A result of F. Albiac and E. Briem [Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 114, No. 4, Paper No. 163, 13 p. (2020; Zbl 1447.46039)] establishes that for a commutative real Banach algebra with unit \(\mathcal{A}\), the inequality \(\|a^2\|\leq \|a^2+b^2\|\) for all \(a,b\in \mathcal{A}\), implies that \(\mathcal{A}\) is isomorphic to the space \(C(K)\) of continuous real-valued functions on a compact space \(K\), and also that the Gelfand transform preserves the norm of squares. The note under review answers negatively the question whether an isomorphism of \(\mathcal{A}\) onto a \(C(K)\) space has to be square norm preserving. This is done by constructing algebra norms on \(C(K)\) spaces, \(K\) of cardinality greater than two, that are \((1+\epsilon)\)-equivalent to the usual sup-norm, with the norm of the identity equal to \(1\) and where the norm of each non-constant square function differs from the corresponding sup-norm one.
The authors consider the case of two-dimensional algebras \(\mathcal{A}\) satisfying the inequality \(\|a^2\|\leq k \|a^2+b^2\|\) for all \(a,b\in \mathcal{A}\) and some \(k\geq 1\). Among other things, they provide examples that, although such an inequality fails for \(k=1,\) it holds for \(k\ge 1+\sqrt{2}\).