Serban Belinschi, Université Toulouse III
"Noncommutative hyperbolic metrics"
Classical
complex
analytic
maps
are
contractions
with
respect
to
certain
distances
on
complex
domains.
The
Kobayashi
distance
and
the
Kobayashi
metric
are
probably
the
best
known
among
them.
Very
roughly,
the
Kobayashi
metric
at
a
given
point
of
a
domain
is
the
reciprocal
of
the
radius
of
the
biggest
complex
disk
that
can
be
embedded
in
the
domain
while
mapping
the
origin
to
the
given
point.
Using
intrinsic
properties
of
noncommutative
sets,
we
define
a
metric
in
a
similar
way
on
noncommutative
domains
in
operator
spaces.
We
show
that
this
metric
satisfies
several
useful
properties
and
can
be
used
to
a
certain
extent
to
classify
noncommutative
domains.
We
conclude
with
an
application
to
free
probability.
The
talk
is
based
on
joint
work
with
Victor
Vinnikov.
MC 5501