Multi-positive linear functionals on locally<Emphasis Type="Italic">C</Emphasis>*-algebras

We will show that ann×n matrix of continuous linear functionals on a locallyC*-algebraA, which satisfies the generalized positivity condition induces a continuous *-representation ofA on a Hilbert space. This generalizes the classical GNS-representation. Also, we give a necessary and sufficient condition such that this representation is irreducible, and determine a certain class of extreme points in the set of all continuous completely positive linear maps fromA toM _n (ℂ) that preserve identity.