Adam Dor On, Department of Pure Mathematics, University of Waterloo
“Matrix convex sets: Inclusions, dilations and completely positive interpolation”
A matrix convex set is a stratified set of the form S = ∪n≥1Sn, where each Sn is comprised of d-tuples of n × n matrices, and is closed under application of unital completely positive maps from Mn to Mk. Our main tools are polar duality in the sense of Effros and Winkler, and constructions of commuting normal dilations in the sense of Helton, Klep, McCullough and Schweighofer.
Given two matrix convex sets S and T , we find geometric conditions on S or on T , to ensurethatS1 ⊂T1 impliesS⊂C·T forsomeconstantC>0.
Under various symmetry conditions on S we can guarantee that C = d, the number of variables. We show that d is sharp for the maximal matrix ball Wmax(B ). We then find and.
Our
results
have
implications
to
spectrahedral
inclusion
problems
studied
by
Helton,
Klep,
McCullough
and
Schweighofer,
and
to
existence
of
interpolating
unital
completely
positive
maps.
The
first
20
minutes
of
this
talk
should
be
accessible
to
graduate
students
outside
of
analysis.
This
is
joint
work
with
Ken
Davidson,
Orr
Shalit
and
Baruch
Solel.
MC 5417