Invertible element in a Banach algebra

Let $A$ be a unital Banach algebra. An element $a\in A$ is invertible if there exists $b\in A$ such that

ab=ba=1_A.

The element $b$ is unique and is denoted $a^{-1}$ . The invertible elements form a group $A^\times$ under multiplication. For a nonunital Banach algebra, internal invertibility is unavailable; spectral questions are instead formulated in the unitization $\widetilde A$ . In particular, for $a\in A$ , one tests whether $\lambda 1_{\widetilde A}-a$ is invertible in $\widetilde A$ , rather than silently treating $A$ as unital.

Openness and the Neumann series

The group $A^\times$ is open in $A$ , and inversion is continuous. If $\|1_A-x\|<1$ , then

x^{-1}=\sum_{n=0}^{\infty}(1_A-x)^n.

More generally, if $a$ is invertible and $\|a^{-1}(b-a)\|<1$ , the same series shows that $b$ is invertible. Thus invertibility is stable under sufficiently small norm perturbations.

One-sided inverses

In a general Banach algebra, a left inverse need not be a right inverse. For example, the unilateral shift on a Hilbert space has a left inverse but no right inverse. The definition therefore requires both equations. In a commutative algebra the distinction disappears, but existence still depends on the ambient unital algebra.

Spectrum and resolvent

The spectrum of $a\in A$ is

\sigma_A(a)=\{\lambda\in\mathbb C: \lambda 1_A-a\notin A^\times\},

and its complement is the resolvent set. For nonunital $A$ , the same formula is evaluated in $\widetilde A$ ; consequently $0$ belongs to the spectrum of every $a\in A$ . This convention makes spectra compatible with representations and functional calculus.

References

Gerard J. Murphy, C-Algebras and Operator Theory*, Academic Press, 1990. DOI record. Relevant: §1.2 on invertibility, spectra, and resolvents.
Theodore W. Palmer, Banach Algebras and the General Theory of -Algebras, Volume I*, Cambridge University Press, 1994. Publisher record. Relevant: Chapter 1 on unital Banach algebras and invertible elements.