A normed algebra A whose radical is isomorphic to C.

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Suppose that A = \{( \begin{matrix}a & b \\ 0 & c \end{matrix} ); a, b, c ∈ C\}. Then A is a subalgebra of M_{2}(C) ≃ B(C^{2}) and the only its characters are f(( \begin{matrix}a & b \\ 0 & c \end{matrix} )) = a and

g(( \begin{matrix}a & b \\ 0 & c \end{matrix} ))) = c, since

\{( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} )  , ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} )  , ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} )\}

is a basis for A. Therefore Rad(A) = \{( \begin{matrix}0 & b \\ 0 & 0 \end{matrix} ); b ∈ C\} is isometrically isomorphic to C.