The p-cones in dimension n≥ 3 are not homogeneous when p≠ 2

Ito, Masaru; Lourenço, Bruno F.

doi:10.1016/j.laa.2017.07.029

Cited by 9 publications

(15 citation statements)

References 12 publications

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“…Proposition 25. [17] Let V be a finite-dimensional ordered vector space with a strictly convex homogeneous positive cone, then V is order isomorphic to a spin-factor, i.e. V ∼ = H⊕R where H is a real finite-dimensional Hilbert space with the order on H ⊕R given by (v, t) ≥ 0 ⇐⇒ t ≥ v 2 .…”

Section: Self-dualitymentioning

confidence: 99%

“…Unfortunately, at this point we still cannot use the characterisation of rank 2 spaces from [17] as this classification only holds for finite-dimensional spaces. It is currently unclear whether the generalisation of this theorem to infinite-dimensional spaces holds, although it seems reasonable.…”

Section: Definition 21mentioning

confidence: 99%

“…As shown in [25] a directed complete sequential product space with normal sequential product is homogeneous. The necessary propositions presented in sections 3.2, 3.3 and 3.4 still hold and since the spaces V p,q are required to be finite-dimensional, the characterisation theorem of [17] applies so that we indeed have symmetry of transition probabilities. Since the atomic effects lie dense in the space (see also [25]) we have then satisfied all the conditions of [1,Theorem 9.38] so that V is indeed a JB-algebra.…”

Section: Definition 21mentioning

confidence: 99%

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Sequential product spaces are Jordan algebras

Wetering

2019

Journal of Mathematical Physics

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We show that finite-dimensional order unit spaces equipped with a continuous sequential product as defined by Gudder and Greechie are homogeneous and self-dual. As a consequence of the Koecher-Vinberg theorem these spaces therefore correspond to Euclidean Jordan algebras. We remark on the significance of this result in the context of reconstructions of quantum theory. In particular, we show that sequential product spaces that have locally tomographic tensor products, i.e. their vector space tensor products are also sequential product spaces, must be C * algebras. Finally we remark on a couple of ways these results can be extended to the infinite-dimensional setting of JB-and JBW-algebras and how changing the axioms of the sequential product might lead to a new characterisation of homogeneous cones.

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Section: Self-dualitymentioning

confidence: 99%

Section: Definition 21mentioning

confidence: 99%

Section: Definition 21mentioning

confidence: 99%

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Sequential product spaces are Jordan algebras

Wetering

2019

Journal of Mathematical Physics

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“…As discussed in Section 1 of [6], an often overlooked point is that K * depends on ·, · . Accordingly, it is entirely plausible that a cone that is not self-dual under the Euclidean inner product might become self-dual if the inner product is chosen appropriately.…”

Section: Introductionmentioning

confidence: 99%

“…This state of affairs brings us to the case of the p-cones. Up until the recent articles [5,6], there was no rigorous proof that the p-cones L n+1 p were not symmetric when p = 2 and n ≥ 2. Now, although we know that L n+1 p is not homogeneous for p = 2 and n ≥ 2, it still remains to investigate whether L n+1 p could become self-dual under an appropriate inner product.…”

Section: Introductionmentioning

confidence: 99%

The automorphism group and the non-self-duality of p-cones

Ito

Lourenço

2019

Journal of Mathematical Analysis and Applications

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In this paper, we determine the automorphism group of the p-cones (p = 2) in dimension greater than two. In particular, we show that the automorphism group of those p-cones are the positive scalar multiples of the generalized permutation matrices that fix the main axis of the cone. Next, we take a look at a problem related to the duality theory of the p-cones. Under the Euclidean inner product it is well-known that a p-cone is self-dual only when p = 2. However, it was not known whether it is possible to construct an inner product depending on p which makes the p-cone self-dual. Our results shows that no matter which inner product is considered, a p-cone will never become self-dual unless p = 2 or the dimension is less than three.

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Amenable cones: error bounds without constraint qualifications

Lourenço

2019

Math. Program.

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We provide a framework for obtaining error bounds for linear conic problems without assuming constraint qualifications or regularity conditions. The key aspects of our approach are the notions of amenable cones and facial residual functions. For amenable cones, it is shown that error bounds can be expressed as a composition of facial residual functions. The number of compositions is related to the facial reduction technique and the singularity degree of the problem. In particular, we show that symmetric cones are amenable and compute facial residual functions. From that, we are able to furnish a new Hölderian error bound, thus extending and shedding new light on an earlier result by Sturm on semidefinite matrices. We also provide error bounds for the intersection of amenable cones, this will be used to prove error bounds for the doubly nonnegative cone. At the end, we list some open problems.

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The p-cones in dimension n≥ 3 are not homogeneous when p≠ 2

Cited by 9 publications

References 12 publications

Sequential product spaces are Jordan algebras

Sequential product spaces are Jordan algebras

The automorphism group and the non-self-duality of p-cones

Amenable cones: error bounds without constraint qualifications

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