Chris Ramsey, Pure Mathematics, University of Waterloo
"Maximal Ideal Space Techniques in Non-Selfadjoint Operator Algebras"
The
following
thesis
is
divided
into
two
main
parts.
In
the
first
part
we
study
the
problem
of
characterizing
algebras
of
functions
living
on
analytic
varieties.
Specifically,
we
consider
the
restrictions
$\cM_V$
of
the
multiplier
algebra
$\cM$
of
Drury-Arveson
space
to
a
holomorphic
subvariety
$V$
of
the
unit
ball
as
well
as
the
algebras
$\cA_V$
of
continuous
multipliers
under
the
same
restriction.
We
find
that
$\cM_V$
is
completely
isometrically
isomorphic
to
$\cM_W$
if
and
only
if
$W$
is
the
image
of
$V$
under
a
biholomorphic
automorphism
of
the
ball.
In
this
case,
the
isomorphism
is
unitarily
implemented.
Furthermore,
when
$V$
and
$W$
are
homogeneous
varieties
then
$\cA_V$
is
isometrically
isomorphic
to
$\cA_W$
if
and
only
if
the
defining
polynomial
relations
are
the
same
up
to
a
change
of
variables.
The
problem
of
characterizing
when
two
such
algebras
are
(algebraically)
isomorphic
is
also
studied.
In
the
continuous
homogeneous
case,
two
algebras
are
isomorphic
if
and
only
if
they
are
similar.
However,
in
the
multiplier
algebra
case
the
problem
is
much
harder
and
several
examples
will
be
given
where
no
such
characterization
is
possible.
In
the
second
part
we
study
the
triangular
subalgebras
of
UHF
algebras
which
provide
new
examples
of
algebras
with
the
Dirichlet
property
and
the
Ando
property.
This
in
turn
allows
us
to
describe
the
semicrossed
product
by
an
isometric
automorphism.
We
also
study
the
isometric
automorphism
group
of
these
algebras
and
prove
that
it
decomposes
into
the
semidirect
product
of
an
abelian
group
by
a
torsion
free
group.
Various
other
structure
results
are
proven
as
well.