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,‖⋅‖), then there exists an equivalent algebra norm |||⋅||| on A such that |||s|||⩽1 for every s∈S. Although associativity is not an essential requirement in this result, it is easy to find examples of nonassociative normed algebras A where such a result fails. Actually, it can fail even if the subset S is reduced to a nonzero idempotent. We prove that it remain true in the nonassociative setting whenever the subset S is assumed to be contained in the nucleus of A. In the particular case that the subset S reduces to a nonzero nuclear idempotent p, we show that the equivalent algebra norm |||⋅||| above can be chosen so that p becomes a strongly exposed point of the closed unit ball of (A,|||⋅|||). We study those (possibly nonassociative) normed algebras A satisfying the “norm-one boundedness property” (in short, NBP), which means that, as happened in the associative case, for every bounded and multiplicatively closed subset S of A, there exists an equivalent algebra norm |||⋅||| on A such that |||s|||⩽1 for every s∈S. We show that absolute-valued algebras, JB-algebras, and nilpotent normed algebras fulfil the NBP. We also show that, if an anti-commutative complete normed algebraic algebra A satisfies the NBP, then there exists n∈N such that for every a∈A, where La denotes the operator of left multiplication by a. It follows from a celebrated theorem of E.I. 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.