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 $\widehat{\bigotimes }_{N,s,\pi} X$ ($N$-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 $X$, ${\mathcal{P}} ^N (X)$, 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 $w^\ast$-slice of the unit ball of ${\mathcal{P}}^{N}_{I} (Y)$ (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.