Order theory and interpolation in operator algebras

David P. Blecher ¹   ; Charles John Read ²

¹ Department of Mathematics University of Houston Houston, TX 77204-3008, U.S.A.
² Department of Pure Mathematics University of Leeds Leeds LS2 9JT, England

Studia Mathematica, Tome 225 (2014) no. 1, pp. 61-95

Résumé

In earlier papers we have introduced and studied a new notion of positivity in operator algebras, with an eye to extending certain $C^*$-algebraic results and theories to more general algebras. Here we continue to develop this positivity and its associated ordering, proving many foundational facts. We also give many applications, for example to noncommutative topology, noncommutative peak sets, lifting problems, peak interpolation, approximate identities, and to order relations between an operator algebra and the $C^*$-algebra it generates. In much of this it is not necessary that the algebra have an approximate identity. Many of our results apply immediately to function algebras, but we will not take the time to point these out, although most of these applications seem new.

Détail

DOI : 10.4064/sm225-1-4

Keywords: earlier papers have introduced studied notion positivity operator algebras eye extending certain * algebraic results theories general algebras here continue develop positivity its associated ordering proving many foundational facts many applications example noncommutative topology noncommutative peak sets lifting problems peak interpolation approximate identities order relations between operator algebra * algebra generates much necessary algebra have approximate identity many results apply immediately function algebras time point these out although these applications seem