Slices in the Unit Ball of the Symmetric Tensor Product of a Banach Space

We prove that every infinite-dimensional C*-algebra X satisfies that every slice of the unit ball of (circle times) over cap (N-fold projective symmetric tensor product of X) has diameter two. We deduce that every infinite-dimensional Banach space X whose dual is an L-1-space satisfies the same result. As a consequence, if X is either a C*-algebra or either a predual of an L-1-space, then the space of all N-homogeneous polynomials on X, P-N (X), is extremely rough, whenever X is infinite-dimensional. If Y is a predual of a von Neumann algebra, then Y is infinite-dimensional if, and only if, every w*-slice of the unit ball of P-I(N)(Y) (the space of integral N-homogeneous polynomials on Y) has diameter two. As a consequence, under the previous assumptions, the N-fold symmetric injective tensor product of Y is extremely rough. Indeed, this isometric condition characterizes infinite-dimensional spaces in the class of preduals of von Neumann algebras.