On multiplicatively closed subsets of normed algebras
It is well-known that, if \(S\) is a bounded and multiplicatively closed subset of an associative normed algebra (\(A,| \cdot |\)), then there exists an equivalent algebra norm \(\|\cdot \|\) on \(A\) such that \(\|s\|\leqslant 1\) for every \(s\in S\). It is not very hard to find examples of nonassociative normed algebras \(A\) where such a result fails. In this paper, the authors prove that a similar result remains true in the nonassociative case whenever the subset \(S\) is assumed to be contained in the nucleus of \(A\). It is also shown that, in the particular case that the subset \(S\) reduces to a nonzero nuclear idempotent \(p\), the equivalent algebra norm \(|| \cdot ||\) above can be chosen so that \(p\) becomes a strongly exposed point of the closed unit ball of (\(A,|| \cdot ||\)). Normed (not necessarily associative) algebras \(A\) satisfying the ``norm-one boundedness property'' (i.e., for every bounded and multiplicatively closed subset \(S\) of \(A\), there exists an equivalent algebra norm \(|| \cdot ||\) on \(A\) such that \(|| s || \leqslant 1\) for every \(s\in S\)) are especially studied. It is shown that absolute-valued algebras, \(J\)B-algebras, and nilpotent normed algebras fulfil the NBP. In the case when an anti-commutative complete normed algebraic algebra \(A\) satisfies the NBP, there exists \(n \in\mathbb N\) such that \(L^n_a = 0\) for every \(a\in A\), where \(L_a\) denotes the operator of left multiplication by \(a\). It follows from a celebrated theorem of E.\,I.\thinspace Zel'manov on the so-called Engel Lie algebras that a complete normed algebraic Lie algebra satisfies the NBP if and only if it is nilpotent.
scientific article; zbMATH DE number 5696921
scientific article