Journal of Convex Analysis 18 (2011), No. 2, 513--528
Copyright Heldermann Verlag 2011
Slices in the Unit Ball of the Symmetric Tensor Product of a Banach Space
María D. Acosta
Universidad de Granada, Facultad de Ciencias, Dep. de Análisis Matemático, 18071 Granada, Spain
dacosta@ugr.es
Julio Becerra Guerrero
Universidad de Granada, Facultad de Ciencias, Dep. de Análisis Matemático, 18071 Granada, Spain
juliobg@ugr.es
[Abstract-pdf]
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.
Keywords: Banach spaces, slice, homogeneous polynomial, integral polynomial, symmetric projective tensor product, symmetric injective tensor product, C-star-algebra.
MSC: 46B20; 46B25, 46B28
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