Pure states, positive matrix polynomials and sums of hermitian squares

Sums of hermitian squares and the BMV conjecture

Recently Lieb and Seiringer showed that the Bessis-Moussa-Villani conjecture from quantum physics can be restated in the following purely algebraic way: The sum of all words in two positive semidefinite matrices where the number of each of the two letters is fixed is always a matrix with nonnegative trace. We show that this statement holds if the words are of length at most 13. This has previously been known only up to length 7. In our proof, we establish a connection to sums of hermitian squares of polynomials in noncommuting variables and to semidefinite programming. As a by-product we obtai…

Connes' embedding conjecture and sums of hermitian squares

We show that Connes' embedding conjecture on von Neumann algebras is equivalent to the existence of certain algebraic certificates for a polynomial in noncommuting variables to satisfy the following nonnegativity condition: The trace is nonnegative whenever self-adjoint contraction matrices of the same size are substituted for the variables. These algebraic certificates involve sums of hermitian squares and commutators. We prove that they always exist for a similar nonnegativity condition where elements of separable II_1-factors are considered instead of matrices. Under the presence of Connes'…

投影的非厄米和之布朗测度

The Brown Measure of Non-Hermitian Sums of Projections

本文通过将两个自由独立厄米投影生成的冯·诺依曼代数非平凡部分建模为 2×2 随机矩阵，计算了非正规算子 X=p+iq 的布朗测度，并发现其支撑在双曲线上，同时分析了其原子与对称性。

Sums of Hermitian Squares as an Approach to the BMV Conjecture

Lieb and Seiringer stated in their reformulation of the Bessis-Moussa-Villani (BMV) conjecture that all coefficients of the polynomial p(t)=Tr[(A+tB)^m], where A and B are positive semidefinite matrices of the same size and m an arbitrary integer, are nonnegative. The coefficient of t^k is the trace of S_{m,k}(A,B), which is the sum of all words of length m in the letters A and B in which B appears exactly k times. We consider the case k=4 and show that S_{m,4}(A,B) is a sum of hermitian squares and commutators. In particular, the trace of S_{m,4}(A,B) is nonnegative.

非厄米投影和的法律的收敛性

Convergence of the Laws of Non-Hermitian Sums of Projections

本文研究由两个独立 Haar 酉旋转厄米矩阵构成的非厄米随机矩阵模型，证明其经验谱分布收敛到相应自由独立厄米元之和的 Brown 测度，并给出逆命题。

Sums of commutators in pure C*-algebras

In a pure C*-algebra (i.e., one having suitable regularity properties in its Cuntz semigroup), any element on which all bounded traces vanish is a sum of 7 commutators.