The Jacobson radical of a non-associative algebra and the uniqueness of the complete norm topology

Abstract

In 1967, Barry E. Johnson proved the uniqueness of the complete norm topology for semisimple Banach algebras as well as the automatic continuity of homomorphisms from a Banach algebra onto a semisimple Banach algebra. In this paper, we show that the associativity of the product is superfluous in these results.

The multiplicative spectrum and the uniqueness of the complete norm topology

“…Particularly, whenever A is associative, we obtain a well known theorem of B. E. Johnson [14] (see also [5,6,20]) that in words of T. Palmer is a "cornerstone of the Banach algebra theory". Moreover, Theorem 3.5 and Corollary 3.6 improve the main results in [17]. Also, because determining the set of elements having m-spectral radius equal to zero is not difficult (in the level of abstraction that we are dealing with) these results can be easily applied, as the Example 3.7 shows.…”

“…Moreover, semisimple algebras were defined as those algebras A such that Rad(A) = {0}. As a main result [17,Theorem 20], it was proved that a semisimple complete normed algebras has a unique complete norm topology. Therefore, associativity is a superfluous assumption in the aforementioned classical result of Johnson [14].…”

“…By the usual convention, Rad(B) = B if there are no primitive ideals of B. In [17], the Jacobson radical Rad(A) of an algebra A (possibly non associative) was defined as the intersection of all primitive ideals of A. There, the primitive ideals of A were defined as the largest ideal contained in a one-sided maximal modular ideal of A.…”

“…Our next goal is to prove that the class of m-semisimple algebras is wider than that of semisimple algebras but, nevertheless, m-semisimple complete normed algebras enjoy the property of the uniqueness of the complete norm topology (which improves [17,Theorem 20]). …”

“…As said before, a non-associative spectral theory for general algebras is still in a preliminary state (see [27] and [17] for an approach). The main obstacle to be faced seems to be that, for these algebras, there is no notion of invertibility with so many good properties such as those in the associative setting.…”