An element x of a Banach algebra such r(x) <∥ x ∥.

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Consider x = [ \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ] in the C^{∗}-algebra M_{2}(C) ≃ B(C^{2}). Then sp(x) = {0}. So r(x) = 0. But ∥ x ∥= 1 (since its associated operator T (z_{1}, z_{2}) = (z_{2}, 0) has norm 1).