Lower bounds for commutative radical Banach algebras

One says a commutative radical Banach algebra A has a lower bound if there is a lower growth condition on $\text{∥x}{}^{\mathrm{n}}\text{∥}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}$ for all nonzero elements x in A. If A is a separable algebra we give necessary and sufficient conditions for A to possess a lower bound.