We study subalgebras of equipped with topologies that generalize both the uniform and the strict topology. In particular, we study the Stone-Weierstrass property and describe the ideal structure of these algebras.
By considering arbitrary mappings from a Banach algebra into the set of all nonempty, compact subsets of the complex plane such that for all , the set lies between the boundary and connected hull of the exponential spectrum of , we create a general framework in which to generalize a number of results involving spectra such as the exponential and singular spectra. In particular, we discover a number of new properties of the boundary spectrum.
In 1971, Grauert and Remmert proved that a commutative, complex, Noetherian Banach algebra is necessarily finite-dimensional. More precisely, they proved that a commutative, complex Banach algebra has finite dimension over ℂ whenever all the closed ideals in the algebra are (algebraically) finitely generated. In 1974, Sinclair and Tullo obtained a non-commutative version of this result. In 1978, Ferreira and Tomassini improved the result of Grauert and Remmert by showing that the statement...
For every closed subset C in the dual space of the Heisenberg group we describe via the Fourier transform the elements of the hull-minimal ideal j(C) of the Schwartz algebra and we show that in general for two closed subsets of the product of and is different from .
This paper may be viewed as having two aims. First, we continue our study of algebras of operators on a Hilbert space which have a contractive approximate identity, this time from a more Banach-algebraic point of view. Namely, we mainly investigate topics concerned with the ideal structure, and hereditary subalgebras (or HSA's, which are in some sense a generalization of ideals). Second, we study properties of operator algebras which are hereditary subalgebras in their bidual, or equivalently which...
The structure of closed ideals of a regular algebra containing the classical A∞ is considered. Several division and approximation results are proved and a characterization of those ideals whose intersection with A∞ is not {0} is obtained. A complete description of the ideals with countable hull is given, with applications to synthesis of hyperfunctions.
On caractérise, à l’aide de la notion algébrique d’idéal réel, les idéaux fermés de type fini de l’anneau des fonctions différentiables sur ayant la propriété des zéros, et les idéaux fermés principaux de ayant la propriété des zéros.
It is proved that a linear surjection , which preserves noninvertibility between semisimple, unital, complex Banach algebras, is automatically injective.
We characterize elements in a semisimple Banach algebra which are quasinilpotent equivalent to maximal finite rank elements.
We give a spectral characterisation of rank one elements and of the socle of a semisimple Banach algebra.
Without the "scarcity lemma", two kinds of "rank one elements" are identified in semisimple Banach algebras.