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]
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 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 .
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]
Michael's Conjecture
The question of whether all linear multiplicative functionals on an -convex Frechet algebra are continuous is known as Michael's Conjecture.[16]. 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.[17]
^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.
^Patel, S. R. (2022-06-28). "On affirmative solution to Michael's acclaimed problem in the theory of Fréchet algebras, with applications to automatic continuity theory".
arXiv:2006.11134 [
math.FA].
Sources
Fragoulopoulou, Maria (2005). "Bibliography". Topological Algebras with Involution. North-Holland Mathematics Studies. Vol. 200. Amsterdam: Elsevier B.V. pp. 451–485.
doi:
10.1016/S0304-0208(05)80031-3.
ISBN978-044452025-8.
Husain, Taqdir (1991). Orthogonal Schauder Bases. Pure and Applied Mathematics. Vol. 143. New York City: Marcel Dekker.
ISBN0-8247-8508-8.
Michael, Ernest A. (1952). Locally Multiplicatively-Convex Topological Algebras. Memoirs of the American Mathematical Society. Vol. 11.
MR0051444.
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.
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]
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 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 .
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]
Michael's Conjecture
The question of whether all linear multiplicative functionals on an -convex Frechet algebra are continuous is known as Michael's Conjecture.[16]. 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.[17]
^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.
^Patel, S. R. (2022-06-28). "On affirmative solution to Michael's acclaimed problem in the theory of Fréchet algebras, with applications to automatic continuity theory".
arXiv:2006.11134 [
math.FA].
Sources
Fragoulopoulou, Maria (2005). "Bibliography". Topological Algebras with Involution. North-Holland Mathematics Studies. Vol. 200. Amsterdam: Elsevier B.V. pp. 451–485.
doi:
10.1016/S0304-0208(05)80031-3.
ISBN978-044452025-8.
Husain, Taqdir (1991). Orthogonal Schauder Bases. Pure and Applied Mathematics. Vol. 143. New York City: Marcel Dekker.
ISBN0-8247-8508-8.
Michael, Ernest A. (1952). Locally Multiplicatively-Convex Topological Algebras. Memoirs of the American Mathematical Society. Vol. 11.
MR0051444.
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.