Consider the Banach algebra l² and the dense subalgebra l₀² of l² consisting of the sequences which vanish out of a finite set. Let A₀ be the vector space direct sum l₀²ÅC. A₀ is an algebra with (x,a)(y,b) = (xy,0), x,y Î l², a,b Î C. Also ||(x,a)|| = max(||x||,|a-å_{n = 1}^¥ x(n)|) is a norm on A₀. Let A is the completion of A₀. Rad(A) = C(0,1). If (x,a) Î A₀ and [x,a] denotes the image of (x,a) in A/Rad(A), then [x,a]® x defines an isometric isomorphism of A₀/Rad A into l₀² which can be extended to an isometric isomorphism of A/Rad A onto l². Suppose that there exists a homeomorphic isomorphism of l² with a subalgebra A₁ of A. Let x_k denotes x_k(n) = d_kn  (k,n Î N) and e_k denotes the corresponding element of A₁. Choose a sequence ((x_n,a_n))_{n Î N} in A₀ such that lim_n(x_n,a_n) = e_k in A. Since e_k² = e_k, we have lim_n(x_n²,0) = e_k. Thus lim_n x_n(k) = 0 or 1 for all k Î N and also e_k Î l₀². The elements e_k are pairwise orthogonal idempotents. If d_n = å_{k = 1}ⁿ [( e_k)/ k] and t_n = å_{k = 1}ⁿ [( x_k)/ k], then (t_n) converges in l². But d_n doesn't converge in A, a contradiction.
Ref.
[Ric] C.E. Rickart, General theory of Banach algebras, Princeton, Van Nastrand, 1960.