Abstract

We prove that every infinite-dimensional C∗C^*-algebra XX satisfies that every slice of the unit ball of ⨂^N,s,πX\widehat{\bigotimes }_{N,s,\pi} X (NN-fold projective symmetric tensor product of XX) has diameter two. We deduce that every infinite-dimensional Banach space XX whose dual is an L1L_1-space satisfies the same result. As a consequence, if XX is either a C∗C^*-algebra or either a predual of an L1L_1-space, then the space of all NN-homogeneous polynomials on XX, PN(X){\mathcal{P}} ^N (X), is extremely rough, whenever XX is infinite-dimensional. If YY is a predual of a von Neumann algebra, then YY is infinite-dimensional if, and only if, every w∗w^\ast-slice of the unit ball of PIN(Y){\mathcal{P}}^{N}_{I} (Y) (the space of integral NN-homogeneous polynomials on YY) has diameter two. As a consequence, under the previous assumptions, the NN-fold symmetric injective tensor product of YY is extremely rough. Indeed, this isometric condition characterizes infinite-dimensional spaces in the class of preduals of von Neumann algebras.

Author information

Contact details are reproduced from the original publication and may be historical.

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

Keywords

  • Banach spaces
  • slice
  • homogeneous polynomial
  • integral polynomial
  • symmetric projective tensor product
  • symmetric injective tensor product
  • C-star-algebra

Mathematics Subject Classification

46B20; 46B25, 46B28

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Suggested citation

M. D. Acosta, J. Becerra Guerrero. “Slices in the Unit Ball of the Symmetric Tensor Product of a Banach Space.” Journal of Convex Analysis 18 (2011), No. 2, 513–528. https://doi.org/10.68381/jca18033

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Published by Heldermann Verlag, 2011. Rights now held by Banach Press.