A short proof that \(M_ n(A)\) is local if \(A\) is local and Fréchet. (English) Zbl 0804.46054
Summary: We give a short and very general proof of the fact that the property of a dense Fréchet subalgebra of a Banach algebra being local, or closed under the holomorphic functional calculus in the Banach algebra, is preserved by tensoring with the \(n\times n\) matrix algebra of the complex numbers.