection (order retract) Retracts
Let E be a vector space and let E + ⊆ E be a convex cone. A subspace F ⊆ E is an order retract if there exists a positive projection E → F . If E is furthermore a topological vector space, thenwe say that F is a topological order retract if there exists a continuous positive projection E → F .For simplicity, we shall speak of retracts and top-retracts , as there is minimal chance ofconfusion with other types of retracts (e.g. from topology). Retracts will play a more prominentrole in the follow-up paper [Dob20b]. Various examples can be found there.Note that a retract provides at the same time an injective pullback (i.e. bipositive map) F , → E and a surjective pushforward (“quotient”) E (cid:16) F . We will show that, although theprojective tensor product does not preserve bipositive maps and the injective tensor product doesnot preserve quotients, retracts are sufficiently rigid to be preserved by both.To illustrate their place in the theory, note that every top-retract is a complemented subspace(after all, it admits a continuous projection ). If E + = { } , then the top-retracts are preciselythe complemented subspaces.As far as we know, order retracts are not a very common notion, and have not received muchattention. However, some special cases already play a role in the theory, such as projection bands in Riesz spaces (see e.g. [Zaa97, § projectionally exposed faces in finite-dimensional cones(see e.g. [BLP87], [ST90]). Positive bilinear maps If E , F , G are vector spaces with convex cones E + , F + , G + , then a bilinear map b : E × F → G is positive if b ( E + , F + ) ⊆ G + .In terms of the isomorphism Bil ( E w × F w ) ∼ = L ( E w , F w ∗ ) (cf. § b ∈ Bil ( E w × F w ) is positive if and only if b ( x, · ) defines a positive linear functional on F for every x ∈ E + , or equivalently, if and only if the corresponding map E w → F w ∗ is positive.Thus, contrary to the topological setting, there is no difference between positive and “separatelypositive” bilinear forms. Some authors require a complemented subspace to be closed, but this is automatic: if P : E → E is a continuousprojection with range F , then F = ker(id E − P ), so F is closed. aces and extremal rays Let E be a vector space and let E + ⊆ E be a convex cone. A face (or extremal set ) of E + is a(possibly empty) convex subset M ⊆ E + such that, if M contains a point in the relative interiorof a line segment in E + , then M also contains the endpoints of that segment. If ϕ is a continuouspositive linear functional, then ker( ϕ ) ∩ E + is a face. Faces of this type are called exposed .Every convex cone has a unique minimal non-empty face (the lineality space lin( E + ), containedin every face) and a unique maximal face (the cone itself, containing every face). Note that E + isa proper cone if and only if { } is a face.Let x ∈ E + \ { } . If M := { λx : λ ∈ R ≥ } is a face, then we say that x is an extremaldirection , and M is an extremal ray . If x is an extremal direction, then so is µx for every µ > E + ) ⊆ E + \ { } denote the set of all extremal directions of E + .If M ⊆ E + is a non-empty subset, then E + ∩ M ⊥ defines a face of E + . Faces of this type arecalled dual faces . In the finite-dimensional case, dual faces are precisely the exposed faces, butthis is not true in locally convex spaces. For more on dual and exposed faces, see Appendix A.3. Order ideals
Let E be a vector space, let E + ⊆ E be a convex cone. A subspace I ⊆ E is called an order ideal if the pushforward of E + along the quotient map E → E/I is a proper cone. If no ambiguity canarise (i.e. if the space does not carry a multiplication), then we call I simply an ideal .A subspace I ⊆ E is an ideal if and only if I ∩ E + is a face of E + (Proposition A.2). Conversely,if M ⊆ E + is a face, then span( M ) is an ideal satisfying span( M ) ∩ E + = M (Proposition A.3).Thus, I I + defines a many-to-one correspondence between ideals and faces. We shall drawheavily upon this correspondence.If K ⊆ E + is a subcone, then every ideal I ⊆ E with respect to E + is also an ideal withrespect to K . More generally, if T : E → F is a positive linear map and if J ⊆ F is an ideal,then T − [ J ] ⊆ E is also an ideal (Proposition A.3). In particular, if F + is a proper cone, then { } ⊆ F + is a face, so ker( T ) ∩ E + is a face of E + . It can be shown that all faces can be writtenin this form (Proposition A.4(b)).We will show in Corollary A.12 that the maximal order ideals are precisely the kernels ofnon-zero positive linear functionals, or in other words, the supporting hyperplanes of E + . Inparticular, not every maximal ideal is closed. (Example: the kernel of a discontinuous positivelinear functional.)For more about ideals and faces, see Appendix A and [Bon54].12 The projective cone
Let
E, F be (real) vector spaces and let E + ⊆ E , F + ⊆ F be convex cones. The simplest way todefine a cone in E ⊗ F is to consider the projective cone E + ⊗ π F + := ( k X i =1 x i ⊗ y i : k ∈ N , x , . . . , x k ∈ E + , y , . . . , y k ∈ F + ) . If E, F are locally convex and if α is a compatible locally convex topology on E ⊗ F , then wedenote by E + ⊗ πα F + and E + ˜ ⊗ πα F + the same cone, but embedded in the topological vectorspaces E ⊗ α F and E ˜ ⊗ α F , respectively. (The topology is denoted in the subscript; the cone inthe superscript.)It is easy to see that E + ⊗ π F + is indeed a (convex) cone. This cone has received a lot ofattention in the literature; see e.g. [Mer64], [PS69], [GL88], [vGK10], [Wor19].In the subsequent sections, we will study the basic properties of the projective cone. We pointout a characteristic property of the projective cone ( § § E + ⊗ π F + to be proper ( § § E + ⊗ π F + will be discussed in § Let
E, F, G be vector spaces equipped with convex cones E + ⊆ E , F + ⊆ F , G + ⊆ G . There is anatural isomorphism Bil( E × F, G ) ∼ = L ( E ⊗ F, G ), which identifies a bilinear map Φ : E × F → G with its linearization Φ L : E ⊗ F → G , Φ L ( P ki =1 x i ⊗ y i ) = P ki =1 Φ( x i , y i ). Proposition 3.1. If E ⊗ F is equipped with the projective cone E + ⊗ π F + , then a linear map Φ L : E ⊗ F → G is positive if and only if its corresponding bilinear map Φ : E × F → G ispositive.Proof. A bilinear map Φ : E × F → G is positive if and only if Φ L ( x ⊗ y ) = Φ( x, y ) ≥ x ∈ E + , y ∈ F + . On the other hand, since E + ⊗ π F + is generated by E + ⊗ F + , we also find thata linear map Φ L : E ⊗ F → G is positive if and only if Φ L ( x ⊗ y ) ≥ x ∈ E + , y ∈ F + . (cid:4) This is the ordered analogue of the characteristic property of the projective topology. It followsthat ( E + ⊗ π F + ) ∗ = Bil( E × F ) + ; ( E, F vector spaces)( E + ⊗ ππ F + ) = ( E + ˜ ⊗ ππ F + ) = B i ‘ ( E × F ) + . ( E, F locally convex)
The projective norm preserves continuous linear maps, quotients, and complemented subspaces(see e.g. [DF93, Propositions 3.2, 3.8, and 3.9(1)], or [K¨ot79, § Proposition 3.2.
Let T ∈ L( E, G ) and S ∈ L( F, H ) .(a) If T [ E + ] ⊆ G + and S [ F + ] ⊆ H + , then ( T ⊗ S )[ E + ⊗ π F + ] ⊆ G + ⊗ π H + .(b) If T [ E + ] = G + and S [ F + ] = H + , then ( T ⊗ S )[ E + ⊗ π F + ] = G + ⊗ π H + . c) If ( E, E + ) and ( F, F + ) are retracts of ( G, G + ) and ( H, H + ) , then ( E ⊗ F, E + ⊗ π F + ) is aretract of ( G ⊗ H, E + ⊗ π F + ) .In summary: the projective cone preserves positive linear maps, pushforwards, and retracts. It follows immediately that the same statements hold for maps between the completions (inthe locally convex case), for the projective cone is contained in the algebraic tensor product.
Proof. (a) Let z ∈ E + ⊗ π F + be given. Then we may write z = P ki =1 x i ⊗ y i with x , . . . , x k ∈ E + , y , . . . , y k ∈ F + . Consequently, we have ( T ⊗ S )( z ) = P ki =1 T ( x i ) ⊗ S ( y i ), which lies in G + ⊗ π H + since T ( x ) , . . . , T ( x k ) ∈ G + , S ( y ) , . . . , S ( y k ) ∈ H + .(b) By (a), T ⊗ S is positive. Now let u ∈ G + ⊗ π H + be given, and write u = P ki =1 v i ⊗ w i with v , . . . , v k ∈ G + and w , . . . , w k ∈ H + . By assumption there are x , . . . , x k ∈ E + , y , . . . , y k ∈ F + such that v i = T ( x i ) and w i = S ( y i ), for all i . Consequently, we have z := P ki =1 x i ⊗ y i ∈ E + ⊗ π F + , and u = ( T ⊗ S )( z ).(c) There are positive linear maps T , T , S , S so that the following two diagrams commute: G HE E, F F. T S T id E S id F Consequently, the following diagram commutes: G ⊗ HE ⊗ F E ⊗ F. T ⊗ S T ⊗ S id E ⊗ id F By (a), the maps in the preceding diagram are all positive for the projective cone, so itfollows that ( E ⊗ F, E + ⊗ π F + ) is a retract of ( G ⊗ H, G + ⊗ π H + ). (cid:4) Next, we prove that the projective tensor product also preservers approximate pushforwards:if T and S are maps whose adjoints are bipositive, then the same is true of T ⊗ S . Lemma 3.3.
Let h E, E i , h F, F i , h G, G i , h H, H i be dual pairs equipped with convex cones E + , F + , G + , H + in the primal spaces. If T ∈ L ( E w , G w ) and S ∈ L ( F w , H w ) are approximatepushforwards, then the map ( T ⊗ S ) : Bil ( G w × H w ) → Bil ( E w × F w ) , (( T ⊗ S ) b )( x, y ) = b ( T x, Sy ) is bipositive. Here ( T ⊗ S ) denotes the adjoint of T ⊗ S : E ⊗ F → G ⊗ H , assuming that E ⊗ F and G ⊗ H are equipped with the largest reasonable duals (cf. § Proof.
Note that ( T ⊗ S ) b is a positive bilinear functional on E × F if and only if b is positiveon T [ E + ] × S [ F + ], so if b is separately weakly continuous, then this is the case if and only if b is positive on T [ E + ] w × S [ F + ] w . (First use weak continuity in the second variable to pass from T [ E + ] × S [ F + ] to T [ E + ] × S [ F + ] w , then use weak continuity in the first variable to proceed to T [ E + ] w × S [ F + ] w .) Analogously, b itself is a positive bilinear functional on G × H if and only if b is positive on G + w × H + w . By assumption, we have T [ E + ] w = G + w and S [ F + ] w = H + w , soit follows that b is positive if and only if ( T ⊗ S ) b is positive. (cid:4) Corollary 3.4.
Let
E, F, G, H be vector spaces with convex cones E + , F + , G + , H + , and let T ∈ L( E, G ) , S ∈ L( F, H ) be linear maps such that T ∗ and S ∗ are bipositive. Then ( T ⊗ S ) ∗ isbipositive with respect to the dual cones ( E + ⊗ π F + ) ∗ ⊆ ( E ⊗ F ) ∗ and ( G + ⊗ π H + ) ∗ ⊆ ( G ⊗ H ) ∗ .Proof. If we understand the primal spaces to belong to the dual pairs h E, E ∗ i , . . . , h H, H ∗ i , thenevery linear map is weakly continuous. Furthermore, ( E ⊗ F ) ∗ = Bil( E × F ) = Bil ( E w × F w ),and the positive cone Bil ( E w × F w ) + coincides with the dual cone ( E + ⊗ π F + ) ∗ ⊆ ( E ⊗ F ) ∗ , byProposition 3.1. Hence the result is a special case of Lemma 3.3. (cid:4) Corollary 3.5.
Let
E, F, G, H be locally convex ( with convex cones ) and let T ∈ L ( E, G ) and S ∈ L ( F, H ) be approximate pushforwards. If T ⊗ α → β S : E ⊗ α F → G ⊗ β H is continuous ( α and β compatible locally convex topologies ) , then T ⊗ α → β S and T ˜ ⊗ α → β S are approximatepushforwards.Proof. Every continuous linear map is also weakly continuous (cf. [K¨ot83, § T ∈ L ( E w , G w ) and S ∈ L ( F w , H w ). Furthermore, since α and β are compatible topologies, wehave ( E ⊗ α F ) ⊆ Bil ( E × F ) = Bil ( E w × F w ) and ( G ⊗ β H ) ⊆ Bil ( F × H ) = Bil ( F w × H w ).It follows that ( T ⊗ α → β S ) is a restriction of the map ( T ⊗ S ) from Lemma 3.3, and therefore itis also bipositive. For the completion, note that ( T ˜ ⊗ α → β S ) = ( T ⊗ α → β S ) . (cid:4) Finally, we show that the projective tensor product does not preserve bipositive maps, even ifall spaces are finite-dimensional and all cones are closed and generating (Example 3.6), or evenclosed, generating and proper (Example 3.7).
Example 3.6.
As a very simple example, let F = G = R with F + = R and G + = R ≥ .Furthermore, let E = span { (1 , − } ⊆ G , and write E + := E ∩ G + = { } . Then the inclusion T : E , → G is bipositive, but E + ⊗ π F + = { } whereas G + ⊗ π F + = G ⊗ F . Since E ⊗ F = { } ,we have ( G + ⊗ π F + ) ∩ ( E ⊗ F ) = E + ⊗ π F + , which shows that T ⊗ id F is not bipositive. Example 3.7.
For a more advanced example, let E be a finite-dimensional space equipped witha proper, generating, polyhedral cone E + which is not a simplex cone. Choose ϕ , . . . , ϕ m ∈ E ∗ such that E + = T mi =1 { x ∈ E : ϕ i ( x ) ≥ } , and let R m be equipped with the standard cone R m ≥ .Then the map T : E → R m , x ( ϕ ( x ) , . . . , ϕ m ( x )) is bipositive.Since E + is not a simplex cone, it follows from [BL75, Proposition 3.1] that E + ⊗ π E ∗ + = E + ⊗ ε E ∗ + . On the other hand, it is well-known that R m ≥ ⊗ π E ∗ + = R m ≥ ⊗ ε E ∗ + , and it followsfrom Theorem 4.16(b) below that T ⊗ id E ∗ is bipositive for the injective cone. Therefore:( T ⊗ id E ∗ ) − [ R m ≥ ⊗ π E ∗ + ] = ( T ⊗ id E ∗ ) − [ R m ≥ ⊗ ε E ∗ + ] = E + ⊗ ε E ∗ + = E + ⊗ π E ∗ + . This shows that T ⊗ id E ∗ is not bipositive for the projective cone.Note that all cones in this example are polyhedral, and therefore closed. In particular, thesituation is not resolved by taking closures. The finite-dimensional techniques used in Example 3.7 will be discussed in more detail in thefollow-up paper [Dob20b].Despite the preceding counterexamples, bipositivity can be preserved under certain additionalconditions. First, if E ⊆ G and F ⊆ H are retracts, then E ⊗ F ⊆ G ⊗ H is also a retract (byProposition 3.2(c)), so in particular the inclusion E ⊗ F , → G ⊗ H is bipositive. Furthermore, weprove in Proposition 3.19 that the projective cone also preserves ideals of proper cones bipositively.15 .3 When is the projective cone proper? There is a simple necessary and sufficient condition for E + ⊗ π F + to be proper, which we provein Theorem 3.10 below. This result was first proved (in three different ways) by Dermenjian andSaint-Raymond [DS70], and recently rediscovered by Wortel [Wor19]. (The original proof seemsto have been forgotten, and before Wortel only special cases were known in the literature.) Theproof given here is different from each of the existing proofs. Further methods of proof will bediscussed in Remark 3.12.We proceed via reduction to the finite-dimensional case, using the following lemmas. Lemma 3.8.
A convex cone E + ⊆ E is generating if and only if its algebraic dual cone E ∗ + isproper.Proof. Note that E ∗ + is not proper if and only if there is some ϕ ∈ E ∗ \ { } such that both ϕ and − ϕ are positive linear functionals, or equivalently, ϕ ( x ) = 0 for all x ∈ E + . This is in turnequivalent to E + being contained in a (linear) hyperplane, which happens if and only if E + is not generating. (cid:4) Corollary 3.9. If E is finite-dimensional, then a closed convex cone E + ⊆ E is proper if andonly if its dual cone E ∗ + is generating.Proof. Set F := E ∗ and F + := E ∗ + . Under the canonical isomorphism E ∼ = E ∗∗ , we have F ∗ + = E + ,by the bipolar theorem (here we use that E + is closed). The result follows from Lemma 3.8,applied to the cone F + ⊆ F . (cid:4) We are now ready to state and prove the main theorem of this section.
Theorem 3.10 (cf. [DS70]) . Let E and F be vector spaces with convex cones E + ⊆ E , F + ⊆ F .Then the projective cone E + ⊗ π F + is proper if and only if E + = { } , or F + = { } , or both E + and F + are proper.Proof. Suppose first that E + , F + = { } and E + is not proper. Then we may choose x ∈ E \ { } such that x, − x ∈ E + , and y ∈ F + \ { } . Both x ⊗ y and − x ⊗ y belong to E + ⊗ π F + , but wehave x ⊗ y = 0, so we see that E + ⊗ π F + is not a proper cone.For the converse, if E + = { } , then E + ⊗ π F + = { } regardless of any properties of F + (andsimilarly if F + = { } ). So assume now that both E + and F + are proper (not necessarily = { } ).Let z ∈ E ⊗ F be given such that z, − z ∈ E + ⊗ π F + . Then we may choose integers n ≥ k ≥ x , . . . , x n ∈ E + , y , . . . , y n ∈ F + such that z = P ki =1 x i ⊗ y i and − z = P ni = k +1 x i ⊗ y i .Consequently, we have P ni =1 x i ⊗ y i = 0.Now set X := span( x , . . . , x n ) ⊆ E and Y := span( y , . . . , y n ) ⊆ F , and let X + ⊆ X and Y + ⊆ Y be the convex cones generated by x , . . . , x n and y , . . . , y n , respectively. Note that X + is a closed proper cone in the finite-dimensional vector space X , since it is finitely generated(hence closed; cf. [AT07, Lemma 3.19]) and contained in the proper cone X ∩ E + (hence alsoproper). Similarly, Y + is a closed proper cone in Y .It follows from Corollary 3.9 that X ∗ + and Y ∗ + are generating cones in X ∗ and Y ∗ , respectively.Therefore clearly X ∗ + ⊗ π Y ∗ + is generating in X ∗ ⊗ Y ∗ . Since h x ⊗ y, ϕ ⊗ ψ i = h x, ϕ ih y, ψ i ≥ x ∈ X + , y ∈ Y + , ϕ ∈ X ∗ + , ψ ∈ Y ∗ + , we have X ∗ + ⊗ π Y ∗ + ⊆ ( X + ⊗ π Y + ) ∗ . It follows that( X + ⊗ π Y + ) ∗ is also generating, and therefore ( X + ⊗ π Y + ) ∗∗ = X + ⊗ π Y + is a proper cone, byLemma 3.8. Since z, − z ∈ X + ⊗ π Y + ⊆ ( X + ⊗ π Y + ) ∗∗ , it follows that z = 0. (cid:4) Remark 3.11.
The final steps in the proof of Theorem 3.10 can be simplified with well-knownresults from the finite-dimensional theory, but we didn’t need that. The dual of the projective cone X + ⊗ π Y + is the injective cone X ∗ + ⊗ ε Y ∗ + , and X + ⊗ π Y + is automatically closed by [Tam77].16 emark 3.12. In the proof of Theorem 3.10, we reduced the problem to finitely generatedproper cones. There are many ways to prove this special case. Apart from the method used hereand the proofs given in [DS70] and [Wor19], we could also have applied either one of the sufficientcriteria from [PS69, Proposition 2.4]. Yet another method is mentioned in [Dob20b, Remark 4.4].Theorem 3.10 will be extended in Corollary 3.17 below, where we determine the linealityspace of E + ⊗ π F + for arbitrary convex cones E + , F + . Furthermore, a result very similar toTheorem 3.10, giving criteria for E + ⊗ π F + to be semisimple (i.e. contained in a weakly closedproper cone), will be given in Corollary 5.11. As a simple application of the theory developed so far, we show two ways to combine faces of E + and F + to form a face of E + ⊗ π F + . The general construction is given in Theorem 3.13; moreconvenient formulas and special cases will be studied afterwards. Theorem 3.13.
Let
E, F be vector spaces, let E + ⊆ E , F + ⊆ F be convex cones, and let M ⊆ E + , N ⊆ F + be non-empty faces. Define M (cid:62) π N := ( M ⊗ π F + ) + ( E + ⊗ π N ); M (cid:63) π N := ( M ⊗ π N ) + (lin( E + ) ⊗ π F + ) + ( E + ⊗ π lin( F + )) . Then:(a) M (cid:62) π N and M (cid:63) π N are faces of E + ⊗ π F + .(b) The face lattice of E + ⊗ π F + contains the following sublattice: M (cid:62) π NM (cid:62) π lin( F + ) = M (cid:63) π F + lin( E + ) (cid:62) π N = E + (cid:63) π NM (cid:63) π N Furthermore, M (cid:62) π N is not just the face generated by M (cid:63) π F + and E + (cid:63) π N , but eventhe sum of these faces, so we have M (cid:62) π N = ( M (cid:62) π lin( F + )) + (lin( E + ) (cid:62) π N ) = ( M (cid:63) π F + ) + ( E + (cid:63) π N ); M (cid:63) π N = ( M (cid:62) π lin( F + )) ∩ (lin( E + ) (cid:62) π N ) = ( M (cid:63) π F + ) ∩ ( E + (cid:63) π N ) . Assume furthermore that E , F , and E ⊗ F belong to the dual pairs h E, E i , h F, F i , and h E ⊗ F, G i ,where G is a compatible dual ( i.e. E ⊗ F ⊆ G ⊆ Bil ( E × F )) . Then:(c) If M and N are dual ( resp. exposed ) faces, then M (cid:62) π N is a dual ( resp. exposed ) face of E + ⊗ π F + .(d) If M and N as well as lin( E + ) and lin( F + ) are dual ( resp. exposed ) faces, then M (cid:63) π N isa dual ( resp. exposed ) face of E + ⊗ π F + .
17 mnemonic for the chosen notation: M (cid:62) π N is generated by the elements x ⊗ y ∈ E + ⊗ s F + with x ∈ M or y ∈ N , whereas M (cid:63) π N is generated by the elements x ⊗ y ∈ E + ⊗ s F + with x ∈ M and y ∈ N , together with what turns out to be the lineality space of E + ⊗ π F + (seeCorollary 3.18 below). Proof of Theorem 3.13. (a) Let I ⊆ E be an order ideal such that M = I ∩ E + (e.g. I = span( M ); cf. Proposition A.3(a)).Then the quotient cone ( E/I ) + ⊆ E/I is proper, the natural map π I : E → E/I is positive,and M = ker( π I ) ∩ E + . Similarly, let J ⊆ F be an ideal such that N = J ∩ F + ; then π J : F → F/J is a positive map to a space with a proper cone, and N = ker( π J ) ∩ F + .Now consider the linear map π I ⊗ π J : E ⊗ F → E/I ⊗ F/J . It follows from Proposition 3.2that π I ⊗ π J is positive, and it follows from Theorem 3.10 that ( E/I ) + ⊗ π ( F/J ) + is aproper cone in E/I ⊗ F/J . As such, it follows that ker( π I ⊗ π J ) ∩ ( E + ⊗ π F + ) is a face of E + ⊗ π F + (cf. Proposition A.4(b)). We claim thatker( π I ⊗ π J ) ∩ ( E + ⊗ π F + ) = M (cid:62) π N. (3.14)Indeed, if z = P ki =1 x i ⊗ y i with x , . . . , x k ∈ E + , y , . . . , y k ∈ F + is such that ( π I ⊗ π J )( z ) =0, then we must have ( π I ⊗ π J )( x i ⊗ y i ) = 0 for all i (since ( E/I ) + ⊗ π ( F/J ) + is proper).As such, for each i we must have x i ∈ ker( π I ) = I or y i ∈ ker( π J ) = J , or possibly both.Equivalently: x i ∈ I ∩ E + = M or y i ∈ J ∩ F + = N . This proves our claim (3.14), and weconclude that M (cid:62) π N is a face of E + ⊗ π F + .To see that M (cid:63) π N is a face, we proceed analogously, where the linear map π I ⊗ π J isreplaced by the linear map Q I,J : E ⊗ F → ( E/I ⊗ F/ lin( F + )) ⊕ ( E/ lin( E + ) ⊗ F/J ) ,x ⊗ y ( π I ( x ) ⊗ π lin( F + ) ( y )) ⊕ ( π lin( E + ) ( x ) ⊗ π J ( y )) . If z = P ki =1 x i ⊗ y i with x , . . . , x k ∈ E + , y , . . . , y k ∈ F + and Q I,J ( z ) = 0, then again wemust have Q I,J ( x i ⊗ y i ) = 0 for all i (since Q I,J is positive and the cone in the codomain isproper). Then either x i ∈ ‘ ( E + ) ⊆ M , or y i ∈ ‘ ( F + ) ⊆ N , or x i / ∈ ‘ ( E + ) and y i / ∈ ‘ ( F + ).In the latter case, we must have x i ∈ M and y i ∈ N . This way we findker( Q I,J ) ∩ ( E + ⊗ π F + ) = M (cid:63) π N. It follows that M (cid:63) π N is also a face of E + ⊗ π F + .(b) Using the notation from the proof of (a), note thatker( Q I,J ) = ker( π I ⊗ π lin( F + ) ) ∩ ker( π lin( E + ) ⊗ π J ) . It follows that M (cid:63) π N = ( M (cid:62) π lin( F + )) ∩ (lin( E + ) (cid:62) π N ) . The other formulas follow straight from the definitions: we have( M (cid:62) π lin( F + )) + (lin( E + ) (cid:62) π N ) = ( M ⊗ π F + ) + ( E + ⊗ π lin( F + ))+ (lin( E + ) ⊗ π F + ) + ( E + ⊗ π N )= ( M ⊗ π F + ) + ( E + ⊗ π N )= M (cid:62) π N, E + ) ⊆ M and lin( F + ) ⊆ N . Likewise, M (cid:63) π F + = ( M ⊗ π F + ) + (lin( E + ) ⊗ π F + ) + ( E + ⊗ π lin( F + ))= ( M ⊗ π F + ) + ( E + ⊗ π lin( F + ))= M (cid:62) π lin( F + ) , and the formula E + (cid:63) π N = lin( E + ) (cid:62) π N follows analogously.(c) If M = (cid:5) M and N = (cid:5) N , then it is routinely verified that M (cid:62) π N = (cid:5) ( M ⊗ s N ). If M and N are furthermore exposed, then we may take M and N to be singletons; consequently, M ⊗ s N is also a singleton.(d) This follows from (c) and the intersection formula from (b). (cid:4) Remark 3.15.
In Theorem 3.13(d), it is required that lin( E + ) and lin( F + ) are exposed/dualfaces. Sometimes this is automatically the case. If E + is weakly closed, then lin( E + ) = lin( E + w ) = ⊥ ( E + ) = (cid:5) ( E + ), so in this case lin( E + ) is always a dual face. Likewise, if E is a separable normedspace and E + is closed, then lin( E + ) is automatically exposed; see Corollary A.19.To see that this assumption cannot be omitted, let E := R with the lexicographical cone,and let F := R with the standard cone. Then the unique one-dimensional face M ⊆ E + and thetrivial face N := { } ⊆ R are both exposed (hence dual), but M (cid:63) π N = { } is neither exposednor dual in E + ⊗ π F + ∼ = E + . Remark 3.16.
By dualizing the example from Example 4.51 below, one can show that not everyfacet of E + ⊗ π F + is necessarily of the form M (cid:62) π N or M (cid:63) π N . In follows that, in general,not every face of E + ⊗ π F + can be written as an intersection of faces of the type M (cid:62) π N or M (cid:63) π N .We proceed to point out the consequences of Theorem 3.13. First of all, it allows us to extendTheorem 3.10, giving a direct formula for the lineality space of E + ⊗ π F + . Corollary 3.17 (The lineality space of the projective cone) . Let E and F be vector spaces, andlet E + ⊆ E and F + ⊆ F be convex cones. Then one has lin( E + ⊗ π F + ) = (lin( E + ) ⊗ π F + ) + ( E + ⊗ π lin( F + ))= (lin( E + ) ⊗ span( F + )) + (span( E + ) ⊗ lin( F + )) . Proof. If x ∈ lin( E + ) and y ∈ F + , then ± x ⊗ y ∈ E + ⊗ π F + , so x ⊗ y ∈ lin( E + ⊗ π F + ). Similarly,if x ∈ E + and y ∈ lin( F + ), then x ⊗ y ∈ lin( E + ⊗ π F + ), so we have(lin( E + ) ⊗ π F + ) + ( E + ⊗ π lin( F + )) ⊆ lin( E + ⊗ π F + ) . On the other hand, it follows from Theorem 3.13(a) that lin( E + ) (cid:62) π lin( F + ) = (lin( E + ) ⊗ π F + ) +( E + ⊗ π lin( F + )) is a face of E + ⊗ π F + , so it must contain the minimal face lin( E + ⊗ π F + ). Thefirst equality follows.For the second equality, we claim that lin( E + ) ⊗ π F + is equal to lin( E + ) ⊗ span( F + ). Indeed,for x ∈ lin( E + ) and y ∈ span( F + ) we may write y = u − v (for some u, v ∈ F + ), so we have x ⊗ y = ( x ⊗ u ) + (( − x ) ⊗ v ) ∈ E + ⊗ π F + . Taking positive linear combinations proves our claim.Analogously, we have E + ⊗ π lin( F + ) = span( E + ) ⊗ lin( F + ), and the second equality follows. (cid:4) This direct formula for the lineality space also simplifies the formula for the lower face M (cid:63) π N .19 orollary 3.18. Let
E, F be vector spaces, let E + ⊆ E , F + ⊆ F be convex cones, and let M ⊆ E + , N ⊆ F + be non-empty faces. Then one has M (cid:63) π N = ( M ⊗ π N ) + lin( E + ⊗ π F + ) , and this defines a face of E + ⊗ π F + .In particular, if E + and F + are proper cones, then M ⊗ π N is a face of E + ⊗ π F + , and thesublattice from Theorem 3.13(b) reduces to ( M ⊗ π F + ) + ( E + ⊗ π N ) M ⊗ π F + E + ⊗ π NM ⊗ π N As a final application, we note that Theorem 3.13 is also a statement about preservation ofbipositive maps.
Proposition 3.19.
Let E and F be vector spaces, and let E + ⊆ E , F + ⊆ F be convex cones. If E + and F + are proper and if I ⊆ E , J ⊆ F are ideals, then the inclusion I ⊗ J , → E ⊗ F isbipositive ( with respect to the projective cone ) .Proof. Let Q I,J : E ⊗ F → ( E/I ⊗ F ) ⊕ ( E ⊗ F/J ) be the map from the proof of Theorem 3.13(a).It follows from said proof (and Corollary 3.18) that I + ⊗ π J + = ker( Q I,J ) ∩ ( E + ⊗ π F + ). Tocomplete the proof, note that ker( Q I,J ) = I ⊗ J . (cid:4) Example 3.6 shows that this is not true if one of the cones is not proper.
The results from § E + ⊗ π F + are obtained in this way.Recall that rext( E + ) ⊆ E + \ { } denotes the set of extremal directions, and M ⊗ s N denotesthe entry-wise tensor product { x ⊗ y : x ∈ M, y ∈ N } . Theorem 3.20 (The extremal rays of the projective cone) . Let E , F be vector spaces equippedwith convex cones E + ⊆ E , F + ⊆ F . Then rext( E + ⊗ π F + ) = rext( E + ) ⊗ s rext( F + ) . Proof. “ ⊆ ”. Suppose that z ∈ ( E + ⊗ π F + ) \ { } defines an extremal ray. Write z = P ki =1 x i ⊗ y i with x , . . . , x k ∈ E + , y , . . . , y k ∈ F + , and x i ⊗ y i = 0 for all i ∈ [ k ]. By extremality of z thereare λ , . . . , λ k ∈ R > such that λ i x i ⊗ y i = z ( i ∈ [ k ]). In particular, z = λ x ⊗ y . Now supposethat 0 ≤ v ≤ x , then 0 ≤ λ v ⊗ y ≤ z , so by extremality of z we must have µλ v ⊗ y = z forsome µ ∈ R ≥ . Since y = 0 and λ = 0, it follows that µv = x , so we see that x defines anextremal ray of E + . Analogously, y defines an extremal ray of F + . This proves the inclusionrext( E + ⊗ π F + ) ⊆ rext( E + ) ⊗ s rext( F + ).“ ⊇ ”. Let x ∈ E + \ { } and y ∈ F + \ { } define extremal rays in E + and F + , respectively.Then M := { λx : λ ≥ } defines a face of E + . Every face contains the lineality space, but20 does not contain a non-zero subspace, so it follows that E + is a proper cone. Analogously, N := { µy : µ ≥ } defines a face of F + , so F + is proper. Now it follows from Corollary 3.18 that M ⊗ π N is a face of E + ⊗ π F + . In other words: x ⊗ y defines an extremal ray of E + ⊗ π F + . (cid:4) Remark 3.21.
Remarkably, Theorem 3.20 has no corner cases: it is true for every pair of convexcones. In particular, if rext( E + ) or rext( F + ) is empty, then rext( E + ⊗ π F + ) is empty as well.Conversely, if each of E + and F + has an extremal ray, then so does E + ⊗ π F + . We conclude our study of the projective cone with an application in convex geometry. Using aslight modification of the construction from § M and N determine faces of their tensor product M ⊗ π N := conv { x ⊗ y : x ∈ M, y ∈ N } . This application is based on the following general principle, giving sufficient conditions forthe sum of faces M (cid:63) π N and M (cid:63) π N (cf. § E + ⊗ π F + . (This is a vast generalization of the method of [BCG13, Example 3.7].) Proposition 3.22.
Let E , F be vector spaces, let E + ⊆ E , F + ⊆ F be convex cones, and let M , M ⊆ E + and N , N ⊆ F + be faces. If M ∩ M = lin( E + ) and N ∩ N = lin( F + ) , then ( M (cid:63) π N ) + ( M (cid:63) π N ) = ( M (cid:62) π N ) ∩ ( M (cid:62) π N ) . In particular, in this case ( M (cid:63) π N ) + ( M (cid:63) π N ) is a face of E + ⊗ π F + .Proof. “ ⊆ ”. It follows from Theorem 3.13(b) that M (cid:63) π N ⊆ M (cid:63) π F + = M (cid:62) π lin( F + ) ⊆ M (cid:62) π N . Three analogous inclusions prove the forward inclusion.“ ⊇ ”. Let z ∈ ( M (cid:62) π N ) ∩ ( M (cid:62) π N ), and write z = P ki =1 x i ⊗ y i with x , . . . , x k ∈ E + and y , . . . , y k ∈ F + . Since z ∈ M (cid:62) π N , it follows from the proof of Theorem 3.13(a) that for all i we have x i ∈ M or y i ∈ N , or possibly both. Likewise, for all i we have x i ∈ M or y i ∈ N , orpossibly both.If x i ∈ lin( E + ) or y i ∈ lin( F + ), then x i ⊗ y i ∈ lin( E + ⊗ π F + ) ⊆ ( M (cid:63) π N ) ∩ ( M (cid:63) π N ),since every face contains the lineality space. So assume x i / ∈ lin( E + ) and y i / ∈ lin( F + ). Then, byassumption, x i (resp. y i ) is contained in at most one of M and M (resp. N and N ). Combinedwith earlier constraints, this show that we must either have x i ∈ M \ M and y i ∈ N \ N , orotherwise x i ∈ M \ M and y i ∈ N \ N . Either way, x i ⊗ y i ∈ ( M (cid:63) π N ) + ( M (cid:63) π N ). (cid:4) If E is a vector space and C ⊆ E is a convex subset, then the homogenization C ( C ) of C isthe convex cone generated by C ⊕ { } ⊆ E ⊕ R . Note that C ( C ) is always a proper cone, andthat the faces of C are in bijective correspondence with the faces of C ( C ).Since we are working over the real numbers, a convex set C ⊆ E is absolutely convex if andonly if C = − C . For sets of this kind, there is a simple way to identify the projective tensorproduct of the homogenizations C ( C ) and C ( D ) with the homogenization of conv( C ⊗ s D ): Proposition 3.23.
Let E and F be ( real ) vector spaces and let C ⊆ E , D ⊆ F be absolutelyconvex sets. Under the natural isomorphism ( E ⊕ R ) ⊗ ( F ⊕ R ) = ( E ⊗ F ) ⊕ E ⊕ F ⊕ R , one has ( C ( C ) ⊗ π C ( D )) ∩ (cid:0) ( E ⊗ F ) ⊕ { } ⊕ { } ⊕ { } (cid:1) = { ( z, , ,
1) : z ∈ conv( C ⊗ s D ) } . It should be noted that many standard cones in infinite-dimensional spaces do not have sufficiently manyextremal rays to generate the cone. For instance, the positive cone of C [0 ,
1] has no extremal rays at all. Some authors define the projective tensor product of convex sets to be the closed convex hull of M ⊗ s N (e.g. [AS17, § roof. Under the aforementioned natural isomorphism, we have ( x, λ ) ⊗ ( y, µ ) = ( x ⊗ y, µx, λy, λµ ).“ ⊆ ”. Let ( z, , , ∈ C ( C ) ⊗ π C ( D ) be given, and write ( z, , ,
1) = P ki =1 λ i · ( x i , ⊗ ( y i , λ , . . . , λ k ≥ x , . . . , x k ∈ C and y , . . . , y k ∈ D . Then ( z, , ,
1) = P ki =1 λ i · ( x i ⊗ y i , x i , y i , P ki =1 λ i = 1 and z = P ki =1 λ i x i ⊗ y i ∈ conv( C ⊗ s D ).“ ⊇ ”. Let z ∈ conv( C ⊗ s D ) be given, and write z = P ki =1 λ i x i ⊗ y i with x , . . . , x k ∈ C and y , . . . , y k ∈ D . Note that ( x i , ⊗ ( y i ,
1) + ( − x i , ⊗ ( − y i ,
1) = 2( x i ⊗ y i , , , z, , ,
1) = k X i =1 12 λ i · (cid:0) ( x i , ⊗ ( y i ,
1) + ( − x i , ⊗ ( − y i , (cid:1) . (3.24)Since C and D are absolutely convex, we have ( ± x i , ∈ C ( C ) and ( ± y i , ∈ C ( D ) for all i . (cid:4) Theorem 3.25.
Let E and F be ( real ) vector spaces, let C ⊆ E , D ⊆ F be absolutely convex,and let M ⊂ C , N ⊂ D be proper faces. Then conv( M ⊗ s N ) is a face of conv( C ⊗ s D ) .Proof. By symmetry, − M ⊆ C and − N ⊆ D also define faces of C and D . First we provethat M ∩ − M = ∅ . Suppose that x ∈ M ∩ − M . Then also − x ∈ M ∩ − M , so by convexity0 ∈ M ∩ − M . But then for every y ∈ C we must have y, − y ∈ M , since 0 belongs to the relativeinterior of the line segment joining y and − y . This contradicts our assumption that M is a properface, so we conclude that M ∩ − M = ∅ . Analogously, N ∩ − N = ∅ .Let M ⊆ C ( C ) be the face of C ( C ) associated with M , and let M ⊆ C ( C ) be the faceassociated with − M . Since M ∩ − M = ∅ , it follows that M ∩ M = { } . Similarly, let N and N be the faces of C ( D ) associated with N and − N , respectively; then N ∩ N = { } .It follows from Proposition 3.22 that ( M (cid:63) π N ) + ( M (cid:63) π N ) is a face of C ( C ) ⊗ π C ( D ).To complete the proof, we show that (cid:0) ( M (cid:63) π N ) + ( M (cid:63) π N ) (cid:1) ∩ (cid:0) ( E ⊗ F ) ⊕ { } ⊕ { } ⊕ { } (cid:1) = { ( z, , ,
1) : z ∈ conv( M ⊗ s N ) } . We proceed analogously to the proof of Proposition 3.23.“ ⊆ ”. Let ( z, , , ∈ ( M (cid:63) π N ) + ( M (cid:63) π N ) be given. Then we may choose integers n ≥ k ≥
0, scalars λ , . . . , λ n ≥ x , . . . , x n ∈ M , y , . . . , y n ∈ N such that( z, , ,
1) = P ki =1 λ i · ( x i , ⊗ ( y i ,
1) + P ni = k +1 λ i · ( − x i , ⊗ ( − y i , P ni =1 λ i = 1 and z = P ni =1 λ i x i ⊗ y i , which shows that z ∈ conv( M ⊗ s N ).“ ⊇ ”. Let z ∈ conv( M ⊗ s N ) be given, and write z = P ki =1 λ i x i ⊗ y i with x , . . . , x k ∈ M and y , . . . , y k ∈ N . Then (3.24) shows that ( z, , , ∈ ( M (cid:63) π N ) + ( M (cid:63) π N ). (cid:4) Corollary 3.26.
Let E and F be ( real ) vector spaces, let C ⊆ E , D ⊆ F be absolutely convex,and let x ∈ C , y ∈ D be extreme points. Then x ⊗ y is an extreme point of conv( C ⊗ s D ) . Remark 3.27.
Theorem 3.25 fails if one of the faces is not proper. Indeed, if M = C , then0 ∈ M ⊗ s N , so now conv( M ⊗ s N ) is a face only if conv( M ⊗ s N ) = conv( C ⊗ s D ).Furthermore, Theorem 3.25 and Corollary 3.26 do not hold for arbitrary (non-symmetric)convex sets. (Example: 1 ⊗ − , ⊗ s [2 , ⊆ R ⊗ R = R .) Remark 3.28.
In many applications it is natural to start with closed absolutely convex sets, andtake the closed convex hull of their tensor product (e.g. [PTT11, Remark 3.19], [AS17, § E , F are finite-dimensional and if C , D are compact, then conv( C ⊗ s D ) is automaticallycompact, so here taking closures is not necessary. In particular:22 orollary 3.29. Let E and F be ( real ) finite-dimensional normed spaces. Then the closed unitball of the projective norm preserves proper faces: if M ⊂ B E , N ⊂ B F are proper faces, then conv( M ⊗ s N ) is a face of B E ⊗ π F . This had already been known for extreme points. More generally, if E and F are Banachspaces, then it follows from a result of Tseitlin ([Tse76], see also [RS82]) that the closed unit ballof the completed projective tensor product E ˜ ⊗ π F preserves extreme points if E or F has theapproximation property and if E or F has the Radon–Nikodym property. (The cited resultsrelate to extreme points in duals of operator spaces. Our assumptions on E and F ensure that E ˜ ⊗ π F ∼ = ( E ˜ ⊗ ε F ) isometrically; see [DF93, Theorem 16.6].) In particular, this settles thefinite-dimensional case, proving Corollary 3.29 for extreme points.We do not know whether the closed unit ball of the projective norm always preserves extremepoints, even in the algebraic tensor product. (This does not follow from Corollary 3.26, for theclosed unit ball of E ⊗ π F is the closure of conv( B E ⊗ s B F ).) Known results in this directionusually start with something stronger than an extreme point, such as a denting point ([RS86,Theorem 5], [Wer87, Corollary 4]).We should point out that the injective norm does not preserve extreme points; see Remark 4.52.23 The injective cone
Let h E, E i , h F, F i be dual pairs of (real) vector spaces, and let E + ⊆ E , F + ⊆ F be convexcones in the primal spaces. The injective cone in E ⊗ F is defined as E + ⊗ ε F + := (cid:8) u ∈ E ⊗ F : h u, ϕ ⊗ ψ i ≥ ϕ ∈ E + , ψ ∈ F + (cid:9) . The notation causes some ambiguity: E + ⊗ ε F + does not only depend on E + and F + , but alsoon the dual pairs h E, E i and h F, F i . To be fully precise, the injective cone should be denoted assomething like ( h E, E i , E + ) ⊗ ε ( h F, F i , F + ). We forego this cumbersome notation for the sakeof clarity; it will always be clear what is meant.If E and F are locally convex and if E ⊗ F is equipped with a compatible topology α (in thesense of Grothendieck [Gro55, p. 89], see also [K¨ot79, § ϕ ∈ E , ψ ∈ F thetensor product ϕ ⊗ ψ : E ⊗ α F → R is continuous, and as such has a unique extension to E ˜ ⊗ α F .In this setting we may likewise define the injective cone as E + ˜ ⊗ εα F + := (cid:8) u ∈ E ˜ ⊗ α F : ( ϕ ˜ ⊗ α ψ )( u ) ≥ ϕ ∈ E + , ψ ∈ F + (cid:9) . Clearly E + ⊗ ε F + = ( E + ˜ ⊗ εα F + ) ∩ ( E ⊗ F ). Note that, unlike the projective cone, the injectivecone typically becomes larger when passing from the algebraic tensor product E ⊗ F to thecompletion E ˜ ⊗ α F . Remark 4.1.
Let G be any reasonable dual of E ⊗ F (cf. p. 9). It is clear from the definitionthat E + ⊗ ε F + is the dual cone of E + ⊗ π F + under the dual pairing h E ⊗ F, G i . Likewise, E + ˜ ⊗ εα F + ⊆ E ˜ ⊗ α F is the dual cone of E + ⊗ π F + ⊆ ( E ˜ ⊗ α F ) .An immediate consequence is that the injective cone is always weakly closed. Furthermore,by the bipolar theorem, the dual cone of E + ⊗ ε F + with respect to the dual pair h E ⊗ F, G i is the σ ( G, E ⊗ F )-closure of E + ⊗ π F + . (Note that this need not be contained in E ⊗ F .)Similarly, in the locally convex setting, the dual cone of E + ˜ ⊗ εα F + is the weak- ∗ closure of E + ⊗ π F + ⊆ ( E ˜ ⊗ α F ) .What follows is a detailed study of the properties of the injective cone. We start by pointingout the characteristic property of the injective cone in § § § § § E + and F + determine faces of E + ⊗ ε F + . We show that the injective cone can be identified with a cone of positive bilinear forms. Let E (cid:126) F denote the space of separately weak- ∗ continuous bilinear forms on E × F : E (cid:126) F := Bil (cid:0) E w ∗ × F w ∗ ) . (K¨othe [K¨ot79, § (cid:2) instead of (cid:126) .)We shall understand E (cid:126) F to be equipped with the cone it inherits from Bil( E × F ). Inother words, b ∈ E (cid:126) F is positive if and only if b ( ϕ, ψ ) ≥ ϕ ∈ E + , ψ ∈ F + .The characteristic property of the injective cone is that it is given by a bipositive map to E (cid:126) F (algebraic case) or ˜ E (cid:126) ˜ F (completed locally convex case). A note about terminology: in the literature, E + ⊗ ε F + is usually called the biprojective cone (see e.g. [Mer64],[PS69], [Bir76], etc.). The results in this section show that this cone is in many ways analogous to the injectivetopology, and as such deserves the name injective cone . The only prior use of this name (that we are aware of) isin [Wit74] and [Mul97]. emark 4.2. Statements about positive bilinear forms can be turned into equivalent statementsabout positive linear operators in the following way. Recall that
Bil ( E w ∗ × F w ∗ ) is naturallyisomorphic to L ( E w ∗ , F w ). Under this correspondence, the positive cone of Bil ( E w ∗ × F w ∗ ) isthe cone of approximately positive operators E w ∗ → F w , i.e. those operators T that satisfy T [ E + ] ⊆ F + w . In particular, if F + is weakly closed, then this is just the cone of positive operators E w ∗ → F w . Similarly, Bil ( E w ∗ × F w ∗ ) ∼ = L ( F w ∗ , E w ), and the positive cone of Bil ( E w ∗ × F w ∗ )corresponds with the approximately positive cone of L ( F w ∗ , E w ).The advantage of sticking to bilinear forms is twofold: it keeps the theory symmetric in E and F , and it avoids the nuisance of having to take the weak closure of F + (or E + ).We proceed to prove the characteristic property in three settings: the algebraic tensor product,the completed injective tensor product, and arbitrary completed tensor products. Situation I: the algebraic tensor product
Let h E, E i and h F, F i be dual pairs. Equip E and F with their respective weak- ∗ topologies,and denote these spaces as E w ∗ and F w ∗ . The dual pairing h E ⊗ F, E ⊗ F i yields a naturalmap E ⊗ F , → ( E ⊗ F ) ∗ ∼ = Bil( E × F ). Note that the elements of E ⊗ F give rise to jointlycontinuous bilinear maps E w ∗ × F w ∗ → R . Indeed, an elementary tensor x ⊗ y ∈ E ⊗ F definesthe bilinear map ( ϕ, ψ )
7→ h x , ϕ ih y , ψ i , which is easily seen to be jointly continuous (use that ϕ
7→ h x , ϕ i and ψ
7→ h y , ψ i are continuous). Consequently, finite sums of elementary tensorsalso define jointly continuous bilinear maps, and the claim follows. This gives us natural inclusions E ⊗ F ⊆ B i ‘ ( E w ∗ × F w ∗ ) ⊆ E (cid:126) F ⊆ Bil( E × F ) . (4.3)From left to right, these are the spaces of (continuous) finite rank, jointly continuous, separatelycontinuous, and all bilinear forms on E w ∗ × F w ∗ . Proposition 4.4.
The elements of E + ⊗ ε F + are precisely those elements in E ⊗ F which definea positive bilinear map E × F → R ; that is: E + ⊗ ε F + = B i ‘ ( E w ∗ × F w ∗ ) + ∩ ( E ⊗ F ) . Proof.
By Remark 4.1, E + ⊗ ε F + is the dual cone of E + ⊗ π F + with respect to the dual pair h E ⊗ F, E ⊗ F i , so we have E + ⊗ ε F + = ( E ⊗ F ) ∗ + ∩ ( E ⊗ F ). It follows from Proposition 3.1 that u ∈ E ⊗ F belongs to E + ⊗ ε F + if and only if u defines a positive bilinear map E × F → R . (cid:4) Corollary 4.5.
All inclusions in (4.3) are bipositive.
Situation II: injective topology, completed
Let E and F be locally convex. Let E (cid:126) ε F denote the space E (cid:126) F (= Bil ( E w ∗ × F w ∗ )) equippedwith the bi-equicontinuous (or injective ) topology ε , that is, the locally convex topology given bythe family of seminorms p M,N ( b ) = sup ϕ ∈ M,ψ ∈ N | b ( ϕ, ψ ) | , ( M ⊆ E and N ⊆ F equicontinuous) . If E and F are complete, then E (cid:126) ε F is also complete (cf. [K¨ot79, § E ˜ ⊗ ε F with the closure of E ⊗ ε F in E (cid:126) ε F , and we have the following inclusionsof vector spaces: E ⊗ F ⊆ E ˜ ⊗ ε F ⊆ E (cid:126) ε F ⊆ Bil( E × F ) , ( E and F complete) . (4.6)25his may fail if E or F is not complete. (In particular, E ⊗ R = E (cid:126) R = E , but E ˜ ⊗ ε R = ˜ E .)However, in general we have E ˜ ⊗ ε F = ˜ E ˜ ⊗ ε ˜ F (cf. [K¨ot79, § E ⊗ F ⊆ E ˜ ⊗ ε F = ˜ E ˜ ⊗ ε ˜ F ⊆ ˜ E (cid:126) ε ˜ F ⊆ Bil( E × F ) . (4.7) Proposition 4.8.
Let E , F be locally convex. Then the natural inclusion E ˜ ⊗ ε F , → ˜ E (cid:126) ε ˜ F isbipositive; that is: E + ˜ ⊗ εε F + = Bil (cid:16) E σ ( E , ˜ E ) × F σ ( F , ˜ F ) (cid:17) + ∩ ( E ˜ ⊗ ε F ) . Proof.
Continuous linear functionals ϕ ∈ E and ψ ∈ F define a functional on E ˜ ⊗ ε F in twodifferent ways: either as the (unique) extension of ϕ ⊗ ψ to the completion E ˜ ⊗ ε F , or as therestriction of the evaluation functional f ϕ,ψ : Bil( E × F ) → R , b b ( ϕ, ψ ) to the subspace E ˜ ⊗ ε F .We claim that these two functionals coincide on E ˜ ⊗ ε F . The inclusion E ⊗ F , → Bil( E × F ) issuch that ( ϕ ⊗ ψ )( u ) = u ( ϕ, ψ ), so the functionals coincide on E ⊗ F . Furthermore, the functional f ϕ,ψ is easily seen to be continuous on ˜ E (cid:126) ε ˜ F (use that the sets { ϕ } ⊆ E , { ψ } ⊆ F areequicontinuous). Hence ϕ ⊗ ψ = f ϕ,ψ on E ⊗ F , and by continuity also on E ˜ ⊗ ε F , which provesour claim.It follows from the claim and the definition of E + ˜ ⊗ εε F + that an element u ∈ E ˜ ⊗ ε F belongsto E + ˜ ⊗ εε F + if and only if it defines a positive bilinear form E × F → R . (cid:4) Corollary 4.9.
All inclusions in (4.6) and (4.7) are bipositive.
We only needed the bi-equicontinuous topology on E (cid:126) F for the proof of Proposition 4.8.From here on out we can forget about it. Situation III: arbitrary compatible topology, completed
Now let α be an arbitrary compatible topology on E ⊗ F ( E and F locally convex). Since theinjective topology is the weakest compatible topology, we have a natural map E ˜ ⊗ α F → E ˜ ⊗ ε F ,so here the picture is as follows: E ⊗ F , → E ˜ ⊗ α F → E ˜ ⊗ ε F , → ˜ E (cid:126) ˜ F , → Bil( E × F ) . (4.10)The map E ˜ ⊗ α F → E ˜ ⊗ ε F need not be injective (this is related to the approximation property;see e.g. [DF93, Theorem 5.6]). However, it remains bipositive. Proposition 4.11.
Let E , F be locally convex, and let α be a compatible topology on E ⊗ F .Then the natural map Φ α → ε : E ˜ ⊗ α F → E ˜ ⊗ ε F is bipositive; that is: E + ˜ ⊗ εα F + = Φ − α → ε [ E + ˜ ⊗ εε F + ] . Proof.
Note that ϕ ˜ ⊗ α ψ = ( ϕ ˜ ⊗ ε ψ ) ◦ Φ α → ε , as they coincide on E ⊗ F . Hence: u ∈ E + ˜ ⊗ εα F + if and only if Φ α → ε ( u ) ∈ E + ˜ ⊗ εε F + . (cid:4) Corollary 4.12.
All maps in (4.10) are bipositive. .2 Mapping properties of the injective cone We show that the injective cone preserves all positive maps, bipositive maps (provided thecones are closed), and retracts, and show that it fails to preserve quotients, pushforwards, andapproximate pushforwards.Let h E, E i , h F, F i , h G, G i , h H, H i be dual pairs, equipped with convex cones E + , F + , G + , H + in the primal spaces. Given T ∈ L ( E w , G w ) and S ∈ L ( F w , H w ), we define T (cid:2) S : Bil( E × F ) → Bil( G × H ) by b (cid:16) ( ϕ, ψ ) b ( T ϕ, S ψ ) (cid:17) , where T ∈ L ( G w ∗ , E w ∗ ), S ∈ L ( H w ∗ , F w ∗ ) denote the respective adjoints.Note that ( T (cid:2) S ) b is separately weak- ∗ continuous whenever b is, so T (cid:2) S restricts to a map T (cid:126) S : E (cid:126) F → G (cid:126) H . Proposition 4.13.
The following diagram commutes: E ⊗ F E (cid:126) F Bil( E × F ) G ⊗ H G (cid:126) H Bil( G × H ) . T ⊗ S T (cid:126)
S T (cid:2) S Proof.
The rightmost square commutes by definition ( T (cid:126) S is the restriction of T (cid:2) S ).For the leftmost square, note that x ⊗ y ∈ E ⊗ F defines the bilinear map ( ϕ, ψ )
7→ h x, ϕ ih y, ψ i ,and T x ⊗ Sy defines the bilinear map ( ϕ, ψ )
7→ h
T x, ϕ ih Sy, ψ i = h x, T ϕ ih y, S ϕ i . (cid:4) Proposition 4.14. If E , F , G , H are locally convex, if T ∈ L ( E, G ) , S ∈ L ( F, H ) , and if α and β are compatible topologies on E ⊗ F and G ⊗ H for which the map T ⊗ α → β S : E ⊗ α F → G ⊗ β H is continuous, then the following diagram commutes: E ⊗ F E ˜ ⊗ α F E ˜ ⊗ ε F ˜ E (cid:126) ˜ F Bil( E × F ) G ⊗ H G ˜ ⊗ β H G ˜ ⊗ ε H ˜ G (cid:126) ˜ H Bil( G × H ) . T ⊗ S T ˜ ⊗ α → β S T ˜ ⊗ ε S ˜ T (cid:126) ˜ S T (cid:2) S Here the horizontal maps are the ones from (4.10), which are bipositive by Corollary 4.12.
Proof.
The rightmost square commutes since T (cid:2) S = ˜ T (cid:2) ˜ S (use that T : E → G and itscompletion ˜ T : ˜ E → ˜ G have the same adjoint T = ˜ T : G → E ), and ˜ T (cid:126) ˜ S is a restriction of˜ T (cid:2) ˜ S . (However, ˜ T (cid:126) ˜ S = T (cid:126) S , as the domain and codomain are different!)The other squares (and the triangles) commute because the respective compositions agree onthe dense subspace E ⊗ F (or G ⊗ H ). (cid:4) Lemma 4.15.
Let h E, E i , h F, F i , h G, G i , h H, H i be dual pairs, and let T ∈ L ( E w , G w ) and S ∈ L ( F w , H w ) .(a) If T and S are positive, then T (cid:2) S is positive.(b) If E + w = T − [ G + w ] and F + w = S − [ H + w ] , then T (cid:126) S is bipositive. roof. (a) Let b ∈ Bil( E × F ) be positive. If ϕ ∈ G + and ψ ∈ H + , then ϕ ◦ T ∈ E + and ψ ◦ S ∈ F + (the composition of positive linear maps is positive), so ( T (cid:2) S )( b )( ϕ, ψ ) ≥
0. It follows that( T (cid:2) S )( b ) is a positive bilinear map on G × H , so T (cid:2) S is positive.(b) It follows from the assumptions and duality (cf. [Dob20c, Proposition 2]) that the adjoints T ∈ L ( G w ∗ , E w ∗ ), S ∈ L ( H w ∗ , F w ∗ ) are weak- ∗ approximate pushforwards. But note that T (cid:126) S is precisely the map ( T ⊗ S ) from Lemma 3.3, so it follows from said lemma that T (cid:126) S is bipositive. (cid:4) Theorem 4.16.
Let T ∈ L ( E w , G w ) and S ∈ L ( F w , H w ) .(a) If T and S are positive, then ( T ⊗ S )[ E + ⊗ ε F + ] ⊆ G + ⊗ ε H + .(b) If E + w = T − [ G + w ] and F + w = S − [ H + w ] , then E + ⊗ ε F + = ( T ⊗ S ) − [ G + ⊗ ε H + ] .In summary: the algebraic injective cone preserves continuous positive maps and ( continuous ) approximately bipositive maps.Proof. All horizontal arrows in the diagram from Proposition 4.13 are bipositive (by Corollary 4.5),so (a) and (b) follow easily from Lemma 4.15. For the summary, recall from Remark 4.1 that E + ⊗ ε F + and G + ⊗ ε H + are weakly closed, so in (b) we find that T ⊗ S is approximatelybipositive (in addition to being bipositive). (cid:4) Theorem 4.17.
Let E , F , G , H be locally convex, let T ∈ L ( E, G ) and S ∈ L ( F, H ) , andlet α and β be compatible topologies on respectively E ⊗ F and G ⊗ H for which the map T ⊗ α → β S : E ⊗ α F → G ⊗ β H is continuous.(a) If T and S are positive, then ( T ˜ ⊗ α → β S )[ E + ˜ ⊗ εα F + ] ⊆ G + ˜ ⊗ εβ H + .(b) If E and F are complete and if E + = T − [ G + ] and F + = S − [ H + ] , then E + ˜ ⊗ εα F + =( T ˜ ⊗ α → β S ) − [ G + ˜ ⊗ εβ H + ] .In summary: the completed injective cone preserves continuous positive maps, and ( continuous ) approximately bipositive maps if E and F are complete.Proof. (a) All horizontal arrows in the diagram from Proposition 4.14 are bipositive (by Corollary 4.12),so the result follows from Lemma 4.15(a).(b) Recall: in a locally convex space, the weak closure and original closure of a convex conecoincide. Moreover, note that we may assume without loss of generality that G and H arealso complete. (Extend T to the map ˜ T : E → ˜ G , and let g G + denote the closure of G + in˜ G . Then ˜ T − [ g G + ] = T − [ G + ], since ran( ˜ T ) ⊆ G .)We refer again to the diagram from Proposition 4.14. All horizontal arrows in are bipositive,and the vertical arrow T (cid:126) S = ˜ T (cid:126) ˜ S is bipositive by Lemma 4.15(b). The result is easilydeduced. (cid:4) By our definition, approximately bipositive maps are already required to be continuous. emark 4.18. We get one of the characteristic properties of the injective topology for free: if E , F , G , H are locally convex, E and F complete, and if T ∈ L ( E, G ) and S ∈ L ( F, H ) are injective,then so is T ˜ ⊗ ε S ∈ L ( E ˜ ⊗ ε F, G ˜ ⊗ ε H ). Indeed, equip all spaces with the trivial cone { } , thenevery dual cone is the entire dual space, so Bil( E × F ) + = { } . Therefore E ˜ ⊗ ε F and G ˜ ⊗ ε H are also equipped with the zero cone (since E ˜ ⊗ ε F → Bil( E × F ) is bipositive and injective).Since T ˜ ⊗ ε S is bipositive, we have ( T ˜ ⊗ ε S ) − [ { } ] = { } , so T ˜ ⊗ ε S is injective.This shows immediately that the completeness assumptions in Theorem 4.17(b) cannot beomitted. (After all, T ˜ ⊗ ε id R : E ˜ ⊗ ε R → G ˜ ⊗ ε R is simply the completion ˜ T : ˜ E → ˜ G , which mayfail to be injective even if T is injective.)A similar argument shows that the weak closures in Lemma 4.15(b) and subsequent theoremscannot be omitted: the map T ⊗ ε id R : E ⊗ R → G ⊗ R is simply T , but with the positive cones E + , G + replaced by their weak closures. But one does not necessarily have T − [ G + w ] = E + w whenever T − [ G + ] = E + . (Concrete example: let G = R with G + = { ( x, y ) : x > } ∪ { (0 , } ,let E := span { (0 , } ⊆ G with E + := G + ∩ E , and let T be the inclusion E , → G .) Remark 4.19.
A topological order retract G ⊆ E is given by two continuous positive linearmaps E (cid:16) G , → E , so it follows at once that the injective cone (in all its incarnations) preservesall topological order retracts, without any assumptions on completeness or weak closures. Theargument is analogous to that of Proposition 3.2(c).The following example shows that the injective cone does not preserve pushforwards, not evenapproximately. Example 4.20 (Dual to Example 3.7) . Let E be a finite-dimensional space equipped with aproper, generating, polyhedral cone which is not a simplex cone. Let x , . . . , x m be representativesof the extremal rays of E + , and let R m be equipped with the standard cone R m ≥ . Then the map T : R m → E , ( λ , . . . , λ m ) λ x + . . . + λ m x m is a pushforward (i.e. T [ R m ≥ ] = E + ).Since E + is not a simplex cone, it follows from [BL75, Proposition 3.1] that E + ⊗ π E ∗ + = E + ⊗ ε E ∗ + . On the other hand, R m ≥ ⊗ π E ∗ + = R m ≥ ⊗ ε E ∗ + , and it follows from Proposition 3.2(b)that T ⊗ id E ∗ is a pushforward for the projective cone. Therefore:( T ⊗ id E ∗ )[ R m ≥ ⊗ ε E ∗ + ] = ( T ⊗ id E ∗ )[ R m ≥ ⊗ π E ∗ + ] = E + ⊗ π E ∗ + = E + ⊗ ε E ∗ + . This shows that T ⊗ id E ∗ is not a pushforward for the injective cone.Note that all cones in this example are polyhedral, and therefore closed. In particular, thesituation is not resolved by adding closures, which shows that the injective cone does not preserveapproximate pushforwards. The finite-dimensional techniques used in Example 4.20 will be discussed in more detail inthe follow-up paper [Dob20b].
We determine the lineality space of E (cid:126) F , and we use this to give necessary and sufficientconditions for the injective cone (in all its incarnations) to be proper. Direct formulas for thelineality space (under certain topological assumptions) will be given in § h E, E i , h F, F i be dual pairs, equipped with convex cones E + ⊆ E , F + ⊆ F inthe primal spaces. Proposition 4.21.
The lineality space of ( E (cid:126) F ) + is the set of those bilinear forms in E (cid:126) F that vanish on span( E + ) w ∗ × span( F + ) w ∗ = lin( E + w ) ⊥ × lin( F + w ) ⊥ . roof. If b ∈ E (cid:126) F vanishes on span( E + ) w ∗ × span( F + ) w ∗ , then in particular it vanishes on E + × F + , so evidently both b and − b define positive bilinear forms. Conversely, if b ∈ lin(( E (cid:126) F ) + ),then both b and − b are positive on E + × F + , so it follows that b must vanish on E + × F + . Therefore b also vanishes on span( E + ) × span( F + ), and consequently on span( E + ) w ∗ × span( F + ) w ∗ . (Useweak- ∗ continuity in one variable at a time, as we did in the proof of Lemma 3.3.)Since lin( E + w ) = ⊥ ( E + ) (see § E + ) w ∗ = lin( E + w ) ⊥ . (cid:4) Direct formulas for the lineality space of the injective cone will be given in Corollary 4.37(c)(in E ⊗ F ) and Corollary 4.41(b) (in E (cid:126) F ). For now, we focus on conditions for the injectivecone to be proper. Theorem 4.22.
The following are equivalent:(i) E + ⊗ ε F + is a proper cone;(ii) For every subspace E ⊗ F ⊆ G ⊆ E (cid:126) F , the cone G + := G ∩ ( E (cid:126) F ) + is proper.(iii) ( E (cid:126) F ) + is a proper cone;(iv) E = { } , or F = { } , or both E + w and F + w are proper cones.In particular, the injective tensor product of weakly closed proper cones is a proper cone. Note that the equivalence (i) ⇐⇒ (iv) is very similar to Theorem 3.10. However, we shouldpoint out that the corner case is slightly different now. In Theorem 3.10, the corner case is whenone of the cones is trivial; here the corner case is when one of the spaces is trivial. Proof of Theorem 4.22. (iii) = ⇒ (ii) . Trivial. (ii) = ⇒ (i) . Immediate, since E + ⊗ ε F + = ( E ⊗ F ) ∩ ( E (cid:126) F ) + . (iv) = ⇒ (iii) . If E = { } , then clearly E (cid:126) F = { } , so ( E (cid:126) F ) + is a proper cone regardlessof any properties of F + (and similarly if F = { } ). If E + w and F + w are proper cones, thenlin( E + w ) = lin( F + w ) = { } , so it follows from Proposition 4.21 that lin(( E (cid:126) F ) + ) = { } . (i) = ⇒ (iv) . We prove the contrapositive: suppose that E, F = { } and that E + w is not aproper cone. Then we may choose x ∈ E \ { } with ± x ∈ E + w . Note that ( E + w ) = E + , so forevery ϕ ∈ E + we have ϕ ( x ) , ϕ ( − x ) ≥
0, and therefore ϕ ( x ) = 0. Now choose any y ∈ F \ { } (herewe use that F = { } ), then for all ϕ ∈ E + , ψ ∈ F + we have h x ⊗ y, ϕ ⊗ ψ i = ϕ ( x ) ψ ( y ) = 0 · ψ ( y ) = 0,so we find ± x ⊗ y ∈ E + ⊗ ε F + . Since x and y are non-zero, we have x ⊗ y = 0, and we concludethat E + ⊗ ε F + fails to be proper. (cid:4) To tell whether E + ˜ ⊗ εα F + is a proper cone, we need to assume that E and F are complete.(In the case where E and F are not complete, an answer can be found by first passing to thecompletions ˜ E , ˜ F .) Corollary 4.23.
Let
E, F be complete locally convex spaces, E + ⊆ E , F + ⊆ F convex cones,and α a compatible locally convex topology on E ⊗ F . Then the following are equivalent:(i) E + ˜ ⊗ εα F + ⊆ E ˜ ⊗ α F is a proper cone;(ii) E = { } , or F = { } , or both E + and F + are proper cones and the natural map E ˜ ⊗ α F → E ˜ ⊗ ε F is injective. roof. First of all, recall that E + = E + w , since E + is convex and E is locally convex. Likewise, F + = F + w .For the injective topology, recall from (4.6) that we have E ⊗ F ⊆ E ˜ ⊗ ε F ⊆ E (cid:126) F , since E and F are complete. Hence for α = ε the result follows from Theorem 4.22.For general α , recall that E ˜ ⊗ α F → E ˜ ⊗ ε F is bipositive. Therefore: (i) = ⇒ (ii) . If E + ˜ ⊗ εα F + is proper, then the bipositive map E ˜ ⊗ α F → E ˜ ⊗ ε F is automaticallyinjective (cf. Remark A.7), and the subcone E + ⊗ ε F + ⊆ E + ˜ ⊗ εα F + is also proper. It followsfrom Theorem 4.22 that (ii) holds. (ii) = ⇒ (i) . It follows from the assumptions that E + ˜ ⊗ εε F + is a proper cone and that E ˜ ⊗ α F → E ˜ ⊗ ε F is injective. (The latter statement is trivially true if E = { } or F = { } ;otherwise it holds by assumption.) Since E ˜ ⊗ α F → E ˜ ⊗ ε F is bipositive and injective, it followsthat E + ˜ ⊗ εα F + is also proper. (cid:4) Remark 4.24.
In Corollary 4.23, the assumption that E and F are complete cannot be omitted.Under the natural isomorphism E ˜ ⊗ α R ∼ = ˜ E , the injective cone E + ˜ ⊗ εα R + corresponds with f E + (the closure of E + in ˜ E ). However, it can happen that E + is proper but f E + is not (e.g. [Dob20a,Example 6.4]). Remark 4.25.
The natural map E ˜ ⊗ α F → E ˜ ⊗ ε F is not always injective; this is related tothe approximation property. Further remarks along this line can be found in § In this section, we present a general way to construct faces of the space E (cid:126) F = Bil ( E w ∗ × F w ∗ )of separately weak- ∗ continuous bilinear forms. This will be used in § E ⊗ F → E (cid:126) F and E ˜ ⊗ α F → ˜ E (cid:126) ˜ F (see § E (cid:126) F ) + (resp. ( ˜ E (cid:126) ˜ F ) + ) immediately gives us a facein E + ⊗ ε F + (resp. E + ˜ ⊗ εα F + ). Therefore we focus on faces in E (cid:126) F . For ideals in E ⊗ F and E ˜ ⊗ α F , see § Definition 4.26.
Let h E, E i , h F, F i be dual pairs, and let E + ⊆ E , F + ⊆ F be convex cones.Given b ∈ E (cid:126) F and subsets M ⊆ E , N ⊆ F , let us write b ( M , · ) := { b ( ϕ, · ) : ϕ ∈ M } ⊆ ( F w ∗ ) = F ; b ( · , N ) := { b ( · , ψ ) : ψ ∈ N } ⊆ ( E w ∗ ) = E. Given subsets M ⊆ E , M ⊆ E , N ⊆ F , N ⊆ F , we define M (cid:110) N := (cid:8) b ∈ E (cid:126) F : b ( M , · ) ⊆ N (cid:9) ; M (cid:111) N := (cid:8) b ∈ E (cid:126) F : b ( · , N ) ⊆ M (cid:9) . Under the natural isomorphism E (cid:126) F = Bil ( E w ∗ × F w ∗ ) ∼ = L ( E w ∗ , F w ), the set M (cid:110) N is simplythe set of operators T : E w ∗ → F w satisfying T [ M ] ⊆ N . Likewise, M (cid:111) N corresponds withthe set of operators S : F w ∗ → E w satisfying S [ N ] ⊆ M .Note that the positive cone can be described as ( E (cid:126) F ) + = E + (cid:110) F + w = E + w (cid:111) F + . Lemma 4.27. If M ⊆ E + , N ⊆ F + are subsets of the dual cones and if M ⊆ E + w , N ⊆ F + w are faces, then ( M (cid:110) N ) ∩ ( E (cid:126) F ) + and ( M (cid:111) N ) ∩ ( E (cid:126) F ) + are faces of ( E (cid:126) F ) + . roof. Given ϕ ∈ E , let L ϕ : E (cid:126) F → ( F w ∗ ) = F denote the map b b ( ϕ, · ). If ϕ ∈ E + , then L ϕ is a positive linear map in the sense that L ϕ [( E (cid:126) F ) + ] ⊆ F + w . Therefore L − ϕ [ N ] ∩ ( E (cid:126) F ) + defines a face of ( E (cid:126) F ) + . Since ( M (cid:110) N ) ∩ ( E (cid:126) F ) + can be written as an intersection of faces,( M (cid:110) N ) ∩ ( E (cid:126) F ) + = \ ϕ ∈ M L − ϕ [ N ] ∩ ( E (cid:126) F ) + , it also a face of ( E (cid:126) F ) + . The conclusion for ( M (cid:111) N ) ∩ ( E (cid:126) F ) + follows by symmetry. (cid:4) Lemma 4.27 presents a very general way of defining faces of the injective cone, and will befundamental to most of what follows. As a first application, we study a construction of faces inthe injective cone that is dual to the construction in the projective cone (cf. § § Theorem 4.28.
Let M ⊆ E + w , N ⊆ F + w be faces, and define M (cid:62) ε N := ( M (cid:111) N (cid:5) ) ∩ ( M (cid:5) (cid:110) N ) ∩ ( E (cid:126) F ) + ; M (cid:63) ε N := ( M (cid:111) F + ) ∩ ( E + (cid:110) N ) . Then:(a) M (cid:62) ε N and M (cid:63) ε N are faces of ( E (cid:126) F ) + .(b) The face lattice of ( E (cid:126) F ) + contains the following partially ordered subset: M (cid:62) ε NM (cid:62) ε lin( F + w ) = M (cid:63) ε F + w = M (cid:111) F + lin( E + w ) (cid:62) ε N = E + w (cid:63) ε N = E + (cid:110) NM (cid:63) ε N This subset respects meets from the face lattice: M (cid:63) ε N = ( M (cid:62) ε lin( F + w )) ∩ (lin( E + w ) (cid:62) ε N ) = ( M (cid:63) ε F + w ) ∩ ( E + w (cid:63) ε N ) . (c) If M and N are dual faces, then so are M (cid:62) ε N and M (cid:63) ε N , and one has M (cid:62) ε N = (cid:5) ( M (cid:5) (cid:63) π N (cid:5) ) = ( M (cid:111) N (cid:5) ) ∩ ( E (cid:126) F ) + = ( M (cid:5) (cid:110) N ) ∩ ( E (cid:126) F ) + ; M (cid:63) ε N = (cid:5) ( M (cid:5) (cid:62) π N (cid:5) ) . If this is the case, then the subset from (b) respects meets and joins from the lattice of h ( E (cid:126) F ) + , E + ⊗ π F + i -dual faces ( as defined in Appendix A.3 ) .(d) If M and N are exposed faces, then so is M (cid:62) ε N .(e) If M and N as well as lin( E + w ) and lin( F + w ) are exposed faces, then so is M (cid:63) ε N . Note: in the finite-dimensional case, the conclusion in (c) is simply that the four-elementsubset from (b) respects the operations of the lattice of exposed faces. (Here we use that( E + ⊗ ε F + ) ∗ = E ∗ + ⊗ π F ∗ + because E ∗ + ⊗ π F ∗ + is closed; see [Dob20b, Corollary 4.11(b)].)32 roof of Theorem 4.28. (a) Note that everything in M (cid:111) F + is automatically positive, for if b ( · , F + ) ⊆ M then certainly b ( · , F + ) ⊆ E + w . This shows that M (cid:111) F + = ( M (cid:111) F + ) ∩ ( E (cid:126) F ) + . Now the result followsfrom Lemma 4.27, since the intersection of two faces is again a face.(b) If b ∈ M (cid:111) F + , then b ( · , F + ) ⊆ M , so in particular b vanishes on M (cid:5) × F + . Therefore b ( M (cid:5) , · ) ⊆ ⊥ ( F + ) = lin( F + w ), which shows that M (cid:111) F + ⊆ M (cid:5) (cid:110) lin( F + w ). Since we alsohave M (cid:111) F + ⊆ ( E (cid:126) F ) + (see (a)), it follows from the definition that M (cid:62) ε lin( F + w ) = ( M (cid:111) F + ) ∩ ( M (cid:5) (cid:110) lin( F + w )) ∩ ( E (cid:126) F ) + = M (cid:111) F + . Similarly, since E + (cid:110) F + w = ( E (cid:126) F ) + , it follows again from the definition that M (cid:63) ε F + w = ( M (cid:111) F + ) ∩ ( E + (cid:110) F + w ) ∩ ( E (cid:126) F ) + = M (cid:111) F + . The equality lin( E + w ) (cid:62) ε N = E + w (cid:63) ε N = E + (cid:110) N follows analogously. As a consequence,the intersection formula follows immediately from the definition of M (cid:63) ε N . Finally, theupwards inclusions follow by noting that if M ⊆ M ⊆ E + w and N ⊆ N ⊆ F + w arefaces, then M (cid:62) ε N ⊆ M (cid:62) ε N .(c) If b ∈ M (cid:111) N (cid:5) , then b ( · , N (cid:5) ) ⊆ M , so in particular b vanishes on M (cid:5) × N (cid:5) . Conversely, if b ∈ ( E (cid:126) F ) + vanishes on M (cid:5) × N (cid:5) , then b ( · , N (cid:5) ) ⊆ (cid:5) ( M (cid:5) ) = M , so b ∈ M (cid:111) N (cid:5) . Thisproves that( M (cid:111) N (cid:5) ) ∩ ( E (cid:126) F ) + = (cid:8) b ∈ ( E (cid:126) F ) + : b ( ϕ, ψ ) = 0 for all ϕ ∈ M (cid:5) , ψ ∈ N (cid:5) (cid:9) = (cid:5) ( M (cid:5) ⊗ s N (cid:5) ) . By symmetry, the same is true of M (cid:5) (cid:110) N , so we find M (cid:62) ε N = ( M (cid:111) N (cid:5) ) ∩ ( E (cid:126) F ) + = ( M (cid:5) (cid:110) N ) ∩ ( E (cid:126) F ) + = (cid:5) ( M (cid:5) ⊗ s N (cid:5) ) . Since M (cid:5) (cid:63) π N (cid:5) is the face (of E + ⊗ π F + ) generated by M (cid:5) ⊗ s N (cid:5) , it follows that M (cid:62) ε N = (cid:5) ( M (cid:5) (cid:63) π N (cid:5) ). This shows that M (cid:62) ε N is a dual face.Since lin( E + w ) = (cid:5) ( E + ) and lin( F + w ) = (cid:5) ( F + ) are dual faces, it follows from the intersectionformula from (b) that M (cid:63) ε N is also a dual face. Furthermore, since lin( E + w ) (cid:5) = E + andlin( F + w ) (cid:5) = F + , it follows that M (cid:63) ε N = ( M (cid:62) ε lin( F + w )) ∩ (lin( E + w ) (cid:62) ε N )= (cid:5) ( M (cid:5) (cid:63) π F + ) ∩ (cid:5) ( E + (cid:63) π N (cid:5) )= (cid:5) (cid:0) ( M (cid:5) (cid:63) π F + ) + ( E + (cid:63) π N (cid:5) ) (cid:1) = (cid:5) ( M (cid:5) (cid:62) π N (cid:5) ) , where the last step uses that M (cid:5) (cid:62) π N (cid:5) = ( M (cid:5) (cid:63) π F + ) + ( E + (cid:63) π N (cid:5) ), by Theorem 3.13(b).That the diagram from (b) respects joins from the lattice of h ( E (cid:126) F ) + , E + ⊗ π F + i -dualfaces follows from duality. Indeed, by Theorem 3.13(c) and Theorem 3.13(d), M (cid:5) (cid:62) π N (cid:5) and M (cid:5) (cid:63) π N (cid:5) are h E + ⊗ π F + , ( E (cid:126) F ) + i -dual faces (use that lin( E + ) and lin( F + ) are33utomatically dual faces, because E + and F + are weak- ∗ closed; cf. Remark 3.15), so itfollows that ( M (cid:62) ε N ) (cid:5) = M (cid:5) (cid:63) π N (cid:5) ;( M (cid:63) ε N ) (cid:5) = M (cid:5) (cid:62) π N (cid:5) . Therefore the join of M (cid:62) ε lin( F + w ) and lin( E + w ) (cid:62) ε N in the lattice of h ( E (cid:126) F ) + , E + ⊗ π F + i -dual faces is given by (cid:5) (cid:16)(cid:0) M (cid:62) ε lin( F + w ) (cid:1) (cid:5) ∩ (cid:0) lin( E + w ) (cid:62) ε N (cid:1) (cid:5) (cid:17) = (cid:5) (cid:0) ( M (cid:5) (cid:63) π F + ) ∩ ( E + (cid:63) π N (cid:5) ) (cid:1) = (cid:5) (cid:0) M (cid:5) (cid:63) π N (cid:5) (cid:1) = M (cid:62) ε N. (d) Suppose that M = (cid:5) { ϕ } and N = (cid:5) { ψ } . Then in particular M and N are dual faces, soby (c) we have M (cid:62) ε N = (cid:5) ( M (cid:5) (cid:63) π N (cid:5) ) = ( M (cid:111) N (cid:5) ) ∩ ( E (cid:126) F ) + = ( M (cid:5) (cid:110) N ) ∩ ( E (cid:126) F ) + . We prove that M (cid:62) ε N = (cid:5) { ϕ ⊗ ψ } . Evidently one has { ϕ ⊗ ψ } ⊆ M (cid:5) (cid:63) π N (cid:5) , so (cid:5) { ϕ ⊗ ψ } ⊇ (cid:5) ( M (cid:5) (cid:63) π N (cid:5) ) = M (cid:62) ε N . For the converse, suppose that b ∈ ( E (cid:126) F ) + is suchthat b ( ϕ , ψ ) = 0. Then b ( · , ψ ) ∈ (cid:5) { ϕ } = M , so b vanishes on M (cid:5) × { ψ } . It follows that b ( M (cid:5) , · ) ⊆ (cid:5) { ψ } = N , so b ∈ ( M (cid:5) (cid:110) N ) ∩ ( E (cid:126) F ) + = M (cid:62) ε N .(e) This follows from (d) and the intersection formula from (b). (cid:4) Remark 4.29.
In Theorem 4.28(e), it is required that lin( E + w ) and lin( F + w ) are exposed.Recall that this is automatically the case if E and F are separable normed spaces; see Remark 3.15and Corollary A.20.Much as in the projective case, this assumption on lin( E + w ) and lin( F + w ) cannot be omitted.The example runs along the same lines as the example in Remark 3.15, except we need a muchlarger space. Let E + be a weakly closed proper cone for which { } is not exposed (cf. Example A.21,Example A.22), and let F := R with the standard cone, so that E (cid:126) F ∼ = E . Take some exposedface M ⊆ E + , and let N := { } ⊆ R be the minimal face. Then M (cid:63) ε N = { } , which is notexposed by assumption. Remark 4.30.
Theorem 4.28(c) presents a duality between the four-element sublattices fromTheorem 3.13(b) and Theorem 4.28(b). In the projective diagram, the top face M (cid:62) π N is notmerely the join, but even the sum of the left and right faces M (cid:63) π F + and E + (cid:63) π N . Given thatthe injective diagram is dual to the projective diagram, could the same be true here?Unfortunately, this is not the case, and it already fails for proper, generating, polyhedralcones in finite-dimensional spaces. In this setting, all faces are exposed, so by Theorem 4.28(c) anequivalent question is the following: if f : E ∗ → F is positive with f [ M (cid:5) ] ⊆ N , then can f bewritten as f = g + h with g and h positive and g [ M (cid:5) ] = { } and h [ E ∗ + ] ⊆ N ?Counterexample: let F + be a proper, generating, polyhedral cone with a facet N ⊆ F + suchthat at least two extremal rays of F + are not contained in N . Furthermore, let E + := F ∗ + with M := N (cid:5) , and let f : E ∗ = F → F be the identity. Then one has f [ M (cid:5) ] ⊆ N . However, if f = g + h is the desired decomposition, then rank( g ) ≤
1, because ker( g ) contains a facet, so g [ F + ] is either a ray or { } . But now every x ∈ F + can be written as x = g ( x ) + h ( x ) ∈ g [ F + ] + N ,contradicting our assumption that at least two extremal rays of F + are not contained in N .34 .5 Order ideals for the injective cone Recall that I I + defines a surjective many-to-one correspondence between order ideals andfaces (see Appendix A.1). In order to get more convenient formulas for the faces of the injectivecone, it is helpful to formulate these results in terms of ideals. The main aim in this section isto provide sufficient conditions so that I ⊗ J and ( I ⊗ F ) + ( E ⊗ J ) are ideals for the injectivecone, given that I ⊆ E and J ⊆ F are ideals in the base spaces. (Similar questions in E (cid:126) F and E ˜ ⊗ α F are also addressed.)Recall from § E ⊗ F , → E (cid:126) F and E ˜ ⊗ α F → ˜ E (cid:126) ˜ F . Given subsets X ⊆ E (cid:126) F and Y ⊆ ˜ E (cid:126) ˜ F , we denote by X ∩ ( E ⊗ F ) and Y ∩ ( E ˜ ⊗ α F ) the inverse images of X and Y under these maps. (This is a slight abuse of notation,for the map E ˜ ⊗ α F → ˜ E (cid:126) ˜ F might fail to be injective in the absence of the approximationproperty, but this will cause no confusion.) It is not hard to see that the inverse image of anideal (resp. face) under a bipositive map is again an ideal (reps. face) (cf. Proposition A.3(b)), so( E ⊗ F ) ∩ X and ( E ˜ ⊗ α F ) ∩ Y are ideals (resp. faces) whenever X and Y are ideals (resp. faces).This is the approach that we will take: we establish ideals in E (cid:126) F and restrict these to ideals inthe algebraic/completed tensor product.In order to obtain ideals in E (cid:126) F , we note that the faces obtained in Lemma 4.27 cansometimes be written as the positive part of a linear subspace. Lemma 4.31.
In the notation from § M ⊆ E and N ⊆ F are subsets and if M ⊆ E and N ⊆ F are linear subspaces, then M (cid:110) N and M (cid:111) N are linear subspaces.(b) If I ⊆ E and J ⊆ F are subspaces and if I is weakly closed, then I ⊥ (cid:110) J ⊆ I (cid:111) J ⊥ .(c) If I ⊆ E and J ⊆ F are weakly closed subspaces, then I ⊥ (cid:110) J = I (cid:111) J ⊥ = ⊥ ( I ⊥ ⊗ J ⊥ ) ,where the orthogonal complement is taken with respect to the dual pair h E (cid:126) F, E ⊗ F i . Note that ⊥ ( I ⊥ ⊗ J ⊥ ) ⊆ E (cid:126) F is the set of separately weak- ∗ continuous bilinear forms E × F → R that vanish on I ⊥ × J ⊥ . Proof of Lemma 4.31. (a) If T , T : E w ∗ → F w map the subset M ⊆ E in the subspace N ⊆ F , and if λ, µ ∈ R arearbitrary, then λT + µT also maps M in N .(b) If b ( I ⊥ , · ) ⊆ J , then b ( I ⊥ , J ⊥ ) = { } , hence b ( · , J ⊥ ) ⊆ ⊥ ( I ⊥ ) = I , since I is weakly closed.(c) Since J is weakly closed, one has b ( I ⊥ , · ) ⊆ J if and only if b ( I ⊥ , J ⊥ ) = { } , i.e. b vanisheson I ⊥ × J ⊥ . Therefore I ⊥ (cid:110) J = ⊥ ( I ⊥ ⊗ J ⊥ ). The other equality follows analogously. (cid:4) We can now formulate the following “linearization” of Lemma 4.27.
Lemma 4.32.
Let M ⊆ E + be a set of positive linear functionals, and let N ⊆ F + w be a face.(a) If J ⊆ F is a weakly closed subspace such that J ∩ F + w = N , then ( M (cid:110) N ) ∩ ( E (cid:126) F ) + = (cid:0) span( M ) w ∗ (cid:110) J (cid:1) ∩ ( E (cid:126) F ) + . In particular, span( M ) w ∗ (cid:110) J is an ideal in E (cid:126) F . b) If J ⊆ F is a subspace such that J ∩ F + w = N , then ( M (cid:110) N ) ∩ ( E (cid:126) F ) + ∩ ( E ⊗ F ) = (cid:0) span( M ) w ∗ (cid:110) J (cid:1) ∩ ( E (cid:126) F ) + ∩ ( E ⊗ F ) . In particular, (cid:0) span( M ) w ∗ (cid:110) J (cid:1) ∩ ( E ⊗ F ) is an ideal in E ⊗ F .Interchanging E and F yields corresponding statements for ideals of the form I (cid:111) span( N ) w ∗ and ( I (cid:111) span( N ) w ∗ ) ∩ ( E ⊗ F ) .Proof. (a) “ ⊆ ”. If b ∈ M (cid:110) N , then we have b ( M , · ) ⊆ N ⊆ J , so it follows by linearity and continuitythat b (span( M ) w ∗ , · ) ⊆ J . This shows that M (cid:110) N ⊆ span( M ) w ∗ (cid:110) J .“ ⊇ ”. If b ∈ (cid:0) span( M ) w ∗ (cid:110) J (cid:1) ∩ ( E (cid:126) F ) + , then b ( M , · ) ⊆ b (span( M ) w ∗ , · ) ⊆ J , but also b ( M , · ) ⊆ b ( E + , · ) ⊆ F + w by positivity, so we find b ( M , · ) ⊆ J ∩ F + w = N .To conclude that span( M ) w ∗ (cid:110) J is an ideal, note that it is a linear subspace (byLemma 4.31(a)) whose positive part is a face (by Lemma 4.27).(b) “ ⊆ ”. If b ∈ ( M (cid:110) N ) ∩ ( E ⊗ F ), then b ( M , · ) ⊆ N ⊆ J . But b has finite rank, sothere is a finite-dimensional (hence closed) subspace Y ⊆ J such that b ( M , · ) ⊆ Y . Bylinearity and continuity, it follows that b (span( M ) w ∗ , · ) ⊆ Y ⊆ J , which shows that( M (cid:110) N ) ∩ ( E ⊗ F ) ⊆ span( M ) w ∗ (cid:110) J .The reverse inclusion “ ⊇ ” and the conclusion follow as in (a). (cid:4) Recall that we call a convex cone E + ⊆ E in a topological vector space semisimple if E + w is a proper cone, or equivalently, if span( E + ) is weak- ∗ dense in E (see § I ⊆ E is a weakly closed subspace, then the quotient E/I belongs to the dualpair h E/I, I ⊥ i , the weak topology of E/I coincides with the quotient topology E w /I , and theweak- ∗ topology on ( E/I ) = I ⊥ ⊆ E coincides with the relative σ ( E , E )-topology (see § ∗ topology on ( E/I ) ∼ = I ⊥ . Theorem 4.33.
Let h E, E i , h F, F i be dual pairs, and let E + ⊆ E , F + ⊆ F be convex cones.Given subspaces I ⊆ E and J ⊆ F , we define I (cid:62) J := ( I ⊥ (cid:110) J ) ∩ ( I (cid:111) J ⊥ ); I (cid:63) J := (lin( E + w ) ⊥ (cid:110) J ) ∩ ( I (cid:111) lin( F + w ) ⊥ ) . Suppose that I and J are ideals with respect to E + w and F + w , respectively. Then:(a) ( I (cid:63) J ) ∩ ( E ⊗ F ) is an ideal in E ⊗ F ( with respect to the injective cone ) ;(b) If I and J are weakly closed, then I (cid:63) J is an ideal in E (cid:126) F ;(c) If I is weakly closed and ( E/I ) + is semisimple, or if J is weakly closed and ( F/J ) + issemisimple, then ( I (cid:62) J ) ∩ ( E ⊗ F ) is an ideal in E ⊗ F ( with respect to the injective cone ) ;(d) If I and J are weakly closed, and if at least one of ( E/I ) + and ( F/J ) + is semisimple, then I (cid:62) J is an ideal in E (cid:126) F . In other words, I ∩ E + w and J ∩ F + w are faces of E + w and F + w , respectively. roof. (a) Since lin( E + w ) = ⊥ ( E + ) (see § E + ) w ∗ = lin( E + w ) ⊥ . Hence it followsfrom Lemma 4.32(b) that (lin( E + w ) ⊥ (cid:110) J ) ∩ ( E ⊗ F ) is an ideal in E ⊗ F . Analogously,( I (cid:111) lin( F + w ) ⊥ ) ∩ ( E ⊗ F ) is an ideal in E ⊗ F . The conclusions follows since the intersectionof two ideals is an ideal.(b) Analogous to (a), using Lemma 4.32(a) instead of Lemma 4.32(b).(c) Assume that I is weakly closed and ( E/I ) + is semisimple (the other case is analogous).Since I is weakly closed, it follows from Lemma 4.31(b) that I (cid:62) J = I ⊥ (cid:110) J . Furthermore,by duality, the adjoint of the pushforward E → E/I is the pullback (bipositive map)(
E/I ) ∼ = I ⊥ → E (see [Dob20a, Proposition 2]), so we have ( E/I ) + = I ⊥ ∩ E + . Since( E/I ) + is semisimple, its dual cone ( E/I ) + separates points on E/I . Equivalently, thesubspace span((
E/I ) + ) = span( I ⊥ ∩ E + ) is weak- ∗ dense in I ⊥ . Hence it follows fromLemma 4.32(b) that ( I ⊥ (cid:110) J ) ∩ ( E ⊗ F ) is an ideal in E ⊗ F .(d) Analogous to (c), using Lemma 4.32(a) instead of Lemma 4.32(b). (cid:4) Remark 4.34.
In terms of the mapping properties, it is not surprising that the semisimplicity of(
E/I ) + and ( F/J ) + plays a role in Theorem 4.33. Let π I : E → E/I and π J : F → F/J denotethe canonical maps. If both (
E/I ) + and ( F/J ) + are semisimple, then ( E/I ) (cid:126) ( F/J ) is a propercone (by Theorem 4.22), so now evidently ker( π I (cid:126) π J ) = ⊥ ( I ⊥ ⊗ J ⊥ ) is an ideal in E (cid:126) F .What is surprising in Theorem 4.33 is that it is sufficient for only one of ( E/I ) + and ( F/J ) + to be semisimple. This could not have been predicted solely on the basis of the mapping properties.The following example shows that we need at least one of the quotients to be semisimple, even inthe finite-dimensional case. Example 4.35.
Let E := R and let E + ⊆ E be the second-order cone E + = { ( x , x , x ) : p x + x ≤ x } . The injective cone E ∗ + ⊗ ε E + can be identified with the cone L + ( E, E ) ofpositive linear operators E → E . If we identify E ∗ with R via the standard inner product, then E + is self-dual. The vectors (1 , , , ( − , , ∈ R define extremal rays of E + , so the subspaces I := span { ( − , , } ⊆ E ∗ and J := span { (1 , , } ⊆ E are ideals (cf. Proposition A.3(a)). Itfollows from Lemma 4.31(c) that I (cid:62) J = I ⊥ (cid:110) J . We show that this is not an ideal.Let b ∈ E ∗ ⊗ E = Bil( E, E ∗ ) ∼ = L ( E, E ) correspond to the identity E → E , and let b ∈ E ∗ ⊗ E be the bilinear form E × E ∗ → R corresponding with the linear map ( x , x , x ) ( x , − x , x ).Clearly b and b are positive. However, since dim( I ⊥ ) = 2 and dim( J ) = 1, maps in I ⊥ (cid:110) J cannot be invertible, so in particular we have b , b / ∈ I ⊥ (cid:110) J .It is not hard to see that b + b ∈ I ⊥ (cid:110) J , and evidently we have 0 ≤ b ≤ b + b . This showsthat I ⊥ (cid:110) J is not an ideal. We conclude this section by providing more convenient direct formulas for the ideals I (cid:62) J and I (cid:63) J and their restrictions to E ⊗ F or E ˜ ⊗ α F . Roughly speaking, under certain topologicalassumptions we have I (cid:62) J = ( I (cid:126) F ) + ( E (cid:126) J ) and I (cid:63) J = I (cid:126) J . Ideals in the algebraic tensor product
We show that the ideals ( I (cid:63) J ) ∩ ( E ⊗ F ) and ( I (cid:62) J ) ∩ ( E ⊗ F ) from Theorem 4.33 are alwaysequal to ( I ⊗ J ) + lin( E + ⊗ ε F + ) and ( I ⊗ F ) + ( E ⊗ J ), respectively Lemma 4.36. If I ⊆ E and J ⊆ F are subspaces, then ( I ⊥ (cid:110) J ) ∩ ( I (cid:111) J ⊥ ) ∩ ( E ⊗ F ) = ( I ⊗ F ) + ( E ⊗ J ) + ( I w ⊗ J w ) . roof. Choose an algebraic decomposition E ∼ = E ⊕ E ⊕ E with E ∼ = I and E ⊕ E ∼ = I w ,and likewise for F ∼ = F ⊕ F ⊕ F . Then E ⊗ F ∼ = L i =1 L j =1 ( E i ⊗ F j ). Under this identification,( I ⊥ (cid:110) J ) ∩ ( E ⊗ F ) corresponds with those elements that are zero in the E ⊗ F and E ⊗ F components. Likewise, ( I (cid:111) J ⊥ ) ∩ ( E ⊗ F ) corresponds with those elements that are zero in the E ⊗ F and E ⊗ F components, and the conclusion follows. (cid:4) Corollary 4.37.
Let I ⊆ E and J ⊆ F be subspaces.(a) If at least one of I and J is weakly closed, then ( I ⊥ (cid:110) J ) ∩ ( I (cid:111) J ⊥ ) ∩ ( E ⊗ F ) = ( I ⊗ F ) + ( E ⊗ J ) . (b) If both I and J are weakly closed, then ( ⊥ ( I ⊥ ⊗ J ⊥ )) ∩ ( E ⊗ F ) = ( I ⊗ F ) + ( E ⊗ J ) . (c) The lineality space of the injective cone ( in E ⊗ F ) is lin( E + ⊗ ε F + ) = (lin( E + w ) ⊗ F ) + ( E ⊗ lin( F + w )) . Proof. (a) If I is weakly closed, then I w ⊗ J w ⊆ I ⊗ F , so the result follows from Lemma 4.36.(b) Immediate, for now we have ⊥ ( I ⊥ ⊗ J ⊥ ) = I ⊥ (cid:110) J = I (cid:111) J ⊥ , by Lemma 4.31(c).(c) By Proposition 4.21, we have lin( E + ⊗ ε F + ) = ( ⊥ (lin( E + w ) ⊥ ⊗ lin( F + w ) ⊥ )) ∩ ( E ⊗ F ),where we note that lin( E + w ) and lin( F + w ) are weakly closed subspaces. (cid:4) Theorem 4.38.
Let I ⊆ E and J ⊆ F be ideals with respect to E + w and F + w .(a) One has ( I (cid:63) J ) ∩ ( E ⊗ F ) = ( I ⊗ J ) + lin( E + ⊗ ε F + ) , and this is an ideal in E ⊗ F ( withrespect to the injective one ) ;(b) If I is weakly closed and ( E/I ) + is semisimple, or if J is weakly closed and ( F/J ) + issemisimple, then one has ( I (cid:62) J ) ∩ ( E ⊗ F ) = ( I ⊗ F ) + ( E ⊗ J ) , and this is an ideal in E ⊗ F ( with respect to the injective cone ) .Proof. (a) Every ideal contains the lineality space, so we may choose a decomposition E ∼ = E ⊕ E ⊕ E with E ∼ = lin( E + w ) and E ⊕ E ∼ = I , and likewise for F ∼ = F ⊕ F ⊕ F . With respect tothe decomposition E ⊗ F ∼ = L i =1 L j =1 ( E i ⊗ F j ), the subspace (lin( E + w ) ⊥ (cid:110) J ) ∩ ( E ⊗ F )corresponds with those elements that are zero in the E ⊗ F and E ⊗ F components, and( I (cid:111) lin( F + w ) ⊥ ) ∩ ( E ⊗ F ) corresponds with those elements that are zero in the E ⊗ F and E ⊗ F components. Since lin( E + ⊗ ε F + ) = ( E ⊗ F ) + ( E ⊗ F ) (by Corollary 4.37(c)) and I ⊗ J = ( E ⊕ E ) ⊗ ( F ⊕ F ), the conclusion follows. (This is an ideal by Theorem 4.33(a).)(b) The formula follows from Corollary 4.37(a), and this is an ideal by Theorem 4.33(c). (cid:4) deals in the space of separately weak- ∗ continuous bilinear forms Let I ⊆ E and J ⊆ F be subspaces, and write I + := I ∩ E + w and J + := J ∩ F + w . If we let I and J belong to the dual pairs h I, E /I ⊥ i , h J, F /J ⊥ i , then the inclusions T : I , → E , S : J , → F are weakly continuous (weak homomorphisms in fact; see § T (cid:126) S : I (cid:126) J → E (cid:126) F is injective and bipositive, by Lemma 4.15(b). In other words,we may interpret I (cid:126) J as a subspace of E (cid:126) F , and moreover ( I (cid:126) J ) + = ( I (cid:126) J ) ∩ ( E (cid:126) F ) + . Lemma 4.39.
The image of I (cid:126) J under the natural inclusion T (cid:126) S : I (cid:126) J , → E (cid:126) F is equalto ( E (cid:110) J ) ∩ ( I (cid:111) F ) .Proof. By definition (cf. § E (cid:126) S : E (cid:126) J , → E (cid:126) F is given by ((id E (cid:126) S ) b )( ϕ, ψ ) = b ( ϕ, S ψ ). Therefore the following diagram commutes: E (cid:126) J Bil ( E w ∗ × J w ∗ ) L ( E w ∗ , J w ) E (cid:126) F Bil ( E w ∗ × F w ∗ ) L ( E w ∗ , F w ) . ∼ id E (cid:126) S R SR ∼ An operator T ∈ L ( E w ∗ , F w ) lies in the image of L ( E w ∗ , J w ) if and only if T [ E ] ⊆ J . Thereforea bilinear form b ∈ E (cid:126) F is the extension of a bilinear form in E (cid:126) J if and only if b ∈ E (cid:110) J .By the same argument, I (cid:126) J = ( E (cid:126) J ) ∩ ( I (cid:111) F ), and the conclusion follows. (cid:4) We will henceforth identify I (cid:126) J with the subspace ( E (cid:110) J ) ∩ ( I (cid:111) F ) ⊆ E (cid:126) F .Next, we turn to complementary decompositions. We say that a subspace E ⊆ E is weaklycomplemented if it is complemented in the weak topology. (Recall that complemented subspacesand their complements are automatically closed: if P : E → E is a continuous projection, thenker( P ) and ran( P ) = ker(id E − P ) are closed.) Lemma 4.40. If E ⊆ E and F ⊆ F are weakly complemented subspaces with complements E ⊆ E and F ⊆ F , respectively, then E (cid:126) F decomposes as the internal ( algebraic ) direct sum E (cid:126) F = ( E (cid:126) F ) ⊕ ( E (cid:126) F ) ⊕ ( E (cid:126) F ) ⊕ ( E (cid:126) F ) . Proof.
The complementary pairs give rise to weakly continuous complementary decompositions E w ∼ = ( E ) w × ( E ) w and F w ∼ = ( F ) w × ( F ) w (topological products). Dualizing the first of these,we obtain a weak- ∗ continuous complementary decomposition E w ∗ ∼ = ( E ⊥ ) w ∗ ⊕ ( E ⊥ ) w ∗ (locallyconvex sum). Using the mapping properties of locally convex sums and topological products(see e.g. [K¨ot79, § E (cid:126) F = Bil ( E w ∗ × F w ∗ ) ∼ = L ( E w ∗ , F w )= L (cid:0) ( E ⊥ ) w ∗ ⊕ ( E ⊥ ) w ∗ , ( F ) w × ( F ) w (cid:1) ∼ = Y i ∈{ , } Y j ∈{ , } L (cid:0) ( E ⊥ i ) w ∗ , ( F j ) w (cid:1) ∼ = Y i ∈{ , } Y j ∈{ , } ( E i ) w (cid:126) ( F j ) w . (cid:4) That T (cid:126) S is injective follows from Remark 4.18. This is a classical result; see also [K¨ot79, § Note that the indices are reversed when passing to the dual: we have ( E ) w ∗ ∼ = E w ∗ /E ⊥ ∼ = ( E ⊥ ) w ∗ and viceversa. orollary 4.41. (a) If E ⊆ E and F ⊆ F are weakly complemented subspaces with complements E ⊆ E and F ⊆ F , respectively, then ⊥ ( E ⊥ ⊗ F ⊥ ) = ( E (cid:126) F ) ⊕ ( E (cid:126) F ) ⊕ ( E (cid:126) F ) , where the orthogonal complement is taken with respect to the dual pair h E (cid:126) F, E ⊗ F i .(b) If lin( E + w ) and lin( F + w ) are weakly complemented with complements X and Y , then lin(( E (cid:126) F ) + ) = (cid:0) lin( E + w ) (cid:126) lin( F + w ) (cid:1) ⊕ (cid:0) lin( E + w ) (cid:126) Y (cid:1) ⊕ (cid:0) X (cid:126) lin( F + w ) (cid:1) . Now we come to concrete descriptions of the subspaces I (cid:62) J and I (cid:63) J from Theorem 4.33. Theorem 4.42.
Let I ⊆ E and J ⊆ F be weakly closed ideals with respect to E + w and F + w .(a) If lin( E + w ) and lin( F + w ) are weakly complemented, then I (cid:63) J = ( I (cid:126) J ) + lin(( E (cid:126) F ) + ) ,and this is an ideal in E (cid:126) F .(b) If I and J are weakly complemented, then I (cid:62) J = ( I (cid:126) F ) + ( E (cid:126) J ) . This is an ideal in E (cid:126) F if at least one of ( E/I ) + and ( F/J ) + is semisimple.Proof. (a) “ ⊇ ”. It follows from Theorem 4.33(b) that I (cid:63) J is an ideal. Since every ideal contains thelineality space, we have lin(( E (cid:126) F ) + ) ⊆ I (cid:63) J . Furthermore, we have E (cid:110) J ⊆ lin( E + w ) ⊥ (cid:110) J and I (cid:111) F ⊆ I (cid:111) lin( F + w ) ⊥ , and therefore I (cid:126) J ⊆ I (cid:63) J (by Lemma 4.39).“ ⊆ ”. The orthogonal complement of a weakly complemented subspace is weak- ∗ complementedin the dual, so we may choose weak- ∗ continuous projections P : E → lin( E + w ) ⊥ , → E and Q : F → lin( F + w ) ⊥ , → F . Let b ∈ I (cid:63) J be given, and define b ( ϕ, ψ ) = b ( P ϕ, Qψ ).Evidently b is separately weak- ∗ continuous, so b ∈ E (cid:126) F . Furthermore, b and b agreeon lin( E + w ) ⊥ × lin( F + w ) ⊥ = span( E + ) w ∗ × span( F + ) w ∗ , so b is positive and b − b ∈ lin(( E (cid:126) F ) + ) (by Proposition 4.21). Finally, since b ∈ (lin( E + w ) ⊥ (cid:110) J ) ∩ ( I (cid:111) lin( F + w ) ⊥ ),we have b ∈ ( E (cid:110) J ) ∩ ( I (cid:111) F ) = I (cid:126) J . Therefore, b = b +( b − b ) ∈ ( I (cid:126) F )+lin(( E (cid:126) F ) + ).(b) By Lemma 4.31(c), we have I (cid:62) J = ⊥ ( I ⊥ ⊗ J ⊥ ), so the direct formula follows fromCorollary 4.41(a). The conditions for I (cid:62) J to be an ideal follow from Theorem 4.33(d). (cid:4) Corollary 4.43. If E + w and F + w are proper cones, and if I ⊆ E and J ⊆ F are weakly closedideals with respect to E + w and F + w , then I (cid:126) J is an ideal in E (cid:126) F . Ideals in completed locally convex tensor products
Finally, we turn our attention to ideals in the completed tensor product E ˜ ⊗ α F . The ideals I (cid:62) J and I (cid:63) J obtained in Theorem 4.33 can be restricted to ideals in E ˜ ⊗ α F (with respectto the injective cone). However, although we were able to find more convenient formulas for theintersections of I (cid:62) J and I (cid:63) J with the algebraic tensor product E ⊗ F (cf. Theorem 4.38),there are no similar formulas for the intersections with E ˜ ⊗ α F . To illustrate the obstruction, wefirst rephrase the problem in the more common terminology of normed tensor products.Let E and F be Banach spaces, let E + ⊆ E and F + ⊆ F be closed proper cones, and let J ⊆ F be a closed order ideal. Then E (cid:126) F ∼ = L ( E w ∗ , F w ) ⊆ L ( E , F ) is the subspace of thoseoperators T : E → F for which the range of the adjoint T : F → E is contained in E . By40heorem 4.33, the subspace { } (cid:62) J = E (cid:63) J = E (cid:110) J is an ideal in E (cid:126) F . The elements of thisideal are simply the weak- ∗ -to-weak continuous operators E → F whose range is contained in J . In particular, if α is a tensor norm, then ( E (cid:110) J ) ∩ ( E ˜ ⊗ α F ) = L ( E , J ) ∩ ( E ˜ ⊗ α F ). It iswell-known that this can be different from E ˜ ⊗ α J . We give two examples. Example 4.44. If F has the approximation property but J does not, then one has E ˜ ⊗ ε J = K ( E, J ) for some appropriate Banach space E , but also E ˜ ⊗ ε F = K ( E, F ) (cf. [DF93, § L ( E, J ) ∩ ( E ˜ ⊗ ε F ) = K ( E, J ) is strictly larger than E ˜ ⊗ ε F . Example 4.45.
It is well-known that the operator ideal of nuclear operators is not injective:if J ⊆ F is a closed subspace and if T : E → J is nuclear as a map E → F , then it does notnecessarily follow that T is also nuclear as a map E → J (cf. [DF93, § E ˜ ⊗ π F = N ( E, F ) and E ˜ ⊗ π J = N ( E, J ), it canhappen that L ( E, J ) ∩ N ( E, F ) = N ( E, J ), so that L ( E, J ) ∩ ( E ˜ ⊗ π F ) = E ˜ ⊗ π J . The obstruction is a purely topological one, and has nothing to do with cone-theoretic issues.Therefore we only sketch the proofs of the following special cases, where a convenient formulacan be obtained.
Theorem 4.46 (Injective topology; approximation property) . Let E and F be complete locallyconvex spaces, let E + ⊆ E , F + ⊆ F be closed proper cones, and let I ⊆ E , J ⊆ F be closed ideals.If I or J has the approximation property, then I ˜ ⊗ ε J is an ideal in E ˜ ⊗ ε F .Proof sketch. The ε -product E ε F is the subspace of E (cid:126) F consisting of those operators b ∈ Bil ( E w ∗ , F w ∗ ) ∼ = L ( E w ∗ , F w ) ∼ = L ( F w ∗ , E w ) that map equicontinuous subsets of E in relativelycompact sets in F , or equivalently, that map equicontinuous subsets of F in relatively compactsets in E (cf. [K¨ot79, § E ε F ) ∩ ( I (cid:126) J ) ⊆ I ε J .Since E and F are complete, we have E ˜ ⊗ ε F ⊆ E ε F (cf. [K¨ot79, § I and J are complete and I or J has the approximation property, we have I ˜ ⊗ ε J = I ε J (cf. [K¨ot79, § I (cid:126) J ) ∩ ( E ˜ ⊗ ε F ) ⊆ ( I (cid:126) J ) ∩ ( E ε F ) ⊆ I ε J = I ˜ ⊗ ε J. On the other hand, one clearly has I ˜ ⊗ ε J ⊆ ( I (cid:126) J ) ∩ ( E ˜ ⊗ ε F ), so we have equality. ByCorollary 4.43, I (cid:126) J is an ideal in E (cid:126) F , so it follows that I ˜ ⊗ ε J is an ideal in E ˜ ⊗ ε F . (cid:4) If E and F are Banach spaces, then the ε -product in the proof of Theorem 4.46 can bereplaced by a suitable space of compact operators.The second situation where a more convenient formula can be obtained is if the subspaces arecomplemented. Let us say that a locally convex tensor topology is a locally convex topology α defined on E ⊗ F for every pair ( E, F ) of locally convex spaces such that:(i) α is a compatible topology on E ⊗ F for every pair ( E, F );(ii) α satisfies the continuous mapping property : if T : E → G and S : F → H are continuous,then T ⊗ S : E ⊗ α F → G ⊗ α H is also continuous.Examples of locally convex tensor topologies include the projective topology π and the injectivetopology ε . More generally, every tensor norm gives rise to a locally convex tensor topology thateven satisfies the equicontinuous mapping property; see [DF93, § Theorem 4.47 (Arbitrary topology; complemented subspaces) . Let E and F be complete locallyconvex spaces, let E + ⊆ E , F + ⊆ F be convex cones, let α be a locally convex tensor topology,and let I ⊆ E , J ⊆ F be closed ideals with respect to E + and F + . a) If I , J , lin( E + ) and lin( F + ) are complemented, then ( I (cid:63) J ) ∩ ( E ˜ ⊗ α F ) = ( I ˜ ⊗ α F ) +lin( E + ˜ ⊗ εα F + ) , and this is an ideal in E ˜ ⊗ α F ( with respect to the injective cone ) .(b) If I and J are complemented, then ( I (cid:62) J ) ∩ ( E ˜ ⊗ α F ) = ( I ˜ ⊗ α F ) + ( E ˜ ⊗ α J ) . This is anideal in E ˜ ⊗ α F ( with respect to the injective cone ) if at least one of ( E/I ) + and ( F/J ) + issemisimple.Proof sketch. Given closed subspaces E , . . . , E n ⊆ E , we say that E ∼ = L ni =1 E i topologically if the canonical map L ni =1 E i → E is a topological isomorphism. Equivalently, if E ∼ = L ni =1 E i algebraically, then one has E ∼ = L ni =1 E i topologically if and only if every projection E → E i iscontinuous (cf. [Sch99, Theorem 2.2]).If E ∼ = L ni =1 E i topologically and F ∼ = L mj =1 F j topologically, then E ˜ ⊗ α F ∼ = L i,j ( E i ˜ ⊗ α F j )topologically. Analogously, Lemma 4.40 can be extended to prove that E (cid:126) F ∼ = L i,j ( E i (cid:126) F j ),and the following diagram commutes: E ˜ ⊗ α F E ˜ ⊗ ε F E (cid:126) F L i,j ( E i ˜ ⊗ α F j ) L i,j ( E i ˜ ⊗ ε F j ) L i,j ( E i (cid:126) F j ) . ∼ ∼ ∼ In particular, for every subset A ⊆ [ n ] × [ m ] we have M ( i,j ) ∈ A ( E i (cid:126) F j ) ! ∩ ( E ˜ ⊗ α F ) = M ( i,j ) ∈ A ( E i ˜ ⊗ α F j ) . Therefore the result follows from Theorem 4.42, using the following decompositions:(a) E ∼ = E ⊕ E ⊕ E and F ∼ = F ⊕ F ⊕ F (topologically), where E = lin( E + ), E ⊕ E = I , F = lin( F + ), F ⊕ F = J ; and A = { (1 , , (1 , , (1 , , (2 , , (2 , , (3 , } .(b) E ∼ = E ⊕ E and F ∼ = F ⊕ F , where E = I , F = J ; and A = { (1 , , (1 , , (2 , } . (cid:4) As an application of the results from § § Proposition 4.48. If x ∈ E + w \ { } and y ∈ F + w \ { } define extremal rays of E + w and F + w , then x ⊗ y ∈ E ⊗ F ⊆ E (cid:126) F defines an extremal ray of ( E (cid:126) F ) + . In other words: rext( E + w ) ⊗ s rext( F + w ) ⊆ rext(( E (cid:126) F ) + ) . Proof.
Let M := { λx : λ ≥ } denote the ray generated by x . Then M is an extremal ray, soin particular a face. Every face contains the minimal face lin( E + w ), but M does not containa non-trivial subspace, so E + w is a proper cone. Furthermore, I := span( M ) = span( x ) is anideal by Proposition A.3(a), and is weakly closed because it is finite-dimensional. Analogously, J := span( y ) is a weakly closed ideal in F , so it follows from Corollary 4.43 that I ⊗ J defines anideal in E (cid:126) F . To complete the proof, note that x ⊗ y ∈ ( E (cid:126) F ) + , and that − x ⊗ y / ∈ ( E (cid:126) F ) + because ( E (cid:126) F ) + is a proper cone. In other words, ( I ⊗ J ) + is the ray generated by x ⊗ y . (cid:4) orollary 4.49. If h E, E i , h F, F i are dual pairs and if E + ⊆ E , F + ⊆ F are convex cones,then rext( E + w ) ⊗ s rext( F + w ) ⊆ rext( E + ⊗ ε F + ) . Corollary 4.50. If E , F are complete locally convex spaces, if E + ⊆ E , F + ⊆ F are convexcones, and if α is a compatible locally convex topology on E ⊗ F for which the natural map E ˜ ⊗ α F → E ˜ ⊗ ε F is injective, then rext( E + w ) ⊗ s rext( F + w ) ⊆ rext( E + ˜ ⊗ εα F + ) . Note: if E ˜ ⊗ α F → E ˜ ⊗ ε F is not injective, then E + ˜ ⊗ εα F + does not have extremal rays, sinceit is not a proper cone (cf. Corollary 4.23).In Theorem 3.20, we found that the extremal rays of the projective cone are precisely thetensor products of the extremal rays of the base cones. This is not true for the injective cone; thefollowing example shows that the inclusion from Corollary 4.49 can be strict. Example 4.51.
Let E be finite-dimensional, and let E + ⊆ E be a proper, generating, polyhedralcone which is not a simplex cone. Then both E ∗ + ⊗ π E + and E ∗ + ⊗ ε E + are proper, generating,polyhedral cones (use that the class of proper, generating, polyhedral cones is closed undertaking duals and projective tensor products). As such, they are generated by their extremal rays.However, it follows from [BL75, Proposition 3.1] that E ∗ + ⊗ π E + = E ∗ + ⊗ ε E + , so in particularrext( E ∗ + ⊗ ε E + ) = rext( E ∗ + ⊗ π E + ) = rext( E ∗ + ) ⊗ s rext( E + ).It will follow from Corollary 5.4(b) below that the additional extremal directions of theinjective cone must necessarily have rank ≥ Remark 4.52.
It is somewhat remarkable that the injective cone preserves extremal rays, becausethe injective norm does not preserve extreme points (of the closed unit ball). Indeed, if E = F = R with the Euclidean norm, then E ⊗ ε F ∼ = R × with the operator norm (i.e. the Schatten ∞ -norm).But the extreme points of the unit ball for the operator norm are the orthogonal matrices, whichin particular have full rank. In other words, no rank 1 operator is an extreme point, so in thiscase ext( B E ⊗ ε F ) is disjoint from ext( B E ) ⊗ s ext( B F ).This discrepancy can be explained as follows. In § E and F , we considered the respective “ice cream cones”in E ⊕ R and F ⊕ R . However, the tensor product ( E ⊕ R ) ⊗ ( F ⊕ R ) ∼ = ( E ⊗ F ) ⊕ E ⊕ F ⊕ R is larger than ( E ⊗ F ) ⊕ R , so the projective cone is larger than the homogenization of theprojective unit ball. In order to recover an extreme point of the projective unit ball, we had towork with a two-dimensional face of the projective cone. Thus, extremal rays of the projectivecone do not correspond directly with extreme points of the projective unit ball. Apparently, thetwo-dimensional face used in this argument has no analogue in the injective cone.43 Applications to reasonable cones
In this section, we give two applications of the results from § § reasonable cones (analogous to reasonable crossnorms). Definition 5.1. If h E, E i and h F, F i are dual pairs, and if E + ⊆ E and F + ⊆ F are convexcones, then we say that a convex cone K ⊆ E ⊗ F is reasonable if it satisfies the following criteria:(i) For all x ∈ E + and y ∈ F + one has x ⊗ y ∈ K ;(ii) For all ϕ ∈ E + and ψ ∈ F + , one has ϕ ⊗ ψ ∈ K .Here K denotes the dual cone of K with respect to any reasonable dual G of E ⊗ F (that is, E ⊗ F ⊆ G ⊆ Bil ( E × F ); see § ϕ ⊗ ψ ∈ E ⊗ F .Reasonable cones in the completed tensor product E ˜ ⊗ α F ( E and F locally convex, α acompatible locally convex topology on E ⊗ F ) are defined analogously. Proposition 5.2.
Let h E, E i , h F, F i be dual systems, and let E + ⊆ E , F + ⊆ F be convexcones. Then E + ⊗ π F + and E + ⊗ ε F + are reasonable cones, and one has E + ⊗ π F + ⊆ E + ⊗ ε F + .Furthermore, a convex cone K ⊆ E ⊗ F is reasonable if and only if E + ⊗ π F + ⊆ K ⊆ E + ⊗ ε F + .Proof. For x ∈ E + , y ∈ F + , ϕ ∈ E + , ψ ∈ F + we have h x, ϕ i ≥ h y, ψ i ≥
0, and therefore h x ⊗ y, ϕ ⊗ ψ i = h x, ϕ i · h y, ψ i ≥
0. It follows that E + ⊗ π F + and E + ⊗ ε F + are reasonable andthat E + ⊗ π F + ⊆ E + ⊗ ε F + .For a general convex cone K ⊆ E ⊗ F , clearly Definition 5.1(i) is equivalent to E + ⊗ π F + ⊆ K ,and Definition 5.1(ii) is equivalent to K ⊆ E + ⊗ ε F + . (cid:4) Likewise, a convex cone
K ⊆ E ˜ ⊗ α F is reasonable if and only if E + ˜ ⊗ πα F + ⊆ K ⊆ E + ˜ ⊗ εα F + .If this is the case, then in particular K ∩ ( E ⊗ F ) is a reasonable cone in the algebraic tensorproduct E ⊗ F . The definition of reasonable cones is based on two criteria regarding rank one tensors in E ⊗ F and E ⊗ F . We show that, in certain cases, all reasonable cones contain the same rank onetensors, and we use this to classify all rank one tensors in the projective and injective cones.If h E, E i is a dual pair, then we say that a convex cone E + ⊆ E is approximately generating if span( E + ) is weakly dense in E .If K ⊆ E ⊗ F is a convex cone, then we understand K to be the dual cone with respectto some reasonable dual G of E ⊗ F . The choice of G does not matter, for we will restrict ourattention to K ∩ ( E ⊗ F ).The following result is an extension of [Bar76, Theorem 3.3]. Proposition 5.3.
Let h E, E i , h F, F i be dual systems, let E + ⊆ E , F + ⊆ F be convex cones,and let K ⊆ E ⊗ F be a reasonable cone.(a) If E + and F + are weakly closed proper cones, then a rank one tensor x ⊗ y ∈ E ⊗ F belongs to K if and only if either x ∈ E + and y ∈ F + or − x ∈ E + and − y ∈ F + .(b) If E + and F + are approximately generating, then a rank one tensor ϕ ⊗ ψ ∈ E ⊗ F defines a positive linear functional on K if and only if either ϕ ∈ E + and ψ ∈ F + or − ϕ ∈ E + and − ψ ∈ F + . roof. (a) “ ⇐ =”. If x ∈ E + and y ∈ F + , then x ⊗ y ∈ K by definition. If − x ∈ E + and − y ∈ F + ,note that x ⊗ y = − x ⊗ − y ∈ K .“= ⇒ ”. Let x ⊗ y ∈ K be of rank one (i.e. with x , y = 0). Weakly closed proper conesare semisimple, so the dual cones E + and F + separate points on E + and F + , respectively.Choose ϕ ∈ E + , ψ ∈ F + such that h x , ϕ i 6 = 0 and h y , ψ i 6 = 0. Then ϕ ⊗ ψ definesa positive linear functional on K , so we have h x , ϕ ih y , ψ i = h x ⊗ y , ϕ ⊗ ψ i ≥
0. Itfollows that h x , ϕ i and h y , ψ i have the same sign. Since − x ⊗ − y = x ⊗ y , we mayassume without loss of generality that h x , ϕ i , h y , ψ i > ϕ ∈ E + is arbitrary, then ϕ ⊗ ψ is a positive linear functional on K , so we have h x , ϕ ih y , ψ i = h x ⊗ y , ϕ ⊗ ψ i ≥
0. Since h y , ψ i >
0, it follows that h x , ϕ i ≥ ϕ ∈ E + , which shows that x ∈ ( E + ) = E + w = E + . Analogously, we find y ∈ F + .(b) In this case E + and F + separate points on E and F , so the same proof can be carriedout. (If ϕ ⊗ ψ ∈ K has rank one, then we may choose x ∈ E + , y ∈ F + such that h x , ϕ ih y , ψ i >
0, and subsequently use these to show that ϕ ∈ E + and ψ ∈ F + or − ϕ ∈ E + and − ψ ∈ F + .) (cid:4) Corollary 5.4.
Let h E, E i , h F, F i be dual systems, let E + ⊆ E , F + ⊆ F be convex cones.(a) If E + and F + are weakly closed proper cones, then all reasonable cones in E ⊗ F agree onthe rank one tensors.(b) The set of rank one extremal directions of the injective cone E + ⊗ ε F + ⊆ E ⊗ F is givenby rext( E + w ) ⊗ s rext( F + w ) .(c) If E + and F + are weakly closed proper cones, and if K ⊆ E ⊗ F is a reasonable cone, thenthe set of rank one extremal directions of K is given by rext( E + ) ⊗ s rext( F + ) .Proof. (a) Immediate from Proposition 5.3(a).(b) If x ∈ E + w and y ∈ F + w are extremal directions, then x ⊗ y is an extremal directionof E + ⊗ ε F + , by Corollary 4.49. For the converse, suppose that x ⊗ y is a rank oneextremal direction of E + ⊗ ε F + . Then E ⊗ F = { } (since there exist rank one tensors),so E = { } and F = { } . Similarly, E + ⊗ ε F + is a proper cone (since it has extremaldirections), so now it follows from Theorem 4.22 that E + w and F + w are proper cones.Since E + ⊗ ε F + = E + w ⊗ ε F + w , it follows from (a) that x ⊗ y ∈ E + w ⊗ π F + w . Clearly x ⊗ y is automatically extremal in this (smaller) cone, so it follows from Theorem 3.20that x and y (or − x and − y ) are extremal directions of E + w and F + w .(c) By (a), every rank one extremal direction of a reasonable cone is also an extremal directionof every smaller reasonable cone. By (b) and Theorem 3.20, the projective and injectivecones have the same rank one extremal directions. (cid:4) Remark 5.5.
In general E + ⊗ π F + and E + ⊗ ε F + do not agree on the rank one tensors. Forexample, if E + is not weakly closed, then all non-zero tensors in E ⊗ R ∼ = E have rank one, but E + ⊗ π R ≥ = E + whereas E + ⊗ ε R ≥ = E + w . As a more extreme example, let E + = E and F + = { } ; then E + ⊗ π F + = { } , whereas E + ⊗ ε F + = E ⊗ F .Using Proposition 5.3, we can determine exactly which rank one tensors belong to the projectiveand injective cones (without additional assumptions on E + and F + ).45 roposition 5.6. Let h E, E i , h F, F i be dual pairs and let E + ⊆ E , F + ⊆ F be convex cones.(a) A rank one tensor x ⊗ y ∈ E ⊗ F belongs to the projective cone E + ⊗ π F + if and only ifat least one of the following applies:(i) x ∈ lin( E + ) and y ∈ span( F + ) ;(ii) x ∈ span( E + ) and y ∈ lin( F + ) ;(iii) x ∈ E + and y ∈ F + ;(iv) − x ∈ E + and − y ∈ F + .(b) A rank one tensor x ⊗ y ∈ E ⊗ F belongs to the injective cone E + ⊗ ε F + if and only if atleast one of the following applies:(i) x ∈ lin( E + w ) ;(ii) y ∈ lin( F + w ) ;(iii) x ∈ E + w and y ∈ F + w ;(iv) − x ∈ F + w and − y ∈ F + w .Proof. (a) “ ⇐ =”. If x ∈ E + and y ∈ F + , then clearly x ⊗ y ∈ E + ⊗ π F + . If x ∈ lin( E + ) and y ∈ span( F + ), then it follows from Corollary 3.17 that x ⊗ y ∈ lin( E + ⊗ π F + ) ⊆ E + ⊗ π F + .The other two cases are analogous.“= ⇒ ”. Let x ⊗ y ∈ E + ⊗ π F + be of rank one (i.e. with x , y = 0), and write x ⊗ y = P ki =1 x i ⊗ y i with x , . . . , x k ∈ E + , y , . . . , y k ∈ F + . Note that we must have y ∈ span( F + ):choose ϕ ∈ E such that ϕ ( x ) = 1, then y = ( ϕ ⊗ id F )( x ⊗ y ) = ( ϕ ⊗ id F ) k X i =1 x i ⊗ y i ! = k X i =1 ϕ ( x i ) y i ∈ span( F + ) . Analogously, x ∈ span( E + ).Let π lin( E + ) : E → E/ lin( E + ) and π lin( F + ) : F → F/ lin( F + ) be the canonical maps. Sincelin( E + ) and lin( F + ) are ideals, the quotient cones are proper (cf. Appendix A.1). Fornotational convenience, let x , . . . , x k and y , . . . , y k denote the images of x , . . . , x k and y , . . . , y k in the respective quotients. Now x ⊗ y ∈ ( E/ lin( E + )) + ⊗ π ( F/ lin( F + )) + hasrank at most one. If it has rank zero, then x ∈ lin( E + ) or y ∈ lin( F + ), so we are done.Assume therefore that x ⊗ y has rank one.Define X := span { x , . . . , x k } ⊆ E/ lin( E + ), and let X + ⊆ X ∩ ( E/ lin( E + )) + be the convexcone generated by x , . . . , x k . Then X is finite-dimensional and X + is closed (because itis finitely generated) and proper (since it is contained in the proper cone ( E/ lin( E + )) + ).Define Y + ⊆ F/ lin( F + ) and Y ⊆ F/ lin( F + ) analogously.Since x , . . . , x k and y , . . . , y k belong to X and Y , it follows that x ⊗ y = P ki =1 x i ⊗ y i holds in X ⊗ Y , so we have x ⊗ y ∈ X + ⊗ π Y + . Since X + and Y + are closed and proper,it follows from Proposition 5.3(a) that x ∈ X + and y ∈ Y + or − x ∈ X + and − y ∈ Y + .Since the quotient maps π lin( E + ) and π lin( F + ) are bipositive (cf. Proposition A.6), it followsthat x ∈ E + and y ∈ F + or − x ∈ E + and − y ∈ F + .46b) “ ⇐ =”. If x ∈ E + w and y ∈ F + w , then x ⊗ y ∈ E + w ⊗ π F + w ⊆ E + w ⊗ ε F + w = E + ⊗ ε F + . If x ∈ lin( E + w ), y ∈ F , then x ⊗ y ∈ lin( E + ⊗ ε F + ) ⊆ E + ⊗ ε F + , by Corollary 4.37(c).The other two cases are analogous.“= ⇒ ”. Let x ⊗ y ∈ E + ⊗ ε F + be of rank one (i.e. with x , y = 0). If x ∈ lin( E + w )or y ∈ lin( F + w ), then we are done, so assume x / ∈ lin( E + w ) and y / ∈ lin( F + w ). Sincelin( E + w ) = ⊥ ( E + ), this means that we may choose ϕ ∈ E + , ψ ∈ F + such that h x , ϕ i 6 = 0and h y , ψ i 6 = 0. Now it follows from the argument of Proposition 5.3 that either x ∈ E + w and y ∈ F + w , or − x ∈ E + w and − y ∈ F + w . (cid:4) The previous proposition can be paraphrased as follows: every rank one tensor in the projectiveor injective cone is either positive for obvious reasons (conditions (iii) and (iv) ) or belongs to thelineality space (conditions (i) and (ii) ). An ideal (resp. face) of the injective cone is also an ideal (resp. face) of every subcone. Thereforethe results from § Proposition 5.7.
Let h E, E i , h F, F i be dual pairs, let E + ⊆ E , F + ⊆ F be convex cones, andlet K ⊆ E ⊗ F be a reasonable cone. Then:(a) If E + and F + are weakly closed and if I ⊆ E , J ⊆ F are ideals, then ( I ⊗ J )+lin( E + ⊗ ε F + ) is an ideal with respect to K . Additionally, if I is weakly closed and ( E/I ) + is semisimple,or if J is weakly closed and ( F/J ) + is semisimple, then ( I ⊗ F ) + ( E ⊗ J ) is an ideal withrespect to K .(b) The lineality space of K satisfies (lin( E + ) ⊗ span( F + )) + (span( E + ) ⊗ lin( F + )) ⊆ lin( K ) ⊆ (lin( E + w ) ⊗ F ) + ( E ⊗ lin( F + w )) . (c) If E + and F + are weakly closed and if x ∈ E + , y ∈ F + define extremal rays, then x ⊗ y defines an extremal ray of K . Recall that a convex cone E + is semisimple if it is contained in a weakly closed proper cone, orequivalently, if E + separates points on E . In this section, we prove that every reasonable conein E ⊗ F is semisimple if E + and F + are semisimple, and we determine necessary and sufficientcriteria for the projective and injective cones to be semisimple. Similar results in completed locallyconvex tensor products will be discussed in § Proposition 5.8.
Let h E, E i , h F, F i be dual pairs, and let E + ⊆ E , F + ⊆ F be convex cones.If G is a reasonable dual of E ⊗ F and if K ⊆ E ⊗ F is a reasonable cone, then E + w ⊗ π F + w ⊆ K w . (Here K w denotes the σ ( E ⊗ F, G )-closure of K .)47 roof. Let T i denote the finest compatible topology on the tensor product E w ⊗ F w (knownas the inductive topology, not to be confused with the injective topology). Then the naturalmap E × F → E ⊗ F is separately continuous as a map E w × F w → ( E ⊗ F, T i ), and thedual of ( E ⊗ F, T i ) is Bil ( E w × F w ) = Bil ( E × F ); cf. [K¨ot79, § G ⊆ Bil ( E × F ), it follows that w = σ ( E ⊗ F, G ) is weaker than T i . Therefore: K i ⊆ K w .To complete the proof, we show that E + w ⊗ π F + w ⊆ K i . Since K is reasonable, we have E + ⊗ π F + ⊆ K ⊆ K i . Since E w × F w → ( E ⊗ F, T i ) is separately continuous, for every x ∈ E + one has x ⊗ F + w ⊆ K i . (The inverse image of K i under the map y x ⊗ y contains F + , andtherefore F + w .) Then, by the same argument, for every y ∈ F + w we have E + w ⊗ y ⊆ K i . Itfollows that E + w ⊗ s F + w ⊆ K i , and the result follows by taking positive combinations. (cid:4) For clarity, let us say that K is G -semisimple if it is semisimple for the dual pair h E ⊗ F, G i (i.e. if K σ ( E ⊗ F,G ) is a proper cone), where G is a reasonable dual of E ⊗ F . Theorem 5.9.
Let h E, E i , h F, F i be dual pairs, let E + ⊆ E , F + ⊆ F be convex cones, let G bea reasonable dual of E ⊗ F , and let K ⊆ E ⊗ F be a reasonable cone.(a) If E + and F + are semisimple, then K is G -semisimple.(b) If E + = { } and F + = { } , and if K is G -semisimple, then E + and F + are semisimple.Proof. (a) Semisimplicity means that E + w and F + w are proper cones, so it follows from Theorem 4.22that E + ⊗ ε F + is a proper cone. Furthermore, E + ⊗ ε F + is weakly closed (cf. Remark 4.1),so it follows that K is contained in a weakly closed proper cone.(b) It follows from Proposition 5.8 that E + w ⊗ π F + w ⊆ K w , where K w is a proper cone (bysemisimplicity). In particular, E + w ⊗ π F + w is a proper cone. By assumption, we have E + w , F + w = { } , so it follows from Theorem 3.10 that E + w and F + w must be propercones as well. Equivalenty: E + and F + are semisimple. (cid:4) Remark 5.10.
We note that the partial converse given in Theorem 5.9(b) is the best we can do.If one of the cones is trivial, then the outcome depends on the other cone. Indeed, let
E, F = { } with convex cones E + ⊆ E , F + ⊆ F , such that E + = { } and F + is not semisimple. Then E + ⊗ π F + = { } , which is semisimple, but E + ⊗ ε F + is not semisimple by Theorem 4.22.More can be said if we choose the cone beforehand. The injective cone is already weaklyclosed with respect to any reasonable dual, so Theorem 4.22 tells us exactly when E + ⊗ ε F + issemisimple. For the projective cone, we obtain necessary and sufficient criteria very similar tothose in Theorem 3.10. Corollary 5.11.
Let h E, E i , h F, F i be dual pairs, let E + ⊆ E , F + ⊆ F be convex cones, andlet G be a reasonable dual of E ⊗ F . Then E + ⊗ π F + is G -semisimple if and only if E + = { } ,or F + = { } , or both E + and F + are semisimple.Proof. If E + = { } or F + = { } , then E + ⊗ π F + = { } , which is semisimple. The rest followsfrom Theorem 5.9. (cid:4) Remark 5.12.
Barring corner cases, we find that E + ⊗ π F + is semisimple if and only if E + ⊗ ε F + is a proper cone. It is tempting to conjecture that the projective cone is always dense in theinjective cone. For locally convex lattices, Birnbaum [Bir76, Proposition 3] found a positive answer,48ut in general this is far from being true. Counterexamples have been known for a long time(e.g. [Bir76, Example following Proposition 3]; [BL75, Proposition 3.1]). Very recently, Aubrun,Lami, Palazuelos and Pl´avala [ALPP19] proved that this fails for all closed, proper, generatingcones in finite-dimensional spaces, unless at least one of the cones is a simplex cone. We willprove some special cases of this result in the follow-up paper [Dob20b]. Remark 5.13.
Fremlin [Fre72] developed a theory of tensor products of Archimedean Rieszspaces, which was further developed by Grobler and Labuschagne [GL88], and Van Gaans andKalauch [vGK10] to a theory of tensor products of Archimedean cones. In this setting, thechallenge is to extend the projective cone to a proper Archimedean cone. Van Gaans and Kalauch[vGK10] showed that the projective tensor product of two generating Archimedean cones is alwayscontained in a proper Archimedean cone (cf. [vGK10, Lemma 4.2]).Our results are parallel to this. If the given cones E + ⊆ E and F + ⊆ F are not onlyArchimedean but also closed in some locally convex topology (this is a stronger assumption),then their projective tensor product is contained in a closed (hence Archimedean) proper cone. Inother words, we start with a stronger assumption, and end up with a stronger conclusion.The preceding results are no substitute for the methods developed in [vGK10]. For example,the space L p [0 ,
1] with p ∈ (0 ,
1) does not admit a non-trivial positive linear functional, so here wehave an Archimedean cone which fails to be semisimple in a rather dramatic way. Consequently,our results fail to prove that the projective tensor product L p + [0 , ⊗ π L p + [0 ,
1] is contained in aproper Archimedean cone, which we know to be true by the results of [vGK10]. (In fact, since L p + [0 ,
1] is a lattice cone, this follows already from Fremlin’s original result [Fre72, Theorem 4.2]).
In the completed setting, semisimplicity turns out to be more subtle. This is because there is oneadditional requirement for the injective cone to be proper: not only do E + w and F + w need tobe proper, but the natural map E ˜ ⊗ α F → E ˜ ⊗ ε F must be injective. (See Corollary 4.23.) Thisleads to the following analogue of Theorem 5.9. Theorem 5.14.
Let E , F be complete locally convex spaces, E + ⊆ E , F + ⊆ F convex cones, α a compatible locally convex topology on E ⊗ F , and K ⊆ E ˜ ⊗ α F a reasonable cone.(a) If E + and F + are semisimple and if E ˜ ⊗ α F → E ˜ ⊗ ε F is injective, then K is semisimple.(b) If E + = { } and F + = { } , and if K is semisimple, then E + and F + are semisimple.Proof. (a) It follows from the assumptions and Corollary 4.23 that the injective cone E + ˜ ⊗ εα F + isproper, so K is contained in a closed proper cone.(b) If K is semisimple, then in particular K ∩ ( E ⊗ F ) is semisimple, so the result follows fromTheorem 5.9(b). (cid:4) The gap between the necessary and sufficient conditions in Theorem 5.14 is even larger thanit was in Theorem 5.9. We show that this gap is related to the approximation property. Forsimplicity, we restrict our attention to Banach spaces.We recall some generalities. Let α be a finitely generated tensor norm, then we say (following[DF93, § E has the α -approximation property if for all Banach spaces F the natural map E ˜ ⊗ α F → E ˜ ⊗ ε F is injective. The π -approximation property (where π denotesthe projective tensor norm) is simply called the approximation property . If a Banach space E E also has the α -approximation property for every finitelygenerated tensor norm α (cf. [DF93, Proposition 17.20]).Some tensor norms α have the property that every Banach space has the α -approximationproperty (and therefore E ˜ ⊗ α F → E ˜ ⊗ ε F is always injective). One of these is the injectivetensor norm ε , for obvious reasons. More generally, this is true for every totally accessible tensornorm α ; see [DF93, Proposition 21.7(2)]. This includes all tensor norms which are (left and right)injective; see [DF93, Proposition 21.1(3)]. Corollary 5.15.
Let E and F be Banach spaces, let E + ⊆ E , F + ⊆ F be convex cones, andlet α be a finitely generated tensor norm. If E or F has the α -approximation property, then theprojective cone E + ˜ ⊗ πα F + ⊆ E ˜ ⊗ α F is semisimple if and only if E + = { } , or F + = { } , orboth E + and F + are semisimple.Proof. The α -approximation property guarantees that E ˜ ⊗ α F → E ˜ ⊗ ε F is injective. If E + = { } or F + = { } , then E + ˜ ⊗ ππ F + = { } . The other cases follow from Theorem 5.14. (cid:4) The proofs of Corollary 5.11 and Corollary 5.15 rely on the injective cone to draw conclusionsabout the projective cone. However, in general these two can be far apart (cf. Remark 5.12). Ifthe map E ˜ ⊗ α F → E ˜ ⊗ ε F is not injective, then the injective cone E + ˜ ⊗ εα F + is not proper, butthat does not mean that the projective cone E + ˜ ⊗ πα F + cannot be semisimple. This leaves openthe following interesting question, to which we do not know the answer: Question 5.16.
Let
E, F be real Banach spaces, and let E + ⊆ E , F + ⊆ F be closed proper cones.Is the projective cone E + ˜ ⊗ ππ F + in the completed projective tensor product E ˜ ⊗ π F necessarilycontained in a closed proper cone? Equivalently: if the positive continuous linear functionals separate points on E and F , thendo the positive continuous bilinear forms E × F → R separate points on E ˜ ⊗ π F ?By Corollary 5.11, the positive continuous bilinear forms separate points on E ⊗ π F , but thatis not enough. Furthermore, if E or F has the approximation property, then the positive bilinearforms of rank one already separate points on E ˜ ⊗ π F , but this technique does not work in theabsence of the approximation property. 50 Ideals, faces, and duality
In this appendix, we discuss some of the basic properties of faces and ideals in preordered vectorspaces. Although many of these results are known in some form, the connections between theseconcepts are not particularly well-known. The main body of this paper draws heavily on theseconnections (especially the results from § A.1).In § A.1, we show that order ideals (as defined by Kadison in the 1950s) are closely related tofaces of the positive cone. This is very useful, as it allows us to quotient out a face. (This is oneof the main tools in the construction of faces of the projective cone in § § A.2, we outline the homomorphism and isomorphism theorems for ideals in ordered vectorspaces. As an application, we show that the maximal order ideals are precisely the supportinghyperplanes of the positive cone. This shows that general (non-maximal) order ideals can bethought of as being the “supporting subspaces” of the positive cone.Finally, in § A.3, we extend the theory of dual faces to cones in infinite-dimensional spaces. Weshow that not every dual face is necessarily exposed, although this is true in separable normedspaces.
A.1 Faces and ideals
Let E be a preordered vector space with positive cone E + ⊆ E , and let ≤ be the linear preordercorresponding with E + . A subset M ⊆ E is full (or order-convex ) if x ≤ y ≤ z with x, z ∈ M implies y ∈ M . A non-empty subset M ⊆ E + that is a convex cone in its own right is called a subcone . Recall that a face (or extremal set ) of E + is a (possibly empty) convex subset M ⊆ E + such that, if M intersects the relative interior of a line segment in E + , then M contains bothendpoints of that segment. Proposition A.1.
A non-empty subset M ⊆ E + is a face if and only if it is a full subcone.Proof. “= ⇒ ”. Suppose that M is a face. First we show that M is a convex cone. If x ∈ M and λ >
1, then the line segment from 0 to λx contains x in its relative interior, so the endpoints 0and λx must also belong to M . Then, since M is convex, for all λ ∈ [0 ,
1] we also have λx ∈ M .Since a face is convex by assumption, we conclude that M is a convex cone.To see that M is full, suppose that x ≤ y ≤ z with x, z ∈ M . Then x, z ∈ E + , so in particularwe have y ≥ x ≥
0, or in other words, y ∈ E + . Furthermore, we have z − y ∈ E + (since y ≤ z ),so it follows that y + 2( z − y ) ∈ E + . Since z = y + ( z − y ) is in the relative interior of the linesegment from y to y + 2( z − y ), we must have y ∈ M , which proves that M is full.“ ⇐ =”. Suppose that M ⊆ E + is a full subcone, and suppose that x, z ∈ E + and λ ∈ (0 ,
1) aresuch that y := λx + (1 − λ ) z belongs to M . If x = z , then evidently x = z = y ∈ M , so assume x = z . Then y lies in the relative interior of the line segment between x and z , so for small enough µ < µx + (1 − µ ) y also lies on this line segment. In particular, µx + (1 − µ ) y ≥
0, orequivalently, y ≥ − µ − µ x . But we have − µ − µ > x ≥
0, so we find 0 ≤ − µ − µ x ≤ y . Since M isfull, it follows that − µ − µ x ∈ M , and therefore x ∈ M . Analogously, z ∈ M . (cid:4) Note that the lineality space lin( E + ) = E + ∩ − E + = { x ∈ E + : 0 ≤ x ≤ } and the cone E + itself are full subcones, and therefore faces of E + . Furthermore, clearly every face containslin( E + ) and is contained in E + , so these are the unique minimal and maximal faces of E + . Next we come to the subject of ideals. If I ⊆ E is a subspace, then we define the quotientcone ( E/I ) + to be the image of E + under the canonical map E → E/I . In order-theoretic terms, these are the least and the greatest element in the set of faces (ordered by inclusion). roposition A.2. For a linear subspace I ⊆ E , the following are equivalent:(i) I is full;(ii) if − y ≤ x ≤ y and y ∈ I , then x ∈ I ;(iii) if ≤ x ≤ y and y ∈ I , then x ∈ I ;(iv) I + := I ∩ E + is a face of E + ;(v) the quotient cone ( E/I ) + is proper.Proof. (i) = ⇒ (ii) . Clear. (ii) = ⇒ (iii) . Suppose that 0 ≤ x ≤ y and y ∈ I . Then we also have − y ≤ ≤ x , so we find − y ≤ x ≤ y . It follows that x ∈ I . (iii) = ⇒ (iv) . Every linear subspace is a convex cone, and the intersection of two convexcones is a convex cone, so I + ⊆ E + is a subcone. If x ≤ y ≤ z with x, z ∈ I + , then in particular0 ≤ y ≤ z with z ∈ I , so we have y ∈ I . Furthermore, we have y ≥ x ≥
0, so y ∈ I + , which showsthat I + is full. By Proposition A.1, I + is a face of E + . (iv) = ⇒ (v) . Let z ∈ ( E/I ) + ∩ − ( E/I ) + be given, then we may choose x, y ∈ E + such that z = π ( x ) = π ( − y ). It follows that π ( x + y ) = 0, so x + y ∈ I . As such, we have 0 ≤ x ≤ x + y and 0 ≤ y ≤ x + y with 0 , x + y ∈ I + , so we find x, y ∈ I + (since I + is full). It follows that z = 0,which shows that ( E/I ) + is a proper cone. (v) = ⇒ (i) . Clearly the natural map π : E → E/I is positive. Suppose that x ≤ y ≤ z with x, z ∈ I , then 0 = π ( x ) ≤ π ( y ) ≤ π ( z ) = 0, so it follows that π ( y ) = 0 (since ( E/I ) + is a propercone). Therefore: y ∈ I . (cid:4) A subspace I satisfying any one (and therefore all) of the conditions of Proposition A.2 iscalled an order ideal , or simply ideal if no ambiguity can arise (i.e. if the space does not haveadditional algebraic structure). Order ideals have been studied since the 1950s (e.g. [Kad51],[Bon54]), but the link between ideals and faces does not appear to be well-known.We give a few useful ways to obtain ideals or faces: Proposition A.3.
Let
E, F be vector spaces and let E + ⊆ E , F + ⊆ F be convex cones.(a) If M ⊆ E + is a non-empty face, then span( M ) is an ideal satisfying M = span( M ) ∩ E + .(b) If T : E → F is a positive linear map and if J ⊆ F is an ideal, then T − [ J ] ⊆ E is an ideal.Proof. (a) Clearly M ⊆ span( M ) ∩ E + . Moreover, since M is a convex cone, every x ∈ span( M ) can bewritten as x = m − m with m , m ∈ M . If furthermore x ∈ E + , then we find 0 ≤ x ≤ m (because m − x = m ≥ x ∈ M (because M is full). This shows that M = span( M ) ∩ E + . It follows from Proposition A.2(iv) that span( M ) is an ideal.(b) If x ≤ y ≤ z and x, z ∈ T − [ J ], then T ( x ) ≤ T ( y ) ≤ T ( z ) with T ( x ) , T ( z ) ∈ J . Since J isfull, it follows at once that T ( y ) ∈ J , which shows that T − [ J ] is also full. (cid:4) It follows from Proposition A.2(iv) and Proposition A.3(a) that the map I I + defines asurjective many-to-one correspondence between the ideals and the non-empty faces.A first (and rather important) application of this correspondence is given in Proposition A.4(b)below. If ϕ : E → R is a positive linear functional, then ker( ϕ ) ∩ E + is easily seen to be a face,and faces of this type are called exposed . This can be generalized in the following way: if F is any52ector space with a proper cone F + ⊆ F , and if T : E → F is a positive linear map, then it isstill relatively easy to see that ker( T ) ∩ E + is a face. (It is crucial that F + is proper!) Althoughnot every face is exposed, the following result shows that this slight extension already capturesall faces. Proposition A.4.
Let E be a vector space and let E + ⊆ E be a convex cone.(a) (cf. [Bon54, §
2, p. 403]) A subspace I ⊆ E is an ideal if and only if it occurs as the kernelof a positive linear map T : E → F with F + proper.(b) A non-empty subset M ⊆ E + is a face if and only if it can be written as M = ker( T ) ∩ E + with T : E → F positive and F + proper.Proof. (a) If I ⊆ E is an ideal, then ( E/I ) + is a proper cone (by Proposition A.2(v)), the map T : E → E/I is positive, and I = ker( T ).Conversely, if T : E → F is a positive linear map with F + ⊆ F a proper cone, then { } ⊆ F + is an ideal (because F + is proper), so it follows from Proposition A.3(b) that ker( T ) is anideal in E .(b) If M ⊆ E + is a face, then I := span( M ) is an ideal with M = I ∩ E + (by Proposition A.3(a)),so ( E/I ) + is a proper cone, the map T : E → E/I is positive, and M = ker( T ) ∩ E + .Conversely, if T : E → F is a positive linear map with F + ⊆ F a proper cone, then itfollows from (a) that ker( T ) is an ideal, so ker( T ) ∩ E + is a face. (cid:4) Remark A.5.
Just as lin( E + ) and E + are the smallest and the largest face of E + , the smallestand the largest ideals of E are lin( E + ) and E . Apart from this, the maximal ideals = E are ofsome interest; see Corollary A.12 below.For now, we show that the smallest ideal has the following special property. Proposition A.6.
Let E be a vector space, E + ⊆ E a convex cone, and I ⊆ E a subspace. Thenthe quotient π I : E → E/I is bipositive if and only if I ⊆ lin( E + ) .In particular, the only ideal I ⊆ E for which the quotient π I : E → E/I is bipositive is theminimal ideal I = lin( E + ) .Proof. Bipositivity of the quotient E → E/I means that, if x ∈ E + and x + I = y + I , then y ∈ E + . Equivalently: if x ∈ E + and z ∈ I , then x + z ∈ E + . Evidently this is the case if and onlyif I ⊆ E + (use that 0 ∈ E + ). But I is a subspace, so we have I ⊆ E + if and only if I ⊆ lin( E + ).If I is an ideal, then we have lin( E + ) ⊆ I (every ideal contains the minimal ideal), so thesecond conclusion follows immediately. (cid:4) Remark A.7. If E + is proper and if F + is arbitrary, then every bipositive map T : E → F isautomatically injective, since ker( T ) = T − [ { } ] ⊆ T − [ F + ] = E + is a subspace contained in E + ,which must therefore be { } . The preceding proposition shows that this is no longer true if E + isnot proper. A.2 The homomorphism and isomorphism theorems
In connection with the ideal theory, we investigate to which extent the homomorphism andisomorphism theorems hold for ordered vector spaces.The homomorphism theorem and the third isomorphism theorem hold true for ordered vectorspaces. 53 roposition A.8 (Homomorphism theorem) . Let E , F be vector spaces, E + ⊆ E , F + ⊆ F convex cones, T : E → F a positive linear map, and I ⊆ E a subspace with I ⊆ ker( T ) . Thenthere is a unique positive linear map ˜ T : E/I → F for which the following diagram commutes: E F.E/I Tπ I ˜ T Proof.
Since I ⊆ ker( T ), there is a unique linear map ˜ T : E/I → F for which the diagramcommutes. This map is automatically positive: if y ∈ ( E/I ) + , then there is some x ∈ E + suchthat y = π I ( x ), and it follows that ˜ T ( y ) = T ( x ) ∈ T [ E + ] ⊆ F + . (cid:4) Proposition A.9 (Third isomorphism theorem) . Let E be a vector space, E + ⊆ E a convexcone, and I ⊆ J ⊆ E subspaces. Then the natural isomorphism ( E/I ) / ( J/I ) ∼ = E/J is bipositivefor the respective quotient cones. Furthermore, the bijective correspondence J J/I between thesubspaces I ⊆ J ⊆ E and the subspaces of E/I restricts to a bijective correspondence of orderideals ( in other words, J is an ideal in E if and only if J/I is an ideal in
E/I ) .Proof. We have the following commutative diagram of linear maps:
E E/J.E/I π J π I π J/I
To see that the natural isomorphism (
E/I ) / ( J/I ) ∼ = E/J is bipositive, note that pushforwardscommute: an element of
E/J belongs to either one of the pushforward cones (
E/J ) + and(( E/I ) / ( J/I )) + if and only if it has a positive element of E in its preimage.Since a subspace is an ideal if and only if the quotient cone is proper, it follows immediatelythat J is an ideal in E if and only if J/I is an ideal in
E/I . (cid:4) Analogous results hold for closed ideals in ordered topological vector spaces. (We assume nocompatibility between the positive cone and the topology, so questions of continuity and positivityare completely separate from one another.)Contrary to the preceding results, the first and second isomorphism theorems fail for orderedvector spaces. We only have the following weaker statements, of which the (simple) proofs areomitted.
Proposition A.10 (Partial first isomorphism theorem) . Let E , F be vector spaces, E + ⊆ E , F + ⊆ F convex cones, and T : E → F a positive linear map. Then the natural linear isomorphism E/ ker( T ) ∼ −→ ran( T ) is positive, but not necessarily bipositive. Counterexample against bipositivity: E = F and T = id E , but E + strictly contained in F + . Proposition A.11 (Partial second isomorphism theorem) . Let F be a vector space, F + ⊆ F a convex cone, E ⊆ F a subspace, and I ⊆ F an order ideal. Then E + I is a subspace of F , E ∩ I is an order ideal of E , and the natural linear isomorphism E/ ( E ∩ I ) ∼ −→ ( E + I ) /I , x + ( E ∩ I ) x + I is positive, but not necessarily bipositive. Counterexample against bipositivity: F = R with standard cone, and E, I ⊆ F two differentone-dimensional subspaces, each of which meets F + only in 0. Then E + := E ∩ F + = { } , so thecone of E/ ( E ∩ I ) is { } , whereas the cone of ( E + I ) /I = R /I is generating.54 lassification of maximal order ideals As an application of the preceding results, we show that the third isomorphism theorem gives ageometric characterization of the maximal order ideals.Following common terminology from algebra, we say that an order ideal I ⊆ E is proper if I = E , and maximal if it is not contained in another proper ideal. Furthermore, we say that apreordered vector space E is simple if E + is proper and if E has exactly two order ideals (namely,the trivial ideals { } and E ). Bonsall [Bon54, Theorem 2] proved that an ordered vector space issimple if and only if it is one-dimensional (with either the standard cone or the zero cone). Combining this with Proposition A.9, we find:
Corollary A.12.
The maximal order ideals of E are precisely the supporting hyperplanes of E + .Proof. It is easy to see that the supporting hyperplanes of E + are precisely the kernels of thenon-zero positive linear functionals. (For a proof, see e.g. [Dob20a, Proposition 4.1].) Furthermore,it follows from Proposition A.9 that an ideal I ⊆ E is maximal if and only if E/I is simple.If ϕ : E → R is a non-zero positive linear functional, then ker( ϕ ) is an ideal, which is maximalsince E/ ker( ϕ ) is one-dimensional and therefore simple.Conversely, if I ⊆ E is a maximal ideal, then E/I is simple, so dim(
E/I ) = 1 and the quotientcone (
E/I ) + is either { } or isomorphic to the standard cone R ≥ . Either way, we can choosea linear isomorphism E/I ∼ −→ R which is positive (but not necessarily bipositive), so that thecomposition ϕ : E → E/I → R is a non-zero positive linear functional with I = ker( ϕ ). (cid:4) For more on maximal ideals, see [Bon54, § A.3 Dual and exposed faces
In the finite-dimensional setting (with closed cones), dual faces are well studied in the literature(see e.g. [Bar78a], [Wei12]). We outline a theory of face duality in dual pairs.A positive pairing is a dual pair h E, F i of (real) preordered vector spaces such that h x, y i ≥ x ∈ E + , y ∈ F + . In this case we say h E + , F + i is a positive pair . Given a positive pair h E + , F + i and a non-empty subset N ⊆ F + , we define the ( pre ) dual face (cid:5) N := E + ∩ ⊥ N = (cid:8) x ∈ E + : h x, y i = 0 for all y ∈ N (cid:9) . Analogously, for a non-empty subset M ⊆ E + we define the dual face M (cid:5) := F + ∩ M ⊥ = (cid:8) y ∈ F + : h x, y i = 0 for all x ∈ M (cid:9) . Note that (cid:5) N (resp. M (cid:5) ) depends not only on N (resp. M ), but also implicitly on E + (resp. F + ). Proposition A.13.
Let h E + , F + i be a positive pair.(a) If N ⊆ F + is non-empty, then (cid:5) N is a face of E + .(b) If N ⊆ N ⊆ F + are non-empty, then (cid:5) N ⊇ (cid:5) N .(c) If N ⊆ F + is non-empty, then N ⊆ ( (cid:5) N ) (cid:5) and (cid:5) N = (cid:5) (( (cid:5) N ) (cid:5) ) .Similar statements hold with N and (cid:5) N replaced by M and M (cid:5) . Bonsall also includes { } among the simple ordered spaces, but we require exactly two ideals. (Similarly, webelieve that 1 is not prime, the empty topological space is not connected, etc.) This is just a matter of convention. There is a slight abuse of notation here, for if E + and F + are not generating, then the positive pair dependsnot only on E + and F + , but also on E and F (but this will cause no confusion). roof. (a) Every y ∈ F + defines a positive linear functional ϕ y : E → R , x
7→ h x, y i . As such, the set E + ∩ ker( ϕ y ) is a face, by Proposition A.3(b). Since we can write (cid:5) N = \ y ∈ N ( E + ∩ ker( ϕ y )) , it follows that (cid:5) N is also a face of E + .(b) This follows from the definition, since ⊥ N ⊇ ⊥ N .(c) If y ∈ N , then by definition one has h x, y i = 0 for all x ∈ (cid:5) N , so it follows that y ∈ ( (cid:5) N ) (cid:5) .This proves the inclusion N ⊆ ( (cid:5) N ) (cid:5) .Write M := (cid:5) N . It follows from the preceding argument that M ⊆ (cid:5) ( M (cid:5) ) = (cid:5) (( (cid:5) N ) (cid:5) ). Onthe other hand, combining the inclusion N ⊆ ( (cid:5) N ) (cid:5) with (b), we find M = (cid:5) N ⊇ (cid:5) (( (cid:5) N ) (cid:5) ),so we conclude that equality holds. (cid:4) A face M ⊆ E + is said to be an h E + , F + i -dual face if M = (cid:5) N for some non-empty subset N ⊆ F + , and an h E + , F + i -exposed face if there is some y ∈ F + such that M = E + ∩ ker( ϕ y ),or equivalently, if M is the h E + , F + i -dual face of a singleton. Likewise, the faces N ⊆ F + of theform N = M (cid:5) (resp. N = { x } (cid:5) ) are the h F + , E + i -dual (resp. h F + , E + i -exposed) faces of F + .The operations M M (cid:5) and N (cid:5) N define a so-called Galois connection (see e.g. [Ber15, § h E + , F + i -dual faces, ordered by inclusion, formsa complete lattice, which we denote as F h E + ,F + i .If h E + , G + i is a positive pair and if F ⊆ G and F + ⊆ F ∩ G + , then evidently one has F h E + ,F + i ⊆ F h E + ,G + i , but the inclusion F h E + ,F + i , → F h E + ,G + i should not be expected to be alattice homomorphism.Given a dual pair h E, E i and a convex cone E + ⊆ E , the most natural lattice of dual facesin E + is the lattice F h E + ,E + i , where E + ⊆ E is the dual cone of E + . (This is the largest of alllattices F h E + ,F + i with F + ⊆ E .) The h E + , E + i -dual (resp. h E + , E + i -exposed) faces will simplybe called the dual (resp. exposed ) faces of E + . The difference between dual and exposed faces If E is finite-dimensional and if E + is closed, then every dual face is exposed, so F h E + ,E ∗ + i issimply the lattice of exposed faces (see e.g. [Bar78a]). We intend to show that things becomemore complicated in the infinite-dimensional case. We illustrate these subtleties by establishingvarious equivalent definitions of dual and exposed faces.For notational simplicity, we formulate the results in the remainder of this appendix not fordual pairs but for locally convex spaces. We recall some basic theory. If T is a locally convextopology on E that is compatible with the dual pair h E, E i , then a subspace I ⊆ E is T -closed ifand only if it is weakly closed. If this is the case, then the quotient E/I is once again a (Hausdorff)locally convex space, and (
E/I ) ∼ = I ⊥ as vector spaces. Furthermore, if T is the weak topology σ ( E, E ), then E/I carries the weak topology σ ( E/I, I ⊥ ) (see e.g. [Con07, § V.2] or [K¨ot83, § E + ⊆ E is quasi-semisimple if E + separates points on E + ;that is, if for every x ∈ E + there is some ϕ x ∈ E + such that h x, ϕ x i >
0. This is equivalent tothe (geometric) requirement that E + ∩ lin( E + ) = { } , since lin( E + ) = ⊥ ( E + ). It follows thata quasi-semisimple cone is automatically proper. Clearly every semisimple cone (in particular,every closed proper cone in a locally convex space) is quasi-semisimple.56 roposition A.14. Let E be locally convex. For a face M ⊆ E + the following are equivalent:(i) M is exposed.(ii) There exists some ϕ ∈ M (cid:5) such that for all x ∈ E + \ M one has h x, ϕ i > .(iii) M = E + ∩ span( M ) , and the quotient ( E/ span( M )) + admits a strictly positive continuouslinear functional.Proof. (i) = ⇒ (iii) . Choose ϕ ∈ E + such that M = E + ∩ ker( ϕ ). Then span( M ) ⊆ ker( ϕ ), soit follows that M = E + ∩ span( M ) and that ϕ factors through E/ span( M ): E R E/ span( M ) ϕ π ψ If E/ span( M ) is equipped with the quotient cone, then ψ : E/ span( M ) → R is strictly positive. (iii) = ⇒ (ii) . We have ( E/ span( M )) ∼ = span( M ) ⊥ = M ⊥ . So if ψ : E/ span( M ) → R iscontinuous and strictly positive, then the composition ϕ : E π −→ E/ span( M ) ψ −→ R is continuousand positive, and belongs to M ⊥ . Therefore ϕ ∈ E + ∩ M ⊥ = M (cid:5) . Furthermore, every x ∈ E + \ M satisfies h x, ϕ i = h πx, ψ i >
0, since ψ is strictly positive. (ii) = ⇒ (i) . The requirement { ϕ } ⊆ M (cid:5) ensures that M ⊆ (cid:5) ( M (cid:5) ) ⊆ (cid:5) { ϕ } , and theassumption that h x, ϕ i > x ∈ E + \ M guarantees that (cid:5) { ϕ } ⊆ M . (cid:4) Proposition A.15.
Let E be locally convex. For a face M ⊆ E + the following are equivalent:(i) M is a dual face.(ii) For every x ∈ E + \ M there is some ϕ x ∈ M (cid:5) such that h x, ϕ x i > .(iii) M = E + ∩ span( M ) , and the quotient ( E/ span( M )) + is quasi-semisimple.Proof. (i) = ⇒ (iii) . Choose some non-empty N ⊆ E + such that M = (cid:5) N = E + ∩ ⊥ N . Thenspan( M ) ⊆ ⊥ N , so it follows that M = E + ∩ span( M ) and that every ϕ ∈ N factors through E/ span( M ): E R E/ span( M ) ϕπ ψ Write K for the positive cone of E/ span( M ), and let y ∈ K be such that h y, ψ i = 0 for all ψ ∈ K .Choose x ∈ E + such that y = π ( x ), then for every ϕ ∈ N we may choose some ψ ∈ K such that ϕ = ψ ◦ π , and therefore h x, ϕ i = h y, ψ i = 0. It follows that x ∈ M , and therefore y = 0, showingthat K is quasi-semisimple. (iii) = ⇒ (ii) . Let x ∈ E + \ M be given. Then x is mapped to a non-zero positive vector in E/ span( M ), so there is a positive continuous linear functional ψ x : E/ span( M ) → R such that h πx, ψ x i > ϕ x : E π −→ E/ span( M ) ψ x −→ R iscontinuous and positive, and h x, ϕ x i = h πx, ψ x i > (ii) = ⇒ (i) . We have M ⊆ (cid:5) ( M (cid:5) ), and the assumption that every x ∈ E + \ M admits some ϕ x ∈ M (cid:5) such that h x, ϕ x i > (cid:5) ( M (cid:5) ) ⊆ M . (cid:4) Theorem A.16 (Compare [Sch60, Proposition 15.2]) . Let E be locally convex, and let M ⊆ E + be a face of the form E + ∩ I , where I ⊆ E is a closed subspace. If E/I admits a separable normcompatible with the dual pair h E/I, ( E/I ) i ( = h E/I, I ⊥ i ) , then M is a dual face if and only if M is exposed.Proof. Every exposed face is dual. For the converse, suppose that M is a dual face, and let k · k be a separable norm compatible with the dual pair h E/I, I ⊥ i . We shall understand E/I and(
E/I ) to be equipped with the respective norm topologies. Since E/I is separable, its dual (
E/I ) and every subset thereof is weak- ∗ separable (this is because it can be written as the union of acountable family of separable metrizable spaces; see [K¨ot83, § ∗ dense countable subset N = { ϕ k } ∞ k =1 in the dual cone ( E/I ) + . Define ψ := P ∞ k =1 ϕ k k k ϕ k k ;this is well-defined because ( E/I ) is a Banach space. Since (
E/I ) + is a closed convex cone, wehave ψ ∈ ( E/I ) + .We claim that ψ is a strictly positive functional. To that end, let x ∈ ( E/I ) + be such that ψ ( x ) = 0. For all k we have ϕ k ( x ) ≥
0, but ψ ( x ) = P ∞ k =1 ϕ k ( x )2 k k ϕ k k = 0, so we must have ϕ k ( x ) = 0.It follows that x ∈ ⊥ N = ⊥ (( E/I ) + ) = lin( ( E/I ) + ). Since M is a dual face, the quotient face( E/I ) + is quasi-semisimple, so ( E/I ) + ∩ lin( ( E/I ) + ) = { } . It follows that x = 0, which showsthat ψ is a strictly positive functional. We conclude that M is exposed. (cid:4) Corollary A.17.
A face of finite codimension is dual if and only if it is exposed.
Corollary A.18.
In a separable normed space, the dual faces are precisely the exposed faces.
Corollary A.19. If E is a separable normed space and if E + is closed, then lin( E + ) is exposed. (After all, lin( E + ) = lin( E + ) = ⊥ ( E + ) = (cid:5) ( E + ) is a dual face.) Corollary A.20.
Every quasi-semisimple cone ( in particular, every closed proper cone ) in aseparable normed space admits a strictly positive continuous linear functional. In general, not every dual face is exposed. As a generic example, let E + be a cone that issemisimple but does not admit a strictly positive functional. Then { } is a dual face, for bysemisimplicity, E + separates points, so (cid:5) ( E + ) = ⊥ ( E + ) = { } . However, { } is not exposed, sincethere is no strictly positive functional.We give two concrete realizations of this generic example: one in an inseparable Hilbert space,and one in a separable Fr´echet space. These examples show that the preceding corollaries cannoteasily be extended beyond the setting of separable normed spaces. Example A.21.
Let Ω be an uncountable set, and consider the Hilbert space E = ‘ R (Ω) withthe non-negative cone E + = { x ∈ ‘ R (Ω) : x ω ≥ ω ∈ Ω } . Then E + is semisimple, so { } is a dual face. However, E + does not admit a strictly positive functional, since every vector in E = ‘ R (Ω) is zero in all but at most countably many coordinates. Technically this is not completely well-defined; if ϕ k = 0, then we must replace ϕ k k k ϕ k k by 0. xample A.22. Let s be the space of all (real) sequences with the topology of pointwiseconvergence. Then s is a separable Fr´echet space with topological dual s = c , the space ofsequences of finite support. (The last statement is a special case of duality between productsand locally convex direct sums; see [K¨ot83, § s is closed andproper, so { } is a dual face. However, there is no strictly positive functional. Remark A.23.
There are certain faces which stand no chance of being either exposed or a dualface. If M ⊆ E + is a face, then we know from Proposition A.3(a) that M = E + ∩ span( M ), butit may happen that E + ∩ span( M ) is larger than M . If this is the case, then M can never be adual (resp. exposed) face, in light of Proposition A.15(iii) (resp. Proposition A.14(iii)).As a concrete example, let E = ‘ ∞ R with its usual cone and norm, and let M = E + ∩ c be theset of all non-negative sequences with finite support. Then M is a face, but E + ∩ span( M ) = E + ∩ c is the (larger) set of all non-negative sequences converging to 0. Therefore M is not exposed or adual face. Acknowledgements
I am grateful to Onno van Gaans, Marcel de Jeu, Dion Gijswijt, Martijn Caspers, and Remy vanDobben de Bruyn for helpful discussions and comments. Part of this work was carried out whilethe author was (partially) supported by the Dutch Research Council (NWO), project number613.009.127.
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