Essential amenability of Fréchet algebras
UDC 517.98 Essential amenability of Banach algebras have been defined and investigated. Here, this concept will be introduced for Frechet algebras. Then a number of well-known results of essential amenability of Banach algebras are generalized for Fréchet algebras. Moreover, related results about Se...
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Essential amenability of Banach algebras have been defined and investigated. Here, this concept will be introduced for Frechet algebras. Then a number of well-known results of essential amenability of Banach algebras are generalized for Fréchet algebras. Moreover, related results about Segal–Fréchet algebras are provided. As the main result, it is provedthat if $(\mathcal{A} , p_{\ell})$ is an amenable Fréchet algebra with a uniformly bounded approximate identity, then every symmetric Segal – Fréchet algebra in $(\mathcal{A} , p_{\ell})$ is essentially amenable. |
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DOI: 10.37863/umzh.v72i7.830
UDC 517.98
F. Abtahi, S. Rahnama (Dep. Pure Math., Univ. Isfahan, Iran)
ESSENTIAL AMENABILITY OF FRÉCHET ALGEBRAS
СУТТЄВА АМЕНАБЕЛЬНIСТЬ АЛГЕБР ФРЕШЕ
Essential amenability of Banach algebras have been defined and investigated. Here, this concept will be introduced for
Fréchet algebras. Then a number of well-known results of essential amenability of Banach algebras are generalized for
Fréchet algebras. Moreover, related results about Segal – Fréchet algebras are provided. As the main result, it is proved
that if (\scrA , p\ell ) is an amenable Fréchet algebra with a uniformly bounded approximate identity, then every symmetric
Segal – Fréchet algebra in (\scrA , p\ell ) is essentially amenable.
Суттєву аменабельнiсть банахових алгебр було визначено та вивчено ранiше. Тут цю концепцiю визначено для
алгебр Фреше. Пiсля цього ряд вiдомих результатiв iз аменабельностi банахових алгебр узагальнено на випадок
алгебр Фреше. Також наведено результати, якi стосуються алгебр Сiгала – Фреше. Основним є твердження про те,
що у випадку, коли (\scrA , p\ell ) — аменабельна алгебра Фреше з рiвномiрно обмеженою наближеною тотожнiстю, всi
симетричнi алгебри Сiгала – Фреше у (\scrA , p\ell ) є суттєво аменабельними.
1. Introduction. Amenability of Fréchet algebras was introduced by Helemskii in [7] (Defini-
tion 2.16) and was studied by A. Yu. Pirkovskii in [10]. Also in [8], P. Lawson and C. J. Read intro-
duced and studied approximate amenability and approximate contractibility of Fréchet algebras. In
[1], we studied the notion of weak amenability of Fréchet algebras and obtained some results on weak
amenability of Fréchet algebras. Also according to the basic definition of Segal algebras and abstract
Segal algebras, we introduced the concept of a Segal – Fréchet algebra in the Fréchet algebra (\scrA , p\ell ).
We then showed that every continuous linear left multiplier on a Fréchet algebra (\scrA , p\ell ) is a continu-
ous linear left multiplier on any Segal – Fréchet algebra in (\scrA , p\ell ). Moreover, we showed that if \scrA is
a commutative Fréchet Q-algebra, then the space of all modular maximal closed ideals of \scrA and any
Segal – Fréchet algebra (\scrB , qm) in (\scrA , p\ell ) are homeomorphic. In particular, we proved that (\scrA , p\ell )
is semisimple if and only if (\scrB , qm) is semisimple (see [2]). Recently, we introduced the concept of
character contractibility of Fréchet algebras, according to its definition in the Banach case [3]. We
then verified available results about right \varphi -contractibility and right character contractibility of Ba-
nach algebras for Fréchet algebras. Finally, we provided related results about Segal – Fréchet algebras.
In [5], Ghahramani and Loy introduced and investigated the notion of essential amenability of
Banach algebra. Then Samea [12] continued this verification and as a main result, he generalized
[5] (Theorem 7.1). In fact he proved that any symmetric abstract Segal algebra with respect to an
amenable Banach algebra is essentially amenable [12] (Theorem 4.4).
In the present work, we first introduce the concept of essential amenability of Fréchet algebras.
Our definition of essential amenability coincides with the Banach algebra case, whenever Banach
algebra \scrA is considered as a Fréchet algebra. Then we verify most of the available results in the
Banach algebra case, for Fréchet algebras. The last section contains the main results of this paper.
We first recall from [2], the concept of Segal – Fréchet algebra in a Fréchet algebra (\scrA , p\ell ). We
c\bigcirc F. ABTAHI, S. RAHNAMA, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7 867
868 F. ABTAHI, S. RAHNAMA
show that a proper Segal – Fréchet algebra in (\scrA , p\ell ) cannot contain a right (left, two-sided) locally
bounded approximate identity. This result is a stronger result in comparison with the known result
about abstract Segal algebras [4] (Theorem 1.2). Then we study the results of [12] for Segal – Fréchet
algebras. As the main result we verify [12] (Theorem 4.4) for the Fréchet algebra case and prove
that if (\scrA , p\ell ) is an amenable Fréchet algebra with a uniformly bounded approximate identity, then
every symmetric Segal – Fréchet algebra in (\scrA , p\ell ) is essentially amenable. At the end, we provide
some examples of essentially amenable Fréchet algebras, which are in fact Segal – Fréchet algebras
in some Banach algebras.
2. Preliminaries. A topological algebra \scrA is an algebra, that is also a topological vector space
and the multiplication \scrA \times \scrA - \rightarrow \scrA , defined by (a, b) \mapsto \rightarrow ab is separately continuous. Moreover, a
Fréchet algebra is a complete topological algebra, such that its topology is defined by the countable
family of increasing submultiplicative seminorms; see, for example, [6]. Note that our definition of
a Fréchet algebra is presented hear, with a slight modification of that given in [10]. Indeed by [10],
a Fréchet algebra is a complete topological algebra, such that its topology is given by a countable
family of increasing (not necessarily submultiplicative) seminorms. In fact the definition of a Fréchet
algebra in this paper coincides with the concept of Fréchet – Arens – Michael (or m-convex) algebras,
given in [10].
Note that a closed subalgebra F of a Fréchet algebra (\scrA , p\ell ), is always a Fréchet algebra under
the restricted seminorms p\ell on F. Moreover, if I is a proper closed ideal of \scrA , then \scrA /I is a Fréchet
space and its topology is defined by the seminorms
\^p\ell (a+ I) = \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
p\ell (a+ b) : b \in I
\bigr\}
.
In fact (\scrA /I, \^p\ell ) is a Fréchet algebra (see [6], 3.2.10).
A net (e\alpha )\alpha in a Fréchet algebra (\scrA , p\ell ) is called right approximate identity if a = \mathrm{l}\mathrm{i}\mathrm{m}\alpha ae\alpha ,
for all a \in \scrA . Left approximate identities are defined similarly. A net (e\alpha )\alpha is called two-sided
approximate identity (or just an approximate identity) if it is both left and right approximate identity.
An approximate identity (e\alpha )\alpha (right, left, or two-sided) is bounded if the set \{ e\alpha \} is a bounded set
in \scrA . Furthermore, an approximate identity (e\alpha ) in a Fréchet algebra is called uniformly bounded if
\mathrm{s}\mathrm{u}\mathrm{p}
\ell \in \BbbN
\mathrm{s}\mathrm{u}\mathrm{p}
\alpha
p\ell (e\alpha ) < \infty .
Note that most of the Fréchet algebras with a bounded uniformly approximate identity are unital.
However, C\infty (\Omega ) is a non-unital Fréchet algebra which possesses a bounded uniformly approximate
identity. It is obvious that both concepts of bounded approximate identity and uniformly bounded
approximate identity are the same, in Banach algebras.
Let (\scrA , p\ell ) be a Fréchet algebra. A Fréchet \scrA -bimodule is a Fréchet space X together with the
structure of an \scrA -bimodule, such that the corresponding mappings are separately continuous. We
call X a Banach \scrA -bimodule, in the case where X is a Banach space. A Banach \scrA -bimodule X is
called neo-unital if
X = \scrA .X.\scrA = \{ a.x.b : a, b \in \scrA , x \in X\} .
Note that if I is a closed ideal of \scrA , then (\scrA /I, \^p\ell ) is a Fréchet \scrA -bimodule, with the following
module actions:
b.(a+ I) = ba+ I and (a+ I).b = ab+ I
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
ESSENTIAL AMENABILITY OF FRÉCHET ALGEBRAS 869
for all a, b \in \scrA (see [16] for more details). Also the quotient map q : \scrA \rightarrow \scrA /I, defined by
a \mapsto \rightarrow a+ I, is a continuous \scrA -bimodule epimorphism.
Now consider X\ast , the dual space of X, with the module actions given by
\langle a.f, x\rangle = \langle f, x.a\rangle , \langle f.a, x\rangle = \langle f, a.x\rangle
for all a \in \scrA , x \in X and f \in X\ast . As it is mentioned in [10], by [16] (3.1) if X is a Banach
\scrA -bimodule, then so is X\ast .
3. Essential amenability of Fréchet algebras. Let (\scrA , p\ell ) be a Fréchet algebra and X be a
Banach \scrA -bimodule. According to [7] a continuous derivation of \scrA into X is a continuous linear
mapping D from \scrA into X such that
D(ab) = a.D(b) +D(a).b
for all a, b \in \scrA . For each x \in X the mapping Dx : \scrA \rightarrow X defined by
Dx(a) = a.x - x.a, a \in \scrA ,
is a continuous derivation and is called the inner derivation associated with x.
Many concepts related to Banach algebras, have been introduced and studied for Fréchet algebras.
In all of these generalizations, it has been observed that these definitions become compatible, whe-
never a Banach algebra is considered as a Fréchet algebra. Here, we introduce the concept of essential
amenability for Fréchet algebras, according to its definition for the Banach algebra case. Recall that
a Banach algebra \scrA is called essentially amenable if for every neo-unital Banach \scrA -bimodule X,
each continuous derivation of \scrA into X\ast is inner.
Definition 3.1. Let (\scrA , p\ell ) be a Fréchet algebra. We call \scrA , essentially amenable if for every
neo-unital Banach \scrA -bimodule X, each continuous derivation of \scrA into X\ast is inner.
It is obvious that if \scrA is an essentially amenable Banach algebra then \scrA is essentially amenable,
when considered as a Fréchet algebra. Moreover by [10] (Theorem 9.8), (\scrA , p\ell ) is amenable if and
only if for each Banach \scrA -bimodule X, every continuous derivation from \scrA to X\ast is inner. It
follows that every amenable Fréchet algebra is essentially amenable.
By [11] (Proposition 2.1.5), if a Banach algebra \scrA has a bounded approximate identity, then
\scrA is amenable if and only if \scrA is essentially amenable. In fact both concepts of amenability and
essential amenability are coincided. The main key in the proof of [11] (Proposition 2.1.5) is Cohen
factorization theorem. It is worth noting that Cohen factorization theorem is also valid for the
Fréchet algebras, having a uniformly bounded approximate identity (see [17]). Thus one can prove
the following result, with the same arguments as in the proof of [11] (Proposition 2.1.5), and so the
proof is left to the reader.
Theorem 3.1. Let (\scrA , p\ell ) be a Fréchet algebra with a uniformly bounded approximate identity.
Then \scrA is amenable if and only if \scrA is essentially amenable.
The following result is a generalization of [5] (Proposition 2.2), for the Fréchet algebra case. The
proof is similar and is left to the reader.
Proposition 3.1. Let (\scrA , p\ell ) and (\scrB , qm) be Fréchet algebras and \Phi : \scrA \rightarrow \scrB be a continuous
epimorphism. If \scrA is essentially amenable then \scrB is essentially amenable.
Since for every closed two-sided ideal I of a Fréchet algebra (\scrA , p\ell ), the quotient map q :
\scrA \rightarrow \scrA /I is a continuous epimorphism, thus the following result is obtained.
Corollary 3.1. Let (\scrA , p\ell ) be an essentially amenable Fréchet algebra and I be a closed two-
sided ideal of \scrA . Then (\scrA /I, \^p\ell ) is essentially amenable.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
870 F. ABTAHI, S. RAHNAMA
4. Essential amenability of Segal – Fréchet algebras. Let (\scrA , p\ell ) be a Fréchet algebra. Ac-
cording to [2], a Fréchet algebra (\scrB , qm) is a Segal – Fréchet algebra in (\scrA , p\ell ), if the following
conditions are satisfied:
(i) \scrB is a dense left ideal in \scrA ;
(ii) the map
i : (\scrB , qm) - \rightarrow (\scrA , p\ell ), a \mapsto \rightarrow a a \in \scrB ,
is continuous;
(iii) the map
(\scrB , p\ell )\times (\scrB , qm) - \rightarrow (\scrB , qm), (a, b) \mapsto \rightarrow ab, a, b \in \scrB , (4.1)
is jointly continuous.
It is not hard to see that the implication (4.1) implies that the map
(\scrA , p\ell )\times (\scrB , qm) - \rightarrow (\scrB , qm), (a, b) \mapsto \rightarrow ab, a \in \scrA , b \in \scrB ,
is also jointly continuous. Moreover (\scrB , qm) is a symmetric Segal – Fréchet algebra in (\scrA , p\ell ) if \scrB
is a dense two-sided ideal in \scrA , and the map
(\scrB , qm)\times (\scrB , p\ell ) - \rightarrow (\scrB , qm), (a, b) \mapsto \rightarrow ab, a, b \in \scrB ,
is jointly continuous.
Note that the concept of Segal – Fréchet algebra corresponds to the concept of abstract Segal
algebra, in the case where \scrA and \scrB are Banach algebras.
We investigate the results of [12], for the Segal – Fréchet algebras. Recall that for Fréchet algebra
(\scrA , p\ell ),
\scrA .\scrA = \{ a.b : a, b \in \scrA \} .
Moreover, \scrA 2 is the linear span of \scrA .\scrA . We also denote by \scrA 2
\scrA
, the closure of \scrA 2 in \scrA . We
commence with the following lemma which is a generalization of [12] (Lemma 3.1). The proof is
similar and is left to the reader.
Lemma 4.1. Let (\scrA , p\ell ) be a Fréchet algebra such that \scrA .\scrA is dense in \scrA , and (\scrB , qm) be a
Segal – Fréchet algebra in (\scrA , p\ell ). Then B2
\scrB
is a Segal – Fréchet algebra in (\scrA , p\ell ).
We recall from [10], the concept of locally bounded approximate identity. A topological algebra
\scrA has a right (respectively, left) locally bounded approximate identity if for each neighborhood U
of the zero element of \scrA , there exists CU > 0 such that for each finite subset F of \scrA , there exists
bF \in CUU with a - abF \in U (respectively, a - bFa \in U ) for all a \in F. We say that \scrA has a
locally bounded approximate identity, if for each neighborhood U of the zero element of \scrA , there
exists CU > 0 such that for each finite subset F of \scrA , there exists bF \in CUU with a - abF \in U
and a - bFa \in U for all a \in F.
It is not hard to see that if \scrA has a locally bounded approximate identity, then \scrA .\scrA is dense in
\scrA . Thus, we have the following result from Lemma 4.1, immediately.
Corollary 4.1. Let (\scrA , p\ell ) be a Fréchet algebra with a locally bounded approximate identity, and
(\scrB , qm) be a Segal – Fréchet algebra in (\scrA , p\ell ). Then B2
\scrB
is a Segal – Fréchet algebra in (\scrA , p\ell ).
Remark 4.1. Let (\scrA , p\ell ) be a Fréchet algebra.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
ESSENTIAL AMENABILITY OF FRÉCHET ALGEBRAS 871
1. By [10] (Proposition 5.2), \scrA has a right (respectively, left) approximate identity if and only
if, for each finite subset F \subseteq \scrA and each zero neighborhood U \subseteq \scrA , there exists b \in \scrA such that
a - ab \in U (a - ba \in U) for all a \in F. Moreover, \scrA has a right (respectively, left) bounded
approximate identity if and only if there exists a bounded subset B \subseteq \scrA such that for each finite
subset F \subseteq \scrA and each zero neighborhood U \subseteq \scrA , there exists b \in B such that a - ab \in U
(a - ba \in U) for all a \in F.
2. By [10] (Remark 6.4), the existence of a bounded right (respectively, left, two-sided) ap-
proximate identity implies the existence of a right (respectively, left, two-sided) locally bounded
approximate identity, which, in turn, implies the existence of a right (respectively, left, two-sided)
approximate identity. But the existence of a right (respectively, left, two-sided) locally bounded
approximate identity does not imply, in general, the existence of a bounded right (respectively, left,
two-sided) approximate identity (see [10], Proposition 10.4).
The following lemma is useful in application. One can easily prove it, similar to [10] (Proposi-
tion 5.2) and also [10] (Remark 5.3).
Lemma 4.2. Let (\scrA , p\ell ) be a Fréchet algebra and C \subseteq \scrA . Then \scrA has a right (respectively,
left) approximate identity with the elements in C if and only if for each finite subset F \subseteq \scrA and each
zero neighborhood U \subseteq \scrA there exists c \in C such that a - ac \in U (a - ca \in U) for all a \in F.
Moreover, \scrA has a two-sided approximate identity with the elements in C if and only if for each
finite subset F \subseteq \scrA and each zero neighborhood U \subseteq \scrA there exists c \in C such that a - ac \in U
and a - ca \in U for all a \in F.
The following result is known for Banach algebras, that we generalize it for Fréchet algebras.
Proposition 4.1. Let (\scrA , p\ell ) be a Fréchet algebra with a right (respectively, left, two-sided)
approximate identity and (\scrB , qm) be a Segal – Fréchet algebra in (\scrA , p\ell ). Then (\scrA , p\ell ) has a right
(respectively, left, two-sided) approximate identity with elements in \scrB .
Proof. We only prove the right version. By Lemma 4.2, it is sufficient to show that for each
finite subset F \subseteq \scrA and each zero neighborhood U \subseteq \scrA there exists b \in \scrB such that a - ab \in U
for all a \in F. Let F = \{ a1, . . . , an\} be a finite subset of \scrA and U = p - 1
\ell ([0, \varepsilon )), where \varepsilon > 0 is
arbitrary. By the hypothesis and also Remark 4.1, part 1, for V = p - 1
\ell
\biggl( \biggl[
0,
\varepsilon
2
\biggr) \biggr)
there is aF \in \scrA
such that ai - aiaF \in V for all i = 1, . . . , n. Suppose that
K = \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
p\ell (ai) : i = 1, . . . , n
\bigr\}
.
By the density of \scrB in \scrA , there is bF \in \scrB such that
p\ell (aF - bF ) \leq
\varepsilon
2K
.
Now, for each i = 1, . . . , n, we have
p\ell (aibF - ai) \leq p\ell (aibF - aiaF ) + p\ell (aiaF - ai) \leq
\leq p\ell (ai)p\ell (bF - aF ) +
\varepsilon
2
\leq
\leq K
\varepsilon
2K
+
\varepsilon
2
= \varepsilon .
It follows that aibF - ai \in U for all i = 1, . . . , n.
Proposition 4.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
872 F. ABTAHI, S. RAHNAMA
The following result is obtained immediately from Remark 4.1 and Proposition 4.1.
Corollary 4.2. Let (\scrA , p\ell ) be a Fréchet algebra with a locally bounded approximate identity,
and (\scrB , qm) be a Segal – Fréchet algebra in (\scrA , p\ell ). Then (\scrA , p\ell ) has an approximate identity with
elements in \scrB .
In the next proposition, we extend [12] (Lemma 3.2) to the Fréchet algebra case.
Proposition 4.2. Let (\scrA , p\ell ) be a Fréchet algebra with a uniformly bounded left approximate
identity, and (\scrB , qm) be a Segal – Fréchet algebra in (\scrA , p\ell ). Then there is a left approximate identity
for (\scrA , p\ell ) with elements in B2
\scrB
, which is also a left approximate identity for the Segal – Fréchet
algebra
\Bigl(
B2
\scrB
, qm
\Bigr)
.
Proof. Let (e\alpha ) be a uniformly bounded left approximate identity for \scrA . By the Cohen’s
factorization theorem [17] (Theorem 3), \scrA .\scrA = \scrA . It is easily verified that \scrB .\scrB is dense in \scrA . So,
for each \ell \in \BbbN , \varepsilon > 0 and \alpha , there exists e(\alpha ,\ell ,\varepsilon ) \in \scrB .\scrB such that
p\ell (e\alpha - e(\alpha ,\ell ,\varepsilon )) < \mathrm{m}\mathrm{i}\mathrm{n}\{ 1, \varepsilon \} . (4.2)
Since (p\ell )\ell \in \BbbN is an increasing sequence of seminorms, thus, for all k \leq \ell ,
pk(e\alpha - e(\alpha ,\ell ,\varepsilon )) \leq p\ell (e\alpha - e(\alpha ,\ell ,\varepsilon )) < \mathrm{m}\mathrm{i}\mathrm{n}\{ 1, \varepsilon \} (4.3)
and so for each k \leq \ell
pk(e(\alpha ,\ell ,\varepsilon )) \leq p\ell (e\alpha ) + \varepsilon . (4.4)
Then (e(\alpha ,\ell ,\varepsilon ))(\alpha ,\ell ,\varepsilon ) is a directed net in \scrB .\scrB with (\alpha 1, \ell 1, \varepsilon 1) \preceq (\alpha 2, \ell 2, \varepsilon 2) if and only if \alpha 1 \preceq \alpha 2,
\ell 1 \leq \ell 2 and \varepsilon 2 \leq \varepsilon 1. By the hypothesis, there is K > 0 such that
\mathrm{s}\mathrm{u}\mathrm{p}
\ell
\mathrm{s}\mathrm{u}\mathrm{p}
\alpha
p\ell (e\alpha ) \leq K. (4.5)
Let n0 \in \BbbN and a \in \scrA be fixed. There is \alpha 0 such that, for all \alpha \geq \alpha 0,
pn0(e\alpha a - a) < \varepsilon .
By using (4.2) and (4.3), for all \alpha \geq \alpha 0 and \ell \geq n0, we have
pn0(e(\alpha ,\ell ,\varepsilon )a - a) \leq pn0(e(\alpha ,\ell ,\varepsilon )a - e\alpha a) + pn0(e\alpha a - a) <
< \varepsilon + pn0(a)pn0(e(\alpha ,\ell ,\varepsilon ) - e\alpha ) < \varepsilon + pn0(a)\varepsilon .
Since \varepsilon > 0 is arbitrary, it follows that (e(\alpha ,\ell ,\varepsilon ))(\alpha ,\ell ,\varepsilon ) is a left approximate identity for \scrA . Now let
m \in \BbbN be fixed. Since \scrB is a Segal – Fréchet algebra, by (4.1), there exist Cm > 0 and \ell 0,m0 \in \BbbN
such that
qm(b1b2) \leq Cmp\ell 0(b1)qm0(b2) (4.6)
for all b1, b2 \in \scrB . Thus,
qm(e(\alpha ,\ell ,\varepsilon )b1b2 - b1b2) \leq Cmp\ell 0(e(\alpha ,\ell ,\varepsilon )b1 - b1)qm0(b2) - - - - \rightarrow
(\alpha ,\ell ,\varepsilon )
0.
Consequently, for each b \in \scrB 2,
qm(e(\alpha ,\ell ,\varepsilon )b - b) - - - - \rightarrow
(\alpha ,\ell ,\varepsilon )
0. (4.7)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
ESSENTIAL AMENABILITY OF FRÉCHET ALGEBRAS 873
Now suppose that b \in \scrB 2
\scrB
and \delta > 0. By the density of \scrB 2 in \scrB 2
\scrB
, there is b\delta \in \scrB 2 such that
qmax\{ m,m0\} (b - b\delta ) < \delta .
Therefore, by using (4.6), we obtain
qm(b - e(\alpha ,\ell ,\varepsilon )b) \leq qm(b - b\delta ) + qm(b\delta - e(\alpha ,\ell ,\varepsilon )b\delta ) + qm(e(\alpha ,\ell ,\varepsilon )b\delta - e(\alpha ,\ell ,\varepsilon )b) \leq
\leq \delta + qm(b\delta - e(\alpha ,\ell ,\varepsilon )b\delta ) + Cmp\ell 0(e(\alpha ,\ell ,\varepsilon ))\delta .
Consequently, by (4.3) – (4.5) and (4.7) we get
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
(\alpha ,\ell ,\varepsilon )
qm(b - e(\alpha ,\ell ,\varepsilon )b) \leq \delta (1 + Cm \mathrm{s}\mathrm{u}\mathrm{p}
(\alpha ,\ell ,\varepsilon )
(p\ell 0(e\alpha ) + 1)) \leq
\leq \delta (1 + Cm(K + 1)).
Since \delta > 0 is arbitrary, it follows that, for each m \in \BbbN ,
\mathrm{l}\mathrm{i}\mathrm{m}
(\alpha ,\ell ,\varepsilon )
qm(b - e(\alpha ,\ell ,\varepsilon )b) = 0,
which implies that (e(\alpha ,\ell ,\varepsilon )) is a left approximate identity for \scrB 2.
Proposition 4.2 is proved.
Theorem 4.1. Let (\scrA , p\ell ) be a Fréchet algebra with a uniformly bounded approximate identity,
and (\scrB , qm) be a symmetric Segal – Fréchet algebra in (\scrA , p\ell ). Then B2
\scrB
is a symmetric Segal –
Fréchet algebra in (\scrA , p\ell ) such that there exists an approximate identity for B2
\scrB
, which is also an
approximate identity for (\scrA , p\ell ).
Proof. By similar methods, one can show that if in Corollary 4.1, (\scrB , qm) is a symmetric
Segal – Fréchet algebra in (\scrA , p\ell ), then B2
\scrB
is also symmetric Segal – Fréchet algebra in (\scrA , p\ell ).
Moreover, (e(\alpha ,\ell ,\varepsilon ))(\alpha ,\ell ,\varepsilon ), constructed in the proof of Proposition 4.2, is also a right approximate
identity for both (\scrA , p\ell ) and also
\Bigl(
B2
\scrB
, qm
\Bigr)
.
Theorem 4.1 is proved.
A known result due to Burnham asserts that a proper abstract Segal algebra cannot contain a
bounded (right, left, two-sided) approximate identity [4] (Theorem 1.2). In the following result, we
prove a stronger result about Segal – Fréchet algebras. In fact we show that a proper Segal – Fréchet
algebra does not contain even a right (left, two-sided) locally bounded approximate identity.
Proposition 4.3. Let (\scrA , p\ell ) be a Fréchet algebra and (\scrB , qm) be a proper Segal – Fréchet al-
gebra in (\scrA , p\ell ). Then (\scrB , qm) does not contain a right (left, two-sided) locally bounded approximate
identity.
Proof. Suppose on the contrary that \scrB contains a right locally bounded approximate identity.
For a \in \scrA , there is a sequence (bn) in \scrB such that bn \rightarrow a, in the topology of \scrA . We show that
(bn) is a Cauchy sequence in \scrB . For m \in \BbbN by (4.1), there exist \ell 0,m0 \in \BbbN and Km > 0 such that
qm(bc) \leq Kmp\ell 0(b)qm0(c) (4.8)
for all b, c \in \scrB . For \varepsilon > 0, set
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
874 F. ABTAHI, S. RAHNAMA
U = q - 1
m
\bigl(
[0, \varepsilon )
\bigr)
\cap q - 1
m0
\bigl(
[0, \varepsilon )
\bigr)
.
Thus by the hypothesis, there is Cm,\varepsilon > 0 such that for each F = \{ b\} \subseteq \scrB , there is bF \in Cm,\varepsilon U
with
b - bbF \in U.
Thus,
qm(b - bbF ) < \varepsilon and qm0(bF ) < \varepsilon Cm,\varepsilon . (4.9)
By (4.8) and (4.9), we obtain
qm(b) < qm(bbF ) + \varepsilon \leq Kmp\ell 0(b)qm0(bF ) + \varepsilon \leq
\bigl(
KmCm,\varepsilon p\ell 0(b) + 1
\bigr)
\varepsilon .
Note that the above inequalities are independent from b. It follows that for each b \in \scrB
qm(b) \leq
\bigl(
KmCm,\varepsilon p\ell 0(b) + 1
\bigr)
\varepsilon .
Since (bn) is convergent in \scrA , thus for \ell 0 and \eta =
1
KmCm,\varepsilon
, there is N \in \BbbN such that, for all
n \geq N,
p\ell 0(b - bn) < \eta .
Thus, for each n \geq N, we have
qm(b - bn) \leq (KmCm,\varepsilon \eta + 1)\varepsilon = 2\varepsilon .
It follows that (bn) is a Cauchy sequence with respect to the seminorm qm. Thus, (bn) is Cauchy
with respect to all the seminorm qm, which implies that (bn) is Cauchy in the topology of \scrB . Thus,
(bn) is convergent in the topology of \scrB . Since this sequence is convergent to a, in the topology of
\scrA , it follows that a \in \scrB . Therefore, \scrB = \scrA , which is a contradiction. Similarly, \scrB can not contain
left or two-sided locally bounded approximate identity.
Proposition 4.3 is proved.
Lemma 4.3. Let (\scrA , p\ell ) be a Fréchet algebra with a locally bounded approximate identity, and
(\scrB , qm) be a Segal – Fréchet algebra in (\scrA , p\ell ). Then the following assertions are equivalent:
(i) (\scrB , qm) has a locally bounded approximate identity;
(ii) \scrB = \scrA , as sets;
(iii) (\scrB , qm) is a Fréchet algebra isomorphic to (\scrA , p\ell ).
Proof. (i) \Rightarrow (ii). It is obtained by Proposition 4.3.
(ii) \Rightarrow (iii). Consider the identity map \iota : \scrB \rightarrow \scrA . By the definition of a Segal – Fréchet algebra,
\iota is a continuous bijection. Now open mapping theorem for Fréchet spaces implies that \iota is an
isomorphism.
(iii) \Rightarrow (i). It is trivial.
The following proposition is a generalization of [12] (Proposition 4.1). The proof is similar and
is left to the reader.
Proposition 4.4. Let (\scrA , p\ell ) be a Fréchet algebra and I be a closed subalgebra of \scrA that
contains \scrA .\scrA . If I is essentially amenable, then so is \scrA .
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ESSENTIAL AMENABILITY OF FRÉCHET ALGEBRAS 875
We state here the main result of the present work. In fact we generalize [12] (Theorem 4.4) to
the Fréchet algebra case.
Theorem 4.2. Let (\scrA , p\ell ) be an amenable Fréchet algebra with a uniformly bounded approxi-
mate identity and (\scrB , qm) be a symmetric Segal – Fréchet algebra in (\scrA , p\ell ). Then \scrB is essentially
amenable. Moreover, (\scrB , qm) is amenable if and only if \scrA = \scrB , as two sets.
Proof. Suppose that (\scrB , qm) is amenable. Then by [10] (Theorem 9.7), \scrB has a locally
bounded approximate identity. By Proposition 4.3 and Lemma 4.3, \scrB = \scrA . Conversely suppose that
\scrA = \scrB , as two sets. Then by Lemma 4.3, (\scrB , qm) is a Fréchet algebra isomorphic to (\scrA , p\ell ). It
follows that \scrB is amenable. Now suppose that \scrB is not amenable. We show that \scrB is essentially
amenable. By Theorem 4.1, J = B2
\scrB
is a symmetric Segal – Fréchet algebra in \scrA . Moreover, J has
an approximate identity (e\alpha )\alpha , which is an approximate identity for \scrA . Now let X be a neo-unital
Banach J -bimodule and also let D : J \rightarrow X\ast be a continuous derivation. Some arguments similar
to the proof of [5] (Theorem 7.1), show that X is an \scrA -bimodule. In fact for each a \in \scrA and
x \in X, we define a.x = (ab1).y1 and also x.a = y2.(b2a), where x = b1.y1 and x = y2.b2, for
some b1, b2 \in J and y1, y2 \in X. Then one can use the closed graph theorem [9] (Theorem 8.8), to
show that the module actions are continuous. Take x \in X and a \in \scrA and write x = b.y, for some
b \in J and y \in X. Then as the proof of [5] (Theorem 7.1) we obtain\bigl\langle
D(ab) - a.D(b), y
\bigr\rangle
= \mathrm{l}\mathrm{i}\mathrm{m}
\alpha
\bigl\langle
D(e\alpha a), x
\bigr\rangle
.
Define \widetilde D : \scrA \rightarrow X\ast by \bigl\langle \widetilde D(a), x
\bigr\rangle
=
\bigl\langle
D(ab) - a.D(b), y
\bigr\rangle
,
where x = b.y, for some b \in J and y \in X. With the reasons exactly similar to those given
in [5] (Theorem 7.1), such as using the uniform boundedness theorem and closed graph theorem
[9] (Theorem 8.8), we obtain that \widetilde D is a well defined continuous linear map, which is in fact an
extension of D to \scrA . Since J is dense in \scrA it follows that \widetilde D must also be a derivation. But then by
the amenability of \scrA , \widetilde D is inner. It follows that D is inner. Therefore, J is essentially amenable.
Since J is a closed two-sided ideal of \scrB that contains \scrB .\scrB , thus \scrB is essentially amenable, by
Proposition 4.4.
Theorem 4.2 is proved.
We end this paper with some examples of essentially amenable Fréchet algebras.
Examples 4.1. 1. We explain here part (b) of [13] (Example 3.3), which is a nice example in
the field of Fréchet algebras. Let X be a countable set. A function \sigma : X \rightarrow [1,\infty ) is called a scale
on X. If \sigma = \{ \sigma n\} \infty n=0 is a family of scales on X, define the Fréchet space
\scrS \infty
\sigma (X) =
\bigl\{
\varphi : X \rightarrow \BbbC , \| \varphi \| \infty n < \infty , n \in \BbbN
\bigr\}
,
where
\| \varphi \| \infty n = \mathrm{s}\mathrm{u}\mathrm{p}
x\in X
\bigl\{
\sigma n(x)| \varphi (x)|
\bigr\}
.
Then \scrS \infty
\sigma (X) is called the sup-norm \sigma -rapidly vanishing functions on X. The family \sigma will satisfy
\sigma 0 \leq \sigma 1 \leq . . . \leq \sigma n \leq . . . , so that the families of norms
\bigl\{
\| .\| \infty n
\bigr\} \infty
n=0
are increasing. Moreover,
it is easy to see that all of them are submultiplicative under pointwise multiplication. Consequently,
\scrS \infty
\sigma (X) is a Fréchet algebra. Now let \scrA = c0(X) be the commutative Banach algebra of complex-
valued sequences which vanish at infinity, with pointwise multiplication and sup-norm \| .\| \scrA = \| .\| \infty .
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
876 F. ABTAHI, S. RAHNAMA
Also let \scrB be \scrS \infty
\sigma (X), topologized by the sup-norms
\| f\| n = \| \sigma nf\| \infty .
Then it is easy to see that the inequalities \| fg\| n \leq \| f\| n\| g\| \infty are satisfied for all n \in \BbbN . Moreover,
\scrB is a dense Fréchet ideal in \scrA . In fact \scrB is a Segal – Fréchet algebra in \scrA . It is known that c0(X)
is a commutative C\ast -algebra. All commutative C\ast -algebras are amenable and admit an approximate
identity bounded by 1 (see [11]). So c0(X) is amenable in the sense of a Fréchet algebra and has
uniformly bounded approximate identity. Now Theorem 4.2 implies that \scrS \infty
\sigma (X) is an essentially
amenable Segal – Fréchet algebra.
2. Let G be a compact connected Lie group. Also let C\infty (G) be the space consisting of all
infinitely differentiable functions on G. We refer to [15] for a full information about the construction
of C\infty (G). Consider the general group algebra L1(G) under convolution product. It is known that
L1(G) has a bounded approximate identity and since G is compact, L1(G) is an amenable Banach
algebra. Thus L1(G) is amenable as a Fréchet algebra [10] (Theorem 9.6), which has a uniformly
bounded approximate identity. By [13] (Example 8.4) and [14], C\infty (G) is a Fréchet algebra under
convolution product, which satisfies all the conditions of a Segal – Fréchet algebra in \scrA = L1(G).
Now Theorem 4.2 implies that C\infty (G) is an essentially amenable Segal – Fréchet algebra.
References
1. F. Abtahi, S. Rahnama, A. Rejali, Weak amenability of Fréchet algebras, U.P.B. Sci. Bull., Ser. A, 77, № 4, 93 – 104
(2015).
2. F. Abtahi, S. Rahnama, A. Rejali, Semisimple Segal – Fréchet algebras, Period. Math. Hungar., 71, 146 – 154 (2015).
3. F. Abtahi, S. Rahnama, \varphi –Contractibility and character contractibility of Fréchet algebras, Ann. Funct. Anal., 8,
№ 1, 75 – 89 (2017).
4. J. T. Burnham, Closed ideals in subalgebras of Banach algebras, Proc. Amer. Math. Soc., 32, № 2, 551 – 555 (1972).
5. F. Ghahramani, R. J. Loy, Generalized notions of amenability, J. Funct. Anal., 208, 229 – 260 (2004).
6. H. Goldmann, Uniform Fréchet algebras, North-Holland Math. Stud., 162, North-Holand, Amsterdam; New York
(1990).
7. A. Ya. Helemskii, The homology of Banach and topological algebras, Kluwer Acad. Publ., Dordrecht (1989).
8. P. Lawson, C. J. Read, Approximate amenability of Fréchet algebras, Math. Proc. Cambridge Phil. Soc., 145, 403 – 418
(2008).
9. R. Meise, D. Vogt, Introduction to functional analysis, Oxford Sci. Publ. (1997).
10. A. Yu. Pirkovskii, Flat cyclic Fréchet modules, amenable Fréchet algebras, and approximate identities, Homology,
Homotopy and Appl., 11, № 1, 81 – 114 (2009).
11. V. Runde, Lectures on amenability, Springer-Verlag, Berlin; Heidelberg (2002).
12. H. Samea, Essential amenability of abstract Segal algebras, Bull. Aust. Math. Soc., 79, 319 – 325 (2009).
13. L. B. Schweitzer, Dense nuclear Fréchet ideals in C\ast -algebras, Univ. California, San Francisco, preprint (2013).
14. H. H. Schaefer, Topological vector spaces, Third printing corrected, Grad. Texts Math., 3, Springer, New York (1971).
15. M. Sugiura, Fourier series of smooth functions on compact Lie groups, Osaka J. Math., 8, 33 – 47 (1971).
16. J. L. Taylor, Homology and cohomology for topological algebras, Adv. Math., 9, 137 – 182 (1972).
17. J. Voigt, Factorization in Fréchet algebras, J. London Math. Soc. (2), 29, 147 – 152 (1984).
Received 07.06.17,
after revision — 15.10.19
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
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| id | umjimathkievua-article-830 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:04:09Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/ef/74e15c3d7d65832a12b19d8adcbbb2ef.pdf |
| spelling | umjimathkievua-article-8302022-03-26T11:01:53Z Essential amenability of Fréchet algebras Essential amenability of Fréchet algebras Abtahi , F. Rahnama, S. Abtahi , F. Rahnama, S. Fr´echet algebra Banach algebra UDC 517.98 Essential amenability of Banach algebras have been defined and investigated. Here, this concept will be introduced for Frechet algebras. Then a number of well-known results of essential amenability of Banach algebras are generalized for Fréchet algebras. Moreover, related results about Segal–Fréchet algebras are provided. As the main result, it is provedthat if $(\mathcal{A} , p_{\ell})$ is an amenable Fréchet algebra with a uniformly bounded approximate identity, then every symmetric Segal – Fréchet algebra in $(\mathcal{A} , p_{\ell})$ is essentially amenable. УДК 517.98 Суттєва аменабельнiсть алгебр Фреше Суттєву аменабельнiсть банахових алгебр було визначено та вивчено ранiше. Тут ця концепцiя визначається для алгебр Фреше. Пiсля цього ряд вiдомих результатiв з аменабельностi банахових алгебр узагальнюється на випадок алгебр Фреше. Також наводяться результати, якi стосуються алгебр Сiгала – Фреше. Головним є твердження про те, що у випадку, коли $(\mathcal{A} , p_{\ell})$ є аменабельною алгеброю Фреше з рiвномiрно обмеженою наближеною тотожнiстю, всi симетричнi алгебри Сiгала – Фреше у $(\mathcal{A} , p_{\ell})$ є суттєво аменабельними. &nbsp; Institute of Mathematics, NAS of Ukraine 2020-07-15 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/830 10.37863/umzh.v72i7.830 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 7 (2020); 867-876 Український математичний журнал; Том 72 № 7 (2020); 867-876 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/830/8724 |
| spellingShingle | Abtahi , F. Rahnama, S. Abtahi , F. Rahnama, S. Essential amenability of Fréchet algebras |
| title | Essential amenability of Fréchet algebras |
| title_alt | Essential amenability of Fréchet algebras |
| title_full | Essential amenability of Fréchet algebras |
| title_fullStr | Essential amenability of Fréchet algebras |
| title_full_unstemmed | Essential amenability of Fréchet algebras |
| title_short | Essential amenability of Fréchet algebras |
| title_sort | essential amenability of fréchet algebras |
| topic_facet | Fr´echet algebra Banach algebra |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/830 |
| work_keys_str_mv | AT abtahif essentialamenabilityoffrechetalgebras AT rahnamas essentialamenabilityoffrechetalgebras AT abtahif essentialamenabilityoffrechetalgebras AT rahnamas essentialamenabilityoffrechetalgebras |