<s>
In	O
mathematics	O
,	O
Jordan	B-Algorithm
operator	I-Algorithm
algebras	I-Algorithm
are	O
real	O
or	O
complex	O
Jordan	O
algebras	O
with	O
the	O
compatible	O
structure	O
of	O
a	O
Banach	O
space	O
.	O
</s>
<s>
The	O
axioms	O
for	O
complex	O
Jordan	B-Algorithm
operator	I-Algorithm
algebras	I-Algorithm
,	O
first	O
suggested	O
by	O
Irving	O
Kaplansky	O
in	O
1976	O
,	O
require	O
an	O
involution	O
and	O
are	O
called	O
JB*	O
algebras	O
or	O
Jordan	O
C*	B-Algorithm
algebras	I-Algorithm
.	O
</s>
<s>
By	O
analogy	O
with	O
the	O
abstract	O
characterisation	O
of	O
von	O
Neumann	O
algebras	O
as	O
C*	B-Algorithm
algebras	I-Algorithm
for	O
which	O
the	O
underlying	O
Banach	O
space	O
is	O
the	O
dual	O
of	O
another	O
,	O
there	O
is	O
a	O
corresponding	O
definition	O
of	O
JBW	B-Algorithm
algebras	I-Algorithm
.	O
</s>
<s>
Those	O
that	O
can	O
be	O
realised	O
using	O
ultraweakly	B-Algorithm
closed	I-Algorithm
Jordan	O
algebras	O
of	O
self-adjoint	O
operators	O
with	O
the	O
operator	O
Jordan	O
product	O
are	O
called	O
JW	O
algebras	O
.	O
</s>
<s>
The	O
JBW	B-Algorithm
algebras	I-Algorithm
with	O
trivial	O
center	O
,	O
so-called	O
JBW	O
factors	O
,	O
are	O
classified	O
in	O
terms	O
of	O
von	O
Neumann	O
factors	O
:	O
apart	O
from	O
the	O
exceptional	O
27	O
dimensional	O
Albert	O
algebra	O
and	O
the	O
spin	B-Algorithm
factors	I-Algorithm
,	O
all	O
other	O
JBW	O
factors	O
are	O
isomorphic	O
either	O
to	O
the	O
self-adjoint	O
part	O
of	O
a	O
von	O
Neumann	O
factor	O
or	O
to	O
its	O
fixed	O
point	O
algebra	O
under	O
a	O
period	O
two	O
*	O
-anti-automorphism	O
.	O
</s>
<s>
Jordan	B-Algorithm
operator	I-Algorithm
algebras	I-Algorithm
have	O
been	O
applied	O
in	O
quantum	O
mechanics	O
and	O
in	O
complex	O
geometry	O
,	O
where	O
Koecher	O
's	O
description	O
of	O
bounded	O
symmetric	O
domains	O
using	O
Jordan	O
algebras	O
has	O
been	O
extended	O
to	O
infinite	O
dimensions	O
.	O
</s>
<s>
A	O
Jordan	B-Algorithm
operator	I-Algorithm
algebra	I-Algorithm
is	O
a	O
norm-closed	O
subspace	O
of	O
the	O
space	O
of	O
operators	O
on	O
a	O
complex	O
Hilbert	O
space	O
,	O
closed	O
under	O
the	O
Jordan	O
product	O
a	O
∘	O
b	O
=	O
(	O
ab	O
+	O
ba	O
)	O
and	O
closed	O
in	O
the	O
operator	O
norm	O
.	O
</s>
<s>
A	O
JW	O
algebra	O
is	O
a	O
Jordan	O
subalgebra	O
of	O
the	O
Jordan	O
algebra	O
of	O
self-adjoint	O
operators	O
on	O
a	O
complex	O
Hilbert	O
space	O
that	O
is	O
closed	O
in	O
the	O
weak	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
A	O
JBW	B-Algorithm
algebra	I-Algorithm
is	O
a	O
JB	O
algebra	O
that	O
,	O
as	O
a	O
real	O
Banach	O
space	O
,	O
is	O
the	O
dual	O
of	O
a	O
Banach	O
space	O
called	O
its	O
predual	O
.	O
</s>
<s>
If	O
a	O
unital	O
JB	O
algebra	O
is	O
associative	O
,	O
then	O
its	O
complexification	O
with	O
its	O
natural	O
involution	O
is	O
a	O
commutative	O
C*	B-Algorithm
algebra	I-Algorithm
.	O
</s>
<s>
In	O
particular	O
a	O
Jordan	O
algebra	O
is	O
closed	O
in	O
the	O
weak	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
if	O
and	O
only	O
if	O
it	O
is	O
closed	O
in	O
the	O
ultraweak	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
For	O
a	O
JBW	B-Algorithm
algebra	I-Algorithm
,	O
the	O
space	O
of	O
positive	O
normal	O
functionals	O
is	O
invariant	O
under	O
the	O
quadratic	O
representation	O
Q(a )	O
b	O
=	O
 { a , b , a } 	O
.	O
</s>
<s>
An	O
idempotent	O
p	O
in	O
a	O
JBW	B-Algorithm
algebra	I-Algorithm
M	O
is	O
called	O
a	O
projection	O
.	O
</s>
<s>
If	O
p	O
is	O
a	O
projection	O
,	O
then	O
Q(p )	O
M	O
is	O
a	O
JBW	B-Algorithm
algebra	I-Algorithm
with	O
identity	O
p	O
.	O
</s>
<s>
If	O
a	O
is	O
any	O
element	O
of	O
a	O
JBW	B-Algorithm
algebra	I-Algorithm
,	O
the	O
smallest	O
weakly	O
closed	O
unital	O
subalgebra	O
it	O
generates	O
is	O
associative	O
and	O
hence	O
the	O
self-adjoint	O
part	O
of	O
an	O
Abelian	O
von	O
Neumann	O
algebra	O
.	O
</s>
<s>
The	O
projections	O
in	O
a	O
JBW	B-Algorithm
algebra	I-Algorithm
are	O
closed	O
under	O
lattice	O
operations	O
.	O
</s>
<s>
The	O
center	O
of	O
a	O
JBW	B-Algorithm
algebra	I-Algorithm
M	O
consists	O
of	O
all	O
z	O
such	O
L(z )	O
commutes	O
with	O
L(a )	O
for	O
a	O
in	O
M	O
.	O
It	O
is	O
an	O
associative	O
algebra	O
and	O
the	O
real	O
part	O
of	O
an	O
Abelian	O
von	O
Neumann	O
algebra	O
.	O
</s>
<s>
A	O
JBW	B-Algorithm
algebra	I-Algorithm
is	O
called	O
a	O
factor	O
if	O
its	O
center	O
consists	O
of	O
scalar	O
operators	O
.	O
</s>
<s>
If	O
A	O
is	O
a	O
JB	O
algebra	O
,	O
its	O
second	O
dual	O
A**	O
is	O
a	O
JBW	B-Algorithm
algebra	I-Algorithm
.	O
</s>
<s>
Moreover	O
,	O
A**	O
is	O
the	O
JBW	B-Algorithm
algebra	I-Algorithm
generated	O
by	O
A	O
.	O
</s>
<s>
A	O
JB	O
algebra	O
is	O
a	O
JBW	B-Algorithm
algebra	I-Algorithm
if	O
and	O
only	O
if	O
,	O
as	O
a	O
real	O
Banach	O
space	O
,	O
it	O
is	O
the	O
dual	O
of	O
a	O
Banach	O
space	O
.	O
</s>
<s>
As	O
a	O
consequence	O
the	O
weak	O
and	O
strong	O
topologies	O
coincide	O
on	O
a	O
JBW	B-Algorithm
algebra	I-Algorithm
.	O
</s>
<s>
In	O
a	O
JBW	B-Algorithm
algebra	I-Algorithm
,	O
the	O
JBW	B-Algorithm
algebra	I-Algorithm
generated	O
by	O
a	O
Jordan	O
subalgebra	O
coincides	O
with	O
its	O
weak	O
closure	O
.	O
</s>
<s>
Moreover	O
,	O
an	O
extension	O
of	O
the	O
Kaplansky	O
density	O
theorem	O
holds	O
:	O
the	O
unit	O
ball	O
of	O
the	O
subalgebra	O
is	O
weakly	O
dense	O
in	O
the	O
unit	O
ball	O
of	O
the	O
JBW	B-Algorithm
algebra	I-Algorithm
it	O
generates	O
.	O
</s>
<s>
Tomita	O
–	O
Takesaki	O
theory	O
has	O
been	O
extended	O
by	O
to	O
normal	O
states	O
of	O
a	O
JBW	B-Algorithm
algebra	I-Algorithm
that	O
are	O
faithful	O
,	O
i.e.	O
</s>
<s>
The	O
JBW	O
factors	O
of	O
Type	O
I2	O
are	O
the	O
spin	B-Algorithm
factors	I-Algorithm
.	O
</s>
<s>
can	O
be	O
realized	O
as	O
Jordan	O
algebras	O
of	O
self-adjoint	O
operators	O
on	O
a	O
Hilbert	O
space	O
closed	O
in	O
the	O
weak	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
.	O
</s>
