Polynomial algebras on classical Banach spaces

The classical Stone-Weierstrass theorem claims that the algebra of all real polynomials on a finite-dimensional real Banach space X is dense, in the topology of uniform convergence on bounded sets (we will always consider this topology, unless otherwise stated), in the space of continuous real functions on X. On the other hand ([12]), on every infinite-dimensional Banach space X there exists a uniformly continuous real function not approximable by continuous polynomials. Moreover, on some spaces (e.g. lp see [12], [5]) a new phenomenon occurs; the closure of the algebra generated by polynomials of degree at most n (An) does not contain all polynomials of higher degree. In our paper we completely clarify this situation for the classical Banach spaces. We also present some partial answers in the general case. With exception of C(K) Asplund spaces, our results are new. Our strategy rests on the same basic idea, used to obtain the previous partial results in [12], [5], that the polynomial P (