Abstract:
We consider uniformly closed symmetric algebras of bounded complex-valued functions defined on compact spaces (including ordered ones) and exhibiting discontinuities of various types. For this class of algebras, explicit descriptions of their maximal ideal spaces are provided, and their topological properties are studied. It is demonstrated how these properties influence the algebraic and topological structure of the algebras under consideration.
It is shown that even for algebras consisting of functions with the simplest types of discontinuities, the maximal ideal spaces turn out to be compact spaces with exotic properties. In particular, one such space is the well-known "Split interval" (double arrow space) introduced by Alexandroff and Urysohn — a classical example of a compact space with an intricate structure. Furthermore, for one type of algebra, the maximal ideal space is shown to be homeomorphic to the Alexandroff duplicate of the initial domain of definition.
In all considered examples, the maximal ideal space is a fibration over the initial domain, where the structure of each fiber reflects the type of discontinuities of the functions contained in the algebra. A theorem is proposed regarding the embedding of the maximal ideal space of a commutative Banach algebra into the pullback (fibered product) of the maximal ideal spaces of the subalgebras generating the given algebra. Sufficient conditions for the coincidence of the maximal ideal space with the indicated pullback are established for the case of algebras of discontinuous functions.
