We introduce and study locally weakly

-algebras (locally weakly Rickart

-algebras) and locally

-algebras (locally Rickart

-algebras) as a locally multiplicatively-convex generalization of weakly Rickart

-algebras and Rickart

-algebras, respectively. Among other basic properties of these algebras, it is established:

� That a locally -algebra is a locally -algebra iff there exists its Arens-Michael decomposition consisting entirely of -algebras;

� That a bounded part of a locally -algebra is a -algebra;

� The Spectral Theorem and The Polar Decomposition Theorem for �locally -algebras.

weakly Rickart

-algebras, Rickart

-algebras, locally weakly Rickart

-algebras, locally Rickart

-algebras, Hausdorff projective limit,

-algebras.