Suppose
is
-convex. Then
is a
-extreme point, if whenever
is a
-convex combination of
with invertible
, then there are unitaries
such that
for
.
Suppose now and let
be compact and
-convex. Let
be the matrix convex hull of
. Then
is a simple compact and matrix convex set in
, such that
([Fis96]). Thus it is possible to conceive a
-convex subset of
as a matrix convex set in
. Now the matrix convex Krein-Milman theorem can be used. Moreover, it follows from the work of Farenick and Morenz that the structural elements of
are exactly the not reducible
-extreme points of
. So following theorem holds: Let
be compact and
-convex, then
is equal to the
-convex hull of its
-extreme points.
In order to get a somewhat more general result, the definition of the extreme points can be changed.
Suppose that is a hyperfinite factor and that
is
-convex. Then
is a
-extreme point, if whenever
is a
-convex combination of
such that all
are positive and invertible, then it follows that
and
forr
.67
With this definition following theorem hold: Let
be
-convex and weak* compact. Then
is equal to the weak* closure of the
-convex hull of its
-extreme points.