A Note on Spectrally Dominant Norms

Throughout this paper A will denote a finite-dimensional associative complex algebra. A vector space norm · on A is said to be stable if there exists a positive constant σ satisfying x ≤ σ x n for every x in A and all natural numbers n. It is well known that stable norms · on A are spectrally dominant (that is, for every x in A, the inequality ρ x ≤ x holds, where ρ · denotes the spectral radius). In the case that A is the algebra of all m×m complex matrices for some m in , a celebrated theorem of Friedland and Zenger [FZ] asserts that, conversely, spectrally dominant norms on A are stable. As a matter of fact, the assertion in the Friedland– Zenger theorem does not remain true for arbitrary A, even if A is commutative and has a unit. An elemental counterexample is the following.