An algebrically semisimple non-commutative Banach algebra.

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We show that B(X), the algebra of bounded linear mappings from normed space X into X is semi-simple: Suppose that x_{0} ≠ 0 is fixed in X. Then I_{x_{0}} = {T ∈ B(X); T x_{0} = 0} is a left ideal in B(X). We shall show that it is maximal. Let J be a left ideal properly containing I_{x_{0}}. Then Jx_{0} = {T x_{0}; T ∈ J} is a nonzero linear subspace of X which is invariant under each S ∈ B(X). If Jx_{0} ≠ X, then there exists a nonzero y ∈ Jx_{0} and an element z ∈ X such that z ∉ Jx_{0}. If S ∈ B(X) such that Sy = z, then z ∈ Jx_{0} for Jx_{0} is invariant under all elements of B(X). Thus Jx_{0} = X. So that there exists U ∈ J such that Ux_{0} = x_{0}. For each T ∈ B(X), T U − UT ∈ I_{x_{0}}. Hence T ∈ J + I_{x_{0}} ⊆ J. Therefore B(X) = J. Thus Rad(B(X)) ⊆ ∩_{0≠x∈X}I_{x} = {0}. Therefore B(X) is algebrically semisimple.