Local principles

  • Steffen Roch
  • Pedro A. Santos
  • Bernd Silbermann
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 
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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

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