A Banach algebra A that is a topological direct sum (as a Banach space) of a pair of its Banach subalgebras

which are isometrically isomorphic to A.

******************************

Consider A = l^{∞}. Define E = {(x_{n}) ∈ l^{∞} ; x_{2n} = 0} and F = {(x_{n}) ∈ l^{∞} ; x_{2n−1} = 0}. Obviously E and F are closed subalgebras of l^{∞}. Moreover A = E + F and E ∩ F = {0}. So E is a complemented subspace of A with F as a complementary subspace. In addition, φ((x_{1}, x_{2}, x_{3}, …)) = (0, x_{1}, 0, x_{2}, 0, x_{3}, …) is an isometrically isomorphism between l^{∞} and E. One can similarly define an isomorphism between l^{∞} and F. Note that if E is a complemented infinite dimensional subspace of l^{∞} then E is isomorphic to l^{∞}. (cf. [J. Lindenstranss, On complemented subspaces of m. Israel J. Math., 5, 1967, 153-156])