Local principles

Part of the Universitext book series (UTX)

Abstract

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.

Keywords

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.

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© Springer-Verlag London Limited 2011

Authors and Affiliations

• Steffen Roch
• 1
Email author
• Pedro A. Santos
• 2
• Bernd Silbermann
• 3 