George F. Nassopoulos
Spectral Decomposition and Duality in Commutative Locally C*- Algebras, Contemporary Mathematics Vol. 427, 2007

In its natural topology, the spectrum-space of a commutative, unital, locally C^*-algebra A is shown to be a compactly generated Tychonoff space, therein admitting a dual decomposition to the Arens-Michael one for the algebras. Due to the multiplicity of the compatible topologies with a given algebra A, this space alone is sufficient to determine A at a purely algebraic level. Therefore, the compound structure of the spectrum - having no non-trivial counterpart in the usual norm case - is further investigated. Filtered spaces are then introduced as an abstraction of it, and a criterion for the completeness of the function algebras involved is obtained simultaneously characterizing filtered spaces. Finally, a self-contained presentation, in functorial form, of the duality between the algebras in question and the spaces considered is given, so extending the classical Gel'fand duality.

[2000 MSC: 46M15, 46J40, 18B30.]