Spectra for solvable Lie algebras of bundle endomorphisms

Abstract.

The aim of the paper is to investigate spectral properties of the Lie algebras corresponding to the symmetry groups of certain flags of vector bundles over a compact space. Under natural hypotheses, such Lie algebras are solvable, being in general infinite dimensional. The spectral theory of finite-dimensional solvable Lie algebras of operators is extended to this natural class of infinite-dimensional solvable Lie algebras. The discussion uses the language of continuous fields of $C^*$-algebras. The flag manifolds in $C^*$-algebraic framework are naturally involved here, they providing the basic method for obtaining flags of vector bundles.