Matrix Convex Sets
by Nirakar Neo
In the previous post, we introduced some objects in operator algebra. Those will occur in the examples given in this post. The purpose of this post is to define matrix convex sets.
We begin by defining what we mean by a matrix convex set. There are two definitions, and we shall show that both are equivalent. Then we shall present some examples of matrix convex sets.
Definition 1. A matrix convex set in a vector space is a collection , where for each is a non-empty set such that
whenever , and satisfying .
The above definition seems like a natural generalisation of convexity. But the next one is usually easier to use in specific cases.
Definition 2. A matrix convex set in a vector space is a collection , where for each is a non-empty set such that
For any , with , we have .
For any , .
We now show the equivalence of both definitions.
Definition 1 Definition 2. The first part of definition 2 follows trivially (using ). To show the second part, let and ; then we have to show that
and clearly, . Thus (2) is also satisfied.
Definition 2 Definition 1. Let , for satisfying . Then by extension of (2) in definition 2, we have
Now let . Then and by definition 1 we have which implies .
An important consequence is that each is a convex set of . Let be a matrix convex set in . Let . We need to show that for . Set , and . Then
Then , which is nothing but .
We present some examples of matrix convex sets. We start with the simplest vector space.
Example 2. Let . On the vector space , consider the collection
where . This collection defines a matrix convex set in . We shall now verify that this indeed satisfies the conditions mentioned above (using definition 2).
Let , with . Let , then . This implies
which means . Thus . Finally let and . Then we show that
But, for , we have
And similarly we can show the other parts. This completes our verification. Conversely, we can show that any matrix convex set in , where is a closed convex subset of is a closed matrix interval.
Theorem 1 Suppose that is a matrix convex set in where is a bounded closed subset of . Then must be a closed matrix interval.
Proof: Since is a bounded closed and convex it must be a closed interval in , say . Let . Then we have to show that . Let us first show that . Let , then
But from property 1, we see that , and thus the last statement is true; hence . Similarly, we can show . Thus .
Conversely, let , then is self-adjoint and hence we can write
where . Then by property 2, we see that , and by property 1, , that is .
Example 3. Let be an operator space. Then the collection of unit balls, where
is a matrix convex set. It is almost trivial to verify the conditions of matrix convexity using the abstract definition of operator space (given in this post).
Example 4. Let be an operator system. The collection of positive cones, where
is a matrix convex set. For let with . Let . Then for we have
since . Similarly, if and , then it is fairly obvious that .
Example 5. Again let be an operator system. Consider the collection of matrix states where
We say is completely positive if canonically amplified maps are positive for all . Then is a matrix convex set in . Here we have identified an as an element in . For verification, let and . Then we have to show that is completely positive and . By definition, . Then
To show complete positivity, let . Then
Thus is completely positive since is completely positive. Likewise, the statement for is analogously proved. is considered as the matricial version of the state space.
Example 2 and example 5 are the ones which we shall focus upon the next sections.