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
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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.