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 {V} be a vector space. {M_{m,n}(V)} is the vector space of all {m\times n} matrices with entries from {V}, under natural addition and scalar multiplication. In particular, if {m=n}, we write {M_{n,n}(V)} as {M_n(V)}. If {V=\mathbb{C}}, then we denote {M_{m,n}(\mathbb{C})} by {\mathbb{M}_{m,n}}, and {M_{n,n}(\mathbb{C})} by {\mathbb{M}_n}. {\mathbb{I}_n} is the identity matrix in {\mathbb{M}_n}.

We shall be using some mathematical objects which we’ll describe here. The simplest to start with are the operator spaces. Let {\mathcal{H}} be any Hilbert space. An operator space {\mathcal{M}} is a closed linear subspace of {\mathfrak{B}(\mathcal{H})} Using {\mathcal{M}} we construct matricial spaces {M_n(\mathcal{M})}. The natural inclusion

\displaystyle M_n(\mathcal{M})\hookrightarrow M_n(\mathfrak{B}(\mathcal{H}))\simeq \mathfrak{B}(\mathcal{H}^n) \ \ \ \ \ (1)

provides a norm on {M_n(\mathcal{M})} using the operator norm on {\mathfrak{B(\mathcal{H}^n)}}. This description is called a concrete definition of an operator space. Ruan characterised operator spaces in the following abstract way.

We say {\mathcal{M}=(V,\{\|.\|_n\})} is an operator space if {(M_n(V),\|.\|_n)} is a normed linear space, with the norms satisfying

  1. {\|\alpha v\beta\| \leq \|\alpha\|\|v\|\|\beta\|}, where {\alpha,\beta\in\mathbb{M}_m} and {v\in M_m(V)}, and we define {\alpha v \beta=[\Sigma_{j,k}\alpha_{ij}v_{jk}\beta_{kl}]\in M_m(V)}.
  2. {\|v\oplus w\|_{m+n}=\text{max}\{\|v\|_m,\|v\|_n\}}, where {v\in M_m(V), w\in M_n(V)}.

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 {\mathcal{R}} is a closed subspace of {\mathfrak{B}(\mathcal{H})}, which is also closed under adjoint operation and contains the identity operator. Again we can form the different matrix levels {M_n(\mathcal{R})}. The important thing here is that the inclusion (1) defines an ordering on {M_n(\mathcal{R})} which is induced by the usual ordering on {\mathfrak{B}(\mathcal{H}^n)}.

The commutative analogue of an operator system is a function system, which is a closed subspace of {C(X)}, where {X} is a compact, and the subspace is closed under conjugation and contains the constant function, 1 ({1(x)=1} for all {x\in X}). Consider the following important example which we shall refer later. We recall that affine functions are those which preserve convex combinations.

Example 1. Let {X} be a locally convex space, and let {K} be a compact convex subset of {X}. Define

\displaystyle A(K)=\{f:K\rightarrow\mathbb{C} \;|\; f \; \text{is continuous and affine}\}

Then {A(K)} is a function system. Clearly, {A(K)\subset C(K)}. 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 ({1(x)=1} for all {x\in K}) is trivially contained in {A(K)}. The positive elements in {A(K)} are precisely those functions which map each {x\in K} into {[0,\infty)}. 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 – {A} – having an element {e}, which we call the order unit, satisfying

  1. (Order Unit property) For each {a\in A}, there is an {r\in \mathbb{R}} such that {a\leq re}.
  2. (Archimedean property) For each {a\in A}, if {a\leq re} for all {r\in (0,\infty)}, then {a\leq 0}.

On an order unit space {(A,e)}, we define a norm as

\displaystyle \|a\|=\inf\{r\in \mathbb{R}:-re\leq a\leq re\}.

Then {A} 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 {V} be {^*}-vector space. Then {M_n(V)} is also {^*}-vector space by {[v_{ij}]^*=[v_{ij}^*]}. We say {V} is a matrix ordered space if there are proper cones {M_n(V)^+\subset M_n(V)_{sa}} for all {n\in \mathbb{N}}, and such that

\displaystyle \alpha^*M_{m}(V)^+\alpha\subset M_n(V)^+

for all {\alpha\in \mathbb{M}_{m,n}}. Then we have the following definition.

A matrix ordered space {(A,M_n(A)^+)} is an operator system if

  1. {A^+} is proper, that is, if {a,-a\in A^+}, then {a=0}
  2. {M_n(A)^+} is Archimedean for all {n\mathbb{N}}
  3. {A} has an order unit.

Given an operator system {\mathcal{R}}, we define the state space of {\mathcal{R}} as

\displaystyle S(\mathcal{R})=\{\varphi:\mathcal{R}\rightarrow \mathbb{C} \;|\; \|\varphi\|=1=\varphi(1)\}.

Let {V,W} are vector spaces and {f:V\rightarrow W} is a linear map, then the canonical amplification of {f} is the sequence of maps {f_n:M_n(V)\rightarrow M_n(W)} defined by {[v_{ij}]\mapsto [f(v_{ij})].}

In the next post, I’ll start by defining matrix convex sets.