Fréchet algebra explained

over the real or complex numbers that at the same time is also a (locally convex) Fréchet space. The multiplication operation

(a,b)\mapstoa*b

for

a,b\inA

is required to be jointly continuous.If

\{\| ⋅ \|_n

	infty
\}
	n=0

is an increasing family of seminorms forthe topology of

, the joint continuity of multiplication is equivalent to there being a constant

C_n>0

and integer

m\gen

for each

such that

for all

a,b\inA

. Fréchet algebras are also called B₀-algebras.

A Fréchet algebra is

-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:
\|ab\|_n\leq\|a\|_n\|b\|_n

for all
a,b\inA.

m

-convex Fréchet algebras may also be called Fréchet algebras.^[1]

A Fréchet algebra may or may not have an identity element

1_A

. If

is unital, we do not require that

\|1_A\|_n=1,

as is often done for Banach algebras.

Properties

Continuity of multiplication. Multiplication is separately continuous if

a_kb\toab

and

ba_k\toba

for every

a,b\inA

and sequence

a_k\toa

converging in the Fréchet topology of

. Multiplication is jointly continuous if

a_k\toa

and

b_k\tob

imply

a_kb_k\toab

. 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.^[2]

Group of invertible elements. If

invA

is the set of invertible elements of

, then the inverse map

\begin invA \to invA \\ u \mapsto u^ \end

is continuous if and only if

invA

is a

G_\delta

set. Unlike for Banach algebras,

invA

may not be an open set. If

invA

is open, then

is called a Q

-algebra. (If

happens to be non-unital, then we may adjoin a unit to

and work with

invA⁺

, or the set of quasi invertibles may take the place of

invA

Conditions for

-convexity. A Fréchet algebra is
m

-convex if and only if for every, if and only if for one, increasing family
\{\| ⋅ \|_n

infty
\}
n=0

of seminorms which topologize
A

, for each
m\in\N

there exists
p\geqm

and
C_m>0

such that $\| a_1 a_2 \cdots a_n \|_m \leq C_m^n \| a_1 \|_p \| a_2 \|_p \cdots \| a_n \|_p,$ for all
a_1,a_2,...,a_n\inA

and
n\in\N

. A commutative Fréchet
Q

-algebra is
m

-convex, but there exist examples of non-commutative Fréchet
Q

-algebras which are not
m

-convex.

Properties of

-convex Fréchet algebras. A Fréchet algebra is
m

-convex if and only if it is a countable projective limit of Banach algebras. An element of
A

is invertible if and only if its image in each Banach algebra of the projective limit is invertible.^[3]

Examples

Zero multiplication. If

is any Fréchet space, we can make a Fréchet algebra structure by setting

e*f=0

for all

e,f\inE

Smooth functions on the circle. Let

S¹

be the 1-sphere. This is a 1-dimensional compact differentiable manifold, with no boundary. Let

A=C^infty(S¹⁾

be the set of infinitely differentiable complex-valued functions on

S¹

. 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

\left\| \varphi \right\|_ = \left \| \varphi^ \right \|_, \qquad \varphi \in A,

where

\left \| \varphi^ \right \|_ = \sup_ \left |\varphi^(x) \right |

denotes the supremum of the absolute value of the

th derivative

\varphi⁽ⁿ⁾

. Then, by the product rule for differentiation, we have

\begin\| \varphi \psi \|_ &= \left \| \sum_^ \varphi^ \psi^ \right \|_ \\&\leq \sum_^ \| \varphi \|_ \| \psi \|_ \\&\leq \sum_^ \| \varphi \|'_ \| \psi \|'_ \\&= 2^n\| \varphi \|'_ \| \psi \|'_,\end

where

= \frac,

denotes the binomial coefficient and

\| \cdot \|'_ = \max_ \| \cdot \|_.

The primed seminorms are submultiplicative after re-scaling by

	n
C
	n=2

Sequences on

. Let
\Complex^\N

be the space of complex-valued sequences on the natural numbers
\N

. Define an increasing family of seminorms on
\Complex^\N

by $\| \varphi \|_n = \max_ |\varphi(k)|.$ With pointwise multiplication,
\Complex^\N

is a commutative Fréchet algebra. In fact, each seminorm is submultiplicative
\|\varphi\psi\|_n\leq\|\varphi\|_n\|\psi\|_n

for
\varphi,\psi\inA

. This
m

-convex Fréchet algebra is unital, since the constant sequence
1(k)=1,k\in\N

is in
A

.

Equipped with the topology of uniform convergence on compact sets, and pointwise multiplication,

C(\Complex)

, the algebra of all continuous functions on the complex plane

\Complex

, or to the algebra

Hol(\Complex)

of holomorphic functions on

\Complex

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

U=\{g₁,...,g_n\}\subseteqG

such that:

\bigcup_^ U^n = G.

Without loss of generality, we may also assume that the identity element

is contained in

. Define a function

\ell:G\to[0,infty)

\ell(g) = \min \.

Then

\ell(gh)\leq\ell(g)+\ell(h)

, and

\ell(e)=0

, since we define

U⁰=\{e\}

. Let

be the

\Complex

-vector space

S(G) = \biggr\,

where the seminorms

\| ⋅ \|_d

are defined by

\| \varphi \|_ = \| \ell^d \varphi \|_ =\sum_ \ell(g)^d |\varphi(g)|.

is an

-convex Fréchet algebra for the convolution multiplication

\varphi * \psi (g) = \sum_ \varphi(h) \psi(h^g),

is unital because

is discrete, and

is commutative if and only if

is Abelian.

Non

-convex Fréchet algebras. The Aren's algebra $A = L^\omega[0,1] = \bigcap_ L^p[0,1]$ is an example of a commutative non-
m

-convex Fréchet algebra with discontinuous inversion. The topology is given by
L^p

norms $\| f \|_p = \left (\int_0^1 | f(t) |^p dt \right)^, \qquad f \in A,$ and multiplication is given by convolution of functions with respect to Lebesgue measure on
[0,1]

.

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

-convex Frechet algebra are continuous is known as Michael's Conjecture.^[4] 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.^[5]

Sources

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Book: Husain, Taqdir . Orthogonal Schauder Bases . 1991 . Marcel Dekker . New York City . 143 . Pure and Applied Mathematics . 0-8247-8508-8.
Book: Michael, Ernest A. . Locally Multiplicatively-Convex Topological Algebras . 1952 . 11 . Memoirs of the American Mathematical Society . 0051444.
Entire functions in B₀-algebras . Mitiagin . B. . Rolewicz . S. . Żelazko . W. . Studia Mathematica . 1962 . 21 . 3 . 291–306 . 10.4064/sm-21-3-291-306 . 0144222 . free.
Book: Palmer, T.W. . Banach Algebras and the General Theory of *-algebras, Volume I: Algebras and Banach Algebras . 1994 . Cambridge University Press . New York City . 49 . Encyclopedia of Mathematics and its Applications . 978-052136637-3.
Book: Rudin, Walter . Functional Analysis . 1973 . McGraw-Hill Book . New York City . 1.8(e) . Series in Higher Mathematics . registration . . 978-007054236-5.
Book: Waelbroeck, Lucien . Topological Vector Spaces and Algebras . 1971 . 230 . Lecture Notes in Mathematics . 10.1007/BFb0061234 . 978-354005650-8 . 0467234.
Metric generalizations of Banach algebras . Żelazko . W. . Rozprawy Mat. (Dissertationes Math.) . 1965 . 47 . Theorem 13.17 . 0193532.
Concerning entire functions in B₀-algebras . Żelazko . W. . Studia Mathematica . 1994 . 110 . 3 . 283–290 . 10.4064/sm-110-3-283-290 . 1292849 . free.
Encyclopedia: Fréchet algebra . Żelazko . W. . . EMS Press . 2001 . 1994.

Notes and References

.
.
See also .
.
Patel . S. R. . On affirmative solution to Michael's acclaimed problem in the theory of Fréchet algebras, with applications to automatic continuity theory . 2022-06-28 . math.FA . 2006.11134.