Related papers: Asymptotic Tensor Powers of Banach Spaces

Asymptotic Tensor Powers of Banach Spaces

URL: http://arxiv.org/abs/2110.12828v1
Date: Mon, 25 Oct 2021 11:51:12 GMT
Title: Asymptotic Tensor Powers of Banach Spaces
Authors: Guillaume Aubrun, Alexander M\"uller-Hermes
Abstract summary: We show that Euclidean spaces are characterized by the property that their tensor radius equals their dimension. We also show that the tensor radius of an operator whose domain or range is Euclidean is equal to its nuclear norm.
Score: 77.34726150561087
License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
Abstract: We study the asymptotic behaviour of large tensor powers of normed spaces and of operators between them. We define the tensor radius of a finite-dimensional normed space $X$ as the limit of the sequence $A_k^{1/k}$, where $A_k$ is the equivalence constant between the projective and injective norms on $X^{\otimes k}$. We show that Euclidean spaces are characterized by the property that their tensor radius equals their dimension. Moreover, we compute the tensor radius for spaces with enough symmetries, such as the spaces $\ell_p^n$. We also define the tensor radius of an operator $T$ as the limit of the sequence $B_k^{1/k}$, where $B_k$ is the injective-to-projective norm of $T^{\otimes k}$. We show that the tensor radius of an operator whose domain or range is Euclidean is equal to its nuclear norm, and give some evidence that this property might characterize Euclidean spaces.

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