Let be a unital
-algebra.
A subset
is a
-convex set, if
A subset
is a
-absolutely convex set, if
In particular -convex sets are convex and
-absolutely convex sets are absolutely convex.
Example: Let be a Hilbert space and
. The n-th matrix range of
is the set
The sets
Loebl and Paulsen introduced -convex sets in [LP81]. At the beginning of the nineties Farenick and Morenz studied
-convex subsets of
([Far92],[FM93]). Eventually Morenz succeeded in proving an analogue of the Krein-Milman theorem for a compact
-convex subset of
([Mor94]). At the end of the nineties Magajna generalized the notion of
-convex sets to the setting of operator modules and proved some separation theorems [Mag00, Th. 1.1] and also an analogue of the Krein-Milman theorem [Mag98, Th. 1.1].