On the norm-preservation of squares in real algebra representation

Abstract One of the main results 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 norm inequality $$\Vert a^{2}\Vert \le \Vert a^{2}+b^{2}\Vert $$ ‖ a 2 ≤ + b $$a,b\in {\mathcal {A}}$$ , ∈ A sufficient a commutative algebra $${\mathcal with unit to be isomorphic space {C}}_{{\mathbb {R}}}({\mathcal {K}})$$ C R ( K ) continuous real-valued functions on compact Hausdorff {K}}$$ . Moreover, this proposition also shown if above condition (which involves all operations algebra) holds, then real-algebra isomorphism given by transform preserves squares. very natural question springing from above-mentioned result whether an onto always norm-preserving This note devoted providing negative answer problem. To end, we construct norms spaces are $$(1+\epsilon )$$ 1 ϵ -equivalent sup-norm and identity function equal 1, where every nonconstant different standard sup-norm. We provide examples two-dimensional normed a^2\Vert k a^2+b^2\Vert k some $$k>1$$ > but fails $$k=1$$ =