1. Rectangular convex sets In the following we will use notations from [4] and [6]. Particularly, if V and V 0 are vector spaces in duality via a bilinar map h·, ·i, x = [xij ] ∈ Mn,m (V ), and ψ = [ψαβ ] ∈ Mr,s (V 0 ), then we let hhx, ψii be the element [hxij , ψαβ i] of Mnr,ms , where the rows of hhx, ψii are indexed by (i, α) and the colums of hhx, ψii are indexed by (j, β). We also let ϕ(n,m) be the map hh·, ψii : Mn,m (V ) → Mnr,ms . 1.1. Rectangular convex sets. Definition 1.1. A rectagular convex set in a vector space V is a collection K = (Kn,m ) of subsets of Mn,m (V ) with the property that for any αi ∈ Mni ,n and βi ∈ Mmi ,m and vi ∈ Kni mi for 1 ≤ i ≤ ` such that kα1∗ α1 + · · · + α`∗ α` k kβ1∗ β1 + · · · + β`∗ β` k ≤ 1 one has that α1∗ v1 β1 + · · · + α`∗ v` β` ∈ Kn,m . When V is a topological vector space, we say that K is compact if Kn,m is compact for every n, m. The following characterization of rectangular convex sets can be easily verified using the Haagerup-Paulsen-Wittstock decomposition theorem for completely contractive maps [5, Theorem 8.4.]. Lemma 1.2. Suppose that K = (Kn,m ) where Kn,m ⊂ Mn,m (V ). The following assertions are equivalent: (1) K is a rectangular convex set; (2) x ⊕ y ∈ Kn+m,r+s for any x ∈ Kn,r and y ∈ Km,s , and α∗ xβ ∈ Kr,s for any x ∈ Kn,m , α ∈ Mn,r and β ∈ Mm,s with kα∗ αk kβ ∗ βk ≤ 1; (3) x ⊕ y ∈ Kn+m,r+s for any x ∈ Kn,r and y ∈ Km,s , and (σ ⊗ idV ) [Kn,m ] ⊂ Kr,s for any completely contractive map σ : Mn,m → Mr,s . It is clear that, if K is a matrix absolutely convex set, then (Kn,n ) is a matrix convex set in the sense of [7]. Furthermore if K and T are rectangular matrix convex sets such that Tn = Kn for every n ∈ N then Tn,m = Kn,m for every n, m ∈ N. If S = (Sn,m ) is a collection of subsets of a (topological) vector space V , the (closed) rectangular convex hull of S is the smallest (closed) rectangular convex set containning S. Example 1.3. Suppose that X is an operator space. Set Kn,m to be the unit ball Ball (Mn,m (X)) of Mn,m (X). Then CBall (X) = (Kn,m ) is a rectangular convex set. 1.2. The rectangular polar theorem. Suppose that V and V 0 are vector spaces in duality. We endow both V and V 0 with the weak topology induced from such a duality. Let and S = (Sn,m ) is a collection of subsets Sn,m ⊂ Mn,m (V ). We define the rectangular polar S ρ to be the compact rectangular convex subset of V 0 such that ρ f ∈ Sn,m if and only if khhv, f iik ≤ 1 for every r, s ∈ N and every v ∈ Kr,s . The same proof as [4, Lemma 5.1] ρ shows that f ∈ Sn,m if and only if khhv, f iik ≤ 1 for every v ∈ Sn,m . If A ⊂ V , then its absolute polar A◦ is the set of f ∈ V 0 such that |hv, f i| ≤ 1 for every v ∈ F . The classical bipolar theorem asserts that the absolute bipolar A◦◦ is the closed absolutely convex hull of A [2, Theorem 8.1.12]. We will prove below the rectangular analog of this fact. The proof is analogous to the one of [4, Theorem 5.4]. Theorem 1.4. If S = (Sn,m ) is a collection of subsets Sn,m ⊂ Mn,m (V ), then the rectangular bipolar S ρρ is the closed rectangular convex hull of S. Date: January 22, 2016. 2000 Mathematics Subject Classification. Primary ; Secondary . 1

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The proof of [3, Theorem C] shows

that if K is a rectangular convex set in a vector space V , and F is a linear functional on Mn,m (V ) satisfying F |Kn,m ≤ 1, then 2

(1) there exist states p on Mn and q on Mm such that |F (α∗ vβ)| ≤ p (α∗ α) q (β ∗ β) for every r, s ∈ N, α ∈ Mn,r , β ∈ Mm,s , and v ∈ Mr,s (V ), and (2) there exist matrices γ ∈ Mn2 ,1 , δ ∈ Mm2 ,1 , snd a map ϕ : V → Mn,m , such that F (w) = γ ∗ hhw, ϕii δ for every w ∈ Mn,m (W ) and khhw, ϕiik ≤ 1 for every r, s ∈ N and w ∈ Kr,s . From this one can easily deduce the following proposition, which gives the rectangular bipolar theorem as an easy consequence. Theorem 1.5. Suppose that V and V 0 are vector spaces in duality, and K is a compact rectangular convex space ρ in V . If v0 ∈ Mn,m (V ) \Kn,m , then there exists ϕ ∈ Kn,m such that khhv0 , ϕiik > 1.

Proof. By the classical biplar theorem there exists a continuous linear functional F on Mn,m (V ) such that F |Kn,m ≤ ρ 1 and |F (v0 )| > 1. By the remarks above there exists ϕ ∈ Kn,m and contractive γ ∈ Mn2 ×1 and δ ∈ Mm2 ×1 such ∗ ∗ that F (v) = γ hhv, ϕii δ. Thus we have khhv0 , ϕiik ≥ kγ hhv0 , ϕii δk = kF (v0 )k > 1. 1.3. Representation of rectangular convex sets. Suppose that K is a rectangular convex set in a vector space V . A rectangular convex combination in a rectangular convex set K is an expression of the form α1∗ v1 β1 +· · ·+α`∗ v` β` for vi ∈ Kni ,mi , αi ∈ Mni ,n , and βi ∈ Mmi ,m . A proper rectangular convex combination is a rectangular convex combination α1∗ v1 β1 + · · · + α`∗ v` β` where furthermore α1 , . . . , α` and β1 , . . . , β` are right invertible, α1∗ α1 + · · · + α`∗ α` = In , and β1∗ β1 + · · · + β`∗ β` = Im . Definition 1.6. A rectangular affine mapping from a rectangular convex set K to a rectangular convex set T is a collection θ of map θn,m : Kn,m → Tn,m that commutes with rectangular convex combinations. When K and T are compact rectangular convex sets, we say that θ is continuous (resp. homeomorphic) when θn,m is continuous (resp. homeomorphic) for every n, m ∈ N. Given a compact rectangular convex set K we let Aρ (K) be the complex vector space of continuous rectangular affine mappings from K to CBall (C). Here CBall (C) is the rectangular convex set defines as in Example 1.3, where C is endowed with its canonical operator space structure. The space Aρ (K) has a natural operator space structure where Mn,m (Aρ (K)) is identified isometrically with a subspace of C (Kn,m , Mn,m ) endowed with the supremum norm. More generally if Y is any operator space, then we define Aρ (K, Y ) to be the operator space of continuous rectangular affine mappings from K to CBall (Y ). Observe that Mn,m (Aρ (K, Y )) is completely isometric to Aρ (K, Mn,m (Y )). 0 Starting from the operator space Aρ (K) one can consider the rectangular convex set CBall Aρ (K) as in Ex 0 ample 1.3. There is a canonical rectangular affine mapping θ from K to CBall Aρ (K) given by point evaluations. It is clear that such a map is injective. It is furthermore surjective in view of the rectangular bipolar theorem. The argument is similar to the one of the proof of [6, Proposition 3.5]. This shows that the map θ is indeed a homeomor0 phic rectangular affine mapping from K onto CBall Aρ (K) . This implies that the assignment X 7→ CBall (X 0 ) is a 1:1 correspondence between operator spaces and rectangular convex sets. It is also not difficult to verify that such a correspondence is in face an equivalence of categories, where morphisms between operator spaces are completely contractive linear maps, and morphsms between rectangular convex sets are continuous rectangular affine mappings. 1.4. The rectangular Krein-Milman theorem. The notion of (proper) rectangular convex combination yields a natural notion of extreme point in a rectangular convex set. Definition 1.7. An element v of a rectangular convex set K is a rectangular extreme point if for any proper rectangular convex combination α1∗ v1 β1 + · · · + α`∗ v` β` = v one has that, for every 1 ≤ i ≤ `, ni = n, mi = m, and v = u∗i vi wi for some unitaries ui ∈ Mn and wi ∈ Mm . We denote by ∂ρ K = (∂ρ Kn,m ) set of rectangular extreme points of K. Recall that the Krein-Milman theorem asserts that, if K ⊂ V is a compact convex subset of a topological vector space V , then K is the closed convex hull of the set of its extreme points. The following is the natural analog of the Krein-Milman theorem for compact rectangular convex sets.

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Theorem 1.8. Suppose that K is a compact rectangular convex set. Then K is the closed rectangular convex hull of ∂ρ K. We now provide a proof of Theorem 1.8 which is analogous to the proof of the Krein-Milan theorem for compact matrix convex sets [6, Theorem 4.3]. In the course of the proof we will use a variant of Paulsen’s trick [1, §1.3.14]. Suppose that K is a compact rectangular convex set. In view of the representation theorem from 1.3, we can assume without loss of generality that K = CBall (X 0 ) for some operator space X. We will assume that X is concretely represented as a subspace of B (H) for some Hilbert space H. ˜ be the space of operators of the form Fix n, m ∈ N. Let X ⊕n λI x y∗ µI ⊕m for λ, µ ∈ C and x, y ∈ Mn,m (X), where I ⊕n and I ⊕m are the identity operator on, respectively, the n-fold and ˜ 0 ) be defined by m-fold Hilbertian sum of H by itself. If ϕ ∈ Mr,s (X 0 ), then we let ϕ˜ ∈ Mnr+ms (X ⊕n λI x λIrn ϕ(n,m) (x) ∗ ∗ ⊕n 7→ (n,m) y µI ϕ (y) µIms where Irn and Ims denote the identity rn × rn and ms × ms matrices. If ξ ∈ Mr,n and η ∈ Ms,m we also let In ⊗ ξ 0 ξ η = . 0 Im ⊗ η ˜ 0 ) of the form (ξ η)∗ ϕ˜ (ξ η) for r, s ∈ N, ϕ ∈ Kr,s , ξ ∈ Mr,n and We let ∆ be the set of elements of Mn2 +m2 (X η ∈ Ms,m such that kξk2 = kηk2 = 1. It is not difficult to verify as in [6, §4] that one can assume without loss of generality that r ≤ n, s ≤ m, and ξ, η are right invertible. The computation below shows that ∆ is convex. If t1 , t2 ∈ [0, 1] are such that t1 + t2 = 1 then ∗

∗

∗

t1 (ξ1 η1 ) ϕ˜1 (ξ1 η1 ) + t2 (ξ1 η1 ) ϕ˜2 (ξ2 η2 ) = (ξ η) ϕ (ξ η) where ξ=

t 1 ξ1 t η ϕ , η = 1 1 , and ϕ = 1 t 2 ξ2 t2 η2 0

0 . ϕ2

˜ to Mn2 +m2 . Consider Thus ∆ is a compact convex subset of the space of unital completely positive maps from X ∗ now an element (ξ η) ϕ˜ (ξ η) of ∆, where ξ ∈ Mr,n and η ∈ Ms,m are right invertible and ϕ ∈ Kn,m . Assume ∗ that (ξ η) ϕ˜ (ξ η) is an extreme point of ∆. We claim that this implies that ϕ is a rectangular extreme point of K. Indeed suppose that, for some sk , rk ∈ N, ϕk ∈ Krk ,sk , δk ∈ Msk ,s , and γk ∈ Mrk ,r , γ1∗ ϕ1 δ1 + · · · + γ`∗ ϕ` δ` is a proper rectangular convex combination in K that equals ϕ . Then we have that ∗

∗

∗

(ξ η) ϕ˜ (ξ η) = (ξγ1 ηδ1 ) ϕ˜1 (ξγ1 ηδ1 ) + · · · + (ξγ1 ηδ1 ) ϕ˜` (ξγ1 ηδ1 ) . Let tk = k(ξγ1 ηδ1 )k2 for k = 1, 2, . . . , `. Observe that ` X k=1

`

t2k =

1 X (rTr (ξ ∗ γk∗ γk ξ) + sTr (η ∗ δk∗ δk η)) = 1. r+s k=1

Therefore we have that, setting ∗

ψk := t−2 ˜k (ξγ1 ηδ1 ) k (ξγ1 ηδ1 ) ϕ ∗

for k = 1, 2, . . . , `, one obtain elements ψ1 , . . . , ψ` of ∆ such that t21 ψ1 + · · · + t2` ψ` = (ξ η) ϕ˜ (ξ η). Since ∗ ∗ by assumption (ξ η) ϕ˜ (ξ η) is an extreme point of ∆, we can conclude that t2k ψk = (ξ η) ϕ˜ (ξ η) for k = 1, 2, . . . , `. The fact that ξ and η are right invertible now easily implies that rk = r, sk = s, γk and δk are unitaries, and γk∗ ϕk δk = ϕ for k = 1, 2, . . . , `; see also the proof of [6, Lemma 4.4]. This conclude the proof that ϕ is a rectangular extreme point of K. We are now ready to conclude the proof that K is the rectangular convex hull of ∂ρ K. In view of the rectangular bipolar theorem, it is enough to prove that if n, m ∈ N and z ∈ Mn,m (X) is such that ϕ(n,m) (z) ≤ 1 for every

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MARTINO LUPINI

r ≤ n, s ≤ m, and ϕ ∈ ∂ρ Ks,t , then ψ (n,m) (z) ≤ 1 for every ψ ∈ Kn,m . If x ∈ Mn,m (X) then we let x ˜ be the element ⊕n I x x∗ I ⊕m ˜ Observe that if ϕ ∈ Mr,s (X 0 ) and x ∈ Mn,m (X), then of X. Inr ϕ(n,m) (x) ϕ˜(n,m) (˜ x) = (n,m) ∈ Mnr+ms . ∗ ϕ (x) Ims If furthermore ξ ∈ Mr,n and η ∈ Ms,m then "

In ⊗ ξ ∗ ξ (ξ η) ϕ˜ (˜ x) (ξ η) = (n,m) ∗ ∗ (ξ ϕη) (x) ∗

# (n,m) (ξ ∗ ϕη) (x) . Im ⊗ η ∗ η

∗

Let now (ξ η) ϕ˜ (ξ η) be an extreme point of ∆, where ξ ∈ Mr,n and η ∈ Ms,m

right invertible and such are that kξk2 = kηk2 = 1, and ϕ ∈ ∂Kr,s . By assumption we have that idMn,m ⊗ ϕ (z) ≤ 1. Thus by [5, Lemma 3.1] we have Inr ϕ(n,m) (z) ≥0 ∗ ϕ(n,m) (z) Ims and hence " # (n,m) In ⊗ ξ ∗ ξ (ξ ∗ ϕη) (z) ∗ (ξ η) ϕ˜ (˜ z ) (ξ η) = ≥ 0. (n,m) ∗ (ξ ∗ ϕη) (z) Im ⊗ η ∗ η It follows from this that and the classica Krein-Milman theorem that ψ (˜ z ) ≥ 0 for any ψ ∈ ∆. Let us fix ϕ ∈ Kn,m . ∗ If ξ = In and η = Im then (ξ η) ϕ˜ (ξ η) ∈ ∆ and " # (n,m) In ⊗ ξ ∗ ξ (ξ ∗ ϕη) (z) ∗ (ξ η) ϕ˜ (˜ z ) (ξ η) = (n,m) ∗ (ξ ∗ ϕη) (z) Im ⊗ η ∗ η Inr ϕ(n,m) (z) = ≥ 0. ∗ ϕ(n,m) (z) Ims

This implies again by [5, Lemma 3.1] that ϕ(n,m) (z) ≤ 1. Since this is valid for an arbitrary element of Kn,m , the proof is concluded. Remark 1.9. The proof of Theorem 1.8 shows something more precise: if K is a rectangular convex set, then for every n, m ∈ N, Kn,m belongs to the rectangular convex hull of Kr,s for r ≤ n and s ≤ m. The following is an immediate corollary of the rectangular Krein-Milman theorem 1.8 as formulated in Remark 1.9. Corollary 1.10.

Suppose that X is an operator space, n, m ∈ N, and x ∈ Mn,m (X). Then kxk is the maximum of ϕ(n,m) (x) where ϕ ranges among all the rectangular extreme points of Kr,s for r ≤ n and s ≤ m. References 1. David P. Blecher and Christian Le Merdy, Operator algebras and their modules—an operator space approach, London Mathematical Society Monographs. New Series, vol. 30, Oxford University Press, Oxford, 2004. (page 3) 2. John B. Conway, A course in functional analysis, second ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990. (page 1) 3. Edward G. Effros and Zhong-Jin Ruan, On the abstract characterization of operator spaces, Proceedings of the American Mathematical Society 119 (1993), no. 2, 579–584. (page 2) 4. Edward G. Effros and Soren Winkler, Matrix convexity: operator analogues of the bipolar and HahnBanach theorems, Journal of Functional Analysis 144 (1997), no. 1, 117–152. (page 1) 5. Vern I. Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002. (page 1, 4) 6. Corran Webster and Soren Winkler, The Krein-Milman theorem in operator convexity, Transactions of the American Mathematical Society 351 (1999), no. 1, 307–322. (page 1, 2, 3) 7. Gerd Wittstock, On matrix order and convexity, Functional analysis: surveys and recent results, III (Paderborn, 1983), North-Holland Math. Stud., vol. 90, North-Holland, Amsterdam, 1984, pp. 175–188. (page 1)

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Mathematics Department, California Institute of Technology, 1200 E. California Blvd, MC 253-37, Pasadena, CA 91125 E-mail address: [email protected].edu URL: http://www.lupini.org/