## ABSTRACT

This chapter provides some results about the structure of LPA (as the triviality of division LPA and the representations of LPA by the projective and inductive limits by Banach algebras, locally bounded algebras (LBA, in short) and F-algebras). In 1938, S. Mazur in Mazur stated (the first proof of this result has been published in the book Zelazko) that every normed division algebra over the field R of real numbers is one of the fields R and C or the skew field H of real quaternions. A topological algebra is called a locally idempotent algebra (LIA, in short) if (A, t) has a base of neighborhoods of zero, consisting of idempotent sets.