<s>
In	O
functional	B-Application
analysis	I-Application
,	O
a	O
state	O
of	O
an	O
operator	O
system	O
is	O
a	O
positive	B-Algorithm
linear	I-Algorithm
functional	I-Algorithm
of	O
norm	O
1	O
.	O
</s>
<s>
States	O
in	O
functional	B-Application
analysis	I-Application
generalize	B-Algorithm
the	O
notion	O
of	O
density	O
matrices	O
in	O
quantum	O
mechanics	O
,	O
which	O
represent	O
quantum	O
states	O
,	O
both	O
.	O
</s>
<s>
Density	O
matrices	O
in	O
turn	O
generalize	B-Algorithm
state	O
vectors	O
,	O
which	O
only	O
represent	O
pure	O
states	O
.	O
</s>
<s>
For	O
M	O
an	O
operator	O
system	O
in	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
A	O
with	O
identity	O
,	O
the	O
set	O
of	O
all	O
states	O
of	O
M	O
,	O
sometimes	O
denoted	O
by	O
S(M )	O
,	O
is	O
convex	O
,	O
weak-*	O
closed	O
in	O
the	O
Banach	O
dual	O
space	O
M*	O
.	O
</s>
<s>
mappings	O
from	O
physical	O
observables	O
(	O
self-adjoint	O
elements	O
of	O
the	O
C*	B-Algorithm
-algebra	I-Algorithm
)	O
to	O
their	O
expected	O
measurement	O
outcome	O
(	O
real	O
number	O
)	O
.	O
</s>
<s>
States	O
can	O
be	O
viewed	O
as	O
noncommutative	O
generalizations	B-Algorithm
of	O
probability	O
measures	O
.	O
</s>
<s>
By	O
Gelfand	O
representation	O
,	O
every	O
commutative	B-Algorithm
C*	I-Algorithm
-algebra	I-Algorithm
A	O
is	O
of	O
the	O
form	O
C0(X )	O
for	O
some	O
locally	O
compact	O
Hausdorff	O
X	O
.	O
</s>
<s>
A	O
bounded	O
linear	O
functional	O
on	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
A	O
is	O
said	O
to	O
be	O
self-adjoint	O
if	O
it	O
is	O
real-valued	O
on	O
the	O
self-adjoint	O
elements	O
of	O
A	O
.	O
Self-adjoint	O
functionals	O
are	O
noncommutative	O
analogues	O
of	O
signed	O
measures	O
.	O
</s>
<s>
For	O
a	O
Hilbert	O
space	O
H	O
and	O
a	O
vector	O
x	O
in	O
H	O
,	O
the	O
equation	O
x(A )	O
:=	O
⟨Ax	O
,	O
x⟩	O
(	O
for	O
A	O
in	O
B(H )	O
)	O
,	O
defines	O
a	O
positive	B-Algorithm
linear	I-Algorithm
functional	I-Algorithm
on	O
B(H )	O
.	O
</s>
<s>
If	O
A	O
is	O
a	O
C*	O
-subalgebra	O
of	O
B(H )	O
and	O
M	O
an	O
operator	O
system	O
in	O
A	O
,	O
then	O
the	O
restriction	O
of	O
x	O
to	O
M	O
defines	O
a	O
positive	B-Algorithm
linear	I-Algorithm
functional	I-Algorithm
on	O
M	O
.	O
The	O
states	O
of	O
M	O
that	O
arise	O
in	O
this	O
manner	O
,	O
from	O
unit	O
vectors	O
in	O
H	O
,	O
are	O
termed	O
vector	O
states	O
of	O
M	O
.	O
</s>
<s>
For	O
any	O
separable	O
C*	B-Algorithm
-algebra	I-Algorithm
,	O
the	O
set	O
of	O
tracial	O
states	O
is	O
a	O
Choquet	O
simplex	O
.	O
</s>
<s>
A	O
factorial	O
state	O
of	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
A	O
is	O
a	O
state	O
such	O
that	O
the	O
commutant	O
of	O
the	O
corresponding	O
GNS	O
representation	O
of	O
A	O
is	O
a	O
factor	O
.	O
</s>
