<s>
In	O
mathematics	O
,	O
an	O
ordered	B-Algorithm
algebra	I-Algorithm
is	O
an	O
algebra	O
over	O
the	O
real	O
numbers	O
with	O
unit	O
e	O
together	O
with	O
an	O
associated	O
order	O
such	O
that	O
e	O
is	O
positive	O
(	O
i.e.	O
</s>
<s>
e	O
0	O
)	O
,	O
the	O
product	O
of	O
any	O
two	O
positive	O
elements	O
is	O
again	O
positive	O
,	O
and	O
when	O
A	O
is	O
considered	O
as	O
a	O
vector	O
space	O
over	O
then	O
it	O
is	O
an	O
Archimedean	B-Algorithm
ordered	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Let	O
A	O
be	O
an	O
ordered	B-Algorithm
algebra	I-Algorithm
with	O
unit	O
e	O
and	O
let	O
C*	O
denote	O
the	O
cone	O
in	O
A*	O
(	O
the	O
algebraic	O
dual	O
of	O
A	O
)	O
of	O
all	O
positive	O
linear	B-Algorithm
forms	I-Algorithm
on	O
A	O
.	O
</s>
<s>
If	O
f	O
is	O
a	O
linear	B-Algorithm
form	I-Algorithm
on	O
A	O
such	O
that	O
f(e )	O
=	O
1	O
and	O
f	O
generates	O
an	O
extreme	O
ray	O
of	O
C*	O
then	O
f	O
is	O
a	O
multiplicative	O
homomorphism	O
.	O
</s>
<s>
Stone	O
's	O
Algebra	O
Theorem	O
:	O
Let	O
A	O
be	O
an	O
ordered	B-Algorithm
algebra	I-Algorithm
with	O
unit	O
e	O
such	O
that	O
e	O
is	O
an	O
order	O
unit	O
in	O
A	O
,	O
let	O
A*	O
denote	O
the	O
algebraic	O
dual	O
of	O
A	O
,	O
and	O
let	O
K	O
be	O
the	O
-compact	O
set	O
of	O
all	O
multiplicative	O
positive	O
linear	B-Algorithm
forms	I-Algorithm
satisfying	O
f(e )	O
=	O
1	O
.	O
</s>
<s>
If	O
in	O
addition	O
every	O
positive	O
sequence	O
of	O
type	O
l1	O
in	O
A	O
is	O
order	B-Algorithm
summable	I-Algorithm
then	O
A	O
together	O
with	O
the	O
Minkowski	O
functional	O
pe	O
is	O
isomorphic	O
to	O
the	O
Banach	O
algebra	O
.	O
</s>
