Uniform Convergence to the Spectral Radius and Related Properties in Banach Algebras (1996)

Supervised by P. G. Dixon

The scope of this report is the structure theory of general Banach algebras. Our point of departure is the spectral radius formula: Gelfand's assertion of the equality of the spectral radius of an element $a$ of a Banach algebra with the limit $$ \lim_{n\rightarrow\infty} \left\| a^n \right\|^{1/n}. $$ We consider Banach algebras for which this convergence satisfies some conditions of uniformity, concentrating on two conditions: when this convergence is uniform on the unit ball (spectral uniformity) and when this uniformity holds merely on the topologically nilpotent elements of the unit ball (topologically bounded index). The latter is a natural topological analogue of a property of interest in the theory of rings.

We consider questions of stability and follow with a detailed investigation of each of these properties. We find a number of simple consequences of spectral uniformity and show that, amongst others, matrix algebras and $\ell_1$ are spectrally uniform. We find that the spectrally uniform von Neumann algebras are exactly those which satisfy some well-known finiteness properties, in particular, injectivity in the sense of Varopoulos.

For topologically bounded index we prove a topological version of a theorem of Jacobson and use this to investigate semigroup algebras and algebras of operators on a Banach space. An algebra of Hadwin et al. is studied in some detail.

In the final chapter we investigate more general relationships between the properties co-inciding with spectral uniformity and topologically bounded index in von Neumann algebras. We study the injectivity of semigroup algebras, construct an injective Banach algebra which does not satisfy a polynomial identity and make some remarks on a question of Le Merdy. We conclude by treating some Banach algebras which can be seen as generalised $Q$-algebras.

Download thesis (pdf, 6 MB).