Constructing dilations of frame decompositions and (non-commutative) operator-valued measures on Banach spaces

Speaker:Rui Liu (Nankai University)

Date:November 4 Thur. 2:00pm -- 4:00pm

Venue:  ZOOM ID：213232241

Abstract:Dilation theory is a natural paradigm for quantitative complemented-embeddings between Banach spaces by way of exhibiting vectors or operators as the complemented-compression of those which are well-behaved in bigger & better Banach spaces.From 2014 (Memoirs A.M.S.) till now, we focus on Banach dilation theoryfrom frame decompositions andoperator-valued measures(OVMs)on(reflexive)Banach spaces tothe latest non-commutativecases onprojection lattices of vN-algebrasandoperators on Banach spaces. Weconstructthe minimal dilation for quantumOVMsfrom projection lattices of finite vN-algebras without type I_2 direct summand to B(X) where the Banach spaceX is the sequence spaces lp (p<2) or has Shur property. It's surprising for us that the non-commutative dilation closely relies on concrete Banach space geometric properties. Bynon-commutative projection-partition treetechnique, weobtain thedilation for quantumOVMswith bounded p-variation, which have natural examples on completely bounded maps and non-commutative Lp spaces (p>2).