Nilpotent elements in Banach algebras

Abstract:

Let $\mathfrak {A}$ be an ${A^\ast }$-algebra such that any maximal abelian $^\ast$-subalgebra is regular and such that any quasinilpotent element x in $\mathfrak {A}$ satisfies ${x^N} = 0$, with $N < \infty$. Then any irreducible Hilbert space $^\ast$-representation is at most N-dimensional. If $\mathfrak {A}$ is a ${C^\ast }$-algebra, $\mathfrak {A}$ possesses transcendental quasinilpotent elements if there exists a $\pi \in \hat {\mathfrak {A}}$ with $\dim \pi = \infty$.