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Pseudo-linear functionals on Banach algebras. (English) Zbl 0976.46035

Let \({\mathcal A}\) be a Banach algebra over \(K\) (\(=\mathbb{R}\) or \(\mathbb{C}\)). A functional \(F:{\mathcal A}\to K\) is said to be pseudo-linear if it is linear when restricted to any commutative subalgebra of \({\mathcal A}\). It is known that if \(\mathcal A\) has \(M_2(\mathbb{C})\) as a quotient, then there exists a continuous pseudo-linear functional on \(\mathcal A\) which is not linear.
In this paper, the authors show that the algebra of upper triangular \(n\times n\) matrices (for \(n\geq 2\)) and the algebra of upper triangular \(n\times n\) matrices which are constant on the diagonal (for \(n\geq 3\)) also have this property, despite the fact that they do not have \(M_2(\mathbb{C})\) as a quotient.

MSC:

46H99 Topological algebras, normed rings and algebras, Banach algebras
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