Matrix extreme points

A part of convexity theory studies the possibility of representing all points of a convex set as convex combinations of special points, the so called extreme points. The well-known results about this are the Krein-Milman theorem and its sharpenings, the Choquet representation theorems. The question arises whether there are analogous results for non-commutative convexity. The sections Matrix extreme points and $C^*$-extreme points give partial answers to this question.

Let $V$ be a vector space. A matrix convex set of matrices over $V$ is called matrix convex set in $V$ for short. Let $A$ be a set of matrices over $V$. The matrix convex hull of $A$ is the smallest matrix convex set in $V$ containing $A$. Its closure is the smallest closed matrix convex set containing $A$ because the closure of matrix convex sets is matrix convex. Two elements $x,y\in M_n(V)$ are unitarily equivalent, if there is a unitary $u\in M_n$ such that $x=u^*yu$. Let $U(S)$ be the set of all elements, that are unitarily equivalent to elements of $S\subset M_n(V)$. $x\in M_n(V)$ is called reducible, if it is unitarily equivalent to some block matrix ${y\: 0 \choose 0\: z} \in M_n(V)$. A matrix convex combination $\sum_{i=1}^{k} \alpha_i^* x_i\alpha_i$ is called proper, if all $\alpha_i$ are square matrices different from 0.

Let $K$ be a matrix convex set in $V$. Then $x\in K_n$ is a structural element⁶⁶ of $K_n$, if whenever $x=\sum_{i=1}^{k} \alpha_i^* x_i\alpha_i$ is a proper matrix convex combination of $x_i\in K_n$, then every $x_i$ is unitarily equivalent to $x$. The set of all structural elements of $K_n$ is denoted by $\mathrm{str}(K_n)$. The set of structural elements of $K$ is the set of matrices over $V$ consisting of $\mathrm{str}(K_n)$ for all $n\in{\mathbb{N}}$.

Example: Let $L$ be an operator system. The generalized state space of $L$ is the matrix convex set $\mathrm{CS}(L)$ in the dual $L^*$, which consists of the matrix states

$$\mathrm{CS}(L)_n =\{\psi\;\vert\;\psi : L\rightarrow M_n$$    completely positive and unital$$\}.$$

The generalized state space is weak*-compact. It follows from [CE77, Lemma 2.2] that the structural elements of $\mathrm{CS}(L)_n$ are exactly those completely positive and unital mappings which are pure. To every compact and matrix convex set $K$ there is an operator system which has $K$ as its generalized state space [WW99, Prop. 3.5].

Let $V$ be a locally convex space and induce the product topology on $M_n(V)$. The matrix convex Krein-Milman Theorem is: Let $K$ be a compact matrix convex set in $V$. Then $K$ is equal to the closed matrix convex hull of the structural elements of $K$. If $V$ has finite dimension, then $K$ is the matrix convex hull of its structural elements.

The converse result is: Let $K$ be a compact matrix convex set in $V$. Let $S$ be a closed set of matrices, such that $S_n\subset K_n$ and $v^*S_lv \subset S_m$ for all partial isometries $v\in M_{lm}$ and for all $n,m,l\in{\mathbb{N}}$, $l\geq m$. If the closed matrix convex hull of $S$ equals $K$, then all structural elements of $K$ are in $S$.([WW99], [Fis96]).

It is possible to sharpen these results for more special matrix convex sets. A matrix convex set $K$ is called simple, if there are $n\in{\mathbb{N}}$ and $A\subset M_n(V)$, such that $K$ is equal to the matrix convex hull of $A$. $K$ is a simple matrix convex set, if and only if there is $n\in{\mathbb{N}}$ such that $\mathrm{str}(K_m)=\emptyset$ for all $m>n$.

Suppose that $K$ is a matrix convex set in $V$. Then $x\in K_m$ is a matrix extreme point, if $x\in\mathrm{str}(K_m)$ and

$$x\notin \cup_{m<l} \mathrm{1\!\!\!\:l}_{lm}^* \mathrm{str}(K_l) \mathrm{1\!\!\!\:l}_{lm}.$$

Let $\mathrm{mext}(K)$ be the set of matrices consisting of all matrix extreme points of $K$.

Suppose that $K$ is a simple compact matrix convex set in $V$. Then $K$ is equal to the closed matrix convex hull of $\mathrm{mext}(K)$. If $V$ has finite dimension, then the closure is not needed, that means $K$ is the matrix convex hull of $\mathrm{mext}(K)$. In this case the following result also holds: Let $S$ be a set of matrices over $V$ not containing reducible elements such that the matrix convex hull of $S$ equals $K$, then ${\mathrm{mext}}(K)_m\subset U(S_m)$ for all $m\in{\mathbb{N}}$ ([Mor94], [Fis96]).

If $K$ is compact and not simple, $\mathrm{mext}(K)$ may be empty. As an example take the generalized state space $\mathrm{CS}(A)$ of a $C^*$-algebra $A$. Its matrix extreme points are exactly the irreducible finite dimensional representations of $A$. These need not exist in general.

Footnotes

... element⁶⁶

The term structural element is defined by Morenz [Mor94]. Another equivalent definition is given in [WW99]. There the structural elements are called matrix extreme points.