Unbounded Disjointness Preserving Linear Functionals

Abstract

Abstract. Let X be a locally compact Hausdorff space and C 0 (X) the Banach space of continuous functions on X vanishing at infinity. In this paper, we shall study unbounded disjointness preserving linear functionals on C 0 (X). They arise from prime ideals of C 0 (X), and we translate it into the cozero set ideal setting. In particular, every unbounded disjointness preserving linear functional of c 0 can be constructed explicitly through an ultrafilter on N complementary to a cozero set ideal. This ultrafilter method can be extended to produce many, but in general not all, such functionals on C 0 (X) for arbitrary X. We also make some remarks where C 0 (X) is replaced by a non-commutative C*-algebra.

Zero product preserving linear maps of CCR C∗-algebras with Hausdorff spectrum

“…For example, abelian C * -algebras C 0 (X) are CCR. In [4], there are many examples of unbounded zero product preserving linear functionals of C 0 (X), provided X is an infinite set. In [12], an unbounded zero product preserving linear map from c onto ∞ is given, where both c, the C * -algebra of convergent scalar sequences, and ∞ are CCR with Hausdorff spectrum.…”

“…As a matter of facts, the structure of unbounded zero product preserving linear functionals of C * -algebras is quite complicated (see [4]). Furthermore, we know that Banach algebra homomorphisms can be unbounded (see, e.g., [7]).…”

Disjointness Preserving and Functional Type Disjointness Preserving Operators

“…There are many articles for construction of unbounded disjointness preserving operators. The paper [1] of Abramovich and Lipecki develops techniques to construct unbounded disjointness preserving linear functionals on any infinite dimensional -lattice, and this is also mentioned in Remark 2.7 of the paper [2]. This paper [2] is devoted to construct unbounded disjointness preserving linear functionals on function spaces 0 ( ), when is a locally compact Hausdorff space.…”

Проблема Каждана - Мильмана Для Полупростых Компактных Групп Ли

“…Родственным темам по-священа статья [118] о строении неограниченных функционалов, сохраняющих дизъюнктность, и статья [119] о функциях и мерах, удовлетворяющих условию аддитивности или мультипликативности на ортогональных элементах, в связи с различными условиями непрерывности и измеримости.…”

“…Другие результаты по теории одномерных квазипредставлений, одномерных псевдопредставлений, квазихарактеров и псевдохарактеров и и некоторые при-ложения этой теории можно найти в [7], [9], [10]- [12], [18]- [24], [28]- [59], [63], [66], [73], [74], [106]- [110], [113], [118], [133], [139], [188]- [191], [193], [196], [197]. § II.…”