Automatic continuity of homomorphisms in non-associative Banach algebras (Q2871238)

Consider an arbitrary complex algebra, that is, a complex linear space \(A\) endowed with a bilinear map, called the product of \(A\) and denoted by juxtaposition. The algebra \(A\) is called a Banach algebra if it is endowed with a complete norm \(\|\cdot \|\) satisfying \(\|ab\| \leq \|a\| \|b\|\) for any \(a,b \in A\).NEWLINENEWLINEThe problem of automatic continuity on different classes of Banach algebras has attracted the interest of many authors and there are many papers on this subject. In the paper under review, the authors study the problem of automatic continuity on arbitrary Banach algebras \(A\). By introducing the concept of a rare operator of a normed vector space \(B\) in terms of a spectral property, the concept of rare element of a normed algebra \(B\) is also introduced, as those elements for which the right and left multiplication operators on \(B\) are rare. Then they prove as their main result that any surjective homomorphism from an arbitrary Banach algebra \(A\) to a unital normed algebra \(B\) with a simple completion is continuous if and only if \(B\) does not have rare elements.