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 for a,b \in A is required to be jointly continuous. If 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 \ge n for each n such that for all a, b \in A. Fréchet algebras are also called B0-algebras. 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: for all a, b \in A. m-convex Fréchet algebras may also be called Fréchet algebras. A Fréchet algebra may or may not have an identity element 1_A. If A is unital, we do not require that as is often done for Banach algebras.

Properties

Examples

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.

Michael's Conjecture

The question of whether all linear multiplicative functionals on an m-convex Frechet algebra are continuous is known as Michael's Conjecture. For a long time, this conjecture was perhaps the most famous open problem in the theory of topological algebras. Michael's Conjecture was solved completely and affirmatively in 2022.

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