Фреше алгебралары: анықтамасы, қасиеттері және түрлері

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.

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.

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 .

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.

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.