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[a] 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 .[b] Fréchet algebras are also called B0-algebras.[1]
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 [c]-convex Fréchet algebras may also be called Fréchet algebras.[2]
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.[3]
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.[4] 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 [d] and work with , or the set of quasi invertibles[e] 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 .[5] A commutative Fréchet -algebra is -convex,[6] but there exist examples of non-commutative Fréchet -algebras which are not -convex.[7]
Properties of -convex Fréchet algebras. A Fréchet algebra is -convex if and only if it is a countableprojective limit of Banach algebras.[8] An element of is invertible if and only if its image in each Banach algebra of the projective limit is invertible.[f][9][10]
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-dimensionalcompactdifferentiable 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 .[g] Then, by the product rule for differentiation, we have
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 .
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 .[h] Let be the -vector space
where the seminorms are defined by
[i] is an -convex Fréchet algebra for the convolution multiplication
[j] 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 .[11]
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[12] or an F-space.[13]
If the requirement that the number of seminorms be countable is dropped, the algebra becomes locally convex (LC) or locally multiplicatively convex (LMC).[14] A complete LMC algebra is called an Arens-Michael algebra.[15]
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.[16]
^Joint continuity of multiplication means that for every absolutely convexneighborhood of zero, there is an absolutely convex neighborhood of zero for which from which the seminorm inequality follows. Conversely,
^In other words, an -convex Fréchet algebra is a topological algebra, in which the topology is given by a countable family of submultiplicative seminorms: and the algebra is complete.
^If is an algebra over a field , the unitization of is the direct sum , with multiplication defined as
^If is non-unital, replace invertible with quasi-invertible.
^To see the completeness, let be a Cauchy sequence. Then each derivative is a Cauchy sequence in the sup norm on , and hence converges uniformly to a continuous function on . It suffices to check that is the th derivative of . But, using the fundamental theorem of calculus, and taking the limit inside the integral (using uniform convergence), we have
^
We can replace the generating set with , so that . Then satisfies the additional property , and is a length function on .
^
To see that is Fréchet space, let be a Cauchy sequence. Then for each , is a Cauchy sequence in . Define to be the limit. Then
where the sum ranges over any finite subset of . Let , and let be such that for . By letting run, we have
for . Summing over all of , we therefore have for . By the estimate
we obtain . Since this holds for each , we have and in the Fréchet topology, so is complete.
Palmer, T.W. (1994). Banach Algebras and the General Theory of *-algebras, Volume I: Algebras and Banach Algebras. Encyclopedia of Mathematics and its Applications. Vol. 49. New York City: Cambridge University Press. ISBN978-052136637-3.
Rudin, Walter (1973). Functional Analysis. Series in Higher Mathematics. New York City: McGraw-Hill Book. 1.8(e). ISBN978-007054236-5 – via Internet Archive.
Waelbroeck, Lucien (1971). Topological Vector Spaces and Algebras. Lecture Notes in Mathematics. Vol. 230. doi:10.1007/BFb0061234. ISBN978-354005650-8. MR0467234.
Żelazko, W. (1965). "Metric generalizations of Banach algebras". Rozprawy Mat. (Dissertationes Math.). 47. Theorem 13.17. MR0193532.