Two dimensional commutative Banach algebras and von Neumann inequality (Q5957199)

John von Neumann's inequality states that \( ||f(T) ||\leq 1 \) for any bounded operator \(T\) on a Hilbert space \(H \) of norm \(\leq 1 \) and any polynomial \( f(z) \) satisfying \(|f(z)|\leq 1,\) \(|z|\leq 1 , z \in {\mathbb C}.\) The authors discuss the so-called \(n\)-von Neuman inequality for a commutative Banach algebra \( {\mathcal B} \) with identity: \(T_1 ,\dots ,T_n \in {\mathcal B} \) with \(||T_k ||\leq 1 \) for \( k=1,\dots ,n \) imply that \( ||f(T_1 ,\dots ,T_n)||\leq 1 \) for any polnomial \( f(z_1 , \dots ,z_n) \) satisfying \( |f(z_1 ,\dots ,z_n) |\leq 1 ,\) \((|z_k |\leq 1 \) for \(k=1,\dots ,n)\). NEWLINENEWLINENEWLINEThe main result states: if \({\mathcal B}\) is a commutative Banach algebra with identity of dimension two which satisfies the von Neumann inequality, then \({\mathcal B}\) satisfies the \(n\)-von Neumann inequality for all natural numbers \(n, \) and \({\mathcal B}\) is isometric to a subalgebra of the algebra of all bounded linear operators on a Hilbert space.