Digraph Algebras over Discrete Pre-ordered Groups
Kai-Cheong Chan
This
thesis
consists
of
studies
in
the
separate
fields
of
operator
algebras
and
non-associative
algebras.
Two
natural
operator
algebra
structures,
A
⊗max
B
and
A
⊗min
B,
exist
on
the
tensor
product
of
two
given
unital
operator
algebras
A
and
B.
Because
of
the
different
properties
enjoyed
by
the
two
tensor
products
in
connection
to
dilation
theory,
it
is
of
interest
to
know
when
they
coincide
(completely
isometrically).
Motivated
by
earlier
work
due
to
Paulsen
and
Power,
we
provide
conditions
relating
an
operator
algebra
B
and
another
family
{Ci
}i
of
operator
algebras
under
which,
for
any
operator
algebra
A,
the
equality
A
⊗max
B
=
A
⊗min
B
either
implies,
or
is
implied
by,
the
equalities
A
⊗max
Ci
=
A
⊗min
Ci
for
every
i.
These
results
can
be
applied
to
the
setting
of
a
discrete
group
G
pre-ordered
by
a
subsemigroup
G
+,
where
B
⊂
C
∗
r
(G)
is
the
subalgebra
of
the
reduced
group
C
∗
-algebra
of
G
generated
by
G
+,
and
Ci
=
A(Qi
)
are
digraph
algebras
defined
by
considering
certain
pre-ordered
subsets
Qi
of
G.
The
16-dimensional
algebra
A4
of
real
sedenions
is
obtained
by
applying
the
Cayley-Dickson
doubling
process
to
the
real
division
algebra
of
octonions.
The
classification
of
subalgebras
of
A4
up
to
conjugacy
(i.e.
by
the
action
of
the
automorphism
group
of
A4
)
was
completed
in
a
previous
investigation,
except
for
the
collection
of
those
subalgebras
which
are
isomorphic
to
the
quaternions.
We
present
a
classification
of
quater-nion
subalgebras
up
to
conjugacy.