Semisimple Banach algebra
= Semisimple Banach algebra
{title2=$J(A)=\{0\}$}
A <Banach algebra> is semisimple when its <Jacobson radical> is zero. This does not assert that every <quasinilpotent element> is zero. For example, the algebra of bounded operators on a <Banach space> of dimension greater than one is semisimple, but contains nonzero square-zero rank-one operators. This notion also does not require the finite-dimensional direct-sum decomposition associated with a <semisimple algebra>.