Fréchet algebra

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.

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