A Banach algebra

A Banach algebra 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.

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Consider A = l^Ą. Define E = {(x_n) Î l^Ą ; x_2n = 0} and F = {(x_n) Î l^Ą ; x_2n-1 = 0}. Obviously E and F are closed subalgebras of l^Ą. Moreover A = E+F and EÇF = {0}. So E is a complemented subspace of A with F as a complementary subspace. In addition, j((x₁,x₂,x₃,Ľ)) = (0,x₁,0,x₂,0,x₃,Ľ) is an isometrically isomorphism between l^Ą and E. One can similarly define an isomorphism between l^Ą and F. Note that if E is a complemented infinite dimensional subspace of l^Ą then E is isomorphic to l^Ą. (cf. [J. Lindenstranss, On complemented subspaces of m. Israel J. Math., 5, 1967, 153-156])

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