In mathematics, especially functional analysis, a Fréchet algebra, named after Maurice René Fréchet, is an associative algebra
A
over the real or complex numbers that at the same time is also a (locally convex) Fréchet space. The multiplication operation
(
a
,
b
)
↦
a
∗
b
for
a
,
b
∈
A
is required to be jointly continuous. If
{
∥
⋅
∥
n
}
n
=
0
∞
is an increasing family of seminorms for the topology of
A
, the joint continuity of multiplication is equivalent to there being a constant
C
n
>
0
and integer
m
≥
n
for each
n
such that
∥
a
b
∥
n
≤
C
n
∥
a
∥
m
∥
b
∥
m
for all
a
,
b
∈
A
. Fréchet algebras are also called B0-algebras (Mitiagin et al. 1962) (Żelazko 2001).
A Fréchet algebra is
m
-convex if there exists such a family of semi-norms for which
m
=
n
. In that case, by rescaling the seminorms, we may also take
C
n
=
1
for each
n
and the seminorms are said to be submultiplicative:
∥
a
b
∥
n
≤
∥
a
∥
n
∥
b
∥
n
for all
a
,
b
∈
A
.
m
-convex Fréchet algebras may also be called Fréchet algebras (Husain 1991) (Żelazko 2001).
A Fréchet algebra may or may not have an identity element
1
A
. If
A
is unital, we do not require that
∥
1
A
∥
n
=
1
, as is often done for Banach algebras.
We can drop the requirement for the algebra to be locally convex, but still a complete metric space. In this case, the underlying space may be called a Fréchet space (Waelbroeck 1971) or an F-space (Rudin 1973, 1.8(e)).
If the requirement that the number of seminorms be countable is dropped, the algebra becomes locally convex (LC) or locally multiplicatively convex (LMC) (Michael 1952) (Husain 1991). A complete LMC algebra is called an Arens-Michael algebra (Fragoulopoulou 2005, Chapter 1).
Perhaps the most famous, still open problem of the theory of topological algebras is whether all linear multiplicative functionals on an
m
-convex Frechet algebra are continuous. The statement that this be the case is known as Michael's Conjecture (Michael 1952,
§
12, Question 1) (Palmer 1994,
§
3.1).
