In the case commutative, consider the familiar Banach algebra C.
In the non-commutative case, consider the algebra M_n(C) of all n ×n matrices with entries in C. Identifying M_n(C) with B(Cⁿ) = K(Cⁿ) we may regard M_n(C) as a noncommutative C^*-algebra.
Suppose that I_ij is the matrix with the ij-entry 1 and 0 elswhere. Then I_ijI_ab = d_jaI_ib, where d denotes Kronecker's d. Let D be a nontrivial two-sided ideal in M_n(C). There is a nonzero element A = å_{i,j = 1}ⁿa_ijI_ij in D, hence a_rs ¹ 0 for some 1 £ r,s £ n. But I_rsAI_sr = (å_{j = 1}ⁿa_rjI_rj)I_sr = a_rsI_rr Î D. Hence I_ij = I_isI_srI_rj Î D for all 1 £ i,j £ n. Therefore D = M_n(C), a contradiction.