  We introduce and study locally AW*-algebras (Baer locally C*-algebras) as a
locally multiplicatively-convex generalization of AW*-algebras of Kaplansky.
Among other basic properties of these algebras, it is established that:
{\bullet} A locally C*-algebra is a locally AW*-algebra iff there exists its
Arens-Michael decomposition consisting entirely of AW*-algebras; {\bullet} A
bounded part of a locally AW*-algebra is an AW*-algebra; {\bullet} The Spectral
Theorem for locally AW*-algebras.
