Abstract

The paper deals with quasi linear maps on two by two matrices over Banach and \(C^{\ast}\)-algebras. Let \(\varphi:A\to X\) be a homogeneous map between Banach algebra \(A\) and a linear space \(X\). Let us take its amplification \(\psi=\varphi^{(2)}\) to two by two matrix structure \(M_{2}(A)\) over \(A\). If \(\psi(x+x^{2})=\psi(x)+\psi(x^{2})\) for all \(x\), then \(\varphi\) is linear. Ramifications for self adjoint parts of Banach \(\ast\)-algebras and \(C^{\ast}\)-algebras as well applications to Mackey–Gleason problem are given.

中文翻译：

---

Banach和算子代数上的映射线性

摘要

本文讨论了Banach和\（C ^ {\ ast} \）-代数上两两矩阵上的拟线性映射。令\（\ varphi：A \ to X \）是Banach代数\（A \）与线性空间\（X \）之间的齐次映射。让我们以它的放大\（\ PSI = \ varphi ^ {（2）} \）到两个两个矩阵结构\（M_ {2}（A）\）超过\（A \） 。如果对于所有\（x \）\（\ psi（x + x ^ {2}）= \ psi（x）+ \ psi（x ^ {2}）\），则\（\ varphi \）是线性的。Banach \（\ ast \）-代数和\（C ^ {\ ast} \）的自伴部分的分支-代数以及对Mackey-Gleason问题的应用。