Invertible element and general linear group of a Banach algebra

Let $A$ be a unital complex Banach algebra (Unital Banach algebra). An element $a\in A$ is invertible when there is $b\in A$ with

$ab=ba=1.$

Such an element $b$ is then unique: if $b$ and $b^{\mathrm{\prime }}$ both satisfy the two equations, then

$b=b1=b(a{b}^{\mathrm{\prime }})=(ba)b^{\mathrm{\prime }}=1b^{\mathrm{\prime }}=b^{\mathrm{\prime }},$

using associativity and the unit law. The unique $b$ is called the inverse of $a$ and is written $a^{-1}$. The set of all invertible elements of $A$ is denoted

$A^{\times }:=\{a\in A:a\text{ is invertible}\}$

and is called the general linear group of $A$. It is a group under multiplication:

• $1\in A^{\times }$ with $1^{-1}=1$;
• if $a,b\in A^{\times }$ then $ab\in A^{\times }$ with $(ab{)}^{-1}=b^{-1}{a}^{-1}$, since $(ab)(b^{-1}{a}^{-1})=a(b{b}^{-1})a^{-1}=a{a}^{-1}=1$ and symmetrically;
• $(a^{-1}{)}^{-1}=a$ by symmetry of the defining equations.

The map $a\mapsto a^{-1}$ is the inversion map of $A^{\times }$.

• Two-sided inverses are required, and one-sided inverses do not suffice. If $ab=1$ and $ba\mathrm{\ne }1$, then $a$ is not invertible by definition, and this situation really occurs in a unital complex Banach algebra: on ${\mathrm{\ell }}^2({\mathbb{N}}_0)$ the right shift $T{e}_n=e_{n-1}$ for $n\ge 1$, $T{e}_0=0$, and the left shift $S{e}_n=e_{n+1}$ satisfy $TS=1$ while $ST=1-P$, where $P$ is the orthogonal projection onto $\mathbb{C}{e}_0$, so $T$ has a right inverse and is not invertible (ex-spectrum-of-the-unilateral-shift). Thus $ab=1$ alone does not force $ba=1$ in a general unital Banach algebra; both inverses are always verified explicitly below.

• The group need not be dense or connected, but it is open. In a Banach algebra $A^{\times }$ is an open subset of $A$ and inversion is continuous there; this is Invertible group is open and inversion is continuous. Openness is what makes the resolvent set of an element open and hence makes the spectrum closed Spectrum and resolvent set in a Banach algebra.

• Nonunital algebras are not covered here. In a Banach algebra without a unit there is no element $1$ to compare with, so invertibility is not defined by this definition. The companion examples introduce the unitization $A\oplus \mathbb{C}1$ and declare that spectra of elements of a nonunital algebra are always computed in that named unitization (ex-unitization-of-a-nonunital-banach-algebra).

• Reading order. The example items named by ID above are homed on later pages of the plan, so they are named rather than hyperlinked: a body link to later material must be declared as a forward reference, and Step-5b closure removes every such declaration. Rehoming those items to an earlier page (an owner-only reading-order change) would make the citations backward and restore the links.