Boundary Representations and Rectangular Hyperrigidity
aa r X i v : . [ m a t h . OA ] S e p BOUNDARY REPRESENTATIONS AND RECTANGULARHYPERRIGIDITY
ARUNKUMAR C.S, SHANKAR P, AND A.K. VIJAYARAJAN
Abstract.
We explore connections between boundary representations of op-erator spaces and those of the associated Paulsen systems. Using the notions offinite representation and separating property which we introduced, boundaryrepresentations for operator spaces is characterised. We also introduce weakboundary for operator spaces. Rectangular hyperrigidity of operator spacesintroduced here is used to establish an analogue of Saskin’s theorem in thesetting of operator spaces in finite dimensions. Introduction
Noncommutative approximation and extremal theories initiated by Arveson [4]in the context of operator systems in C ∗ -algebras have seen tremendous growth inthe recent past. One of the main results of the classical approximation theory isthe famous Korovkin theorem [21] which concerns convergence of positive linearmaps on function algebras. The classical Korovkin theorem is as follows: for each n ∈ N , let Φ n : C [0 , → C [0 ,
1] be a positive linear map. If lim n →∞ || Φ n ( f ) − f || = 0for every f ∈ { , x, x } , then lim n →∞ || Φ n ( f ) − f || = 0 for every f ∈ C [0 , G = { , x, x } is called Korovkin set in C [0 , A ⊂ C ( X ) and a point ξ ∈ X , the point ξ belong to the choquet bound-ary [5] of A if the corresponding evaluation functional admits a unique unitalcompletely positive extension to C ( X ). Arveson [1] introduced the notion of boundary representation for an operator system and proposed that this is thenon-commutative analogue of Choquet boundary of a uniform algebra. Subse-quently several other authors carried forward the program initiated by Arvesonand the articles [3, 12, 14, 16] are worth mentioning in this context.There is a close relation between Korovkin sets and Choquet boundaries in theclassical setting as suggested by Saskin [25]. Saskin’s theorem states that G is aKorovkin set in C [0 ,
1] if and only if the Choquet boundary of G is [0 , hyperrigid set . Arveson studied hyperrigidity in the setting of operator systems in C ∗ -algebras. Along the lines of Saskin’s theorem in the classical setting Arveson Date : September 28, 2020.2010
Mathematics Subject Classification.
Primary 46L07, 47L25; Secondary 46L52, 47A20.
Key words and phrases.
Operator space, operator system, boundary representation, hyper-rigidity . [4] formulated hyperrigidity conjecture as follows. For an operator system S andthe generated C ∗ -algebra A = C ∗ ( S ), if every irreducible representation of A is aboundary representation for S , then S is hyperrigid. The hyperrigidity conjectureinspired several studies in recent years [8, 10, 19, 22]. Arveson showed in [4] thatthe conjecture is valid whenever C ∗ -algebra has countable spectrum. Davidsonand Kennedy [13] verified in the case when C ∗ -algebra is commutative. Somepartial results in other contexts have also appeared in [9, 20].The original conjecture by Arveson which came to be known as “Arveson’sconjecture” [1] also concerns boundary representations for operator systems in C ∗ -algebras which was completely settled by Davidson and Kennedy [13]. Theconjecture states that every operator system and every unital operator algebrahas sufficiently many boundary representations to completely norm it. A naturalgeneralisation of above mentioned theory in the non-selfadjoint setting can bedone in the context of operator spaces in ternary rings of operators. The recentwork by Fuller, Hartz and Lupini [15] introduced the notion of boundary repre-sentations for operator spaces in ternary rings of operators. They established thenatural operator space analogue of Arveson’s conjecture [1] on boundary repre-sentations. Paulsen’s “off-diagonal” technique and associated generalization ofStinespring’s dilation theorem [23] for completely contractive maps played a cen-tral role to etabilish the conjecture. Arvesons approach yields the existence ofunital completely positive non-commutative Choquet boundary. The correspond-ing adaptation of Fuller, Hartz and Lupini yields the existence of a completelycontractive non-commutative Choquet boundary. In the latter case, the anal-ogy with classical theory of function algebra theory is not very satisfactory, sincethe resulting non-commutative Shilov boundary is not an algebra. To overcomethese difficulties, Clouatre and Ramsey [11] developed the completely boundedcounterpart to Choquet boundaries. They used this completely bounded non-commutative Choquet boundary to construct a non-commutative Shilov bound-ary that is still a C ∗ -algebra.In this paper we show that a boundary representation for an operator spaceinduces a boundary representation for the corresponding Paulsen system and weillustrate this with a couple of examples. We extend the notion of weak bound-ary representation introduced in [22] for operator systems to operator spaces andstudy the relation between weak boundary representations of an operator spacesand the corresponding Paulsen system. The notion of finite representation intro-duced by Arveson in [1] is generalised in the context of operator spaces. In oneof the main results of this article, we characterise boundary representations ofoperator spaces in terms of rectangular operator extreme points, finite represen-tations and separating properties of operator spaces. The notion of rectangularhyperrigidity for operator spaces is introduced and a version of Saskin’s theoremis established.This paper is divided into five sections, besides the introduction. In section2, we gather the necessary background materials and results that are requiredthroughout. Section 3 deals with boundary representation for operator spacesand the corresponding operator system. Section 4 introduces weak boundaryrepresentation for operator spaces and prove that weak boundary representation OUNDARY REPRESENTATIONS AND RECTANGULAR HYPERRIGIDITY 3 for an operator space is introduced and prove that weak boundary representationfor operator space induce a weak boundary representation for the correspondingPaulsen system and vice versa. In section 5, finite representation for opera-tor spaces and separating operator spaces are introduced. We prove that φ is aboundary representation for an operator space X if and only if φ is a rectangularoperator extreme point for X , φ is a finite representation for X and X separates φ . In section 6, we introduce the notion of rectangular hyperrigidity for operatorspaces in ternary ring of operators(TRO). We prove that if an operator spaceis rectangular hyperrigid in the TRO generated by the operator space, then ev-ery irreducible representation of the TRO is a boundary representation for theoperator space. Some partial answer for the converse of the above result is alsoprovided which is a version of classical Saskin’s theorem in this setting. A relationbetween rectangular hyperrigidity of an operator space and hyperrigidity of thecorresponding Paulsen system is also established.2. Preliminaries A ternary ring of operators (TRO) between Hilbert spaces H and K is a normclosed subspace T of B ( H, K ) such that xy ∗ z ∈ T for all x, y, z ∈ T . If T is a TROcontained in B ( H, K ), then T has a canonical operator space structure comingfrom the inclusion B ( H, K ) ⊂ B ( K ⊕ H ). A triple morphism between TRO’s T and T is a linear map φ : T → T such that φ ( xy ∗ z ) = φ ( x ) φ ( y ) ∗ φ ( z ) for all x, y, z ∈ T . Corresponding to any TRO T there is a natural notion of a linkingalgebra L ( T ) of T . Any TRO T can be seen as the 1-2 corner of its linking algebra L ( T ). Note that the linking algebra L ( T ) is a C ∗ -algbara. A triple morphismbetween TRO’s can be seen as the 1-2 corner of ∗ -homomorphism between thecorresponding linking algebras.The notions of nondegenerate, irreducible and faithful representations of TRO’sadmit natural generalizations from C ∗ -algebras. A representation of a TRO T isa triple morphism φ : T → B ( H, K ) for some Hilbert spaces H and K . A linearmap φ : T → B ( H, K ) is nondegenerate if, whenever p , q are projections in B ( H )and B ( K ), respectively, such that pφ ( x ) = φ ( x ) q = 0 for every x ∈ T , one has p = 0 and q = 0. A representation φ : T → B ( H, K ) is irreducible if, whenever p , q are projections in B ( H ) and B ( K ), respectively, such that pφ ( x ) = φ ( x ) q for every x ∈ T , one has p = 0 and q = 0, or p = 1 and q = 1. Finally φ is called faithful if it is injective or, equivalently, completely isometric. A TRO T ⊂ B ( H.K ) is said to act nondegenerately or irreducibly if the correspondinginclusion representation is nondegenerate or irreducible, respectively. There isa 1-1 correspondence between the representations of a TRO and its linking al-gebra. A representation of a TRO is nondegenerate (irreducible) if and only ifthe representation of its linking algebra is nondegenerate (irreducible). We referthe reader to [6, 7] for a nice account on TRO’s and the representation theory ofTRO’s.Given an operator space X ⊂ B ( H, K ), we can assign an operator system S ( X ) ⊂ B ( K ⊕ H ). This operator system called the Paulsen system [23, Lemma
ARUNKUMAR, SHANKAR, AND VIJAYARAJAN S ( X ) is defined to be the space of operators (cid:26)(cid:20) λI K xy ∗ µI H (cid:21) : x, y ∈ X, λ, µ ∈ C (cid:27) where I H and I K denote the identity operators on H and K respectively. Anycompletely contractive map φ : X → B ( ˜ H, ˜ K ) on the operator space X extendscanonically to a unital completely postive map S ( φ ) : S ( X ) → B ( ˜ K ⊕ ˜ H ) definedby S ( φ )( (cid:20) λI K xy ∗ µI H (cid:21) ) = (cid:20) λI ˜ K φ ( x ) φ ( y ) ∗ µI ˜ H (cid:21) . The Paulsen system is defined in [23, Chapter 8] and [6, Section 1.3] only in thecase ˜ H = ˜ K .2.1. Dilations of rectangular operator states.
The following definitions andresults are due to Fuller, Hartz and Lupini [15]. Let X be an operator space. A rectangular operator state is a nondegenerate completely contractive linear map φ : X → B ( H, K ) such that || φ || cb = 1. Rectangular operator state ψ : X → B ( ˜ H, ˜ H ) is said to be a dilation of φ if there exist linear isometries v : H → ˜ K and w : K → ˜ K such that w ∗ ψ ( x ) v = φ ( x ) for every x ∈ X .Maximal dilations of completely positive maps on operator systems introducedby Dritschel and McCullough [14] is generalised in [15] for completely contractivemaps as follows. Definition 2.1. [15] Let φ : X → B ( H, K ) be a rectangular operator state andlet ψ : X → B ( ˜ H, ˜ K ) be a dilation of φ . We can assume that H ⊂ ˜ H and K ⊂ ˜ K . Let p be the orthogonal projection from ˜ H onto H and let q be theorthogonal projection from ˜ K onto K . The dilation ψ is trivial if ψ ( x ) = qψ ( x ) p + (1 − q ) ψ ( x )(1 − p )for every x ∈ X . The operator state φ on an operator space X is maximal if ithas no nontrivial dilation.Suppose X is an operator system, ˜ H = ˜ K , q = p and ψ is a unital completelypositive map, then the notion of maximal dilation as above recovers the usualnotion of maximal dilation. Definition 2.2. [15] Suppose that X is a subspace of TRO T such that T isgenerated as a TRO by X . The operator state φ on X has the unique extensionproperty if any rectangular operator state ˜ φ of T whose restriction to X coincideswith φ is automatically a triple morphism.The above definition is an analogue of [3, Definition 2.1]. The equivalence ofunique extension property and maximality of unital completely positive maps onoperator systems is proved in [3, Propositon 2.4]. Here we have: Proposition 2.3. [15]
Suppose that φ : X → B ( H, K ) is a rectangular operatorstate of X , and T is a TRO containing X as a generating subspace. Then φ ismaximal if and only if it has unique extension property. OUNDARY REPRESENTATIONS AND RECTANGULAR HYPERRIGIDITY 5
The following definition is an operator space analogue of boundary representa-tions introduced by Arveson in [1].
Definition 2.4. [15] Suppose that φ : X → B ( H, K ) is a rectangular operatorstate of X , and T is a TRO containing X as a generating subspace. A rectangularoperator state φ : X → B ( H, K ) is a boundary representation for X if φ has theunique extension property and the unique extension of φ to T is an irreduciblerepresentation of T .It is clear that, when X is an operator system, H = K and φ is a unitalcompletely positive map, the notion of bounday representation as above recoversthe notion of boundary representation for an operator system. Remark . One can formulate the definition of boundary representation for anoperator space in a TRO by starting with an irreducible representation on theTRO. In that case we will say that the irreducible representation is a boundaryrepresentation for the operator space if the restriction to the operator space is aboundary representation in the above sense.
Proposition 2.6. [15]
Suppose ω : S ( X ) → B ( L ω ) is a boundary representationof the Paulsen system S ( X ) associated with X . Then one can decompose L ω asan orthogonal sum K ω ⊕ H ω in such a way that ω = S ( ψ ) for some boundaryrepresentation ψ : X → B ( H ω , K ω ) of X . The natural operator space analogue of Arveson’s conjecture [1] formulated inthe context of operator systems and which stands completely settled is as follows:
Theorem 2.7. [15]
Suppose that X is an operator space. Then X is completelynormed by its boundary representations. boundary representations We investigate the possible relation between boundary representations of anoperator space X and the boundary representations of the Paulsen system S ( X ).Fuller, Hartz and Lupini [15, Proposition 2.8] proved that a boundary represen-tation of the Paulsen system induces a boundary representation of the operatorspace. Here, we establish the converse of [15, Proposition 2.8]. Theorem 3.1.
If a rectangular operators state φ : X → B ( H, K ) is a boundaryrepresentation for X then S ( φ ) is a boundary representation for S ( X ) .Proof. Assume that the rectangular operator state φ : X → B ( H, K ) is a bound-ary representation for X . Let θ : T → B ( H, K ) be the irreducible representationsuch that θ | X = φ . Let A be the C ∗ -algebra generated by S ( X ) inside B ( K ⊕ H ).Observe that A is (cid:26)(cid:20) x + λI K x x x + µI H (cid:21) : x ∈ T T ∗ , x ∈ T, x ∈ T ∗ , x ∈ T ∗ T, λ, µ ∈ C (cid:27) . Define ω : T T ∗ → B ( K ) and ω : T ∗ T → B ( H ) by ω ( xy ∗ ) = θ ( x ) θ ( y ) ∗ and ω ( x ∗ y ) = θ ( x ) ∗ θ ( y ) respectively. The maps ω and ω extend linearly to T T ∗ + ARUNKUMAR, SHANKAR, AND VIJAYARAJAN C I K and T ∗ T + C I H respectively. Thus, ω = (cid:20) ω θθ ∗ ω (cid:21) is a unital representation of A on K ⊕ H such that ω | S ( X ) = S ( φ ).We claim that ω is irreducible. Let P be a non zero projection in B ( K ⊕ H )that commutes with ω ( A ). In particular, P commutes with ω (cid:18)(cid:20) (cid:21)(cid:19) = (cid:20) I K
00 0 (cid:21) and ω (cid:18)(cid:20) (cid:21)(cid:19) = (cid:20) I H (cid:21) . Therefore, P = p ⊕ q where p is a projection on K and q is a projection on H .Thus, ( p ⊕ q ) ω ( x ) = ω ( x )( p ⊕ q ) for every x ∈ A which implies that pθ ( a ) = θ ( a ) q for every a ∈ T . Since θ is an irreducible representation of T , it followsthat p = I K and q = I H and therefore P = I K ⊕ H . Hence ω is an irreduciblerepresentation.Now, to prove S ( φ ) is a boundary representation for S ( X ), it is enough toprove the following. If Φ : A → B ( K ⊕ H ) is any unital completely positivemap with the property Φ | S ( X ) = S ( φ ), then Φ = ω . Let Φ be a such a map. ByStinespring’s dilation theoremΦ( a ) = V ∗ ρ ( a ) V, a ∈ A where ρ : A → B ( L ) is the minimal Stinespring representation and V : K ⊕ H → L . Thus, ω | S ( X ) = Φ | S ( X ) = V ∗ ρ ( · ) V | S ( X ) . Since ρ is a unital representation on L ,we can decompose L = K ρ ⊕ H ρ , where K ρ is the range of the orthogonal projec-tion ρ (cid:18)(cid:20) (cid:21)(cid:19) and H ρ is the range of the orthogonal projection ρ (cid:18)(cid:20) (cid:21)(cid:19) .Then with respect to this decomposition one has that ρ = (cid:20) σ ηη ∗ σ (cid:21) where η : T → B ( H ρ , K ρ ) is a triple morphism and σ , σ are unital representa-tions of the respective C ∗ -algebras.We claim that V = (cid:20) v v (cid:21) , for isometries v : K → K ρ and v : H → H ρ .Since V ∗ (cid:20) (cid:21) V = V ∗ (cid:20) σ (1) 00 0 (cid:21) V = V ∗ ρ (cid:18)(cid:20) (cid:21)(cid:19) V =Φ (cid:20) (cid:21) = S ( φ ) (cid:20) (cid:21) = (cid:20) (cid:21) and V is an isometry, we must have that V = (cid:20) v v (cid:21) for isometries v and v .Since S ( θ ) | X = S ( φ ) = Φ | X = V ∗ ρV , we have θ ( x ) = v ∗ η ( x ) v ∀ x ∈ X. Ourassumption θ is a boundary representation for X implies that θ ( t ) = v ∗ η ( t ) v ∀ t ∈ T . Using [15, Proposition 2.6], θ is maximal implies that η is a trivial dilation.We have η ( t ) = qη ( t ) p + (1 − q ) η ( t )(1 − p ) OUNDARY REPRESENTATIONS AND RECTANGULAR HYPERRIGIDITY 7 for every t ∈ T , where p = v v ∗ and q = v v ∗ . The above equation impliesthat η ( T ) v H ⊆ v K and η ( T ) ∗ v K ⊆ v H . By the minimality assumption ofStinespring theorem, we have K ρ is the closed linear span of η ( T ) η ( T ) ∗ v K ∪ η ( T ) v H and H ρ is the closed linear span of η ( T ) ∗ η ( T ) v H ∪ η ( T ) ∗ v K . Since η ( T ) η ( T ) ∗ v K ∪ η ( T ) v H ⊆ v K and η ( T ) ∗ η ( T ) v H ∪ η ( T ) ∗ v K ⊆ v H , we have K ρ = v K H ρ = v H . Thus, v and v are unitary implies that V is unitary.Hence Φ = ω . (cid:3) We give a couple of examples to illustrate the above theorem.
Example 3.2.
Let X ⊂ B ( H, K ) be an operator space such that the TRO T generated by X acts irreducibly and such that T ∩ K ( H, K ) = { } . Then theidentity representation of T is a boundary representation for X if and only ifthe identity representation of C ∗ ( S ( X )) is a boundary representation for S ( X ).To see this, first assume that the identity representation of T is a boundaryrepresentation for X . Then by rectangular boundary theorem [15, Theorem 2.17]the quotient map B ( H, K ) → B ( H, K ) / K ( H, K ) is not completely isometric.Using the same line of argument in the proof of the converse part of [15, Theorem2.17], we see that the quotient map B ( K ⊕ H ) → B ( K ⊕ H ) / K ( K ⊕ H ) is notcompletely isometry on S ( X ). Then by Arveson’s boundary theorem [2, Theorem2.1.1], the identity representation of C ∗ ( S ( X )) is a boundary representation for S ( X ).Conversely, if the identity representation of C ∗ ( S ( X )) is a boundary represen-tation for S ( X ), then by [15, Proposition 2.8], identity representation of T is aboundary representation for X . Example 3.3.
Let R be an operator system in B ( H ) and let A = C ∗ ( R ) bethe C ∗ -algebra generated by R . In particular, R is an operator space and A = C ∗ ( R ) is itself a TRO generated by R . We have C ∗ ( S ( R )) = M ( A ). UsingHopenwasser’s result [17] , we can conclude that if π is a boundary representationof A for R then S ( π ) is a boundary representation of C ∗ ( S ( R )) for S ( R ). Corollary 3.4.
If a rectangular operators state φ : X → B ( H, K ) has the uniqueextension property for X then S ( φ ) has the unique extension property for S ( X ) .Proof. The proof follows from the same line argument in Theorem 3.1 withoutthe irreducibility assumption. (cid:3)
Proposition 3.5.
Suppose ω : S ( X ) → B ( L ω ) has the unique extension propertyon the Paulsen system S ( X ) associated with X . Then one can decompose L ω as anorthogonal direct sum K ω ⊕ H ω in such a way that ω = S ( φ ) for some rectangularoperator state φ : X → B ( H ω , K ω ) and φ on X has the unique extension property.Proof. The proof follows as in [15, Proposition 2.8]. (cid:3)
We study the tensor product of boundary representations of operator spaces.In the usual way one can define tensor products of TROs and representations ofTROs. The norms on tensor product of TROs are considered to be any operatorspace tensor product norms.
ARUNKUMAR, SHANKAR, AND VIJAYARAJAN
Theorem 3.6.
Suppose X i , i = 1 , are operator spaces and T i be the TROgenerated by X i and containing X i for i = 1 , . If φ i : T i → B ( H i , K i ) , i = 1 , benondegenerate representations such that φ ⊗ φ : T ⊗ T → B ( H ⊗ H , K ⊗ K ) has unique extension property for X ⊗ X , then φ i has unique extension propertyfor X i , i = 1 , .Proof. Since each φ i is nondegenreate representation then φ ⊗ φ is nodegeneraterepresentations. If possible let φ does not have unique extension property for X then there is a rectangular operator state ψ : T → B ( H , K ) such that ψ | X = φ but ψ = φ on T . Thus, ψ ⊗ φ will be a completely contractive extension of φ ⊗ φ | X ⊗ X other than φ ⊗ φ . Which is a contradiction to the assumption that φ ⊗ φ has unique extension for X ⊗ X . (cid:3) Corollary 3.7.
Suppose X i , i = 1 , are operator spaces and T i be the TROgenerated by X i and containing X i . If φ i : T i → B ( H i , K i ) , i = 1 , be irreduciblerepresentations such that φ ⊗ φ : T ⊗ T → B ( H ⊗ H , K ⊗ K ) is a boundaryrepresentation for X ⊗ X , then φ i is a boundary representation for X i , i = 1 , . Problem 3.8. If φ and φ are boundary representations for operator spaces X and X respectively, then φ ⊗ φ is a boundary representation for X ⊗ X ?In the case of operator systems the above problem was answered affirmativelywhen one of the component C ∗ -algebras generated is a GCR algebra as proved in[18, 26]. The same proof techniques employed there will not work in the contextof operator spaces and rectangular boundary representations.4. weak boundary representations Recently, Namboodiri, Pramod, Shankar, and Vijayarajan [22] introduced anotion of weak boundary representation, which is a weaker notion than Arve-son’s [1] boundary representation for operator systems. They studied relationsof weak boundary representation with quasi hyperrigidity of operator systems in[22]. Here we introduce the notion of weak boundary representations for operatorspaces as follows:
Definition 4.1.
Let X ⊂ B ( H, K ) be a operator space and T be a TRO contain-ing X as a generating subspace. An irreducible triple morphism ψ : T → B ( H, K )is called a weak boundary representation for X if ψ | X has a unique rectangularoperator state extension of the form v ∗ ψu , namely ψ itself, where v : H → H and u : K → K are isometries.Suppose X is operator system, H = K , v = u , then the above notion of weakboundary representation recovers the weak boundary representation for operatorsystems. We can observe that all the boundary representations are weak boundaryrepresentations for operator spaces.Now, we investigate the relations between weak boundary representation of anoperator space and the weak boundary representation of it’s Paulsen system. Proposition 4.2.
Suppose ω : C ∗ ( S ( X )) → B ( L ω ) is a weak boundary represen-tation for the Paulsen system S ( X ) associated with X . Then one can decompose OUNDARY REPRESENTATIONS AND RECTANGULAR HYPERRIGIDITY 9 L ω as an orthogonal direct sum K ω ⊕ H ω in such a way that ω = S ( θ ) for someweak boundary representation θ : T → B ( H ω , K ω ) for X .Proof. Arguments similar to the one as in the proof of [15, Proposition 2.8], wehave θ : T → B ( H ω , K ω ) is a triple morphism and θ is irreducible.Now, we will prove that θ is weak boundary representation for X . Let u : H ω → H ω and v : K ω → K ω are isometries such that v ∗ θ ( a ) u = θ ( a ) ∀ a ∈ X .For every a, b ∈ X , (cid:20) v ∗ u ∗ (cid:21) ω (cid:20) λ I K ab ∗ λ I H (cid:21) (cid:20) v u (cid:21) = (cid:20) v ∗ u ∗ (cid:21) (cid:20) λ I K θ ( a ) θ ( b ) ∗ λ I H (cid:21) (cid:20) v u (cid:21) = (cid:20) λ I K v ∗ θ ( a ) u ( v ∗ θ ( b ) u ) ∗ λ I H (cid:21) = (cid:20) λ I K θ ( a ) θ ( b ) ∗ λ I H (cid:21) . Since (cid:20) v u (cid:21) is an isometry on K ω ⊕ H ω and ω is a weak boundary representationfor S ( X ) implies that for all a ∈ T , (cid:20) v ∗ u ∗ (cid:21) ω (cid:18)(cid:20) a (cid:21)(cid:19) (cid:20) v u (cid:21) = ω (cid:18)(cid:20) a (cid:21)(cid:19)(cid:20) v ∗ u ∗ (cid:21) (cid:20) θ ( a )0 0 (cid:21) (cid:20) v u (cid:21) = (cid:20) θ ( a )0 0 (cid:21)(cid:20) v ∗ θ ( a ) u (cid:21) = (cid:20) θ ( a )0 0 (cid:21) From the last equality v ∗ θ ( a ) u = θ ( a ), for every a ∈ T . Thus θ a is weak boundaryrepresentation for X . (cid:3) Proposition 4.3. If θ : T → B ( H, K ) is a weak boundary representation for X then, ω = (cid:20) ω θθ ∗ ω (cid:21) : C ∗ ( S ( X )) → B ( K ⊕ H ) is a weak boundary representationfor S ( X ) .Proof. Arguing as in the proof of Theorem 3.1, we have ω is irreducible represen-tation. Let V be an isometry on K ⊕ H such that V ∗ ωV | S ( X ) = ω | S ( X ) . Since V ∗ (cid:20) (cid:21) V = (cid:20) (cid:21) we can factorize V = (cid:20) v v (cid:21) , where v and v are isometries on K and H respectively. Also we have v ∗ θv | X = θ | X . Our assumption, θ is a weak boundaryrepresentation for X implies that v ∗ θ ( t ) v = θ ( t ) for every t ∈ T and we have( v ( H ) , v ( K )) is a pair of θ -invariant subspace of ( H, K ). The irreducibility of θ implies that v ( H ) = H and v ( K ) = K , therefore v and v are unitaries.Consequently, V is a unitary. Thus V ∗ ωV is a representation of C ∗ ( S ( X )). Since V ∗ ωV = ω on S ( X ), we must have that V ∗ ωV = ω on C ∗ ( S ( X )). (cid:3) Characterisation of boundary representations for operatorspaces
Arveson [1] introduced the notion of finite representations in the setting ofsubalgebras of C ∗ -algebras. Namboodiri, Pramod, Shankar and Vijayarajan [22]explored the relation between finite representations and weak boundary repre-sentations in the context of operator systems. Here, we introduce the notion offinite representation in the setting of operator spaces. Definition 5.1.
Let X be an operator space generating a TRO T . Let φ : T → B ( H, K ) be a representation. We say that φ is a finite representation for X iffor every isometries u : H → H and v : K → K , the condition v ∗ φ ( x ) u = φ ( x ),for all x ∈ X implies that u and v are unitaries.It is clear that, when X is operator system, H = K , v = u , the above notionof finite representation recovers the Arveson’s notion of finite representation. Proposition 5.2.
Let ω : C ∗ ( S ( X )) → B ( L ) be a finite representation forPaulsen system S ( X ) associated with X . Then one can decompose L as an orthog-onal direct sum K ⊕ H in such a way that ω = S ( φ ) for some finite representation φ : T → B ( H, K ) for X .Proof. We can get triple morphism φ : T → B ( H, K ) as in the proof of [15,Proposition 2.8]. Now, we will prove that φ is a finite representation for X .For isometries u : H → H and v : K → K the condition v ∗ φ ( x ) u = φ ( x ) issatisfied. Then for all x, y ∈ X (cid:20) v ∗ u ∗ (cid:21) ω (cid:18)(cid:20) λI K xy ∗ µI H (cid:21)(cid:19) (cid:20) v u (cid:21) = (cid:20) v ∗ u ∗ (cid:21) (cid:20) λI K φ ( x ) φ ( y ) ∗ µI H (cid:21) (cid:20) v u (cid:21) = (cid:20) λI K u ∗ φ ( x ) vv ∗ φ ( y ) ∗ u µI H (cid:21) = S ( φ ) (cid:18)(cid:20) λI K xy ∗ µI H (cid:21)(cid:19) . Thus, (cid:20) v ∗ u ∗ (cid:21) ω (cid:20) v u (cid:21) | S ( X ) = ω | S ( X ) . Since ω is a finite representation for S ( X ),we have (cid:20) v u (cid:21) is a unitary and consequently u and v are unitaries. Hence φ isa finite representation. (cid:3) Proposition 5.3. If φ : T → B ( H, K ) is a finite representation for X then, ω = (cid:20) ω φφ ∗ ω (cid:21) : C ∗ ( S ( X )) → B ( K ⊕ H ) is finite representation for S ( X ) .Proof. Arguing as the in the proof of Theorem 3.1, we have ω is a representation.Let V : K ⊕ H → K ⊕ H be an isometry such that V ∗ ω ( a ) V = ω ( a ) for all a ∈ S ( X ). Since V ∗ (cid:20) (cid:21) V = (cid:20) (cid:21) and V ∗ (cid:20) (cid:21) V = (cid:20) (cid:21) . OUNDARY REPRESENTATIONS AND RECTANGULAR HYPERRIGIDITY 11
Then, we can decompose V = (cid:20) v u (cid:21) , where v : K → K and u : H → H areisometries. For a ∈ S ( X ), we have V ∗ ω ( a ) V = (cid:20) v ∗ u ∗ (cid:21) (cid:20) ω φφ ∗ ω (cid:21) ( a ) (cid:20) v u (cid:21) = (cid:20) ω φφ ∗ ω (cid:21) ( a ) = ω ( a ) . Therefore v ∗ φ ( x ) u = φ ( x ) for every x ∈ X . Since φ is a finite representationfor X implies that v and u are unitaries. Thus V is a unitary. Hence ω is a finiterepresentation for S ( X ). (cid:3) The following theorem shows a relation between finite representations withweak boundary representations.
Theorem 5.4.
Let X be an operator space generating a TRO T. Let φ be anirreducible representation of T . Then φ is a finite representation for X if andonly if φ is a weak boundary representation for X Proof.
Assume that φ : T → B ( H, K ) is a finite representation for X . Let v : K → K and u : H → H are isometries such that v ∗ φ ( x ) u = φ ( x ) , ∀ x ∈ X . Ourassumption implies that v and u are unitaries. Therefore, v ∗ φ ( t ) u = φ ( t ) , ∀ t ∈ T .Hence φ is a weak boundary representation for X .Conversely, Assume that φ is weak boundary representation for X . Let v : K → K and u : H → H are isometries such that v ∗ φ ( x ) u = φ ( x ) , ∀ x ∈ X . Ourassumption implies that v ∗ φ ( t ) u = φ ( t ) , ∀ t ∈ T . Thus we have ( u ( H ) , v ( K ))is a pair of φ -invariant subspace of ( H, K ). The irreducibility of φ implies that u ( H ) = H and v ( K ) = K , therefore u and v are unitaries. Hence φ is a finiterepresentation for X . (cid:3) Arveson[1] introduced the notion of separating subalgbras to characterize bound-ary representations in the context of subalgebras of C ∗ -algebas. Pramod, Shankarand Vijayarajan [24] studied the separating notion in the setting of operator sys-tems and explored the relation with boundary representations. Here, we introducethe notion of separting operator space as follows: Definition 5.5.
Let X be an operator space generating a TRO T . Let φ : T → B ( H, K ) be an irreducible representation. We say that X separates φ if for everyirreducible representation ψ : T → B ( ˜ H, ˜ K ) and isometries u : H → ˜ H and v : K → ˜ K , v ∗ ψ ( x ) u = φ ( x ), for all x ∈ X implies that φ and ψ are unitarilyequivalent. Also, the operator space X is called a separating operator space ifit separates every irreducible representations of T .It is clear that, when X is operator system, H = K , v = u , the notion ofseparating operator space recovers the notion of separating operator system. Proposition 5.6.
Suppose ω : C ∗ ( S ( X )) → B ( L ) is an irreducible representationand S ( X ) separates ω . Then one can decompose L as an orthogonal direct sum K ⊕ H in such a way that ω = S ( φ ) for some irreducible representation φ : T → B ( H, K ) and X separates φ . Proof.
Existence and irreduciblity of a triple morphism φ : T → B ( H, K ) followsfrom the proof of [15, Proposition 2.8]. Now we will prove that X separates φ .Let θ : T → B ( H θ , K θ ) be an irreducible representation of T such that v ∗ θ ( x ) v = φ ( x ) for all x ∈ X , where v : K → K θ and v : H → H θ areisometries. Let ρ : C ∗ ( S ( X )) → B ( K θ ⊕ H θ ) be the irreducible representa-tion of C ∗ ( S ( X )) corresponding to θ . Then for the isometry V = (cid:20) v v (cid:21) we have V ∗ ρV | S ( X ) = ω | S ( X ) . Since S ( X ) separates ω , there exists a unitary U : K ⊕ H → K θ ⊕ H θ such that U ∗ ρ ( a ) U = ω ( a ) for all a ∈ C ∗ ( S ( X )). Using U ∗ (cid:20) (cid:21) U = (cid:20) (cid:21) and U ∗ (cid:20) (cid:21) U = (cid:20) (cid:21) . we can factories U as (cid:20) u u (cid:21) . Thus u ∗ θ ( t ) u = φ ( t ) for every t ∈ T . Hence X separates φ . (cid:3) Proposition 5.7. If φ : T → B ( H, K ) is an irreducible representation such that X separates φ . Then ω = (cid:20) ω φφ ∗ ω (cid:21) : C ∗ ( S ( X )) → B ( K ⊕ H ) is an irreduciblerepresentation such that S ( X ) separates ω .Proof. Using the same line of argument in the proof of Theorem 3.1, we have ω is an irreducible representation. We will prove that S ( X ) separates ω .Let ρ : C ∗ ( S ( X )) → B ( L ) be an irreducible representation such that V ∗ ρV | S ( X ) = ω | S ( X ) for some isometry V : K ⊕ H → L . Since ρ is an irreducible representationof C ∗ ( S ( X )), we can decompose L = K ρ ⊕ H ρ such that ρ = (cid:20) ρ θθ ∗ ρ (cid:21) where θ : T → B ( H ρ , K ρ ) is an irreducible representation of T . Also, we have V = (cid:20) v v (cid:21) where v : K → K ρ and v : H → H ρ are isometries. Substituting the expressionsof ρ and V in the equation V ∗ ρ ( · ) V = ω ( · ), we obtain v ∗ θ ( x ) v = φ ( x ) for all x ∈ X . Our assumption that X separates φ implies that there exists unitaries u and u such that u ∗ θ ( t ) u = φ ( t ) for all t ∈ T . Then it follows that ρ is uniatrilyequivalent to ω by the unitary U = (cid:20) u u (cid:21) . (cid:3) Fuller, Hartz and Lupini [15] introduced the notion of rectangular extremepoints. Suppose that X is an operator space, and φ : X → B ( H, K ) is a com-pletely contractive linear map. A rectangular operator convex combination isan expression φ = α ∗ φ β + α ∗ φ β + · · · + α ∗ n φ n β n , where β i : H → H i and α i : K → K i are linear maps, and φ i : X → B ( H i , K i ) are completely con-tractive linear maps for i = 1 , , · · · , n such that α ∗ α + · · · + α ∗ n α n = 1, and β ∗ β + · · · + β ∗ n β n = 1. Such a rectangular convex combination is proper if α i , β i OUNDARY REPRESENTATIONS AND RECTANGULAR HYPERRIGIDITY 13 are surjective, and trivial if α ∗ i α i = λ i I , β ∗ i β i = λ i I , and α ∗ i φ i β i = λ i φ for some λ i ∈ [0 , . A completely contractive linear map φ : X → B ( H, K ) is a rectanagu-lar operator extreme point if any proper rectangular convex combination is trivial.
Proposition 5.8. [15, Proposition 2.12]
Suppose that φ : X → B ( H, K ) is acompletely contarctive map and S ( φ ) : S ( X ) → B ( K ⊕ H ) is the associated unitalcompletely positive map defined on the Paulsen system. The following assertionsare equivalent: (1) S ( φ ) is a pure completely positive map; (2) S ( φ ) is an operator extreme point; (3) φ is a rectangular operator extreme point. The following theorem characterize the boundary representations of TRO’s foroperator spaces.
Theorem 5.9.
Let X be an operator space generating a TRO T . Let φ : T → B ( H, K ) be an irreducible representation of T . Then φ is a boundary represen-tation for X if and only if the following conditions are satisfied: (i) φ | X is a rectangular operator extreme point. (ii) φ is a finite representation for X . (iii) X separates φ .Proof. Assume that φ : T → B ( H, K ) be a boundary representation for X . Let ω : C ∗ ( S ( X )) → B ( K ⊕ H ) be a representation of S ( X ) such that ω | S ( X ) = S ( φ ).By Theorem 3.1, ω is a boundary representation for S ( X ). Using [1, Theorem2.4.5], we have S ( φ ) is a pure UCP map, ω is a finite representation for S ( X )and S ( X ) separates ω . Thus, Proposition 5.8 implies that φ | X is a rectangularoperator extreme point, Proposition 5.2 implies that φ is a finite representationfor X and Proposition 5.6 implies that X separates φ .Conversely, assume that all the three conditions are satisfied. Using, Proposi-tion 5.8, Proposition 5.3 and Proposition 5.7 we have, S ( φ ) is a pure UCP map, ω is a finite representation for S ( X ) and S ( X ) separates ω . Thus [1, Theorem2.4.5] implies that ω is a boundary representation for S ( X ). By Proposition 2.6, φ is a boundary representation for X . (cid:3) Rectangular Hyperrigidity
In this section, we introduce the notion of rectangular hyperrigidity in the con-text of operator spaces in TRO’s. Rectangular hyperrigidity is the generalizationof Arveson’s [4] notion of hyperrigidity in the context of operator systems in C ∗ -algebras. We define rectangular hyperrigidity as follows: Definition 6.1.
A finite or countably infinite set G of generators of a TRO T is said to be rectangular hyperrigid if for every faithful representation from T to B ( H, K ) and every sequence of completely contractive (CC) maps φ n : B ( H, K ) → B ( H, K ) with k φ n k cb = 1, n = 1 , · · · ,lim n →∞ k φ n ( g ) − g k = 0 , ∀ g ∈ G = ⇒ lim n →∞ k φ n ( t ) − t k = 0 , ∀ t ∈ T (6.1)As in Arveson’s [4] notion of hyperrigity, we have lightened the notion of rect-angular hyperrigidity by identifying T with image π ( T ) where π : T → B ( H, K )faithful nondegenerate representation. Significantly, rectangular hyperrigid setof operators implies not only that equation 6.1 should hold for sequences of CCmaps φ n with k φ n k cb = 1, but also that the property should persist for everyother faithful representation of T . Proposition 6.2.
Let T be a TRO and G a generating subset of T . Then G isrectangular hyperrigid if and only if linear span of G is rectangular hyperrigid.Proof. The proof follows directly from the definition of rectangular hyperrigidity. (cid:3)
Thus, the rectangular hyperrigidity is properly thought of as a property ofoperator subspaces of a TRO. In definition 6.1, when T is a C ∗ -algebra, H = K and φ n , n = 1 , , , .. are UCP maps, the notion of rectangular hyperrigidityrecovers the notion of hyperrigidity. Proposition 6.3.
Let A be a C ∗ -algebra and S be an operator system in A suchthat A = C ∗ ( S ) . If S is rectangular hyperrigid, then S is hyperrigid.Proof. The proof follows from the fact that (see [23, Proposition 3.6]), every UCPmap is completely bounded with CB norm 1. (cid:3)
Thus, the rectangular hyperrigity is a stronger notion than hyperrigidity. Indefinition 6.1, suppose T is a C ∗ -algebra, H = K and φ n , n = 1 , , , .. are UCPmaps then by [23, Proposition 2.11] and [23, Proposition 3.6] the both notionsrectangular hyperrigidity and hyperrigidity coincides.Now, we prove a characterization of rectangular hyperrigid operator spaceswhich leads to study the operator space analogue of Saskin’s theorem ([25], [5,Theorem 4]) relating retangular hyperrigity and boundary representations foroperator spaces. Theorem 6.4.
For every separable operator space X that generates a TRO T ,the following are equivalent: (i) X is rectangular hyperrigid. (ii) For every nondegenerate representation π : T → B ( H , K ) on seperableHilbert spaces and every sequence φ n : T → B ( H , K ) of CC maps with k φ n k cb = 1 , n = 1 , , ... lim n →∞ k φ n ( x ) − π ( x ) k = 0 , ∀ x ∈ X = ⇒ lim n →∞ k φ n ( t ) − π ( t ) k = 0 , ∀ t ∈ T. (iii) For every nondegenerate representation π : T → B ( H , K ) on seperableHilbert spaces, π | X has the unique extension property. OUNDARY REPRESENTATIONS AND RECTANGULAR HYPERRIGIDITY 15 (iv)
For every TRO T , every triple morphism of TRO’s θ : T → T with k θ k cb = 1 and every completely contractive map φ : T → T with k φ k cb =1 , φ ( x ) = x, ∀ x ∈ θ ( X ) = ⇒ φ ( t ) = t, ∀ t ∈ θ ( T ) . Proof. ( i ) = ⇒ ( ii ): Let π : T → B ( H , K ) be a nondegenerate representationon separable Hilbert spaces and let φ n : T → B ( H , K ) of be a sequence ofCC maps with k φ n k cb = 1 such that k φ n ( x ) − π ( x ) k → x ∈ X . Let σ : T → B ( H , K ) be a faithful representation of T on separable Hilbert spaces H and K . Then σ ⊕ π : T → B ( H ⊕ H , K ⊕ K ) is faithful representation,so that each of the linear maps ω n : ( σ ⊕ π )( T ) → B ( H ⊕ H , K ⊕ K ) ω n ( σ ( t ) ⊕ π ( t )) = σ ( t ) ⊕ φ n ( t ) , t ∈ T, is a completely contractive map with CB norm 1. By the Haagerup-Paulsen-Wittstocks extension theorem [23, Theorem 8.2], ω n can be extend to a completelycontractive map ˜ ω n : B ( H ⊕ H , K ⊕ K ) → B ( H ⊕ H , K ⊕ K ) with CBnorm 1. Since for each x ∈ X , φ n ( x ) converges to π ( x ), ˜ ω n must converge inpoint-norm to the identity map on ( σ ⊕ π )( X ). So by the hypothesis (i), ˜ ω n mustconverge to the identity map on ( σ ⊕ π )( T ). Thus we can conclude that for every t ∈ T ,lim sup n →∞ k φ n ( t ) − π ( t ) k ≤ lim sup n →∞ k σ ( t ) ⊕ φ n ( t ) − σ ( t ) ⊕ π ( t ) k = lim n →∞ k ˜ ω n ( σ ( t ) ⊕ φ n ( t )) − σ ( t ) ⊕ π ( t ) k = 0 . ( ii ) = ⇒ ( iii ): Let π : T → B ( H , K ) be a nondegenerate representation onseparable Hilbert spaces and let φ : T → B ( H , K ) be a CC map with k φ k cb = 1and π | X = φ | X . Then ( iii ) follows from ( ii ) by taking φ n = φ for all n = 1 , , ... .( iii ) = ⇒ ( iv ): Let θ : T → T be a triple morphism with k θ k cb = 1,and let φ : T → T be a completely contractive map k φ k cb = 1 that satisfies φ ( θ ( x )) = θ ( x ) ∀ x ∈ X . Let T be the separable sub-TRO of T generated by θ ( T ) ∪ φ ( θ ( T )) ∪ φ ( θ ( T )) ∪ · · · . Then φ ( T ) ⊆ T .Since T is also separable, it has a faithful representation in B ( H , K ) say σ , forsome separable Hilbert spaces H and K . Now the CC map σφσ − : range( σ ) → B ( H , K ) with CB norm 1 can be extended to a CC map ˜ φ : B ( H , K ) → B ( H , K ) with CB norm 1 by the Haagerup-Paulsen-Wittstock extension theo-rem [23, Theorem 8.2]. Also for all x ∈ X , ˜ φ ( σθ ( x )) = σφσ − σθ ( x ) = σφθ ( x ) = σθ ( x ). Since σθ is a nondegenerate triple morphism by assumption (iii) σθ | X hasunique extension property and therefore for all t ∈ T ,˜ φ ( σθ ( t )) = σθ ( t ) σφσ − σθ ( t ) = σθ ( t ) σφθ ( t ) = σθ ( t ) φθ ( t ) = θ ( t ) . In the last line we used the injectivity of σ . ( iv ) = ⇒ ( i ): Suppose that T ⊆ B ( H, K ) is faithfully represented on someHilbert spaces. Let φ n : B ( H, K ) → B ( H, K ) be a sequence of CC maps with k φ n k cb = 1, n = 1 , , ... , such thatlim n →∞ k φ n ( x ) − x k = 0 , ∀ x ∈ X. Let ℓ ∞ ( B ( H, K )) be the TRO of all bounded sequences with components in B ( H, K ) and let c ( B ( H, K )) be the TRO-ideal of all sequences in ℓ ∞ ( B ( H, K ))that converge to zero in norm.Consider the completely bounded map φ : ℓ ∞ ( B ( H, K )) → ℓ ∞ ( B ( H, K ))defined by φ ( b , b , b · · · ) = ( φ ( b ) , φ ( b ) , φ ( b ) · · · ) .φ is a CC map with k φ k cb = 1. By the continuity of the map φ , the TRO-ideal c ( B ( H, K )) is invariant under φ and therefore induces a CC map φ with k φ k cb = 1 on the quotient TRO ℓ ∞ ( B ( H, K )) /c ( B ( H, K )) by φ ( y + c ( B ( H, K ))) = φ ( y ) + c ( B ( H, K )) ∀ x ∈ ℓ ∞ ( B ( H, K )) . Consider the map θ : T → ℓ ∞ ( B ( H, K )) /c ( B ( H, K )) defined by θ ( t ) = ( t, t, t, · · · ) + c ( B ( H, K )) . We claim that the assumption in condition ( iv ) is satisfied. Since lim n →∞ k φ n ( x ) − x k = 0 for all x ∈ X , and therefore the sequence { φ n ( x ) − x } ∈ ℓ ∞ ( B ( H, K ))is actually belongs to c ( B ( H, K )). Thus ( φ ( x ) , φ ( x ) , · · · ) + c ( B ( H, K )) =( x, x, · · · ) + c ( B ( H, K )). Now φ ( θ ( x )) = φ (( x, x, · · · ) + c ( B ( H, K )))= φ (( x, x, · · · )) + c ( B ( H, K ))= ( φ ( x ) , φ ( x ) , · · · ) + c ( B ( H, K ))= ( x, x, · · · ) + c ( B ( H, K ))= θ ( x ) . Then applying ( iv ), for all t ∈ T we have φ ( θ ( t )) = θ ( t ) and hence( φ ( t ) , φ ( t ) , · · · ) − ( t, t, · · · ) ∈ c ( B ( H, K )) . Thus lim n →∞ k φ n ( t ) − t k = 0 for all t ∈ T , hence X is rectangular hyperrigid. (cid:3) Example 6.5.
Let H be an infinite dimensional Hilbert space and V be theunilateral right shift operator on H . Then the operator space S = span { I, V, V ∗ } is not rectangular hyperrigid. To see this, take φ n : B ( H ) → B ( H ) as φ n = V ∗ I S ( · ) V for each n = 1 , · · · . Then φ n is a completely contractive linear mapwith k φ n k cb = 1 and φ n is identity on S . Hence lim n →∞ k φ n ( s ) − s k = 0 ∀ s ∈ S but lim n →∞ k φ n ( V V ∗ ) − V V ∗ k = k I − V V ∗ k = 1. Note that the arguments in thisexample carries over to any isometry V which is not a unitary.We deduce the following necessary conditions for rectangular hyperrigidity: OUNDARY REPRESENTATIONS AND RECTANGULAR HYPERRIGIDITY 17
Corollary 6.6.
Let X be a separable operator space generating a TRO T . If X is rectangular hyperrigid then every irreducible representation of T is a boundaryrepresentation for X .Proof. The assertion is an immediate consequence of condition (ii) of Theorem6.4. (cid:3)
Problem 6.7.
If every irreducible representation of TRO T is a boundary repre-sentation for a separable operator space X ⊆ T , then X is rectangular hyperrigid. Proposition 6.8.
Let X be a operator space generating TRO T . Let π i : T → B ( H i , K i ) be an nondegenerate representation such that π i | X has the unique ex-tension property for i = 1 , , ..., n . Then the direct sum of rectangular operatorstates ⊕ ni =1 π i | X : X → B ( ⊕ ni =1 H i , ⊕ ni =1 K i ) has the unique extension property.Proof. Let φ : T → B ( ⊕ ni =1 H i , ⊕ ni =1 K i ) be an extension of ⊕ ni =1 π i | X to a rectan-gular operator state from T to B ( ⊕ ni =1 H i , ⊕ ni =1 K i ). Let φ i : T → B ( H i , K i ) bethe rectangular operator state such that φ i ( t ) = p i φ ( t ) q i , t ∈ T. Where p i is the projection form ⊕ ni =1 H i onto H i and q i is the projection from ⊕ ni =1 K i onto K i . We have φ i ( x ) = p i φ ( x ) q i = π i ( x ) ∀ x ∈ X. Since π i | X has the unique extension propery implies that φ i ( t ) = π i ( t ) ∀ t ∈ T. Thus, ⊕ ni =1 π i ( t ) = ⊕ ni =1 ( p i φ ( t ) q i ) ∀ t ∈ T ⊕ ni =1 π i ( t ) = ⊕ ni =1 p i ( φ ( t )) ⊕ ni =1 q i ∀ t ∈ T, note that, ⊕ ni =1 p i = I H and ⊕ ni =1 q i = I K . Therefore ⊕ ni =1 π ( t ) = φ ( t ) ∀ t ∈ T .Hence ⊕ ni =1 π i | X has unique extension property. (cid:3) Here, we settle the Problem 6.7 when TRO is finite dimensional. Thus whatwe have here is an finite dimensional version of the classical Saskin’s theorem.
Theorem 6.9.
Let X be an operator space whose generating TRO T is finitedimensional, such that every irreducible representation of T is a boundary repre-sentation for X . Then X is rectangular hyperrigid.Proof. Using item (iii) of Theorem 6.4, it is enough to prove that for every non-degenerate representation π : T → B ( H, K ), the rectangular operator state π | X has the unique extension property. Since T finite dimensional, [7, Theorem 3.1.7]implies that every nondegenerate representation of a finite dimensional TRO isthe finite direct sum of irreducible repersentations. By our assumption every irre-ducible representation restricted to X has unique extension property. By Propo-sition 6.8 finite direct sum of irreducible representation restricted to X has uniqueextension property. Therefore every nondegenerate representation restricted to X has the unique extension property. (cid:3) Now, we explore relations between rectangular hyperrigity of an operator spaceand hyperrigidity of the corresponding Paulsen system.
Theorem 6.10.
Let X be a separable operator space generating a TRO T . Paulsensystem S ( X ) is hyperrigid in C ∗ -algebra C ∗ ( S ( X )) if and only if X is rectangularhyperrigid in TRO T .Proof. Assume that Paulsen system S ( X ) is hyperrigid in C ∗ -algebra.Let φ n : B ( H, K ) → B ( H, K ) be CC maps with k φ n k cb = 1, n = 1 , · · · , suchthat lim n →∞ k φ n ( x ) − x k = 0 , ∀ x ∈ X. Then the corresponding maps S ( φ n ) : B ( K ⊕ H ) → B ( K ⊕ H ), n = 1 , · · · areUCP maps. For all x, y ∈ X and λ, µ ∈ C , we have (cid:13)(cid:13)(cid:13)(cid:13) S ( φ n )( (cid:20) λ xy ∗ µ (cid:21) ) − (cid:20) λ xy ∗ µ (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:20) λ φ n ( x ) φ n ( y ) ∗ µ (cid:21) − (cid:20) λ xy ∗ µ (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:20) φ n ( x ) − xφ n ( y ) ∗ − y ∗ (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) ≤ k φ n ( x ) − x k + k φ n ( y ) ∗ − y ∗ k . Since S ( X ) is hyperrigid in C ∗ ( S ( X )), we conclude that for every t ∈ T lim n →∞ (cid:13)(cid:13)(cid:13)(cid:13) S ( φ n )( (cid:20) t (cid:21) ) − (cid:20) t (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) = 0 . Thus, lim n →∞ k φ n ( t ) − t k = 0 ∀ t ∈ T. Conversely, suppose X is rectangular hyperrigid. By item (iii) of Theorem 6.4every nondegenerate representation of T restricted to X has the unique extensionpropery. From [6, Proposition 3.1.2 and Equation 3.1], we have a one to onecorrespondence between the representations of TRO T and its linking algebra.Using Corollary 3.4 we get, every nondegenerate representation of the C ∗ -algebra C ∗ ( S ( X )) restricted to S ( X ) has the unique extension property. Thus, by [4,Theorem 2.1], S ( X ) is hyperrigid. (cid:3) The following remark gives some partial answers to the Problem 6.7 with extraassumptions.
Remark . Suppose that T is a TRO in B ( H, K ) and every irreducible rep-resentation ψ : T → B ( H, K ) is a boundary representation for X . Then byTheorem 3.1, each S ( ψ ) is a boundary representation of C ∗ ( S ( X )) for S ( X ).Suppose either C ∗ ( S ( X )) has countable spectrum [4, Theorem 5.1] or C ∗ ( S ( X ))is Type I C ∗ -algebra with C ∗ ( S ( X )) ′′ is the codomain for UCP maps on C ∗ ( S ( X ))[20, Corollary 3.3]. Then S ( X ) is hyperrigid. Therefore by Theorem 6.10, X isrectangular hyperrigid in T . Acknowledgments.
The authors would like thank Michael Hartz for somevaluable comments and suggestions regarding this manuscript. The research of
OUNDARY REPRESENTATIONS AND RECTANGULAR HYPERRIGIDITY 19 the first named author is supported by NBHM (National Board of Higher Math-ematics, India) Ph.D Scholarship File No. 0203/17/2019/R&D-II/10974. Theresearch of the second named author is supported by NBHM (National Board ofHigher Mathematics, India) Post Doctoral fellowship File No. 0204/26/2019/R&D-II/12037. The second author would like thank Kerala School of Mathamatics(KSoM), Kozhikode for providing funding to conduct discussion meeting on ”Noncommutative function theory”. This work is an outcome of the discussion meet-ing.
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E-mail address : [email protected] Shankar P, Indian Statistical Institute, Statistics and Mathematics Unit, 8thMile, Mysore Road, Bangalore, 560059, India.
E-mail address : [email protected] A.K. Vijayarajan, Kerala School of Mathematics, Kozhikode 673 571, India.
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