<s>
In	O
the	O
theory	O
of	O
C*	B-Algorithm
-algebras	I-Algorithm
,	O
the	O
universal	B-Algorithm
representation	I-Algorithm
of	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
is	O
a	O
faithful	O
representation	O
which	O
is	O
the	O
direct	O
sum	O
of	O
the	O
GNS	O
representations	O
corresponding	O
to	O
the	O
states	O
of	O
the	O
C*	B-Algorithm
-algebra	I-Algorithm
.	O
</s>
<s>
The	O
various	O
properties	O
of	O
the	O
universal	B-Algorithm
representation	I-Algorithm
are	O
used	O
to	O
obtain	O
information	O
about	O
the	O
ideals	O
and	O
quotients	O
of	O
the	O
C*	B-Algorithm
-algebra	I-Algorithm
.	O
</s>
<s>
The	O
close	O
relationship	O
between	O
an	O
arbitrary	O
representation	O
of	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
and	O
its	O
universal	B-Algorithm
representation	I-Algorithm
can	O
be	O
exploited	O
to	O
obtain	O
several	O
criteria	O
for	O
determining	O
whether	O
a	O
linear	O
functional	O
on	O
the	O
algebra	O
is	O
ultraweakly	B-Algorithm
continuous	O
.	O
</s>
<s>
The	O
method	O
of	O
using	O
the	O
properties	O
of	O
the	O
universal	B-Algorithm
representation	I-Algorithm
as	O
a	O
tool	O
to	O
prove	O
results	O
about	O
the	O
C*	B-Algorithm
-algebra	I-Algorithm
and	O
its	O
representations	O
is	O
commonly	O
referred	O
to	O
as	O
universal	B-Algorithm
representation	I-Algorithm
techniques	O
in	O
the	O
literature	O
.	O
</s>
<s>
on	O
the	O
Hilbert	O
space	O
is	O
known	O
as	O
the	O
universal	B-Algorithm
representation	I-Algorithm
of	O
A	O
.	O
</s>
<s>
As	O
the	O
universal	B-Algorithm
representation	I-Algorithm
is	O
faithful	O
,	O
A	O
is	O
*	O
-isomorphic	O
to	O
the	O
C*	O
-subalgebra	O
Φ(A )	O
of	O
B(HΦ )	O
.	O
</s>
<s>
Since	O
the	O
mapping	O
τ	O
→	O
τ	O
∘	O
Φ−1	O
takes	O
the	O
state	B-Algorithm
space	I-Algorithm
of	O
A	O
onto	O
the	O
state	B-Algorithm
space	I-Algorithm
of	O
Φ(A )	O
,	O
it	O
follows	O
that	O
each	O
state	O
of	O
Φ(A )	O
is	O
a	O
vector	O
state	O
.	O
</s>
<s>
As	O
the	O
set	O
of	O
linear	O
functionals	O
determining	O
the	O
weak	O
topologies	O
coincide	O
,	O
the	O
weak-operator	O
topology	O
on	O
Φ(A )	O
−	O
coincides	O
with	O
the	O
ultraweak	B-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
Thus	O
the	O
weak-operator	O
and	O
ultraweak	B-Algorithm
topologies	I-Algorithm
on	O
Φ(A )	O
both	O
coincide	O
with	O
the	O
weak	O
topology	O
of	O
Φ(A )	O
obtained	O
from	O
its	O
norm-dual	O
as	O
a	O
Banach	O
space	O
.	O
</s>
<s>
As	O
α	O
is	O
ultraweakly	B-Algorithm
bicontinuous	O
,	O
the	O
same	O
is	O
true	O
of	O
α0	O
.	O
</s>
<s>
Moreover	O
,	O
ψ	O
is	O
ultraweakly	B-Algorithm
continuous	O
,	O
and	O
is	O
a	O
*	O
-isomorphism	O
if	O
π	O
is	O
a	O
faithful	O
representation	O
.	O
</s>
<s>
Let	O
A	O
be	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
acting	O
on	O
a	O
Hilbert	O
space	O
H	O
.	O
For	O
ρ	O
in	O
A*	O
and	O
S	O
in	O
Φ(A )	O
−	O
,	O
let	O
Sρ	O
in	O
A*	O
be	O
defined	O
by	O
Sρ(a )	O
=	O
(Φ(a )	O
S	O
)	O
for	O
all	O
a	O
in	O
A	O
.	O
</s>
<s>
If	O
P	O
is	O
the	O
projection	O
in	O
the	O
above	O
commutative	O
diagram	O
when	O
π:A	O
→	O
B(H )	O
is	O
the	O
inclusion	O
mapping	O
,	O
then	O
ρ	O
in	O
A*	O
is	O
ultraweakly	B-Algorithm
continuous	O
if	O
and	O
only	O
if	O
ρ	O
=	O
Pρ	O
.	O
</s>
<s>
Each	O
ρ	O
in	O
A*	O
can	O
be	O
uniquely	O
expressed	O
in	O
the	O
form	O
ρ	O
=	O
ρu+ρs	O
,	O
with	O
ρu	O
ultraweakly	B-Algorithm
continuous	O
and	O
ρs	O
singular	O
.	O
</s>
<s>
Let	O
f	O
and	O
g	O
be	O
continuous	O
,	O
real-valued	O
functions	O
on	O
C4m	O
and	O
C4n	O
,	O
respectively	O
,	O
σ1	O
,	O
σ2	O
,	O
...	O
,	O
σm	O
be	O
ultraweakly	B-Algorithm
continuous	O
,	O
linear	O
functionals	O
on	O
a	O
von	O
Neumann	O
algebra	O
R	O
acting	O
on	O
the	O
Hilbert	O
space	O
H	O
,	O
and	O
ρ1	O
,	O
ρ2	O
,	O
...	O
,	O
ρn	O
be	O
bounded	O
linear	O
functionals	O
on	O
R	O
such	O
that	O
,	O
for	O
each	O
a	O
in	O
R	O
,	O
</s>
<s>
Then	O
the	O
above	O
inequality	O
holds	O
if	O
each	O
ρj	O
is	O
replaced	O
by	O
its	O
ultraweakly	B-Algorithm
continuous	O
component	O
( ρj	O
)	O
u	O
.	O
</s>
