<s>
In	O
mathematics	O
,	O
specifically	O
in	O
functional	B-Application
analysis	I-Application
,	O
a	O
C∗	O
-algebra	O
(	O
pronounced	O
"	O
C-star	O
"	O
)	O
is	O
a	O
Banach	O
algebra	O
together	O
with	O
an	O
involution	B-Algorithm
satisfying	O
the	O
properties	O
of	O
the	O
adjoint	O
.	O
</s>
<s>
A	O
particular	O
case	O
is	O
that	O
of	O
a	O
complex	O
algebra	O
A	O
of	O
continuous	B-Algorithm
linear	I-Algorithm
operators	I-Algorithm
on	O
a	O
complex	O
Hilbert	O
space	O
with	O
two	O
additional	O
properties	O
:	O
</s>
<s>
Another	O
important	O
class	O
of	O
non-Hilbert	O
C*	B-Algorithm
-algebras	I-Algorithm
includes	O
the	O
algebra	O
of	O
complex-valued	O
continuous	O
functions	O
on	O
X	O
that	O
vanish	O
at	O
infinity	O
,	O
where	O
X	O
is	O
a	O
locally	O
compact	O
Hausdorff	O
space	O
.	O
</s>
<s>
C*	B-Algorithm
-algebras	I-Algorithm
were	O
first	O
considered	O
primarily	O
for	O
their	O
use	O
in	O
quantum	O
mechanics	O
to	O
model	O
algebras	O
of	O
physical	O
observables	O
.	O
</s>
<s>
These	O
papers	O
considered	O
a	O
special	O
class	O
of	O
C*	B-Algorithm
-algebras	I-Algorithm
which	O
are	O
now	O
known	O
as	O
von	O
Neumann	O
algebras	O
.	O
</s>
<s>
Around	O
1943	O
,	O
the	O
work	O
of	O
Israel	O
Gelfand	O
and	O
Mark	O
Naimark	O
yielded	O
an	O
abstract	O
characterisation	O
of	O
C*	B-Algorithm
-algebras	I-Algorithm
making	O
no	O
reference	O
to	O
operators	O
on	O
a	O
Hilbert	O
space	O
.	O
</s>
<s>
C*	B-Algorithm
-algebras	I-Algorithm
are	O
now	O
an	O
important	O
tool	O
in	O
the	O
theory	O
of	O
unitary	O
representations	O
of	O
locally	O
compact	O
groups	O
,	O
and	O
are	O
also	O
used	O
in	O
algebraic	O
formulations	O
of	O
quantum	O
mechanics	O
.	O
</s>
<s>
Another	O
active	O
area	O
of	O
research	O
is	O
the	O
program	O
to	O
obtain	O
classification	O
,	O
or	O
to	O
determine	O
the	O
extent	O
of	O
which	O
classification	O
is	O
possible	O
,	O
for	O
separable	O
simple	O
nuclear	O
C*	B-Algorithm
-algebras	I-Algorithm
.	O
</s>
<s>
We	O
begin	O
with	O
the	O
abstract	O
characterization	O
of	O
C*	B-Algorithm
-algebras	I-Algorithm
given	O
in	O
the	O
1943	O
paper	O
by	O
Gelfand	O
and	O
Naimark	O
.	O
</s>
<s>
A	O
C*	B-Algorithm
-algebra	I-Algorithm
,	O
A	O
,	O
is	O
a	O
Banach	O
algebra	O
over	O
the	O
field	O
of	O
complex	O
numbers	O
,	O
together	O
with	O
a	O
map	B-Algorithm
for	O
with	O
the	O
following	O
properties	O
:	O
</s>
<s>
It	O
is	O
an	O
involution	B-Algorithm
,	O
for	O
every	O
x	O
in	O
A	O
:	O
</s>
<s>
For	O
history	O
behind	O
the	O
names	O
C*	O
-	O
and	O
B*	B-Algorithm
-algebras	I-Algorithm
,	O
see	O
the	O
history	O
section	O
below	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
C*	B-Algorithm
-algebras	I-Algorithm
,	O
any	O
*	O
-homomorphism	O
π	O
between	O
C*	B-Algorithm
-algebras	I-Algorithm
is	O
contractive	O
,	O
i.e.	O
</s>
<s>
Furthermore	O
,	O
an	O
injective	O
*	O
-homomorphism	O
between	O
C*	B-Algorithm
-algebras	I-Algorithm
is	O
isometric	O
.	O
</s>
<s>
The	O
term	O
B*	B-Algorithm
-algebra	I-Algorithm
was	O
introduced	O
by	O
C	O
.	O
E	O
.	O
Rickart	O
in	O
1946	O
to	O
describe	O
Banach	O
*	O
-algebras	O
that	O
satisfy	O
the	O
condition	O
:	O
</s>
<s>
for	O
all	O
x	O
in	O
the	O
given	O
B*	B-Algorithm
-algebra	I-Algorithm
.	O
</s>
<s>
This	O
condition	O
automatically	O
implies	O
that	O
the	O
*	O
-involution	O
is	O
isometric	O
,	O
that	O
is	O
,	O
.	O
</s>
<s>
Hence	O
,	O
,	O
and	O
therefore	O
,	O
a	O
B*	B-Algorithm
-algebra	I-Algorithm
is	O
also	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
.	O
</s>
<s>
For	O
these	O
reasons	O
,	O
the	O
term	O
B*	B-Algorithm
-algebra	I-Algorithm
is	O
rarely	O
used	O
in	O
current	O
terminology	O
,	O
and	O
has	O
been	O
replaced	O
by	O
the	O
term	O
'	O
C*-algebra	O
 '	O
.	O
</s>
<s>
The	O
term	O
C*	B-Algorithm
-algebra	I-Algorithm
was	O
introduced	O
by	O
I	O
.	O
E	O
.	O
Segal	O
in	O
1947	O
to	O
describe	O
norm-closed	O
subalgebras	O
of	O
B(H )	O
,	O
namely	O
,	O
the	O
space	O
of	O
bounded	O
operators	O
on	O
some	O
Hilbert	O
space	O
H	O
.	O
'	O
C	O
 '	O
stood	O
for	O
'	O
closed	O
 '	O
.	O
</s>
<s>
In	O
his	O
paper	O
Segal	O
defines	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
as	O
a	O
"	O
uniformly	O
closed	O
,	O
self-adjoint	O
algebra	O
of	O
bounded	O
operators	O
on	O
a	O
Hilbert	O
space	O
"	O
.	O
</s>
<s>
C*	B-Algorithm
-algebras	I-Algorithm
have	O
a	O
large	O
number	O
of	O
properties	O
that	O
are	O
technically	O
convenient	O
.	O
</s>
<s>
Some	O
of	O
these	O
properties	O
can	O
be	O
established	O
by	O
using	O
the	O
continuous	B-Algorithm
functional	I-Algorithm
calculus	O
or	O
by	O
reduction	O
to	O
commutative	O
C*	B-Algorithm
-algebras	I-Algorithm
.	O
</s>
<s>
The	O
set	O
of	O
elements	O
of	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
A	O
of	O
the	O
form	O
forms	O
a	O
closed	O
convex	O
cone	O
.	O
</s>
<s>
The	O
set	O
of	O
self-adjoint	O
elements	O
of	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
A	O
naturally	O
has	O
the	O
structure	O
of	O
a	O
partially	O
ordered	O
vector	O
space	O
;	O
the	O
ordering	O
is	O
usually	O
denoted	O
.	O
</s>
<s>
This	O
partially	O
ordered	O
subspace	O
allows	O
the	O
definition	O
of	O
a	O
positive	B-Algorithm
linear	I-Algorithm
functional	I-Algorithm
on	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
,	O
which	O
in	O
turn	O
is	O
used	O
to	O
define	O
the	O
states	B-Algorithm
of	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
,	O
which	O
in	O
turn	O
can	O
be	O
used	O
to	O
construct	O
the	O
spectrum	O
of	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
using	O
the	O
GNS	O
construction	O
.	O
</s>
<s>
Any	O
C*	B-Algorithm
-algebra	I-Algorithm
A	O
has	O
an	O
approximate	O
identity	O
.	O
</s>
<s>
Using	O
approximate	O
identities	O
,	O
one	O
can	O
show	O
that	O
the	O
algebraic	O
quotient	O
of	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
by	O
a	O
closed	O
proper	O
two-sided	O
ideal	O
,	O
with	O
the	O
natural	O
norm	O
,	O
is	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
.	O
</s>
<s>
Similarly	O
,	O
a	O
closed	O
two-sided	O
ideal	O
of	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
is	O
itself	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
.	O
</s>
<s>
The	O
algebra	O
M(n, C )	O
of	O
n	O
×	O
n	O
matrices	B-Architecture
over	O
C	O
becomes	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
if	O
we	O
consider	O
matrices	B-Architecture
as	O
operators	O
on	O
the	O
Euclidean	O
space	O
,	O
Cn	O
,	O
and	O
use	O
the	O
operator	O
norm	O
||||	O
on	O
matrices	B-Architecture
.	O
</s>
<s>
The	O
involution	B-Algorithm
is	O
given	O
by	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
.	O
</s>
<s>
In	O
fact	O
,	O
all	O
C*	B-Algorithm
-algebras	I-Algorithm
that	O
are	O
finite	O
dimensional	O
as	O
vector	O
spaces	O
are	O
of	O
this	O
form	O
,	O
up	O
to	O
isomorphism	O
.	O
</s>
<s>
The	O
self-adjoint	O
requirement	O
means	O
finite-dimensional	O
C*	B-Algorithm
-algebras	I-Algorithm
are	O
semisimple	O
,	O
from	O
which	O
fact	O
one	O
can	O
deduce	O
the	O
following	O
theorem	O
of	O
Artin	O
–	O
Wedderburn	O
type	O
:	O
</s>
<s>
Each	O
C*	B-Algorithm
-algebra	I-Algorithm
,	O
Ae	O
,	O
is	O
isomorphic	O
(	O
in	O
a	O
noncanonical	O
way	O
)	O
to	O
the	O
full	O
matrix	O
algebra	O
M(dim(e )	O
,	O
C	O
)	O
.	O
</s>
<s>
This	O
vector	O
uniquely	O
determines	O
the	O
isomorphism	O
class	O
of	O
a	O
finite-dimensional	O
C*	B-Algorithm
-algebra	I-Algorithm
.	O
</s>
<s>
A	O
†	O
-algebra	O
(	O
or	O
,	O
more	O
explicitly	O
,	O
a	O
†	O
-closed	O
algebra	O
)	O
is	O
the	O
name	O
occasionally	O
used	O
in	O
physics	O
for	O
a	O
finite-dimensional	O
C*	B-Algorithm
-algebra	I-Algorithm
.	O
</s>
<s>
An	O
immediate	O
generalization	O
of	O
finite	O
dimensional	O
C*	B-Algorithm
-algebras	I-Algorithm
are	O
the	O
approximately	O
finite	O
dimensional	O
C*	B-Algorithm
-algebras	I-Algorithm
.	O
</s>
<s>
The	O
prototypical	O
example	O
of	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
is	O
the	O
algebra	O
B(H )	O
of	O
bounded	O
(	O
equivalently	O
continuous	O
)	O
linear	B-Architecture
operators	I-Architecture
defined	O
on	O
a	O
complex	O
Hilbert	O
space	O
H	O
;	O
here	O
x*	O
denotes	O
the	O
adjoint	O
operator	O
of	O
the	O
operator	O
x	O
:	O
H	O
→	O
H	O
.	O
In	O
fact	O
,	O
every	O
C*	B-Algorithm
-algebra	I-Algorithm
,	O
A	O
,	O
is	O
*	O
-isomorphic	O
to	O
a	O
norm-closed	O
adjoint	O
closed	O
subalgebra	O
of	O
B(H )	O
for	O
a	O
suitable	O
Hilbert	O
space	O
,	O
H	O
;	O
this	O
is	O
the	O
content	O
of	O
the	O
Gelfand	O
–	O
Naimark	O
theorem	O
.	O
</s>
<s>
It	O
is	O
also	O
closed	O
under	O
involution	B-Algorithm
;	O
hence	O
it	O
is	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
.	O
</s>
<s>
Concrete	O
C*	B-Algorithm
-algebras	I-Algorithm
of	O
compact	O
operators	O
admit	O
a	O
characterization	O
similar	O
to	O
Wedderburn	O
's	O
theorem	O
for	O
finite	O
dimensional	O
C*	B-Algorithm
-algebras	I-Algorithm
:	O
</s>
<s>
The	O
space	O
of	O
complex-valued	O
continuous	O
functions	O
on	O
X	O
that	O
vanish	O
at	O
infinity	O
(	O
defined	O
in	O
the	O
article	O
on	O
local	O
compactness	O
)	O
form	O
a	O
commutative	B-Algorithm
C*	I-Algorithm
-algebra	I-Algorithm
under	O
pointwise	O
multiplication	O
and	O
addition	O
.	O
</s>
<s>
The	O
involution	B-Algorithm
is	O
pointwise	O
conjugation	O
.	O
</s>
<s>
As	O
does	O
any	O
C*	B-Algorithm
-algebra	I-Algorithm
,	O
has	O
an	O
approximate	O
identity	O
.	O
</s>
<s>
The	O
Gelfand	O
representation	O
states	B-Algorithm
that	O
every	O
commutative	B-Algorithm
C*	I-Algorithm
-algebra	I-Algorithm
is	O
*	O
-isomorphic	O
to	O
the	O
algebra	O
,	O
where	O
is	O
the	O
space	O
of	O
characters	O
equipped	O
with	O
the	O
weak*	O
topology	O
.	O
</s>
<s>
Furthermore	O
,	O
if	O
is	O
isomorphic	O
to	O
as	O
C*	B-Algorithm
-algebras	I-Algorithm
,	O
it	O
follows	O
that	O
and	O
are	O
homeomorphic	O
.	O
</s>
<s>
Given	O
a	O
Banach	O
*	O
-algebra	O
A	O
with	O
an	O
approximate	O
identity	O
,	O
there	O
is	O
a	O
unique	O
(	O
up	O
to	O
C*	O
-isomorphism	O
)	O
C*	B-Algorithm
-algebra	I-Algorithm
E(A )	O
and	O
*	O
-morphism	O
π	O
from	O
A	O
into	O
E(A )	O
that	O
is	O
universal	O
,	O
that	O
is	O
,	O
every	O
other	O
continuous	O
*	O
-morphism	O
factors	O
uniquely	O
through	O
π	O
.	O
</s>
<s>
Of	O
particular	O
importance	O
is	O
the	O
C*	B-Algorithm
-algebra	I-Algorithm
of	O
a	O
locally	O
compact	O
group	O
G	O
.	O
This	O
is	O
defined	O
as	O
the	O
enveloping	O
C*	B-Algorithm
-algebra	I-Algorithm
of	O
the	O
group	O
algebra	O
of	O
G	O
.	O
The	O
C*	B-Algorithm
-algebra	I-Algorithm
of	O
G	O
provides	O
context	O
for	O
general	O
harmonic	O
analysis	O
of	O
G	O
in	O
the	O
case	O
G	O
is	O
non-abelian	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
dual	O
of	O
a	O
locally	O
compact	O
group	O
is	O
defined	O
to	O
be	O
the	O
primitive	O
ideal	O
space	O
of	O
the	O
group	O
C*	B-Algorithm
-algebra	I-Algorithm
.	O
</s>
<s>
See	O
spectrum	O
of	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
.	O
</s>
<s>
Von	O
Neumann	O
algebras	O
,	O
known	O
as	O
W*	O
algebras	O
before	O
the	O
1960s	O
,	O
are	O
a	O
special	O
kind	O
of	O
C*	B-Algorithm
-algebra	I-Algorithm
.	O
</s>
<s>
They	O
are	O
required	O
to	O
be	O
closed	O
in	O
the	O
weak	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
,	O
which	O
is	O
weaker	O
than	O
the	O
norm	O
topology	O
.	O
</s>
<s>
The	O
Sherman	O
–	O
Takeda	O
theorem	O
implies	O
that	O
any	O
C*	B-Algorithm
-algebra	I-Algorithm
has	O
a	O
universal	O
enveloping	O
W*	O
-algebra	O
,	O
such	O
that	O
any	O
homomorphism	O
to	O
a	O
W*	O
-algebra	O
factors	O
through	O
it	O
.	O
</s>
<s>
A	O
C*	B-Algorithm
-algebra	I-Algorithm
A	O
is	O
of	O
type	O
I	O
if	O
and	O
only	O
if	O
for	O
all	O
non-degenerate	O
representations	O
π	O
of	O
A	O
the	O
von	O
Neumann	O
algebra	O
π(A )	O
′′	O
(	O
that	O
is	O
,	O
the	O
bicommutant	O
of	O
π(A )	O
)	O
is	O
a	O
type	O
I	O
von	O
Neumann	O
algebra	O
.	O
</s>
<s>
A	O
locally	O
compact	O
group	O
is	O
said	O
to	O
be	O
of	O
type	O
I	O
if	O
and	O
only	O
if	O
its	O
group	O
C*	B-Algorithm
-algebra	I-Algorithm
is	O
type	O
I	O
.	O
</s>
<s>
However	O
,	O
if	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
has	O
non-type	O
I	O
representations	O
,	O
then	O
by	O
results	O
of	O
James	O
Glimm	O
it	O
also	O
has	O
representations	O
of	O
type	O
II	O
and	O
type	O
III	O
.	O
</s>
<s>
Thus	O
for	O
C*	B-Algorithm
-algebras	I-Algorithm
and	O
locally	O
compact	O
groups	O
,	O
it	O
is	O
only	O
meaningful	O
to	O
speak	O
of	O
type	O
I	O
and	O
non	O
type	O
I	O
properties	O
.	O
</s>
<s>
In	O
quantum	O
mechanics	O
,	O
one	O
typically	O
describes	O
a	O
physical	O
system	O
with	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
A	O
with	O
unit	O
element	O
;	O
the	O
self-adjoint	O
elements	O
of	O
A	O
(	O
elements	O
x	O
with	O
x*	O
=	O
x	O
)	O
are	O
thought	O
of	O
as	O
the	O
observables	O
,	O
the	O
measurable	O
quantities	O
,	O
of	O
the	O
system	O
.	O
</s>
<s>
A	O
state	O
of	O
the	O
system	O
is	O
defined	O
as	O
a	O
positive	B-Algorithm
functional	I-Algorithm
on	O
A	O
(	O
a	O
C-linear	O
map	O
φ	O
:	O
A	O
→	O
C	O
with	O
φ( u*u	O
)	O
≥	O
0	O
for	O
all	O
u	O
∈	O
A	O
)	O
such	O
that	O
φ(1 )	O
=	O
1	O
.	O
</s>
<s>
This	O
C*	B-Algorithm
-algebra	I-Algorithm
approach	O
is	O
used	O
in	O
the	O
Haag-Kastler	O
axiomatization	O
of	O
local	O
quantum	O
field	O
theory	O
,	O
where	O
every	O
open	O
set	O
of	O
Minkowski	O
spacetime	O
is	O
associated	O
with	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
.	O
</s>
