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.
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.
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.[3] - Group of invertible elements. If
is the set of invertible elements of
, then the inverse map ![{\displaystyle {\begin{cases}invA\to invA\\u\mapsto u^{-1}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67dd76a57a05f95b2a51f51692bf17f6aa83bc8b)
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
[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 ![{\displaystyle \|a_{1}a_{2}\cdots a_{n}\|_{m}\leq C_{m}^{n}\|a_{1}\|_{p}\|a_{2}\|_{p}\cdots \|a_{n}\|_{p},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ea2f6134f3b93847c590661da1dcd5c60e099d5)
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.[f][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-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 ![{\displaystyle \left\|\varphi \right\|_{n}=\left\|\varphi ^{(n)}\right\|_{\infty },\qquad \varphi \in A,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e05c2fb8a551e3d0a61e6397270c2daffadf3dd8)
where ![{\displaystyle \left\|\varphi ^{(n)}\right\|_{\infty }=\sup _{x\in {S^{1}}}\left|\varphi ^{(n)}(x)\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32f4947a8d96dbb1fae060c7dc39daee2776b8f5)
denotes the supremum of the absolute value of the
th derivative
.[g] Then, by the product rule for differentiation, we have ![{\displaystyle {\begin{aligned}\|\varphi \psi \|_{n}&=\left\|\sum _{i=0}^{n}{n \choose i}\varphi ^{(i)}\psi ^{(n-i)}\right\|_{\infty }\\&\leq \sum _{i=0}^{n}{n \choose i}\|\varphi \|_{i}\|\psi \|_{n-i}\\&\leq \sum _{i=0}^{n}{n \choose i}\|\varphi \|'_{n}\|\psi \|'_{n}\\&=2^{n}\|\varphi \|'_{n}\|\psi \|'_{n},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a71c8118dcc46a1302cdda46ea739439c9275300)
where ![{\displaystyle {n \choose i}={\frac {n!}{i!(n-i)!}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6eb9f4c5fd4123a75f60dc0d2f838a8a1f6638f1)
denotes the binomial coefficient and ![{\displaystyle \|\cdot \|'_{n}=\max _{k\leq n}\|\cdot \|_{k}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7656684961f1d46ab922f4f8b4ce12f9c3024feb)
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 ![{\displaystyle \|\varphi \|_{n}=\max _{k\leq n}|\varphi (k)|.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7567b89feb5972c1bab81280e5401373a3a4eaa)
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: ![{\displaystyle \bigcup _{n=0}^{\infty }U^{n}=G.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbf52cbea901c3819d54ac9b55ee325388aef965)
Without loss of generality, we may also assume that the identity element
of
is contained in
. Define a function
by ![{\displaystyle \ell (g)=\min\{n\mid g\in U^{n}\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06f847129cf1e0015c536767555e753c4d40b4a8)
Then
, and
, since we define
.[h] Let
be the
-vector space ![{\displaystyle S(G)={\biggr \{}\varphi :G\to \mathbb {C} \,\,{\biggl |}\,\,\|\varphi \|_{d}<\infty ,\quad d=0,1,2,\dots {\biggr \}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce9b1120b899b7657abd1c8f5a6ada7f9e98fc04)
where the seminorms
are defined by ![{\displaystyle \|\varphi \|_{d}=\|\ell ^{d}\varphi \|_{1}=\sum _{g\in G}\ell (g)^{d}|\varphi (g)|.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/015249e423e6360b437c60505cda146576276341)
[i]
is an
-convex Fréchet algebra for the convolution multiplication ![{\displaystyle \varphi *\psi (g)=\sum _{h\in G}\varphi (h)\psi (h^{-1}g),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8b7e22369546ac0b60d8b2e243dc5c0f3c742dc)
[j]
is unital because
is discrete, and
is commutative if and only if
is Abelian. - Non
-convex Fréchet algebras. The Aren's algebra ![{\displaystyle A=L^{\omega }[0,1]=\bigcap _{p\geq 1}L^{p}[0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6072f216127eadb2a64bb5ff671dc4482140861a)
is an example of a commutative non-
-convex Fréchet algebra with discontinuous inversion. The topology is given by
norms ![{\displaystyle \|f\|_{p}=\left(\int _{0}^{1}|f(t)|^{p}dt\right)^{1/p},\qquad f\in A,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23932757d59556cc7423285308b5d832b9165faa)
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.[16]
Notes
- ^ An increasing family means that for each
.
- ^ Joint continuity of multiplication means that for every absolutely convex neighborhood
of zero, there is an absolutely convex neighborhood
of zero for which
from which the seminorm inequality follows. Conversely, ![{\displaystyle {\begin{aligned}&{}\|a_{k}b_{k}-ab\|_{n}\\&=\|a_{k}b_{k}-ab_{k}+ab_{k}-ab\|_{n}\\&\leq \|a_{k}b_{k}-ab_{k}\|_{n}+\|ab_{k}-ab\|_{n}\\&\leq C_{n}{\biggl (}\|a_{k}-a\|_{m}\|b_{k}\|_{m}+\|a\|_{m}\|b_{k}-b\|_{m}{\biggr )}\\&\leq C_{n}{\biggl (}\|a_{k}-a\|_{m}\|b\|_{m}+\|a_{k}-a\|_{m}\|b_{k}-b\|_{m}+\|a\|_{m}\|b_{k}-b\|_{m}{\biggr )}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9086b8087960bb38a935dbfb707bda9c8d66360)
- ^ 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
, then
is a quasi-inverse for
if
. - ^ 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 ![{\displaystyle {\begin{aligned}&{}\psi _{l}(x)-\psi _{l}(x_{0})\\=&{}\lim _{k\to \infty }\left(\varphi _{k}^{(l)}(x)-\varphi _{k}^{(l)}(x_{0})\right)\\=&{}\lim _{k\to \infty }\int _{x_{0}}^{x}\varphi _{k}^{(l+1)}(t)dt\\=&{}\int _{x_{0}}^{x}\psi _{l+1}(t)dt.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7ceae5d8cbb4e87fdb7cc994832028e8e6f0565)
- ^ 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 ![{\displaystyle {\begin{aligned}&\sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi (g)|\\&\leq \sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi _{m}(g)|+\sum _{g\in S}\ell (g)^{d}|\varphi _{m}(g)-\varphi (g)|\\&\leq \|\varphi _{n}-\varphi _{m}\|_{d}+\sum _{g\in S}\ell (g)^{d}|\varphi _{m}(g)-\varphi (g)|,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7dbbe7f9990549b4ead72e308c90b8701150a200)
where the sum ranges over any finite subset
of
. Let
, and let
be such that
for
. By letting
run, we have ![{\displaystyle \sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi (g)|<\epsilon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1243023107709f65eaa210b55985e75bf6e10f12)
for
. Summing over all of
, we therefore have
for
. By the estimate ![{\displaystyle {\begin{aligned}&{}\sum _{g\in S}\ell (g)^{d}|\varphi (g)|\\&{}\leq \sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi (g)|+\sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)|\\&{}\leq \|\varphi _{n}-\varphi \|_{d}+\|\varphi _{n}\|_{d},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c0b60f153a91390363fa442894fdf48cc3e5eca)
we obtain
. Since this holds for each
, we have
and
in the Fréchet topology, so
is complete. - ^
![{\displaystyle {\begin{aligned}&\|\varphi *\psi \|_{d}\\&\leq \sum _{g\in G}\left(\sum _{h\in G}\ell (g)^{d}|\varphi (h)|\left|\psi (h^{-1}g)\right|\right)\\&\leq \sum _{g,h\in G}\left(\ell (h)+\ell \left(h^{-1}g\right)\right)^{d}|\varphi (h)|\left|\psi (h^{-1}g)\right|\\&=\sum _{i=0}^{d}{d \choose i}\left(\sum _{g,h\in G}\left|\ell ^{i}\varphi (h)\right|\left|\ell ^{d-i}\psi (h^{-1}g)\right|\right)\\&=\sum _{i=0}^{d}{d \choose i}\left(\sum _{h\in G}\left|\ell ^{i}\varphi (h)\right|\right)\left(\sum _{g\in G}\left|\ell ^{d-i}\psi (g)\right|\right)\\&=\sum _{i=0}^{d}{d \choose i}\|\varphi \|_{i}\|\psi \|_{d-i}\\&\leq 2^{d}\|\varphi \|'_{d}\|\psi \|'_{d}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be7b8a419f9e72441f665edf0ed7331715fec0c3)
Citations
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