Almost all positive continuous linear functionals can be extended
AAlmost all positive linear functionals can be extended
Josse van Dobben de Bruyn ∗
24 September 2020
Abstract
Let F be an ordered topological vector space (over R ) whose positive cone F + is weaklyclosed, and let E ⊆ F be a subspace. We prove that the set of positive continuous linearfunctionals on E that can be extended (positively and continuously) to F is weak- ∗ dense inthe topological dual wedge E + . Furthermore, we show that this result cannot be generalizedto arbitrary positive operators, even in finite-dimensional spaces. Extension theorems for positive operators have been studied in great detail, and can be foundin many textbooks on ordered vector spaces (e.g. [Sch99, § V.5], [AT07, § Theorem (Kantorovich) . Let F be an ordered vector space ( over R ) , E ⊆ F a majorizing subspace,and G a Dedekind complete Riesz space. Then every positive operator E → G has a positiveextension F → G . For positive linear functionals, the following necessary and sufficient criterion was establishedindependently by Bauer [Bau57, Th´eor`eme 1] and Namioka [Nam57, Theorem 4.4].
Theorem (Bauer–Namioka) . Let F be an ordered topological vector space ( over R ) and let E ⊆ F be a subspace. A positive continuous linear functional ϕ ∈ E + has an extension in F + if and onlyif there is a convex -neighbourhood M ⊆ F such that ϕ is bounded above on E ∩ ( M − F + ) . In this short note, we prove that almost all positive continuous linear functionals can beextended, provided that the positive cone is weakly closed.
Theorem 1.
Let F be a preordered topological vector space ( over R ) , and let E ⊆ F be a subspace.If the topological dual F separates points on F , and if the positive wedge F + is weakly closed,then the set of all positive continuous linear functionals on E that can be extended positively andcontinuously to F is weak- ∗ dense in E + . Note: if F is locally convex, then F + is weakly closed if and only if F + is closed (because F + isa convex set). Additionally, note that, if F + is weakly closed and F + ∩ − F + = { } , then { } is alsoweakly closed, so the weak topology is Hausdorff. Thus, if F + is a cone, then the requirement that F separates points on F is automatically met, and the statement from the abstract is recovered. Mathematics Subject Classification . Primary: 46A40. Secondary: 47B65, 06F20.
Key words and phrases . Convex cone, partially ordered topological vector space, continuous positive linearfunctional, positive extension. ∗ Partially supported by the Dutch Research Council (NWO), project number 613.009.127. a r X i v : . [ m a t h . F A ] S e p he proof of Theorem 1 is deceptively simple, and it is likely that this has been proved as alemma various times across different areas of mathematics (or physics, computer science, economics,etc.). However, the result appears to be unknown in the ordered vector spaces community, and theauthor has not been able to locate an earlier proof (or statement) of the main result.We show in § Proof of Theorem 1.
Define R := { ϕ | E : ϕ ∈ F + } ⊆ E + , and note that R is the wedge of positivecontinuous linear functionals E → R that can be extended positively and continuously to F . Since F + is weakly closed, we have F + = (cid:8) x ∈ F : h x, ϕ i ≥ ϕ ∈ F + (cid:9) . It follows that E + := F + ∩ E = (cid:8) x ∈ E : h x, ϕ i ≥ ϕ ∈ F + (cid:9) . This shows that E + is the predual wedge of R . Hence, by the bipolar theorem, E + is the weak- ∗ closure of R . (cid:3) The main theorem can be seen as a special case of a more general duality, which we sketch here.In this section, a subscript/superscript w or w ∗ refers to the weak or weak- ∗ topology.Let F and G be preodered topological vector spaces (over R ) whose topological duals separatepoints. We say that T ∈ L ( F, G ) is an approximate pushforward if T [ F + ] w = G + w , and an approximate pullback (or approximately bipositive ) if F + w = T − [ G + w ].Recall that the adjoint of a continuous linear operator T ∈ L ( F, G ) restricts to a weak- ∗ continuous operator T ∈ L ( G w ∗ , F w ∗ ). Proposition 2.
Let F and G be as above, and let T ∈ L ( F, G ) be given. Then:(a) T is an approximate pushforward if and only if T : G → F is bipositive;(b) T : G w ∗ → F w ∗ is a weak- ∗ approximate pushforward if and only if T is an approximatepullback. This is a special case of a general result regarding pushforwards and pullbacks of polars(e.g. [Sch99, Proposition IV.2.3(a)]). For completeness, we give the (simple) proof here.
Proof of Proposition 2. (a) By definition one has T ψ = ψ ◦ T , so the dual wedge of T [ F + ] is given by( T [ F + ]) = { ψ ∈ G : ψ ( T x ) ≥ x ∈ F + } = { ψ ∈ G : ( T ψ )( x ) ≥ x ∈ F + } = { ψ ∈ G : T ψ ∈ F + } = ( T ) − [ F + ] . It follows that T is bipositive if and only if T [ F + ] and G + determine the same dual wedge,or equivalently, if and only if T [ F + ] w = G + w .2b) This follows immediately from (a), since T is equal to T , except with its domain and codomainequipped with the wedges F + w and G + w (after all, the bipolar wedge F + ⊆ ( F w ∗ ) = F coincides with the weak closure F + w ⊆ F , by the bipolar theorem). (cid:3) Theorem 1 can be recovered by applying Proposition 2(b) to the inclusion T : E , → F . Remark 3. If F is locally convex and if E ⊆ F is closed, then the adjoint of the inclusion E , → F is the quotient F w ∗ → F w ∗ /E ⊥ ∼ = E w ∗ . By the duality from Proposition 2, for every positivecontinuous linear functional on E to have a positive and continuous extension on F , it is necessaryand sufficient that the pushforward of F + along the quotient F → F /E ⊥ is already weak- ∗ closed.Note that, even in the finite-dimensional case, a quotient (i.e. projection) of a closed convexcone need not be closed, so even here the cone R ⊆ E ∗ + of extendable positive linear functionalsis “only” dense in E ∗ + . (In fact, Mirkil [Mir57, Corollary 1] proved that, for a finite-dimensionalordered vector space with a closed cone E + , all positive linear functionals on all subspaces can beextended if and only if E + is polyhedral; see also Klee [Kle59, Theorem 4.13].) In other words, thereis no exact duality between pushforwards and pullbacks , even if the spaces are finite-dimensionaland the cones are closed. One might ask if similar results can be obtained for arbitrary positive operators. Unfortunately,this is not the case. We will construct a counterexample below.Recall that a wedge
K ⊆ E is a simplex cone (or Yudin cone ) if there is an (algebraic) basis B of E such that K is the wedge generated by B .If E and G are vector spaces, and if ϕ ∈ E ∗ and z ∈ G , then we write z ⊗ ϕ for the linear map E → G , x
7→ h x, ϕ i z . Situation 4.
Let E be a finite-dimensional vector space (over R ), and let E + ⊆ E be a generatingpolyhedral cone that is not a simplex cone. Let ϕ , . . . , ϕ m ∈ E ∗ + be representatives of the extremerays of E ∗ + , so that E + = T mi =1 { x ∈ E : h x, ϕ i i ≥ } and every positive linear functional is apositive combination of ϕ , . . . , ϕ m . Additionally, let F := R m with the standard cone F + := R m ≥ ,so that the map T : E → F , x ( ϕ ( x ) , . . . , ϕ m ( x )) is bipositive. We will identify E with asubspace of F via this map. Proposition 5.
In Situation 4, the positive linear maps E → E that can be extended to a positivelinear map F → E are precisely the maps of the form P ki =1 x i ⊗ ψ i with x , . . . , x k ∈ E + and ψ , . . . , ψ k ∈ E ∗ + .Proof. Let T : E → E be a positive linear map that can be extended to a positive linear map S : F → E . Then we have T ( x ) = S ( ϕ ( x ) , . . . , ϕ m ( x )) = S ( e ) ϕ ( x ) + · · · + S ( e m ) ϕ m ( x ) , so T can be written as T = P mj =1 S ( e j ) ⊗ ϕ j .Conversely, suppose that T = P ki =1 x i ⊗ ψ i with x , . . . , x k ∈ E + and ψ , . . . , ψ k ∈ E ∗ + . Every ψ i can be written as a positive combination of the ϕ , . . . , ϕ m , so after rearranging the termswe may write T = P mj =1 y j ⊗ ϕ j , where the y j are positive combinations of the x i . In particular, y , . . . , y m ∈ E + . Therefore the map S : F → E , e j y j is a positive extension of T . (cid:3) The following theorem of Barker and Loewy tells us that approximation by operators of theform described in Proposition 5 is not always possible.3 heorem 6 (Barker–Loewy, [BL75, Proposition 3.1]) . Let E be a finite-dimensional ordered vectorspace ( over R ) whose positive cone E + is closed and generating. Then the identity id : E → E canbe written as the limit of a sequence of operators of the form P ki =1 x i ⊗ ψ i ( with x , . . . , x k ∈ E + and ψ , . . . , ψ k ∈ E ∗ + ) if and only if E + is a simplex cone. In particular, if E and F are as in Situation 4, then it follows from Proposition 5 and Theorem 6that the identity E → E cannot be approximated by positive operators that can be extended topositive operators F → E . Remark 7.
A more high-level interpretation of the preceding example is that the injective cone,much like the injective norm, does not preserve quotients/pushforwards. (For a detailed discussionof the similarities between the injective norm and the injective cone, see [Dob20].)Let E and F be as in Situation 4. The cone of positive operators F → E coincides with theinjective cone in the tensor product F ∗ ⊗ E , and the adjoint of the inclusion T : E , → F is thequotient T ∗ : F ∗ (cid:16) E ∗ . By Proposition 2(b), T ∗ is an approximate pushforward. (In fact, all conesin this example are polyhedral, so T ∗ is a pushforward.) However, T ∗ ⊗ id : F ∗ ⊗ E → E ∗ ⊗ E is not an approximate pushforward (with respect to the injective cone), since not every positiveoperator E → E can be approximated by restrictions of positive operators F → E .In summary, Theorem 1 cannot be extended to positive operators precisely because the injectivetensor product of cones does not preserve approximate pushforwards. References [AT07] C.D. Aliprantis, R. Tourky,
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Tensor products of convex cones, part I: mapping properties, faces, andsemisimplicity (2020), arXiv preprints.[Kle59] V. Klee,
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