Local principles

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.