<s>
In	O
other	O
words	O
,	O
a	O
positive	B-Algorithm
linear	I-Algorithm
functional	I-Algorithm
is	O
guaranteed	O
to	O
take	O
nonnegative	O
values	O
for	O
positive	O
elements	O
.	O
</s>
<s>
The	O
significance	O
of	O
positive	B-Algorithm
linear	I-Algorithm
functionals	I-Algorithm
lies	O
in	O
results	O
such	O
as	O
Riesz	O
–	O
Markov	O
–	O
Kakutani	O
representation	O
theorem	O
.	O
</s>
<s>
As	O
in	O
the	O
case	O
when	O
is	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
with	O
its	O
partially	O
ordered	O
subspace	O
of	O
self-adjoint	O
elements	O
,	O
sometimes	O
a	O
partial	O
order	O
is	O
placed	O
on	O
only	O
a	O
subspace	O
and	O
the	O
partial	O
order	O
does	O
not	O
extend	O
to	O
all	O
of	O
in	O
which	O
case	O
the	O
positive	O
elements	O
of	O
are	O
the	O
positive	O
elements	O
of	O
by	O
abuse	O
of	O
notation	O
.	O
</s>
<s>
This	O
implies	O
that	O
for	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
,	O
a	O
positive	B-Algorithm
linear	I-Algorithm
functional	I-Algorithm
sends	O
any	O
equal	O
to	O
for	O
some	O
to	O
a	O
real	O
number	O
,	O
which	O
is	O
equal	O
to	O
its	O
complex	O
conjugate	O
,	O
and	O
therefore	O
all	O
positive	B-Algorithm
linear	I-Algorithm
functionals	I-Algorithm
preserve	O
the	O
self-adjointness	O
of	O
such	O
This	O
property	O
is	O
exploited	O
in	O
the	O
GNS	O
construction	O
to	O
relate	O
positive	B-Algorithm
linear	I-Algorithm
functionals	I-Algorithm
on	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
to	O
inner	O
products	O
.	O
</s>
<s>
There	O
is	O
a	O
comparatively	O
large	O
class	O
of	O
ordered	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
on	O
which	O
every	O
positive	O
linear	B-Algorithm
form	I-Algorithm
is	O
necessarily	O
continuous	O
.	O
</s>
<s>
This	O
includes	O
all	O
topological	B-Algorithm
vector	I-Algorithm
lattices	I-Algorithm
that	O
are	O
sequentially	B-Algorithm
complete	I-Algorithm
.	O
</s>
<s>
Then	O
each	O
of	O
the	O
following	O
conditions	O
is	O
sufficient	O
to	O
guarantee	O
that	O
every	O
positive	B-Algorithm
linear	I-Algorithm
functional	I-Algorithm
on	O
is	O
continuous	O
:	O
</s>
<s>
Consider	O
,	O
as	O
an	O
example	O
of	O
the	O
C*	B-Algorithm
-algebra	I-Algorithm
of	O
complex	O
square	B-Algorithm
matrices	I-Algorithm
with	O
the	O
positive	O
elements	O
being	O
the	O
positive-definite	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
The	O
trace	O
function	O
defined	O
on	O
this	O
C*	B-Algorithm
-algebra	I-Algorithm
is	O
a	O
positive	B-Algorithm
functional	I-Algorithm
,	O
as	O
the	O
eigenvalues	O
of	O
any	O
positive-definite	B-Algorithm
matrix	I-Algorithm
are	O
positive	O
,	O
and	O
so	O
its	O
trace	O
is	O
positive	O
.	O
</s>
<s>
Consider	O
the	O
Riesz	B-Algorithm
space	I-Algorithm
of	O
all	O
continuous	O
complex-valued	O
functions	O
of	O
compact	O
support	O
on	O
a	O
locally	O
compact	O
Hausdorff	O
space	O
Consider	O
a	O
Borel	O
regular	O
measure	O
on	O
and	O
a	O
functional	O
defined	O
by	O
Then	O
,	O
this	O
functional	O
is	O
positive	O
(	O
the	O
integral	O
of	O
any	O
positive	O
function	O
is	O
a	O
positive	O
number	O
)	O
.	O
</s>
<s>
Moreover	O
,	O
any	O
positive	B-Algorithm
functional	I-Algorithm
on	O
this	O
space	O
has	O
this	O
form	O
,	O
as	O
follows	O
from	O
the	O
Riesz	O
–	O
Markov	O
–	O
Kakutani	O
representation	O
theorem	O
.	O
</s>
<s>
Given	O
a	O
space	O
,	O
a	O
price	O
system	O
can	O
be	O
viewed	O
as	O
a	O
continuous	O
,	O
positive	O
,	O
linear	B-Algorithm
functional	I-Algorithm
on	O
.	O
</s>
