Publishing house / Banach Center Publications / All volumes

Abstract

Introduction. The aim of this paper is to review some relevant results concerning the geometry of nonassociative normed algebras, without assuming in the first instance that such algebras satisfy any familiar identity, like associativity, commutativity, or Jordan axiom. In the opinion of the author, the most impressive fact in this direction is that most of the celebrated natural geometric conditions that can be required for associative normed algebras, when imposed on a general nonassociative normed algebra, imply that the algebra is actually "nearly associative". We shall explain this idea by selecting four favourite topics, namely: • Nonassociative Vidav-Palmer theorem, • Nonassociative Gelfand-Naimark theorem, • Nonassociative smooth normed algebras, and • One-sided division absolute valued algebras. Although there are classical nice forerunners in this circle of ideas, as for example the Albert-Urbanik-Wright determination of (nonassociative) absolute valued algebras with a unit ([2], [3], [42], and [41]), a systematic treatment of questions of this type has been made only recently, more precisely since 1980 [34].