2017/06/12 by Mohan Ravichandran, Nikhil Srivastava, Ravichandran, Mohan +1
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #math.CO #math.FA #math.OA
paper · pdf · doi:10.48550/arxiv.1706.03737
23 pages. In the previous version, we had erroneously claimed that the main theorem in this paper implies a polylogarithmic bound in the commutator theorem of Johnson, Ozawa and Schechtman. This has been corrected with a weaker bound. The main results in the paper are unchanged
arxiv created 2017/08/23 · arxiv updated 2017/08/24
Anderson's paving conjecture, now known to hold due to the resolution of the Kadison-Singer problem asserts that every zero diagonal Hermitian matrix admits non-trivial pavings with dimension independent bounds. In this paper, we develop a technique extending the arguments of Marcus, Spielman and Srivastava in their solution of the Kadison-Singer problem to show the existence of non-trivial pavings for collections of matrices. We show that given zero diagonal Hermitian contractions A(1), ⋯, A(k) ∈ Mn(ℂ) and ε> 0, one may find a paving X1 \amalg ⋯ \amalg Xr = [n] where r ≤ 18kε-2 such that, λmax (PXi A(j) PXi) < ε, i ∈ [r], j ∈ [k]. As a consequence, we get the correct asymptotic estimates for paving general zero diagonal matrices; zero diagonal contractions can be (O(ε-2),ε) paved. As an application, we give a simplified proof wth slightly better estimates of a theorem of Johnson, Ozawa and Schechtman concerning commutator representations of zero trace matrices.