关于Haemers界的一个跟踪形式
On a tracial version of Haemers bound
摘要：
我们将图的量子独立数和量子香农容量的上界推广到交换算子模型中。我们引入了分数Haemers界（在$\mathbb{C}$上）的von Neumann代数推广，并证明了推广的上界是交换量子独立数.我们称这个界为迹Haemers界，并证明了它关于强积是乘法的。特别是，这使得它成为香农容量的上限。迹线Haemers界限与另一个著名的香农容量上限Lovász theta函数是不可比拟的。我们表明，分离的tracial和分数Haemers界将反驳Connes的嵌入猜想。在此过程中，我们证明了迹秩和迹Haemers界是图的（交换量子）渐近谱的元素（Zuiddam，Combinatorica，2019）。我们还表明，惯性界（上界的量子独立数）上界的交换量子独立数。
英文摘要：
We extend upper bounds on the quantum independence number and the quantum Shannon capacity of graphs to their counterparts in the commuting operator model. We introduce a von Neumann algebraic generalization of the fractional Haemers bound (over $\mathbb{C}$) and prove that the generalization upper bounds the commuting quantum independence number. We call our bound the tracial Haemers bound, and we prove that it is multiplicative with respect to the strong product. In particular, this makes it an upper bound on the Shannon capacity. The tracial Haemers bound is incomparable with the Lovász theta function, another well-known upper bound on the Shannon capacity. We show that separating the tracial and fractional Haemers bounds would refute Connes' embedding conjecture. Along the way, we prove that the tracial rank and tracial Haemers bound are elements of the (commuting quantum) asymptotic spectrum of graphs (Zuiddam, Combinatorica, 2019). We also show that the inertia bound (an upper bound on the quantum independence number) upper bounds the commuting quantum independence number.
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