Non Isomorphism of the Disc Algebra with Spaces of Differentiable Functions

It is proved that the Disc Algebra does not contain a complemented subspace isomorphic to the space C(k)(Td) of k times continuously differentiable functions on the d-dimensional torus ( k = 1, 2, ... ; d = 2, 3, ... ). Introduction. Recall two interesting problems concerning the space q 1)(T2 ) of continuously differentiable functions on the 2-dimensional torus T2 . (I) Is C(1)(T2 ) isomorphic to a subspace of C(K) (K-compact metric) with a separable annihilator? · (II) Does there exist a 1absolutely summing surjection from C(1)(T2 ) onto an infinite dimensional Hilbert space? The negative answer on each of these questions implies the non-isomorphism of the Disc Algebra A with C(1)(T2). In the present paper we prove the latter fact. Precisely our main result (Theorem 2.1) says that the space ql)(T2 ) is not isomorphic to any complemented subspace of A. The result seems to be interesting because of the method of its proof. We show that the natural embedding of C(l) (T2 ) into the Sobolev space L~l) (T2 ) does not factor through the natural embedding of A into H! for any finite Borel measure p, on the circle. 1991 Mathematics Subject Classification. 46E35, 46B20