### 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

But

where

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

where

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.

your blog is awesome! Thanks!

Thanks 🙂