A Banach algebra A such that Rad(A) is a proper subset of the set {x; r(x) = 0} of all quasi-nilpotent elements.

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1.Suppose that H be a Hilbert space with dimH ≥ 2. Let x, y ∈ H − {0} and < x, y >= 0. The norm of rank one operator (x\overline{⊗}y)(z) =< z, y > x is ∥ x ∥∥ y ∥≠ 0. So x\overline{⊗}y ≠ 0. Also (x\overline{⊗}y)^{2}(z) = (x\overline{⊗}y)(< z, y > x) =< z, y >< x, y > x = 0 so (x\overline{⊗}y)^{2} = 0. Hence it is quasi-nilpotent. But B(H) is semi-simple. Therefore x\overline{⊗}y ∉ Rad(B(H)) = \{0\}.

2.Let A = M_{2}(C) ≃ B(C^{2}) . A is a C^{∗}-algebra so Rad(A) = {0}. The element ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) has the spectrum \{0\} and so r(( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} )) = 0. Hence Rad(A) is not equal to {x; r(x) = 0}.