TITLE: Commutative unital Banach algebra with nilpotent elements
QUESTION [6 upvotes]: What would be a concrete example of a commutative unital Banach algebra that contains nilpotent elements?

REPLY [3 votes]: Upper triangular Toeplitz matrices give another example, i.e., the unital subalgebra of the $n$-by-$n$ matrices generated by the matrix with $1$s on the superdiagonal and $0$s elsewhere.
More generally, take any (nonzero) nilpotent element $a$ of a unital Banach algebra $B$, then take the unital Banach subalgebra of $B$ generated by $a$.