<s>
In	O
mathematics	O
,	O
the	O
notion	O
of	O
a	O
cyclic	B-Algorithm
and	I-Algorithm
separating	I-Algorithm
vector	I-Algorithm
is	O
important	O
in	O
the	O
theory	O
of	O
von	O
Neumann	O
algebras	O
,	O
and	O
in	O
particular	O
in	O
Tomita	O
–	O
Takesaki	O
theory	O
.	O
</s>
<s>
A	O
positive	B-Algorithm
linear	I-Algorithm
functional	I-Algorithm
ω	O
on	O
a	O
*	O
-algebra	O
A	O
is	O
said	O
to	O
be	O
faithful	O
if	O
ω(a )	O
=	O
0	O
,	O
where	O
a	O
is	O
a	O
positive	O
element	O
of	O
A	O
,	O
impliesa	O
=	O
0	O
.	O
</s>
<s>
Every	O
element	O
Ω	O
of	O
H	O
defines	O
a	O
positive	B-Algorithm
linear	I-Algorithm
functional	I-Algorithm
ωΩ	O
on	O
a	O
*	O
-algebra	O
A	O
of	O
bounded	O
linear	O
operators	O
in	O
H	O
by	O
the	O
relation	O
ωΩ(a )	O
=(	O
aΩ	O
,	O
Ω	O
)	O
for	O
all	O
a	O
in	O
A	O
.	O
</s>
<s>
If	O
ωΩ	O
is	O
defined	O
in	O
this	O
way	O
and	O
A	O
is	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
then	O
ωΩ	O
is	O
faithful	O
if	O
and	O
only	O
if	O
the	O
vector	O
Ω	O
is	O
separating	O
for	O
A	O
.	O
</s>
<s>
Note	O
that	O
a	O
von	O
Neumann	O
algebra	O
is	O
a	O
special	O
case	O
of	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
.	O
</s>
