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Normal cones and $$C^{*}$$-$$m$$-convex structure. (English) Zbl 1090.46041
The authors consider unital, symmetric and complete $$m$$-convex algebras which are symmetrically spectrally bounded. It is shown that if the cone of positive elements is normal, then Pták’s function is a $$C^*$$-norm: it is stronger than the initial topology and complete in the commutative case. Conditions under which the two topologies coincide are presented. In the further part of the paper, the condition that the algebras are symmetrically spectrally bounded is dropped, but it is assumed that the algebras are commutative. The normal cone for a family of seminorms is defined and it is shown that under these conditions the normal cone characterizes the cone of positive elements of a $$C^*$$-$$m$$-convex algebra.
##### MSC:
 46K05 General theory of topological algebras with involution 46L05 General theory of $$C^*$$-algebras 46L99 Selfadjoint operator algebras ($$C^*$$-algebras, von Neumann ($$W^*$$-) algebras, etc.)
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