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

We prove that every infinite-dimensional

-algebra

X X

satisfies that every slice of the unit ball of

\widehat{\bigotimes }_{N,s,\pi} X ⨂ ^ N , s , π X

(

N N

-fold projective symmetric tensor product of

X X

) has diameter two. We deduce that every infinite-dimensional Banach space

X X

whose dual is an

L_1 L 1

-space satisfies the same result. As a consequence, if

X X

is either a

C^* C ∗

-algebra or either a predual of an

L_1 L 1

-space, then the space of all

N N

-homogeneous polynomials on

X X

,

{\mathcal{P}} ^N (X) P N ( X )

, is extremely rough, whenever

X X

is infinite-dimensional. If

Y Y

is a predual of a von Neumann algebra, then

Y Y

is infinite-dimensional if, and only if, every

w^\ast w ∗

-slice of the unit ball of

{\mathcal{P}}^{N}_{I} (Y) P I N ( Y )

(the space of integral

N N

-homogeneous polynomials on

Y Y

) has diameter two. As a consequence, under the previous assumptions, the

N N

-fold symmetric injective tensor product of

Y Y

is extremely rough. Indeed, this isometric condition characterizes infinite-dimensional spaces in the class of preduals of von Neumann algebras.