c₀ and c are both closed subspaces of l^¥. In addition for each x = (x_n) Î l¹, r_x: c₀ ® C given by (y_n) ® å_{n = 1}^¥ x_ny_n is a bounded linear functional on c₀ with the norm ||r_x || = ||x ||. Clearly c₀^# is isometrically isomorphic to l¹. Also for each x = (x_n) Î l¹, h_x: c ® C given by (y_n) ® x₁ lim_n x_n + å_{n = 1}^¥x_ny_n is a bounded linear functional on c with the norm ||h_x || = ||x ||. Obviously c^# is isometrically isomorphic to l¹. But by BA25.DVI the closed unit ball of c₀ has no extreme point while the closed unit ball c contains at least (1,1,1,¼) as an extreme point (since if 1 = tx_n + (1-t)y_n with |x_n| £ 1 and |y_n| £ 1, then 1 = t Rex_n + (1-t)Rey_n for all n, so that Rex_n = Rey_n = 1 and hence x_n = y_n = 1 for each n). Thus c₀ and c₁ are not isometrically isomorphic.

Now by [Sak1, Corollary 1.13.3], l¹ can not be a W^*-algebra.

Re.
[Sak1] S. Sakai, C^*-algebras and W^*-algebras, Springer-Verlag, 1971.