Ideal Spaces of Banach Algebras

Abstract

Proof. We may suppose that Y k is non-empty, and as mentioned after Proposition 5.1, we may assume, replacing A by A=I, where I X k , that A is a semisimple PI-Banach algebra of degree 2k. Let J be the ideal J PrimAnY k . Lemma 5.2 shows that ZA Ç J separates the points of Y k , which implies that there is a bijective correspondence between Y k and the

Spectral synthesis for Banach algebras, II

“…On the other hand it was shown in [10] that for uniform algebras, τ r is Hausdorff if and only if τ ∞ is Hausdorff. In [24] it was shown that if there is a compact Hausdorff topology on a subspace of Id(A), which is related to the quotient norms in a useful way, then that topology necessarily coincides with the restriction of τ r . Thus for uniform algebras without spectral synthesis, such as the disc algebra, there is no useful compact Hausdorff topology on the space of closed ideals.…”

“…Because τ ∞ is seldom Hausdorff, the second author introduced another topology τ r on the set Id(A) of closed two-sided ideals of a Banach algebra A [24]. This topology is always compact, like τ ∞ , and it is Hausdorff whenever τ ∞ is Hausdorff [24; 3.1.1], and often even when τ ∞ is not.…”

Pseudo perfectly continuous functions and closedness/compactness of their function spaces

“…The technique of change of topology of a space is prevalent all through mathematics and is of considerable significance and widely used in topology, functional analysis and several other branches of mathematics. For example, weak and weak * topology of a Banach space, hull kernel topology and the multitude of other topologies on Id(A) the space of all closed two sided ideals of a Banach algebra A ( [4], [5], [56]). Moreover, to taste the flavour of applications of the technique in topology see ( [11], [15], [17], [30], [60]).…”

Non-regularity for Banach function algebras

“…Indeed it seems, in general, to have few useful properties, and it has not played a prominent part in the general theory of Banach algebras. An attempt to find a more useful topology has been made in [14].…”