Consider some compact set $K$ and the space of continuos functions $C(K,\mathbb{R})$.
We obviously can define an algebra on this space by defining multiplication as $fg(i) = f(i) g(i)$
A band $B$ is defined as a semigroup where every element is idempotent, this means: $x,y \in B \implies xy \in B, x \in B \implies x^2 = x$
But then the bands in $C(K,\mathbb{R})$ can only contain functions in $C(K,\{0,1\})$, or am I missing something here?
I am asking this question because apparently 'band-irreducible-operators', which I understand as operators which for every band have $T(B)$ is not contained in $B$, (but a definition is hard to find in google) are an important tool to study extensions of the Perron Frobenius theorem