Ben Passer, Department of Pure Mathematics, University of Waterloo
"Containment of Matrix Convex Sets"
This
talk
contains
joint
work
with
Orr
Shalit
and
Baruch
Solel.
A
set
$S
\subseteq
\bigcup_{n=1}^\infty
M_n(\mathbb{C})^d$
is
called
matrix
convex
if
it
is
closed
under
direct
sums
and
the
application
of
unital
completely
positive
maps.
We
may
view
$S$
as
a
disjoint
union
of
its
levels
$S_n
\subseteq
M_n(\mathbb{C})^d$,
each
of
which
is
convex
and
is
closed
under
simultaneous
unitary
conjugations.
Following
work
of
Davidson,
Dor-On,
Shalit,
and
Solel,
we
seek
to
determine
how
much
information
about
$S$
is
contained
in
its
first
level,
$S_1
=
K$.
If
$K$
is
a
compact
convex
set
in
$\mathbb{C}^d$,
then
there
could
be
many
matrix
convex
sets
which
have
$K$
as
the
first
level.
However,
there
are
always
two
extremal
options
$\mathcal{W}^\text{min}(K)$
and
$\mathcal{W}^{\text{max}}(K)$.
We
study
the
disparity
between
these
two
matrix
convex
sets
by
searching
for
an
optimal
constant
$C$
with
\begin{equation*}
\mathcal{W}^{\text{max}}(K)
\subseteq
C
\cdot
\mathcal{W}^{\text{min}}(K).
\end{equation*}
Among
our
results,
we
find
an
optimal
scaling
constant
when
$K$
is
an
$\ell^p$
ball,
and
we
completely
characterize
when
the
constant
$C
=
1$
may
be
achieved.
In
doing
so,
we
make
use
of
theorems
in
convex
geometry,
and
we
find
that
it
suffices
to
check
a
single
matrix
level
$n
=
2^{d-1}$.
Time
permitting
(which
occurs
with
probability
zero),
I
will
discuss
related/generalized
problems
concerning
products
of
matrix
convex
sets,
dilation
theory,
and
extreme
points.
MC 5417