New associative and nonassociative Gelfand-Naimark theorems (Q1313592)

We prove that, if \(A\) is a complex Banach algebra with a unit \(\text{\textbf{1}}\) and a conjugate-linear vector space involution * such that \(\text{\textbf{1}}^*= \text{\textbf{1}}\) and \(\| a^* a\|= \| a^*\| \| a\|\) for all \(a\) in \(A\), and if \(\dim(A)\geq 3\), then \(A\) is a \(C^*\)-algebra. The two-dimensional case is also considered and described.