Local principles

Part of the Universitext book series (UTX)


Before we start our walk through the world of local principles, it is useful to give a general idea of what a local principle should be. A local principle will allow us to study invertibility properties of an element of an algebra by studying the invertibility properties of a (possibly large) family of (hopefully) simpler objects. These simpler objects will usually occur as homomorphic images of the given element. To make this more precise, consider a unital algebra \(\mathcal {A}\) and a family \(\mathcal{W} = (\mathsf {W}_{t})_{t \in T}\) of unital homomorphisms \(\mathsf {W}_{t} : \mathcal {A}\to \mathcal {B}_{t}\) from \(\mathcal {A}\) into certain unital algebras \(\mathcal {B}_{t}\). We say that \(\mathcal{W}\) forms a sufficient family of homomorphisms for\(\mathcal {A}\) if the following implication holds for every element \(a \in \mathcal {A}\):
$$\mathsf {W}_t(a) \; \mbox{is invertible in} \; \mathcal {B}_t \; \mbox{for every } t \in T \quad \Longrightarrow \quad a \; \mbox{is invertible in} \; \mathcal {A}$$
(the reverse implication is satisfied trivially). Equivalently, the family \(\mathcal{W}\) is sufficient if and only if
$$\sigma_{\mathcal{A}} (a) \subseteq \cup_{t \in T} \sigma_{\mathcal {B}_t} \qquad \mbox{for all} \; a \in \mathcal{A}$$
(again with the reverse inclusion holding trivially). In case the family \(\mathcal{W}\) is a singleton, {W} say, then \(\mathcal{W}\) is sufficient if and only if W is a symbol mapping in the sense of Section 1.2.1.


Banach Algebra Compact Hausdorff Space Local Algebra Maximal Ideal Space Commutative Banach Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  • Steffen Roch
    • 1
  • Pedro A. Santos
    • 2
  • Bernd Silbermann
    • 3
  1. 1.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Instituto Superior Técnico, Departamento de MatemáticaUniversidade Técnica de LisboaLisboaPortugal
  3. 3.Fakultät für MathematikTechnische Universität ChemnitzChemnitzGermany

Personalised recommendations