<s>
In	O
functional	B-Application
analysis	I-Application
,	O
a	O
branch	O
of	O
mathematics	O
,	O
an	O
operator	B-Algorithm
algebra	I-Algorithm
is	O
an	O
algebra	O
of	O
continuous	O
linear	B-Architecture
operators	I-Architecture
on	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
,	O
with	O
the	O
multiplication	O
given	O
by	O
the	O
composition	B-Application
of	I-Application
mappings	I-Application
.	O
</s>
<s>
The	O
results	O
obtained	O
in	O
the	O
study	O
of	O
operator	B-Algorithm
algebras	I-Algorithm
are	O
phrased	O
in	O
algebraic	O
terms	O
,	O
while	O
the	O
techniques	O
used	O
are	O
highly	O
analytic	O
.	O
</s>
<s>
Although	O
the	O
study	O
of	O
operator	B-Algorithm
algebras	I-Algorithm
is	O
usually	O
classified	O
as	O
a	O
branch	O
of	O
functional	B-Application
analysis	I-Application
,	O
it	O
has	O
direct	O
applications	O
to	O
representation	O
theory	O
,	O
differential	B-Language
geometry	I-Language
,	O
quantum	O
statistical	O
mechanics	O
,	O
quantum	O
information	O
,	O
and	O
quantum	O
field	O
theory	O
.	O
</s>
<s>
Operator	B-Algorithm
algebras	I-Algorithm
can	O
be	O
used	O
to	O
study	O
arbitrary	O
sets	O
of	O
operators	O
with	O
little	O
algebraic	O
relation	O
simultaneously	O
.	O
</s>
<s>
From	O
this	O
point	O
of	O
view	O
,	O
operator	B-Algorithm
algebras	I-Algorithm
can	O
be	O
regarded	O
as	O
a	O
generalization	O
of	O
spectral	O
theory	O
of	O
a	O
single	O
operator	O
.	O
</s>
<s>
In	O
general	O
operator	B-Algorithm
algebras	I-Algorithm
are	O
non-commutative	O
rings	O
.	O
</s>
<s>
An	O
operator	B-Algorithm
algebra	I-Algorithm
is	O
typically	O
required	O
to	O
be	O
closed	O
in	O
a	O
specified	O
operator	O
topology	B-Architecture
inside	O
the	O
whole	O
algebra	O
of	O
continuous	O
linear	B-Architecture
operators	I-Architecture
.	O
</s>
<s>
Though	O
algebras	O
of	O
operators	O
are	O
studied	O
in	O
various	O
contexts	O
(	O
for	O
example	O
,	O
algebras	O
of	O
pseudo-differential	O
operators	O
acting	O
on	O
spaces	O
of	O
distributions	O
)	O
,	O
the	O
term	O
operator	B-Algorithm
algebra	I-Algorithm
is	O
usually	O
used	O
in	O
reference	O
to	O
algebras	O
of	O
bounded	O
operators	O
on	O
a	O
Banach	O
space	O
or	O
,	O
even	O
more	O
specially	O
in	O
reference	O
to	O
algebras	O
of	O
operators	O
on	O
a	O
separable	O
Hilbert	O
space	O
,	O
endowed	O
with	O
the	O
operator	O
norm	O
topology	B-Architecture
.	O
</s>
<s>
In	O
the	O
case	O
of	O
operators	O
on	O
a	O
Hilbert	O
space	O
,	O
the	O
Hermitian	O
adjoint	O
map	O
on	O
operators	O
gives	O
a	O
natural	O
involution	B-Algorithm
,	O
which	O
provides	O
an	O
additional	O
algebraic	O
structure	O
that	O
can	O
be	O
imposed	O
on	O
the	O
algebra	O
.	O
</s>
<s>
In	O
this	O
context	O
,	O
the	O
best	O
studied	O
examples	O
are	O
self-adjoint	O
operator	B-Algorithm
algebras	I-Algorithm
,	O
meaning	O
that	O
they	O
are	O
closed	O
under	O
taking	O
adjoints	O
.	O
</s>
<s>
These	O
include	O
C*	B-Algorithm
-algebras	I-Algorithm
,	O
von	O
Neumann	O
algebras	O
,	O
and	O
AW*	O
-algebra	O
.	O
</s>
<s>
C*	B-Algorithm
-algebras	I-Algorithm
can	O
be	O
easily	O
characterized	O
abstractly	O
by	O
a	O
condition	O
relating	O
the	O
norm	O
,	O
involution	B-Algorithm
and	O
multiplication	O
.	O
</s>
<s>
Such	O
abstractly	O
defined	O
C*	B-Algorithm
-algebras	I-Algorithm
can	O
be	O
identified	O
to	O
a	O
certain	O
closed	O
subalgebra	O
of	O
the	O
algebra	O
of	O
the	O
continuous	O
linear	B-Architecture
operators	I-Architecture
on	O
a	O
suitable	O
Hilbert	O
space	O
.	O
</s>
<s>
Commutative	O
self-adjoint	O
operator	B-Algorithm
algebras	I-Algorithm
can	O
be	O
regarded	O
as	O
the	O
algebra	O
of	O
complex-valued	O
continuous	O
functions	O
on	O
a	O
locally	O
compact	O
space	O
,	O
or	O
that	O
of	O
measurable	O
functions	O
on	O
a	O
standard	O
measurable	O
space	O
.	O
</s>
<s>
Thus	O
,	O
general	O
operator	B-Algorithm
algebras	I-Algorithm
are	O
often	O
regarded	O
as	O
a	O
noncommutative	O
generalizations	O
of	O
these	O
algebras	O
,	O
or	O
the	O
structure	O
of	O
the	O
base	O
space	O
on	O
which	O
the	O
functions	O
are	O
defined	O
.	O
</s>
<s>
This	O
point	O
of	O
view	O
is	O
elaborated	O
as	O
the	O
philosophy	O
of	O
noncommutative	O
geometry	O
,	O
which	O
tries	O
to	O
study	O
various	O
non-classical	O
and/or	O
pathological	O
objects	O
by	O
noncommutative	O
operator	B-Algorithm
algebras	I-Algorithm
.	O
</s>
<s>
Examples	O
of	O
operator	B-Algorithm
algebras	I-Algorithm
that	O
are	O
not	O
self-adjoint	O
include	O
:	O
</s>
<s>
nest	B-Algorithm
algebras	I-Algorithm
,	O
</s>
