A general theory of tensor products of convex sets in Euclidean spaces
- PDF / 476,157 Bytes
- 26 Pages / 439.37 x 666.142 pts Page_size
- 18 Downloads / 247 Views
Positivity
A general theory of tensor products of convex sets in Euclidean spaces Maite Fernández-Unzueta1
· Luisa F. Higueras-Montaño1
Received: 6 August 2019 / Accepted: 31 January 2020 © Springer Nature Switzerland AG 2020
Abstract We introduce both the notions of tensor product of convex bodies that contain zero in the interior, and of tensor product of 0-symmetric convex bodies in Euclidean spaces. We prove that there is a bijection between tensor products of 0-symmetric convex bodies and tensor norms on finite dimensional spaces. This bijection preserves duality, injectivity and projectivity. We obtain a formulation of Grothendieck‘s Theorem for 0-symmetric convex bodies and use it to give a geometric representation (up to the K G -constant) of the Hilbertian tensor product. We see that the property of having enough symmetries is preserved by these tensor products, and exhibit relations with the Löwner and the John ellipsoids. Keywords Convex body · Tensor product of convex sets · Tensor product of banach spaces · Hilbertian tensor norm · Ideals of linear operators · Grothendieck’s inequality Mathematics Subject Classification 46M05 · 52A21 · 47L20 · 15A69
1 Introduction Given two compact convex sets K 1 and K 2 , there are several different manners to construct a new compact convex set that can be considered as a tensorial product of them. Some of these notions were studied in relation with the so-called Choquet Theory,
The first author was partially supported by Consejo Nacional de Ciencia y Tecnología (CONACyT), grant number 284110. The second named author was supported by CONACyT scholarship for Ph.D. studies.
B
Luisa F. Higueras-Montaño [email protected] Maite Fernández-Unzueta [email protected]
1
Centro de Investigación en Matemáticas CIMAT), A.P. 402, Guanajuato, Gto., Mexico
123
M. Fernández-Unzueta et al.
as done by Semadeni [26], Namioka and Phelps [20] or Behrends and Wittstock [4]. More recently, tensor products of convex sets have been studied in relation with the so called quantum information theory, as can be found in Aubrun and Szarek [2]. Other classes of tensor products as well as some properties of those already mentioned can be found in [5,12,18,28]. In a previous paper, we characterized when a centrally symmetric convex body B in the Euclidean space Rd of dimension d = d1 · · · dl , is the unit ball of a Banach space whose norm is a reasonable crossnorm (see [10, Theorem 3.2]). That is, we showed how to determine in purely geometricterms, if B is the unit ball of a reasonable crossnorm on a tensor product space ⊗li=1 Rdi , · i , when the norms on each factor are not determined a priori. In this same line, we develop here a theory of tensor products of centrally symmetric convex bodies, which is consistent with the theory of tensor norms on finite dimensional Banach spaces. Our main result is that there is a bijection between both, tensor norms on finite dimensions and tensor products of 0-symmetric convex bodies. Moreover, this bijection preserves duality, injectivity and proj
Data Loading...
