nLab
idempotent

Definition

An endomorphism $e:B\to B$ in a category is an idempotent if the composition with itself equals itself

$$e\circ e=e.$$e \circ e = e \,.

A splitting of an idempotent $e$ consists of morphisms $s:A\to B$ and $r:B\to A$ such that $r\circ s=1_A$ and $s\circ r=e$. In this case $A$ is a retract of $B$, and we call $e$ a split idempotent.

Of course, we can simply consider the idempotent elements of any monoid.

The algebra of idempotents

Given an abelian monoid $R$, the idempotent elements form a submonoid? $\mathrm{Idem}(R)$.

Given a commutative ring $R$, the idempotent elements of $R$ form a Boolean algebra $\mathrm{Idem}(R)$ with these operations:

• $\top ≔1$,
• $P\wedge Q≔PQ$,
• $\perp ≔0$,
• $P\vee Q≔P-PQ+Q$,
• $\neg P≔1-P$.

This is important in measure theory; if $R$ is the ring $L^{\mathrm{\infty }}(X,ℳ,𝒩)$ of essentially bounded? real-valued measurable functions on some measurable space $(X,ℳ)$ modulo an ideal $𝒩$ of null sets, then $\mathrm{Idem}(R)$ is the Boolean algebra of characteristic functions of measurable sets modulo null sets, which is isomorphic to the Boolean algebra $ℳ/𝒩$ of measurable sets modulo null sets itself.

If $R$ is a commutative $*$-ring?, then we may restrict to the self-adjoint idempotent elements to get the Boolean algebra $\mathrm{Proj}(R)$. In measure theory, if $R$ is the complex-valued version of $L^{\mathrm{\infty }}(X,ℳ,𝒩)$, then $\mathrm{Proj}(R)$ will still reconstruct $ℳ/𝒩$. In operator algebra theory, the self-adjoint idempotent elements of an operator algebra are called projection operator?s, which the origin of the notation $\mathrm{Proj}$. (Sometimes one requires projection operators to be proper: to have norm $1$; the only projection operator that is not proper is $0$.)

The projection operators of a commutative $W^{\star }$-algebra give the link between operator algebra theory and measure theory; in fact, the categories of commutative $W^{\star }$-algebras and of localisable measurable spaces (or measurable locales in constructive mathematics) are dual, and $W^{\star }$-algebra theory in general may be thought of as noncommutative measure theory. In noncommutative measure theory, the projection operators are still important, but they no longer form a Boolean algebra.