Some Objects of Operator Algebra
by Nirakar Neo
This post is a continuation of this previous post. The aim of this post is to introduce some common objects that are encountered in operator algebra. All of the objects will be needed to understand the paper cited in the previous blog. I am bit terse here and examples are almost absent.
All the vector spaces considered here are assumed to be complex. Let be a vector space. is the vector space of all matrices with entries from , under natural addition and scalar multiplication. In particular, if , we write as . If , then we denote by , and by . is the identity matrix in .
We shall be using some mathematical objects which we’ll describe here. The simplest to start with are the operator spaces. Let be any Hilbert space. An operator space is a closed linear subspace of Using we construct matricial spaces . The natural inclusion
provides a norm on using the operator norm on . This description is called a concrete definition of an operator space. Ruan characterised operator spaces in the following abstract way.
We say is an operator space if is a normed linear space, with the norms satisfying
- , where and , and we define .
- , where .
Operator spaces are usually considered as noncommutative analogues of Banach spaces. A special example of operator space is an operator system. Concretely, an operator system is a closed subspace of , which is also closed under adjoint operation and contains the identity operator. Again we can form the different matrix levels . The important thing here is that the inclusion (1) defines an ordering on which is induced by the usual ordering on .
The commutative analogue of an operator system is a function system, which is a closed subspace of , where is a compact, and the subspace is closed under conjugation and contains the constant function, 1 ( for all ). Consider the following important example which we shall refer later. We recall that affine functions are those which preserve convex combinations.
Example 1. Let be a locally convex space, and let be a compact convex subset of . Define
Then is a function system. Clearly, . It is closed since uniform limit of a sequence of continuous functions is again continuous, and the limit function is easily seen to be affine. Moreover it is closed under conjugation, and the constant function ( for all ) is trivially contained in . The positive elements in are precisely those functions which map each into . In particular, 1 is a positive element.
Function systems are abstractly characterised as complete order unit spaces. An order unit space is a real-partially ordered vector space – – having an element , which we call the order unit, satisfying
- (Order Unit property) For each , there is an such that .
- (Archimedean property) For each , if for all , then .
On an order unit space , we define a norm as
Then becomes a normed linear space. It is easy to see that a function system satisfies all these properties.
Parallel to this characterisation, operator systems have been characterised by Choi-Effros in terms of matrix ordered spaces. Let be -vector space. Then is also -vector space by . We say is a matrix ordered space if there are proper cones for all , and such that
for all . Then we have the following definition.
A matrix ordered space is an operator system if
- is proper, that is, if , then
- is Archimedean for all
- has an order unit.
Given an operator system , we define the state space of as
Let are vector spaces and is a linear map, then the canonical amplification of is the sequence of maps defined by
In the next post, I’ll start by defining matrix convex sets.