In mathematics, especially functional analysis, a Fréchet algebra, named after Maurice René Fréchet, is an associative algebra 
over the real or complex numbers that at the same time is also a (locally convex) Fréchet space. The multiplication operation 
for 
is required to be jointly continuous.
If 
is an increasing family of seminorms for
the topology of , the joint continuity of multiplication is equivalent to there being a constant 
and integer 
for each 
such that 
for all . Fréchet algebras are also called B0-algebras.
A Fréchet algebra is 
-convex if there exists such a family of semi-norms for which 
. In that case, by rescaling the seminorms, we may also take 
for each and the seminorms are said to be submultiplicative: 
for all 
-convex Fréchet algebras may also be called Fréchet algebras.
A Fréchet algebra may or may not have an identity element 
. If is unital, we do not require that 
as is often done for Banach algebras.
Properties
-
Continuity of multiplication. Multiplication is separately continuous if
and 
for every and sequence 
converging in the Fréchet topology of . Multiplication is jointly continuous if and 
imply 
. Joint continuity of multiplication is part of the definition of a Fréchet algebra. For a Fréchet space with an algebra structure, if the multiplication is separately continuous, then it is automatically jointly continuous.
-
Group of invertible elements. If
is the set of invertible elements of , then the inverse map 
is continuous if and only if is a 
set. Unlike for Banach algebras, may not be an open set. If is open, then is called a 
-algebra. (If happens to be non-unital, then we may adjoin a unit to and work with 
, or the set of quasi invertibles may take the place of .)
-
Conditions for -convexity. A Fréchet algebra is -convex if and only if for every, if and only if for one, increasing family of seminorms which topologize , for each
there exists 
and 
such that 
for all 
and 
. A commutative Fréchet -algebra is -convex, but there exist examples of non-commutative Fréchet -algebras which are not -convex.
-
Properties of -convex Fréchet algebras. A Fréchet algebra is -convex if and only if it is a countable projective limit of Banach algebras. An element of is invertible if and only if its image in each Banach algebra of the projective limit is invertible.
Examples
-
Zero multiplication. If
is any Fréchet space, we can make a Fréchet algebra structure by setting 
for all 
.
-
Smooth functions on the circle. Let
be the 1-sphere. This is a 1-dimensional compact differentiable manifold, with no boundary. Let 
be the set of infinitely differentiable complex-valued functions on . This is clearly an algebra over the complex numbers, for pointwise multiplication. (Use the product rule for differentiation.) It is commutative, and the constant function 
acts as an identity. Define a countable set of seminorms on by 
where 
denotes the supremum of the absolute value of the th derivative 
. Then, by the product rule for differentiation, we have 
where 
denotes the binomial coefficient and 
The primed seminorms are submultiplicative after re-scaling by 
.
-
Sequences on
. Let 
be the space of complex-valued sequences on the natural numbers . Define an increasing family of seminorms on by 
With pointwise multiplication, is a commutative Fréchet algebra. In fact, each seminorm is submultiplicative 
for 
. This -convex Fréchet algebra is unital, since the constant sequence 
is in .
- Equipped with the topology of uniform convergence on compact sets, and pointwise multiplication,
, the algebra of all continuous functions on the complex plane 
, or to the algebra 
of holomorphic functions on .
-
Convolution algebra of rapidly vanishing functions on a finitely generated discrete group. Let
be a finitely generated group, with the discrete topology. This means that there exists a set of finitely many elements 
such that: 
Without loss of generality, we may also assume that the identity element 
of is contained in 
. Define a function 
by 
Then 
, and 
, since we define 
. Let be the -vector space 
where the seminorms 
are defined by 
is an -convex Fréchet algebra for the convolution multiplication 
is unital because is discrete, and is commutative if and only if is Abelian.
-
Non -convex Fréchet algebras. The Aren's algebra
is an example of a commutative non--convex Fréchet algebra with discontinuous inversion. The topology is given by 
norms 
and multiplication is given by convolution of functions with respect to Lebesgue measure on 
.
Generalizations
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 or an F-space.
If the requirement that the number of seminorms be countable is dropped, the algebra becomes locally convex (LC) or locally multiplicatively convex (LMC). A complete LMC algebra is called an Arens-Michael algebra.
Open problems
Perhaps the most famous, still open problem of the theory of topological algebras is whether all linear multiplicative functionals on an -convex Frechet algebra are continuous. The statement that this be the case is known as Michael's Conjecture.