A note on normed near-algebras (Q1824162)

Let \({\mathbb{K}}\) be one of the fields \({\mathbb{R}}\) or \({\mathbb{C}}\). A near-algebra A over \({\mathbb{K}}\) (i.e. a linear space on which a multiplication is defined such that A forms a semigroup under multiplication, multiplication is right distributive with respect to addition and \(\lambda (ab)=(\lambda a)b\) for all \(a,b\in A\) and \(\lambda\in {\mathbb{K}})\) is called by the authors a normed near-algebra if there exists a submultiplicative norm \(\| \cdot \|\) on A such that \(\| ab- ac\| \leq \| a\| \| b-c\|\) for each a,b,c\(\in A\). It is shown that many properties of Banach algebras with unit valid also in the case of Banach near-algebras with unit. For example, the set Inv A of all invertible elements of a and the set Qinv A of all quasi-invertible elements of A are open, the mapping \(a\to a^{-1}\) is a homeomorphism of Inv A onto Inv A, the set of all topological divisors of zero of A is a subset of \(A\setminus Inv A\) and the spectrum \(\sigma\) (a) of \(a\in A\) is a closed bounded set of \({\mathbb{C}}\).