Inductive limits in the operator system and related categories

Abstract

We present a systematic development of inductive limits in the categories of ordered *-vector spaces, Archimedean order unit spaces, matrix ordered spaces, operator systems and operator C*-systems. We show that the inductive limit intertwines the operation of passing to the maximal operator system structure of an Archimedean order unit space, and that the same holds true for the minimal operator system structure if the connecting maps are complete order embeddings. We prove that the inductive limit commutes with the operation of taking the maximal tensor product with another operator system, and establish analogous results for injective functorial tensor products provided the connecting maps are complete order embeddings. We identify the inductive limit of quotient operator systems as a quotient of the inductive limit, in case the involved kernels are completely biproximinal. We describe the inductive limit of graph operator systems as operator systems of topological graphs, show that two such operator systems are completely order isomorphic if and only if their underlying graphs are isomorphic, identify the C*-envelope of such an operator system, and prove a version of Glimm's Theorem on the isomorphism of UHF algebras in the category of operator systems. ContentsIn order to avoid excessive notation, we will sometimes denote the ordered *-vector space (V, V + , e) simply by V .Let us denote by OU the category whose objects are ordered *-vector spaces with order units and whose morphisms are unital positive maps, and by AOU the category whose objects are AOU spaces and whose morphisms are unital positive maps. Clearly, we have a forgetful functor F : AOU → OU. In [33, Section 3.2], the process of Archimedeanisation is discussed which provides us with a left adjoint to this functor. Let (V, V + , e) be an ordered *-vector space with order unit. Define D = {v ∈ V h : re + v ∈ V + for every r > 0} and (2) N = {v ∈ V : f (v) = 0 for all f ∈ S(V )}. Clearly, D is a cone, while N is a linear subspace of V . Equip V /N with the involution given by (v + N ) * = v * + N , and set (V /N ) + = {v + N : v ∈ D}.

Spherical representations of $C^*$-flows II: Representation system and Quantum group setup

“…which shows that the mapping given by (3.9) in question is well defined thanks to [15,Remark 3.1]. It is obvious that the resulting map is bilinear.…”

“…The concept of inductive limits of operator systems was first introduced and used by Kirchberg [10], but he treated it in the category of norm-complete operator systems. A systematic study of the concept was recently conducted by Mawhinney and Todorov [15] in the category of general (not necessarily norm-complete) operator systems. We will follow this recent study to discuss the inductive limit of (3.4).…”

“…(1) By its construction, S(α t , β) is also the inductive limit in the category OU of ordered * -vactor spaces with order unit (say OU spaces) as well as even in the category of vector spaces. Thus, for any sequence of (unital positive) linear maps Ψ n : Z(A n ) → V with a single (OU) vector space V such that Ψ n+1 • Θ n+1,n = Ψ n holds for every n ≥ 0, the correspondence Θ∞,n (f ) → Ψ n (f ) defines a well-defined (unital positive) linear map from S(α t , β) to V. See the proof of [15,Theorem 3.5]. This property allows us to describe the vector space S(α t , β) itself as follows.…”

“…With the identity operator I on n≥0 Z(A n ), the vector space S(α t , β) is identified with the quotient space n≥0 Z(A n ) Im(P − I) by the range of I − P , and Θ∞,n is given by the composition of the inclusion map Z(A n ) ֒→ n≥0 Z(A n ) and the quotient map. This can be shown by using [15,Remark 3.1]; actually,…”

“…A locally normal state on S(α t , β) is defined similarly with Θ∞,n in place of Θ ∞,n , and the set of those states is denoted by S ln ( S(α t , β)). (2) It is clear, from the above definition, that the canonical bijection ω ∈ S(S(α t , β)) → ω • q S ∈ S( S(α t , β)) between state spaces (see e.g., [15,Theorem 2.6]) sends S ln (S(α t , β)) onto S ln ( S(α t , β)). Thus, no difference between S(α t , β) and S(α t , β) occurs in the study of locally normal states.…”