Later, Webster and Winkler [29] defined the notion of extreme points, called matrix extreme points, in the case of matrix convex sets. Thus, one of the central goals in the study of matrix convexity is to identify the minimal sets of extreme points of the matrix that span the matrix convex sets. By replacing A with the operator system RandB(H) with a finite-dimensional example (Mn(C) for some positive integer), Farenick proved in the finite-dimensional setting of the operator system and matrix algebras [9] that pure CP mappings are exactly matrix endpoints connected non-commutative compact convex sets of all matrix-valued UCP mappings in the operator system.
In [17] Hamana studied inclusions of operator systems in injective operator systems and showed that every operator system is a subsystem of an injective operator system I(R) - the injective envelope of R. Here we give a characterization of boundary representations of an operator system in terms of boundary points of the associated convex matrix set. We also consider bi-convexity of CC maps between operator spaces and show that (Proposition 4.3.1) the notion is equivalent to the notion of purity and operator finiteness of the corresponding Paulsen maps in the special case of embedding of an operator system in its injectable envelope.
C ∗ − algebras
The set of all complex-valued continuous functions defined on the Compact HausdorffX space, denoted by C(X), where vector addition, vector multiplication, involution, and norm are defined as follows. The set of all bounded linear operators B(H) defined on the Hilbert spaceH, where multiplication is defined as the composition of operators and involution as adjoint and the norm is defined as the norm of the operator. Every commutative C∗-algebra with unity is isometrically isomorphic to C(X), for a complex Hausdorff plane Xin.
In general, any C∗-algebra can be identified as isometrically isomorphic to a subalgebra B(H) for some Hilbert spaceH.
Operator systems and spaces
Completely positive maps
The mapping of CP to the inverse of CP is called a complete order isomorphism between two operator systems, which is a natural isomorphism of operator systems. Any abstract operator system can be embedded in B(H), for some Hilbert space H as a concrete operator system [28, Theorem 13.1]. Similarly for operator spaces, each operator space can be considered as an operator space in B(H) for some Hilbert spaceH [28, Theorem 13.4].
Paulsen system
Boundary representation of C ∗ -algebras
Classical convex sets
C ∗ -convexity
C ∗ -extreme points of C ∗ -convex sets
As in the classical theory, it is interesting to study the analogous and corresponding notion of extreme points and the resulting possible Krein-Milman theorems in C∗-convexity. Using Loebl's result [24] that every matrix is a linear extremum of its unitary orbit, Paulsen and Loebl proved [23, Proposition 23] that the C∗-extremities of C∗-convex sets in Mnare are indeed linear extremums. 23, proposition 23] If T is an aC∗-extremity point of an aC∗-convex set K ⊂Mn, then T is a linear extremity point.
Consider B(H), the C∗-extreme point of a closed unit sphere are exactly isometries and co-isometries.
Injective operator systems and spaces
Injective envelope and C ∗ -envelope of operator systems
Hilbert C ∗ -modules
Let B be a C∗-algebra and E a complex vector space and a right B-module with a sesquilinear map⟨., .⟩B : E × E → B which is linear in the first variable and linear in the second variable such as thatx, y ∈ E, a∈B. We can consider Hilbert C∗ modules over possibly different C∗ algebras where together with the above axioms a compatibility condition between the actions of the algebras is needed. Suppose we have sesquilinear forms A⟨., .⟩and ⟨., .⟩B such that E is both a left HilbertA module and a right HilbertB module such that the forms are related by the equation A⟨x, y⟩ z = x⟨y, z⟩B for all x, y, z ∈ E.
Let A be a C∗-algebra, then A is a HilbertC∗ A-A-bimodule, where bimodule multiplication is defined by multiplication in A and A⟨x, y⟩ = xy∗ and ⟨x, y⟩A=x∗ya are inner products. The following definition of a matrix convex set as a non-commutative analogue of a classical convex set was introduced by Wittstock [30]. It is clear from the definition of matrix convex sets that if K = (Kn) is matrix convex, then every set Kn is a C∗-convex set in Mn(V). If you want to see it, consider it.
A compact matrix convex set is a matrix convex set K = (Kn) in a locally convex topological vector space V such that every K is compact in the product topology in Mn(V). Then W(T) = (Wn(T)) is a matrix convex set in C, where Wn(T) ={ϕ(T)|ϕ :B(H)→ Mnunital is a completely positive mapping} then the matrix domain of T.
Matrix extreme points
We also say that v ∈K is an extremum of a matrix if whenever there is a proper matrix convex combination vi ∈Kni for i k, then each =nandv =u∗iviui for some unitoui ∈Mn. A matrix affine mapping to a matrix convex set K= (Kn) in a vector space V is a sequence θ = (θn) of mappings θn:Kn→Mn(W) for some vector space W such that. Let K = (Kn) and K′ = (Kn′) be matrix convex sets in locally convex topological vector spaces V and W, respectively.
A matrix affine homeomorphismθ = (θn) betweenKandK′ is a matrix affine mapping such that eachθ is a homeomorphism of the product topologies on both sides. It is easy to see that matrix extremal points and linear extremal points are preserved under matrix affine homeomorphisms. Given a compact matrix convex set, one can associate an operator system with it in the following way: if K is a compact matrix convex set in a locally convex spaceV, then A(K) = {f = (fn) : K → C|f matrix affine andf1 continuous} is an operator system in the sense of Choi-Effros.
The following proposition gives the relation between a compact convex matrix set and its associated operator system. If K is a compact matrix convex set in a locally convex spaceV, then A(K) is an operator system, and K and CS(A(K)) are matrix affine homeomorohies.
Boundary points
From the definition of a boundary point it is clear that all boundary points are extreme points of the matrix. But, Kleski [21] gives an example to show that all extreme points of the matrix do not have to be boundary points. Here we give an example to show that there are convex matrix groups for which all extreme points of the matrix are boundary points.
The proof uses Farenick's result, which establishes a connection between the extreme points of the Kan matrix and the pure states of the matrix. By Theorem 3.2.1, there is a one-to-one correspondence between the boundary points of CS(R) and the boundary representations of A(CS(R)). Thus, there is a one-to-one correspondence between the boundary representations for R and the boundary points of CS(R).
Then a representation π : C∗(R) → Mn(C) is a boundary representation for Rif if and only ifπ|R is a boundary point of CS(R). For a convex set of matrix K for which the extreme points of the matrix and the boundary points coincide, the following theorem gives a sufficient condition for a finite-dimensional representation to be a boundary representation for A(K). If π is a representation of C∗(A(K)) into a finite-dimensional Hilbert space such that π|A(K) is pure, then π is a boundary representation.
We deduce that for a trivial module, unitaries are bi-extreme points of the closed unit ball, while bi-extreme points of the closed unit ball of the trivial module B(H), where H is a Hilbert space, are shown as either isometries or co-isometries. It is easy to see that even in the trivial case, bi-convex sets and C∗-convex sets do not coincide, although in this case bi-convex sets are C∗-convex sets and not vice versa.
Bi-extreme points
Unitaries are the extreme points of the convex set of the closed unit sphere of any trivial bimodule.
Bi-extreme CC maps on operator spaces
Paulsen map for the case of embedding the Paulsen system into its injective envelope. In this section, we study the extremum of the canonical embedding mapping from the operator space X to its injective envelope I(X). From the minimality of Φ we can see that π is a minimal X-projection and that the image of π is the injective envelope I(X) of the operator space X[18].
In fact, since the required compatibility condition exists between the above two inner products, I(X) is a HilbertC∗-I11(X)-. Since I(S(X)) is injective, we can extend the maps CPS(ϕ)andΨtoA such that S(ϕ)−Ψ is still completely positive. Such a bi-convex combination is called proper ifαi, βi are straight invertible and is called trivial ifα∗iαi =λiI, βi∗βi =λiI, αi∗ϕiβi =λiϕ for someλi ∈[0,1].
Such an operator convex combination is correct if αi are quite invertible for n and trivial ifα∗iαi =λiI andα∗iϕiαi =λiϕfor someλi ∈[0,1]. 2.12] gives us a characterization of the bi-extremity of a CC map from an operator space into its injective envelope with respect to the purity of the corresponding Paulsen map.
Triple ideals and prime TROs
The purity of embedding of an operator system turns out to be equivalent to a certain nature of the generated C∗ algebra, namely deprimenature. In this chapter, we identify a class of TROs for which the embedding of the generating operator space in its injective envelope is bi-extreme. In the next theorem we prove that if the TRO generated by an operator space is prime, then the canonical embedding of the operator space in its injective envelope is bi-extreme.
Theorem 4.2.1 and Theorem 4.4.2 are part of the published work Bi-extremity of embedding operator spaces in their injective envelopes. In this thesis, boundary representations of an operator system in aC∗ algebra and boundary points of the associated convex matrix set are studied and possible relationships between them are investigated. Here we have established a relationship between boundary representations for a finite dimensional operator system and boundary points of the associated matrix convex set.
Using this result, we generated boundary representations of the C∗ algebra by a finite dimensional operator system characterized in terms of boundary points of the associated matrix convex set. In the subsequent part of the thesis, we study bi-convex subsets of Hilbert C∗- bimodules and explore their limbs. Here we introduced the notion of bi-extremal points of bi-convex subsets of HilbertC∗ bimodules and we proved that unitary bi-extremal points of the bi-convex set are closed unit balls of any trivial bimodule.
We further introduced the concept of the operator convex combination of the map between an operator system and its injective envelope. Here we established equivalent conditions for a CC map to be bi-extreme with respect to purity and operator extremity of the corresponding Paulsen map. Finally, using the above results, for an operator space X, we established a sufficient condition for an embedding mapτ :X →I(X) to be bi-extreme with respect to the primal nature of the triple envelopeT of X .
Tessier, Purity of the Embeddings of Operator Systems in their C∗ and Injective Envelopes, preprint, arXiv.
