Let be a vector space. A matrix convex set of matrices over
is called matrix convex set in
for short. Let
be a set of matrices over
. The matrix convex hull of
is the smallest matrix convex set in
containing
. Its closure is the smallest closed matrix convex set containing
because the closure of matrix convex sets is matrix convex.
Two elements
are unitarily equivalent, if there is a unitary
such that
. Let
be the set of all elements, that are unitarily equivalent to elements of
.
is called reducible, if it is unitarily equivalent to some block matrix
. A matrix convex combination
is called proper, if all
are square matrices different from 0.
Let be a matrix convex set in
. Then
is a structural element66 of
, if whenever
is a proper matrix convex combination of
, then every
is unitarily equivalent to
.
The set of all structural elements of
is denoted by
. The set of structural elements of
is the set of matrices over
consisting of
for all
.
Example:
Let be an operator system. The generalized state space of
is the matrix convex set
in the dual
, which consists of the matrix states
Let be a locally convex space and induce the product topology on
. The matrix convex Krein-Milman Theorem is:
Let
be a compact matrix convex set in
. Then
is equal to the closed matrix convex hull of the structural elements of
. If
has finite dimension, then
is the matrix convex hull of its structural elements.
The converse result is:
Let be a compact matrix convex set in
. Let
be a closed set of matrices, such that
and
for all partial isometries
and for all
,
. If the closed matrix convex hull of
equals
, then all structural elements of
are in
.([WW99], [Fis96]).
It is possible to sharpen these results for more special matrix convex sets. A matrix convex set is called simple, if there are
and
, such that
is equal to the matrix convex hull of
.
is a simple matrix convex set, if and only if there is
such that
for all
.
Suppose that is a matrix convex set in
. Then
is a matrix extreme point, if
and
Suppose that is a simple compact matrix convex set in
. Then
is equal to the closed matrix convex hull of
. If
has finite dimension, then the closure is not needed, that means
is the matrix convex hull of
. In this case the following result also holds:
Let
be a set of matrices over
not containing reducible elements such that the matrix convex hull of
equals
, then
for all
([Mor94], [Fis96]).
If is compact and not simple,
may be empty. As an example take the generalized state space
of a
-algebra
. Its matrix extreme points are exactly the irreducible finite dimensional representations of
. These need not exist in general.