Quantum of Banach algebras

Abstract

A variety of Banach algebras is a non-empty class of Banach algebras, for which there exists a family of laws such that its elements satisfy all of the laws. Each variety has a unique core (see [3]) which is generated by it. Each Banach algebra is not a core but, in this paper, we show that for each Banach algebra there exists a cardinal number (quantum of that Banach algebra) which shows the elevation of that Banach algebra for bearing a core. The class of all cores has interesting properties. Also, in this paper, we shall show that each core of a variety is generated by essential elements and each algebraic law of essential elements permeates to all of the elements of all of the Banach algebras belonging to that variety, which shows the existence of considerable structures in the cores.