An (algebrically) simple Banach algebra.

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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(C^{n}) = K(C^{n}) 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_{ij}I_{αβ} = δ_{jα}I_{iβ}, where δ denotes Kronecker’s δ. Let Δ be a nontrivial twosided ideal in M_{n}(C). There is a nonzero element A = ∑_{i,j=1}^{n}a_{ij}I_{ij} in Δ, hence

a_{rs} ≠ 0 for some 1 ≤ r, s ≤ n. But I_{rs}AI_{sr} = (∑_{j=1}^{n}a_{rj}I_{rj})I_{sr} = a_{rs}I_{rr} ∈ Δ.

Hence I_{ij} = I_{is}I_{sr}I_{rj} ∈ Δ for all 1 ≤ i, j ≤ n. Therefore Δ = M_{n}(C), a contradiction.