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 ||a2||≤ ||a2+ b2|| for a, b∈ A is sufficient for a commutative real Banach algebra A with a unit to be isomorphic to the space CR(K) of continuous real-valued functions on a compact Hausdorff space 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 A onto CR(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 CR(K) which are (1 + ϵ) -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 A where ||a2|| ≤ k|| a2+ b2|| for all a, b∈ A, for some k>1 , but the inequality fails for k= 1.