<s>
In	O
mathematics	O
,	O
especially	O
functional	B-Application
analysis	I-Application
,	O
a	O
Fréchet	B-Algorithm
algebra	I-Algorithm
,	O
named	O
after	O
Maurice	O
René	O
Fréchet	O
,	O
is	O
an	O
associative	O
algebra	O
over	O
the	O
real	O
or	O
complex	O
numbers	O
that	O
at	O
the	O
same	O
time	O
is	O
also	O
a	O
(	O
locally	B-Algorithm
convex	I-Algorithm
)	O
Fréchet	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
the	O
topology	B-Algorithm
of	O
,	O
the	O
joint	O
continuity	O
of	O
multiplication	O
is	O
equivalent	O
to	O
there	O
being	O
a	O
constant	O
and	O
integer	O
for	O
each	O
such	O
that	O
for	O
all	O
.	O
</s>
<s>
Fréchet	B-Algorithm
algebras	I-Algorithm
are	O
also	O
called	O
B0-algebras	O
.	O
</s>
<s>
A	O
Fréchet	B-Algorithm
algebra	I-Algorithm
is	O
-convex	O
if	O
there	B-Algorithm
exists	I-Algorithm
such	O
a	O
family	O
of	O
semi-norms	O
for	O
which	O
.	O
</s>
<s>
In	O
that	O
case	O
,	O
by	O
rescaling	O
the	O
seminorms	O
,	O
we	O
may	O
also	O
take	O
for	O
each	O
and	O
the	O
seminorms	O
are	O
said	O
to	O
be	O
submultiplicative	O
:	O
for	O
all	O
-convex	O
Fréchet	B-Algorithm
algebras	I-Algorithm
may	O
also	O
be	O
called	O
Fréchet	B-Algorithm
algebras	I-Algorithm
.	O
</s>
<s>
A	O
Fréchet	B-Algorithm
algebra	I-Algorithm
may	O
or	O
may	O
not	O
have	O
an	O
identity	O
element	O
.	O
</s>
<s>
Multiplication	O
is	O
separately	O
continuous	O
if	O
and	O
for	O
every	O
and	O
sequence	O
converging	O
in	O
the	O
Fréchet	O
topology	B-Algorithm
of	O
.	O
</s>
<s>
Joint	O
continuity	O
of	O
multiplication	O
is	O
part	O
of	O
the	O
definition	O
of	O
a	O
Fréchet	B-Algorithm
algebra	I-Algorithm
.	O
</s>
<s>
For	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
with	O
an	O
algebra	O
structure	O
,	O
if	O
the	O
multiplication	O
is	O
separately	O
continuous	O
,	O
then	O
it	O
is	O
automatically	O
jointly	O
continuous	O
.	O
</s>
<s>
A	O
Fréchet	B-Algorithm
algebra	I-Algorithm
is	O
-convex	O
if	O
and	O
only	O
if	O
for	O
every	O
,	O
if	O
and	O
only	O
if	O
for	B-Algorithm
one	I-Algorithm
,	O
increasing	O
family	O
of	O
seminorms	O
which	O
topologize	O
,	O
for	O
each	O
there	B-Algorithm
exists	I-Algorithm
and	O
such	O
that	O
for	O
all	O
and	O
.	O
</s>
<s>
A	O
commutative	O
Fréchet	O
-algebra	O
is	O
-convex	O
,	O
but	O
there	B-Algorithm
exist	I-Algorithm
examples	O
of	O
non-commutative	O
Fréchet	O
-algebras	O
which	O
are	O
not	O
-convex	O
.	O
</s>
<s>
Properties	O
of	O
-convex	O
Fréchet	B-Algorithm
algebras	I-Algorithm
.	O
</s>
<s>
A	O
Fréchet	B-Algorithm
algebra	I-Algorithm
is	O
-convex	O
if	O
and	O
only	O
if	O
it	O
is	O
a	O
countable	O
projective	O
limit	O
of	O
Banach	O
algebras	O
.	O
</s>
<s>
If	O
is	O
any	O
Fréchet	B-Algorithm
space	I-Algorithm
,	O
we	O
can	O
make	O
a	O
Fréchet	B-Algorithm
algebra	I-Algorithm
structure	O
by	O
setting	O
for	O
all	O
.	O
</s>
<s>
(	O
Use	O
the	O
product	O
rule	O
for	O
differentiation	B-Algorithm
.	O
)	O
</s>
<s>
Define	O
a	O
countable	O
set	O
of	O
seminorms	O
on	O
by	O
where	O
denotes	O
the	O
supremum	O
of	O
the	O
absolute	O
value	O
of	O
the	O
th	O
derivative	B-Algorithm
.	O
</s>
<s>
Then	O
,	O
by	O
the	O
product	O
rule	O
for	O
differentiation	B-Algorithm
,	O
we	O
have	O
where	O
denotes	O
the	O
binomial	O
coefficient	O
and	O
The	O
primed	O
seminorms	O
are	O
submultiplicative	O
after	O
re-scaling	O
by	O
.	O
</s>
<s>
Let	O
be	O
the	O
space	B-Algorithm
of	I-Algorithm
complex-valued	I-Algorithm
sequences	I-Algorithm
on	O
the	O
natural	O
numbers	O
.	O
</s>
<s>
Define	O
an	O
increasing	O
family	O
of	O
seminorms	O
on	O
by	O
With	O
pointwise	O
multiplication	O
,	O
is	O
a	O
commutative	O
Fréchet	B-Algorithm
algebra	I-Algorithm
.	O
</s>
<s>
This	O
-convex	O
Fréchet	B-Algorithm
algebra	I-Algorithm
is	O
unital	O
,	O
since	O
the	O
constant	O
sequence	O
is	O
in	O
.	O
</s>
<s>
Equipped	O
with	O
the	O
topology	B-Algorithm
of	O
uniform	B-Algorithm
convergence	I-Algorithm
on	I-Algorithm
compact	I-Algorithm
sets	I-Algorithm
,	O
and	O
pointwise	O
multiplication	O
,	O
,	O
the	O
algebra	O
of	O
all	O
continuous	O
functions	O
on	O
the	O
complex	O
plane	O
,	O
or	O
to	O
the	O
algebra	O
of	O
holomorphic	O
functions	O
on	O
.	O
</s>
<s>
Let	O
be	O
a	O
finitely	O
generated	O
group	O
,	O
with	O
the	O
discrete	O
topology	B-Algorithm
.	O
</s>
<s>
This	O
means	O
that	O
there	B-Algorithm
exists	I-Algorithm
a	O
set	O
of	O
finitely	O
many	O
elements	O
such	O
that	O
:	O
Without	O
loss	O
of	O
generality	O
,	O
we	O
may	O
also	O
assume	O
that	O
the	O
identity	O
element	O
of	O
is	O
contained	O
in	O
.	O
</s>
<s>
Let	O
be	O
the	O
-vector	O
space	O
where	O
the	O
seminorms	O
are	O
defined	O
by	O
is	O
an	O
-convex	O
Fréchet	B-Algorithm
algebra	I-Algorithm
for	O
the	O
convolution	O
multiplication	O
is	O
unital	O
because	O
is	O
discrete	O
,	O
and	O
is	O
commutative	O
if	O
and	O
only	O
if	O
is	O
Abelian	O
.	O
</s>
<s>
Non	O
-convex	O
Fréchet	B-Algorithm
algebras	I-Algorithm
.	O
</s>
<s>
The	O
Aren	O
's	O
algebra	O
is	O
an	O
example	O
of	O
a	O
commutative	O
non--convex	O
Fréchet	B-Algorithm
algebra	I-Algorithm
with	O
discontinuous	O
inversion	O
.	O
</s>
<s>
The	O
topology	B-Algorithm
is	O
given	O
by	O
norms	O
and	O
multiplication	O
is	O
given	O
by	O
convolution	O
of	O
functions	O
with	O
respect	O
to	O
Lebesgue	O
measure	O
on	O
.	O
</s>
<s>
We	O
can	O
drop	O
the	O
requirement	O
for	O
the	O
algebra	O
to	O
be	O
locally	B-Algorithm
convex	I-Algorithm
,	O
but	O
still	O
a	O
complete	O
metric	O
space	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
the	O
underlying	O
space	O
may	O
be	O
called	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
or	O
an	O
F-space	B-Algorithm
.	O
</s>
<s>
If	O
the	O
requirement	O
that	O
the	O
number	O
of	O
seminorms	O
be	O
countable	O
is	O
dropped	O
,	O
the	O
algebra	O
becomes	O
locally	B-Algorithm
convex	I-Algorithm
(	O
LC	O
)	O
or	O
locally	O
multiplicatively	O
convex	O
(	O
LMC	O
)	O
.	O
</s>
<s>
Perhaps	O
the	O
most	O
famous	O
,	O
still	O
open	O
problem	O
of	O
the	O
theory	O
of	O
topological	O
algebras	O
is	O
whether	O
all	O
linear	O
multiplicative	O
functionals	O
on	O
an	O
-convex	O
Frechet	B-Algorithm
algebra	I-Algorithm
are	O
continuous	O
.	O
</s>
