An Intrinsic Characterization of Pro-C*-Algebras and Its Applications

A complete locally rn-convex *algebra (Imc~*algebra) A is a pro-C*~algebra iff A contains a dense C*~suba1gebra continuously embedded in A iff A is hermitian and every continuous hermitian linear functional on A is a difference of two continuous positive linear functionals, A complete lmc algebra A is a pro-C*-algebra under some involution iff the dual of A is a direct sum of hermitian functionals (defined with respect to numerical ranges). A metrizable pro-C*-algebra is nuclear (as a locally convex space) if A is a countable inverse limit of finite dimensional C*-algebras. Additionally if A is abelian, it is homeomorphically *isomorphic to the algebra of all scalar sequences with pointwise convergence. A *representation π of a pro-C*-algebra into unbounded Hilbert space operators maps hermitian elements to essentially selfadjoint operators; and if it is topologically irreducible, then it is algebraically irreducible, mapping A into the C*-algebra of bounded operators.