论文标题
凸锥的张量产品
Tensor Products of Convex Cones
论文作者
论文摘要
凸锥的张量产物最近在不同领域出现，从功能分析和操作者理论到近似理论和理论物理学。但是，大多数现有文献都集中在阿基米氏晶格锥上，或在有限维空间中的封闭适当的锥体上，从而排除了许多锥体，甚至包括标准锥，例如无限二二次阳性半闪光锥。对于一般锥体，结果很少，而且相距甚远，许多基本问题仍未得到解决。 在这本回忆录中，我们以完全一般性发展了凸锥的张量产物的理论，对锥或环境空间没有任何限制。我们将一些已知的结果概括为一般情况，我们证明了许多新的结果。我们的主要贡献是：（i）我们表明，投影/注射锥具有类似于投影/注射标准的映射属性； （ii）我们为投影/注射锥的直接空间建立了直接公式； （iii）我们证明了两个封闭的锥体的投影/注射量张量产物包含在封闭的圆锥体中； （iv）我们展示了如何从基本锥面部构造投影/注射锥的面孔。作为应用程序，我们还表明，两个对称凸组的张量产品可保留适当的面孔。 （v）对于有限维空间中的封闭式锥体，我们表明投影锥已封闭，几乎总是严格包含在注射锥中，从而确认了几乎所有凸锥的Barker的猜想。当这本手稿被撰写时，这最后的结果被Aubrun，Lami，Palazuelos和Plávala同时发现，他们独立证明了Barker的猜想是完全普遍的。我们使用完全不同的技术恢复了大型锥体的结果。
Tensor products of convex cones have recently come up in different areas, ranging from functional analysis and operator theory to approximation theory and theoretical physics. However, most of the existing literature focuses either on Archimedean lattice cones or on closed proper cones in finite-dimensional spaces, thereby excluding many cones, including even standard cones such as an infinite-dimensional positive semidefinite cone. For general cones, results are few and far between, and many basic questions remain unanswered. In this memoir, we develop the theory of tensor products of convex cones in full generality, with no restrictions on the cones or the ambient spaces. We generalize a few known results to the general case, and we prove many results which are altogether new. Our main contributions are: (i) We show that the projective/injective cone has mapping properties analogous to those of the projective/injective norm; (ii) We establish direct formulas for the lineality space of the projective/injective cone; (iii) We prove that the projective/injective tensor product of two closed proper cones is contained in a closed proper cone; (iv) We show how to construct faces of the projective/injective cone from faces of the base cones. As an application, we also show that the tensor product of two symmetric convex sets preserves proper faces; (v) For closed cones in finite-dimensional spaces, we show that the projective cone is closed, and almost always strictly contained in the injective cone, thereby confirming a conjecture of Barker for nearly all convex cones. As this manuscript was being written, this last result was superseded by simultaneous discovery by Aubrun, Lami, Palazuelos and Plávala, who independently proved Barker's conjecture in full generality. We recover their result for a large class of cones, using completely different techniques.
