= Volterra convolution algebra on a finite interval
{c}
{title2=$(f*g)(t)=\int_0^t f(s)g(t-s)\,ds$}
Truncated <convolution> makes $L^1(0,1)$ a commutative nonunital <Banach algebra>. The <Tonelli theorem> gives $\|f*g\|_1\leq\|f\|_1\|g\|_1$, and the <Fubini theorem> gives associativity over the integration simplex. Its <unitization> is $\mathbb C\oplus L^1(0,1)$ with norm $|\alpha|+\|f\|_1$ and product $(\alpha,f)(\beta,g)=(\alpha\beta,\alpha g+\beta f+f*g)$. The constant function one has convolution powers $t^{n-1}/(n-1)!$, of norm $1/n!$. Thus it is a nonnilpotent <quasinilpotent element> in this unitization.
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