On the norm-preservation of squares in real algebra representation

Abstract

One of the main results of the article Gelfand theory for real Banach algebras, recently published in [Rev R Acad Cienc Exactas Fís Nat Ser A Mat RACSAM 114(4):163, 2020] is Proposition 4.1, which establishes that the norm inequality Vert a^{2}Vert le Vert a^{2}+b^{2}Vert for a,bin {mathcal {A}} is sufficient for a commutative real Banach algebra {mathcal {A}} with a unit to be isomorphic to the space {mathcal {C}}_{{mathbb {R}}}({mathcal {K}}) of continuous real-valued functions on a compact Hausdorff space {mathcal {K}}. Moreover, in this proposition is also shown that if the above condition (which involves all the operations of the algebra) holds, then the real-algebra isomorphism given by the Gelfand transform preserves the norm of squares. A very natural question springing from the above-mentioned result is whether an isomorphism of {mathcal {A}} onto {mathcal {C}}_{{mathbb {R}}}({mathcal {K}}) is always norm-preserving of squares. This note is devoted to providing a negative answer to this problem. To that end, we construct algebra norms on spaces {mathcal {C}}_{{mathbb {R}}}({mathcal {K}}) which are (1+epsilon )-equivalent to the sup-norm and with the norm of the identity function equal to 1, where the norm of every nonconstant function is different from the standard sup-norm. We also provide examples of two-dimensional normed real algebras {mathcal {A}} where Vert a^2Vert le k Vert a^2+b^2Vert for all a,bin {mathcal {A}}, for some k>1, but the inequality fails for k=1.