On Gelfand-Naimark type Theorems for unital abelian complex and real locally C*-, and locally JB-algebras

摘要

A type Λ for the unital locally C*-algebra B is introduced, and the notions of a local * -homomorphisms between two algebras of the same type A is studied. The structure of the spectrum M(B) is discussed and it is shown how to topologize M(B) to turn it to a certain compactly-generated topological space with an extra filtration property, so that B is locally *-isometric to the functional algebra C(M(B)). After that the structure of M(B) is used to obtain versions of Arens-Kaplansky type theorem for real unital abelian locally C* -algebras and Gelfand-Naimark type theorem for unital abelian locally JB-algebras.