Gelfand theory for real Banach algebras
The authors study the properties of real-valued multiplicative \(\mathbb R\)-linear functionals defined on real Banach algebras. In the literature, these properties have already been studied as subcases of complex-valued multiplicative \(\mathbb C\)-linear functionals, defined on complex Banach algebras. The approach of the present paper is different. Namely, the authors do not use complex Banach algebras, complex-valued funcitonals, etc. This approach makes their research independent of the results known for the complex case and enables them to study a much larger class of Banach algebras and a much larger class of functionals, since every \(\mathbb C\)-linear map is also \(\mathbb R\)-linear and every algebra over \(\mathbb C\) is also an algebra over \(\mathbb R\), but the converse is not true in general. The object of study is a commutative real Banach algebra \(\mathcal A\), which, if unital, has unit element with norm \(1\). For example, the authors show that, if the spectral radius \(r:\mathcal A \to\mathcal A\) satisfies the condition \(r(a^2)\leqslant r(a^2+b^2)\) for all \(a, b\in\mathcal A\), then every complex-valued \(\mathbb R\)-linear algebra homomorphism is, in fact, real-valued.
scientific article; zbMATH DE number 7232902
scientific article