A Banach algebra A that cannot be a (vector space) direct sum of its radical Rad(A) and a Banach algebra B that is homeomorphically isomorphic with A/Rad(A).

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Consider the Banach algebra l^{2} and the dense subalgebra l_{0}^{2} of l^{2} consisting of the sequences which vanish out of a finite set. Let A_{0} be the vector space direct sum l_{0}^{2} ⊕ C. A_{0} is an algebra with (x, α)(y, β) = (xy, 0), x, y ∈ l^{2}, α, β ∈ C. Also ||(x, α)|| = max(||x||, |α − ∑_{n=1}^{∞} x(n)|) is a norm on A_{0}. Let A is the completion of A_{0}. Rad(A) = C(0, 1). If (x, α) ∈ A_{0} and [x, α] denotes the image of (x, α) in A/Rad(A), then [x, α] ↦ x defines an isometric isomorphism of A_{0}/RadA into l_{0}^{2} which can be extended to an isometric isomorphism of A/RadA onto l^{2}. Suppose that there exists a homeomorphic isomorphism of l^{2} with a subalgebra A_{1} of A. Let ξ_{k} denotes ξ_{k}(n) = δ_{kn} (k, n ∈ N) and e_{k} denotes the corresponding element of A_{1}. Choose a sequence ((x_{n}, α_{n}))_{n∈N} in A_{0} such that lim_{n}(x_{n}, α_{n}) = e_{k} in A. Since e_{k}^{2} = e_{k}, we have lim_{n}(x_{n}^{2}, 0) = e_{k}. Thus lim_{n} x_{n}(k) = 0 or 1 for all k ∈ N and also e_{k} ∈ l_{0}^{2}. The elements e_{k} are pairwise orthogonal idempotents. If d_{n} = ∑_{k=1}^{n} \frac{e_{k}}{k} and t_{n} = ∑_{k=1}^{n} \frac{ξ_{k}}{k}, then (t_{n}) converges in l^{2}. But d_{n} doesn’t converge in A, a contradiction. Ref.

[Ric] C.E. Rickart, General theory of Banach algebras, Princeton, Van Nastrand, 1960.