Linearity of Maps on Banach and Operator Algebras

Abstract

© 2020, Pleiades Publishing, Ltd. 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.