Locally bounded approximate diagonal modulo an ideal of Frechet algebras

TL;DR

This paper extends the concept of bounded approximate diagonals modulo an ideal from Banach algebras to Frechet algebras, exploring their relation to amenability.

Contribution

It introduces the notion of locally bounded approximate diagonals modulo an ideal for Frechet algebras and studies their connection to amenability.

Findings

Defined locally bounded approximate diagonals modulo an ideal for Frechet algebras

Established the relationship between amenability modulo an ideal and these diagonals

Extended previous Banach algebra results to the Frechet algebra setting

Abstract

For a Banach algebra A and a closed ideal I, the notion of bounded approximate diagonal modulo I has been studied and investigated. In this paper we define the notion of locally bounded approximate diagonal modulo an ideal I for a Frechet algebra A and obtain the relation between amenability modulo an ideal I and the existence of locally bounded approximate diagonals modulo I.

Equations

$$\mathcal{A}\times \mathcal{A}⟶\mathcal{A},(a,b)\mapsto ab$$

$$D(ab)=a.D(b)+D(a).b,$$

$${\delta }_x(a)=a.x-x.a(a\in \mathcal{A}),$$

$$f_{\alpha \gamma }=f_{\alpha \beta }\circ f_{\beta \gamma },(\alpha ,\beta ,\gamma \in \Lambda ,\alpha \le \beta \le \gamma ).$$

$$E=\{x=(x_{\alpha })\in F:x_{\alpha }=f_{\alpha \beta }(x_{\beta }),\alpha \le \beta \}.$$

$${\phi }_{\lambda \gamma }:\frac{\mathcal{A}}{N_{\gamma }}\to \frac{\mathcal{A}}{N_{\lambda }},{\phi }_{\lambda \gamma }(x+N_{\gamma })=x+N_{\lambda },$$

$$\mathcal{A}=\underleftarrow{\lim }\frac{\mathcal{A}}{N_{\lambda }}=\underleftarrow{\lim }{\mathcal{A}}_{\lambda },$$

$$q(a.x)\le p(a)q(x),(a\in \mathcal{A},x\in X).$$

$$D(a)=a.\phi -\phi .a,(a\in \mathcal{A}\setminus I).$$

$${\hat{p}}_n(a+I)=\inf \{p_n(a+b):b\in I\}.$$

$$(a+I,b+I)\mapsto ab+I(a,b\in \mathcal{A}),$$

$$r_n(u)=\inf \{\sum _{i\in \mathbb{N}}{p}_n(a_i)p_n(b_i);u=\sum _{i\in \mathbb{N}}{a}_i\otimes b_i\},$$

$$a.(b\otimes c)=ab\otimes cand(b\otimes c).a=b\otimes ca(a,b,c\in \mathcal{A}).$$

$${\pi }_{\frac{\mathcal{A}}{I}}:\frac{\mathcal{A}}{I}\hat{\otimes }\frac{\mathcal{A}}{I}\to \frac{\mathcal{A}}{I},(a+I)\otimes (b+I)\mapsto ab+I,(a,b\in \mathcal{A}).$$

$$\underset{\alpha }{\lim }(a.{\pi }_{\frac{\mathcal{A}}{I}}(m_{\alpha })-\tilde{a})=0,(a\in \mathcal{A},\tilde{a}=a+I),$$

$$\underset{\alpha }{\lim }(a.m_{\alpha }-m_{\alpha }.a)=0(a\in \mathcal{A}\setminus I).$$

$$\underset{\alpha }{\lim }(a.{\pi }_{\mathcal{A}}(m_{\alpha })-a)=0,and\underset{\alpha }{\lim }(a.m_{\alpha }-m_{\alpha }.a)=0,(a\in \mathcal{A}).$$

$$U\otimes U=\{a\otimes b:a,b\in U\}.$$

$$\underset{\alpha }{\lim }(a.{\pi }_{\frac{\mathcal{A}}{I}}(m_{\alpha })-\tilde{a})=0,(a\in \mathcal{A},\tilde{a}=a+I),$$

$$\underset{\alpha }{\lim }(a.m_{\alpha }-m_{\alpha }.a)=0(a\in \mathcal{A}\setminus I).$$

$${\hat{r}}_n(M)\le C_{n}and{\hat{r}}_n(a.M-M.a)<\epsilon ,(a\in F),$$

$${\hat{p}}_n(a.{\pi }_{\frac{\mathcal{A}}{I}}(M)-\tilde{a})<\epsilon ,(a\in \mathcal{A},\tilde{a}=a+I).$$

$$\bar{\phi }\otimes \bar{\phi }((a+I)\otimes (b+I))$$

$${\hat{S}}_m(\bar{\phi }\otimes \bar{\phi }(M))\le {\alpha }_m{\hat{r}}_{n_1}(M),(M\in \frac{\mathcal{A}}{I}\hat{\otimes }\frac{\mathcal{A}}{I}).$$

$$(\frac{\mathcal{B}}{J}\hat{\otimes }\frac{\mathcal{B}}{J})\times \mathcal{B}\to \frac{\mathcal{B}}{J}\hat{\otimes }\frac{\mathcal{B}}{J},(M^{\prime },b)\mapsto M^{\prime }.b,(M^{\prime }\in \frac{\mathcal{B}}{J}\hat{\otimes }\frac{\mathcal{B}}{J},b\in \mathcal{B}),$$

$$\mathcal{B}\times (\frac{\mathcal{B}}{J}\hat{\otimes }\frac{\mathcal{B}}{J})\to \frac{\mathcal{B}}{J}\hat{\otimes }\frac{\mathcal{B}}{J},(b,M^{\prime })\mapsto b.M^{\prime },(M^{\prime }\in \frac{\mathcal{B}}{J}\hat{\otimes }\frac{\mathcal{B}}{J},b\in \mathcal{B}),$$

$${\hat{S}}_m(M^{\prime }.b)\le {\hat{S}}_m(M^{\prime })q_{m_0}(b),(M^{\prime }\in \frac{\mathcal{B}}{J}\hat{\otimes }\frac{\mathcal{B}}{J},b\in \mathcal{B}),$$

$${\hat{S}}_m(b.M^{\prime })\le {\hat{S}}_m(M^{\prime })q_{m_0}(b),(M^{\prime }\in \frac{\mathcal{B}}{J}\hat{\otimes }\frac{\mathcal{B}}{J},b\in \mathcal{B}).$$

$$\mathcal{B}\times \frac{\mathcal{B}}{J}\to \frac{\mathcal{B}}{J},(b,c+J)\mapsto bc+J,(b,c\in \mathcal{B}),$$

$$\frac{\mathcal{B}}{J}\times \mathcal{B}\to \frac{\mathcal{B}}{J},(c+J,b)\mapsto cb+J,(b,c\in \mathcal{B}),$$

Full text

0. Introduction

Amenability of Fr$\acute{e}$chet algebras was first introduced by Helemskii in [5] and [6]. Also in [10] Pirkovskii studied this notion and obtained some results about amenability of Fr$\acute{e}$chet algebras. In [1] Amini and Rahimi introduced the notion of amenability modulo an ideal of Banach algebras. Also they defined the notions of bounded approximate identities and bounded approximate diagonals modulo an ideal for a Banach algebra. They showed if $\mathcal{A}$ is a Banach algebra and $I$ is a closed ideal of $\mathcal{A}$, $\mathcal{A}$ is amenable modulo $I$ if and only if $\mathcal{A}$ has a bounded approximate diagonal modulo $I$. Furthermore they prove that if $\mathcal{A}$ is amenable modulo $I$, it has a bounded approximate identity modulo $I$.

In this paper we generalize definition of bounded approximate diagonal modulo an ideal in Banach algebras for Fr$\acute{e}$chet algebras. The paper is organized as follows. In section $1$ we present some preliminaries and definitions related to locally convex spaces and Fr$\acute{e}$chet algebras. In section $2$ we define the notion of locally bounded approximate diagonal modulo an ideal for a Fr$\acute{e}$chet algebra. Mainly we show that amenability modulo an ideal for a Fr$\acute{e}$chet algebra is equivalent to the existence of a locally bounded approximate diagonal modulo an ideal for this Fr$\acute{e}$chet algebra. Finaly we prove that $\mathcal{A}$ is amenable modulo an ideal $I$ if and only if ${\mathcal{A}}_{e}$, the unitization of $\mathcal{A}$, is amenable modulo $I$.

1. Preliminaries

In this section, we mention some definitions and properties about locally convex spaces and also Fr$\acute{e}$chet algebras, which will be used throughout the paper. See [7] and [8] for more informations.

A locally convex space is a topological vector space, whose topology is generated by translations of balanced, absorbent, convex sets. Equivalenty a locally convex space is a topological vector space whose topology is defined by a fundamental system of seminorms. We denote by $(E,p_{\alpha})$, a locally convex space $E$ with a fundamental system of seminorms $(p_{\alpha})_{\alpha}$. All locally convex spaces assume to be Hausdorff in this paper.

A topological algebra ${\mathcal{A}}$ is an algebra which is a topological vector space and the multiplication

[TABLE]

is a separately continuous mapping; see [10]. An important class of topological algebra is the class of Fr$\acute{e}$chet algebras. A Fr$\acute{e}$chet algebra, denoted by $(\mathcal{A},p_{n})$, is a complete topological algebra, whose topology is given by the countable family of increasing submultiplicative seminorms. Also every closed subalgebra of a Fr$\acute{e}$chet algebra is clearly a Fr$\acute{e}$chet algebra.

For a Fr$\acute{e}$chet algebra $(\mathcal{A},p_{n})$, a locally convex $\mathcal{A}$-bimodule is a locally convex topological vector space $X$ with an $\mathcal{A}$-bimodule structure such that the corresponding mappings are separately continuous. Let $(\mathcal{A},p_{n})$ be a Fr$\acute{e}$chet algebra and $X$ be a locally convex $\mathcal{A}$-bimodule. Following [4], a continuous derivation of $\mathcal{A}$ into $X$ is a continuous mapping $D$ from $\mathcal{A}$ into $X$ such that

[TABLE]

for all $a,b\in\mathcal{A}$. Furthermore for each $x\in X$ the mapping $\delta_{x}:\mathcal{A}\to X$ defined by

[TABLE]

is a continuous derivation and is called the inner derivation associated with $x$.

We recall definition of an inverse limit from [2]. Let we are given a family $(E_{\alpha})_{{\alpha}\in\Lambda}$ of algebras, where $\Lambda$ is a directed set. Suppose we are given a family $f_{\alpha\beta}$ of homomorphisms such that $f_{\alpha\beta}:E_{\beta}\to E_{\alpha}$ for any ${\alpha,\beta\in\Lambda}$, with $\alpha\leq\beta$. A projective system of algebras is a family $\{(E_{\alpha},f_{\alpha\beta})\}$ as above which in addition to the above relation, satisfies also the following condition

[TABLE]

Now consider the cartesian product algebra $F=\prod_{\alpha\in\Lambda}E_{\alpha}$, as well as the following subset of $F$,

[TABLE]

Then $E$ is the projective (or inverse) limit of the given projective system $\{(E_{\alpha},f_{\alpha\beta})\}$ and is denoted by $E=\underleftarrow{\lim}\{(E_{\alpha},f_{\alpha\beta})\}$ or simply $E=\underleftarrow{\lim}E_{\alpha}$.

Now let $(\mathcal{A},p_{\lambda})$ be a locally convex algebra. For each ${\lambda\in\Lambda}$ consider the ideal $N_{\lambda}=\ker p_{\lambda}$ and the respective normed algebra $\frac{\mathcal{A}}{N_{\lambda}}$. Suppose that $\varphi_{\lambda}:\mathcal{A}\to\mathcal{A}_{\lambda},\varphi_{\lambda}(x)=x_{\lambda}=x+{N_{\lambda}}$ be the corresponding quotient map, which is clearly a continuous surjective homomorphism. Now for any $\lambda,\gamma\in\Lambda$ with $\lambda\leq\gamma$, one has $N_{\lambda}\subseteq N_{\gamma}$. So that the linking maps

[TABLE]

are well defined continuous surjective homomorphisms such that $\varphi_{\lambda\gamma}\circ\varphi_{\gamma}=\varphi_{\lambda}$. Hence $\varphi_{\lambda\gamma}$’s are uniquely extended to continuous homomorphisms between the Banach algebras $\mathcal{A}_{\gamma}$ and $\mathcal{A}_{\lambda}$, we retain the symbol $\varphi_{\lambda\gamma}$ for the extentions too, which $\mathcal{A}_{\gamma}$ is the compeletion of $\frac{\mathcal{A}}{N_{\gamma}}$. The families $(\frac{\mathcal{A}}{N_{\lambda}},\varphi_{\lambda\gamma})$ [respectively, $(\mathcal{A}_{\lambda},\varphi_{\lambda\gamma})$], form inverse system of normed,[ respectively, Banach] algebras. Their corresponding inverse limits denoted by $\underleftarrow{\lim}\frac{\mathcal{A}}{N_{\lambda}}$ and $\underleftarrow{\lim}\mathcal{A}_{\lambda}$. Moreover in the case when the initial algebra $\mathcal{A}$ is complete, one has

[TABLE]

up to topological isomorphisms. Let $\mathcal{A}$ be a Fr$\acute{e}$chet algebra which its topology defined by the family of increasing submultiplicative seminorms $(p_{n})_{n\in\mathbb{N}}$. For each $n\in\mathbb{N}$ let $\varphi_{n}:\mathcal{A}\to\frac{\mathcal{A}}{\ker p_{n}}$ be the quotient map. Then $\frac{\mathcal{A}}{\ker p_{n}}$ is naturally a normed algebra, normed by setting $\|\varphi_{n}(a)\|_{n}=p_{n}(a)$ for each $a\in\mathcal{A}$. The compeletion $(\mathcal{A}_{n},\|.\|_{n})$ is a Banach algebra. Henceforth we consider $\varphi_{n}$ as a mapping from $\mathcal{A}$ into $\mathcal{A}_{n}$. We call it the canonical map. It is important to note that $\varphi_{n}(\mathcal{A})$ is a dense subalgebra of $\mathcal{A}_{n}$ and in general $\mathcal{A}_{n}\neq\varphi_{n}(\mathcal{A})$. Since $p_{n}\leq p_{n+1}$, there is a naturally induced, norm-decreasing homomorphism $d_{n}:\mathcal{A}_{n+1}\to\mathcal{A}_{n}$ such that $d_{n}\circ\varphi_{n+1}=\varphi_{n}$ for each $n\in\mathbb{N}$. One can show that $\mathcal{A}=\underleftarrow{\lim}(\mathcal{A}_{n},d_{n})$.

The above is the Arense-Michael decomposition of $\mathcal{A}$, which expresses Fr$\acute{e}$chet algebra as an reduced inverse limit of Banach algebras.

Now choose an Arens-Michael decomposition $\mathcal{A}=\underleftarrow{\lim}\mathcal{A}_{n}$ and let $I$ be a closed ideal of $\mathcal{A}$. Then it is easy to see that $I=\underleftarrow{\lim}\overline{I_{n}}$ is an Arens-Michael decomposition of $I$, where $\varphi_{n}:I\to I_{n}$ is the canonical map.(see [10]).

According to [10] if $\mathcal{A}$ is a locally convex algebra and $X$ is a left locally convex $\mathcal{A}$-module, then a continuous seminorm $q$ on $X$ is m-compatible if there exists a continuous submultiplicative seminorm $p$ on $\mathcal{A}$ such that

[TABLE]

Also by [9, 3.4] if $\mathcal{A}$ is a Fr$\acute{e}$chet algebra and $X$ is a complete left $\mathcal{A}$-module with a jointly continuous left module action, then the topology on $X$ can be determined by a directed family of m-compatible seminorms.

2. **Locally bounded approximate diagonal modulo an ideal of a Fr$\acute{e}$chet algebra **

Let $\mathcal{A}$ be a Banach algebra and $I$ be a closed ideal of $\mathcal{A}$. According to [1], $\mathcal{A}$ is amenable modulo $I$, if for every Banach $\mathcal{A}$-bimodule $E$ such that $I.E=E.I=0$ and every derivation $D$ from $\mathcal{A}$ into $E^{*}$, there exists $\varphi\in E^{*}$ such that

[TABLE]

In [11, Definition 2.1] we introduced the definition of amenability modulo an ideal for a Fr$\acute{e}$chet algebra. Also we extended some results of [12], for Fr$\acute{e}$chet algebras. In fact from [11], we have the following definition.

Definition 2.1**.**

Let $(\mathcal{A},p_{n})$ be a Fr$\acute{e}$chet algebra and $I$ be a closed ideal of $\mathcal{A}$. $\mathcal{A}$ is called amenable modulo $I$, if for every Banach $\mathcal{A}$-bimodule $E$ such that $E.I=I.E=0$ each continuous derivation from $\mathcal{A}$ into $E^{*}$ is inner on $\mathcal{A}\setminus I$.

In this section we discuss about approximate diagonal modulo an ideal for a Fr$\acute{e}$chet algebra.

Let $(\mathcal{A},p_{n})$ be a Fr$\acute{e}$chet algebra and $I$ be a closed ideal of $\mathcal{A}$. By [8, Lemma 22.9] $\frac{\mathcal{A}}{I}$ endowed with the quotient topology is a Fr$\acute{e}$chet space and the topology is defined by the seminorms

[TABLE]

Moreover the multiplication

[TABLE]

on $\frac{\mathcal{A}}{I}$ is continuous and $(\frac{\mathcal{A}}{I},\hat{p}_{n})$ is a Fr$\acute{e}$chet algebra; see[3, 3.2.10]. Also we recall projective tensor product of Fr$\acute{e}$chet algebra $(\mathcal{A},p_{n})$ which has been introduced in [14]. It will be denoted by $(\mathcal{A}\widehat{\otimes}\mathcal{A},r_{n})$, where

[TABLE]

for each $u\in\mathcal{A}\widehat{\otimes}\mathcal{A}$. By [14] and [15, Theorem 2] and also [17, Theorem 45.1], $(\mathcal{A}\widehat{\otimes}\mathcal{A},r_{n})$ is again a Fr$\acute{e}$chet algebra. Also $(\mathcal{A}\widehat{\otimes}\mathcal{A},r_{n})$ is a Fr$\acute{e}$chet $\mathcal{A}$-bimodule with the following module actions

[TABLE]

Also the corresponding diagonal operator of $\frac{\mathcal{A}}{I}$ is defined by

[TABLE]

Clearly $\pi_{\frac{\mathcal{A}}{I}}$ is an $\mathcal{A}$-bimodule homomorphism.

A particularly important result related with amenability modulo an ideal of a Banach algebra is the existence of a bounded approximate diagonal and virtual diagonal modulo an ideal for a Banach algebra. In fact according to [12, Definition 6], if $\mathcal{A}$ is a Banach algebra and $I$ is a closed ideal of $\mathcal{A}$, a bounded net $(m_{\alpha})\subseteq\frac{\mathcal{A}}{I}\widehat{\otimes}\frac{\mathcal{A}}{I}$ is an approximate diagonal modulo $I$ if

[TABLE]

[TABLE]

Let us now consider the Fr$\acute{e}$chet algebra $(\mathcal{A},p_{n})$. By [10], a bounded net $(m_{\alpha})\subseteq{\mathcal{A}}\widehat{\otimes}{\mathcal{A}}$ is a bounded approximate diagonal for $\mathcal{A}$ if

[TABLE]

Also according to [10, Definition 6.2], if $\mathcal{A}$ is a complete locally convex algebra, $\mathcal{A}$ has a locally bounded approximate diagonal if for each zero neighborhood $U\subseteq\mathcal{A}$ there exists $C>0$ such that for each finite subset $F\subseteq\mathcal{A}$ there exists $M\in\overline{\Gamma(U\otimes U)}$ such that $a.M-M.a\in\overline{\Gamma(U\otimes U)}$ and $a.\pi_{\mathcal{A}}(M)-a\in U$ for all $a\in F$, where $\overline{\Gamma(U\otimes U)}$ denotes the closure of the absolutely convex hull of the set

[TABLE]

According to these definitions we can generalize the definitions as follows.

Definition 2.2**.**

Let $(\mathcal{A},p_{n})$ be a Fr$\acute{e}$chet algebra and $I$ be a closed ideal of $\mathcal{A}$. We say that $\mathcal{A}$ has a bounded approximate diagonal modulo $I$, if there exists a bounded net $(m_{\alpha})_{\alpha}\subseteq\frac{\mathcal{A}}{I}\widehat{\otimes}\frac{\mathcal{A}}{I}$ such that

[TABLE]

[TABLE]

Also we say that $\mathcal{A}$ has a locally bounded approximate diagonal modulo $I$, if there exists a family $\{C_{n}:\;\;\;n\in\mathbb{N}\}$ of positive real numbers such that for each finite set $F\subseteq\mathcal{A}\setminus I$, each $n\in\mathbb{N}$, and each $\varepsilon>0$ there exists $M\in(\frac{\mathcal{A}}{I}\widehat{\otimes}\frac{\mathcal{A}}{I},\hat{r}_{n})$ such that

[TABLE]

[TABLE]

In what follows, we restrict ourselves to the relation between amenability modulo an ideal and the existence of a locally bounded approximate diagonal modulo an ideal for a Fr$\acute{e}$chet algebra, it is to be noted that the role played by reduced inverse limit is essential.

Theorem 2.3**.**

Let $\varphi:(\mathcal{A},p_{n})\to(\mathcal{B},q_{m})$ be a continuous homomorphism of Fr$\acute{e}$chet algebras with dense range and let $I$ be a closed ideal of $\mathcal{A}$ and $J$ be a closed ideal of $\mathcal{B}$ such that $\varphi(I)\subseteq J$. Suppose that $\mathcal{A}$ has a locally bounded approximate diagonal modulo $I$. Then $\mathcal{B}$ has a locally bounded approximate diagonal modulo $J$.

Proof.

Let $q_{\mathcal{B}}:(\mathcal{B},q_{m})\to(\frac{\mathcal{B}}{J},\hat{q}_{m})$ be the quotient map. So $q_{\mathcal{B}}\circ\varphi:\mathcal{A}\to\frac{\mathcal{B}}{J}$ is a continuous homomorphism with dense range. On the other hand $\varphi(I)\subseteq J$, therefore $I\subseteq\ker q_{\mathcal{B}}\circ\varphi$. Thus the function $\bar{\varphi}:(\frac{\mathcal{A}}{I},\hat{p}_{n})\to(\frac{\mathcal{B}}{J},\hat{q}_{m})$ defined by $\bar{\varphi}(a+I)=q_{\mathcal{B}}\circ\varphi(a)=\varphi(a)+J$ is well defined and it is a continuous homomorphism. We can define $\bar{\varphi}{\otimes}\bar{\varphi}:(\frac{\mathcal{A}}{I}\widehat{\otimes}\frac{\mathcal{A}}{I},\hat{r}_{n})\to(\frac{\mathcal{B}}{J}\widehat{\otimes}\frac{\mathcal{B}}{J},\hat{S}_{m})$ by

[TABLE]

Since $\bar{\varphi}\otimes\bar{\varphi}$ is a continuous map, for each $m\in\mathbb{N}$ there exist $\alpha_{m}>0$ and $n_{1}\in\mathbb{N}$ such that

[TABLE]

Without loss of generality we can assume that $\alpha_{m}>1$. On the other hand by the joint continuity of the module actions

[TABLE]

and

[TABLE]

and using [10, Paga 7] for each $m\in\mathbb{N}$, there exists $m_{0}\in\mathbb{N}$ such that

[TABLE]

and

[TABLE]

Also by [16], $\frac{\mathcal{B}}{J}$ is a Fr$\acute{e}$chet $\mathcal{B}$-bimodule by the module actions defined by

[TABLE]

[TABLE]

and these two actions are jointly continuous. So by [10, Page 7] for each $m\in\mathbb{N}$ there exists $m_{2}\in\mathbb{N}$ such that

[TABLE]

and

[TABLE]

Also the map $\bar{\varphi}:\frac{\mathcal{A}}{I}\to\frac{\mathcal{B}}{J}$ is continuous. So for each $m\in\mathbb{N}$ there exist $n_{0}\in\mathbb{N}$ and $k_{m}>0$ such that

[TABLE]

Without loss of generality, we may assume that, $k_{m}>1$. Let $\{C_{n}:\;\;n\in\mathbb{N}\}$ be a family of positive real numbers as in Definition 2.2. Without loss of generality we may assume that $C_{n}>1$ for each $n\in\mathbb{N}$. Given a finite set $F^{\prime}\subseteq\mathcal{B}\setminus J$, $m\in\mathbb{N}$ and each $\varepsilon>0$, find a finite set $F\subseteq\mathcal{A}\setminus I$ such that $q_{m}(\varphi(a)-b)<\frac{\varepsilon}{3\alpha_{m}C_{n_{1}}}$ for all $a\in F$ and $b\in F^{\prime}$. Since $\mathcal{A}$ has a locally bounded approximate diagonal modulo $I$, so there exists $M\in\frac{\mathcal{A}}{I}\widehat{\otimes}\frac{\mathcal{A}}{I}$ such that

[TABLE]

Also

[TABLE]

Put $M^{\prime}=\bar{\varphi}\otimes\bar{\varphi}(M)\in\frac{\mathcal{B}}{J}\widehat{\otimes}\frac{\mathcal{B}}{J}.$ Therefore

[TABLE]

Now take $b\in F^{\prime}$ and choose $a\in F$ satisfying $q_{m}(\varphi(a)-b)<\frac{\varepsilon}{3\alpha_{m}C_{n_{1}}}.$ Thus

[TABLE]

In the above relations we can assume that $m_{0}>m$. In fact if $m_{0}\leq m$, we have

[TABLE]

Furthermore $\pi_{\frac{\mathcal{B}}{J}}$ is a $\mathcal{B}$-bimodule homomorphism and the quotient map is seminorms decreasing, so

[TABLE]

for each $b\in F^{{}^{\prime}}$ and $\tilde{b}=b+J$. Note that in the above relations we can assume that, $m_{0}>m$. Henceforth $\mathcal{B}$ has a locally bounded approximate diagonal modulo $J$. ∎

Proposition 2.4**.**

Let $(\mathcal{A},p_{n})$ be a Fr$\acute{e}$chet algebra and $I$ be a closed ideal of $\mathcal{A}$. $\mathcal{A}$ has a bounded approximate diagonal modulo $I$ if and only if there exists a bounded set $B\subseteq\frac{\mathcal{A}}{I}\widehat{\otimes}\frac{\mathcal{A}}{I}$ such that for each finite set $F\subseteq\mathcal{A}\setminus I$, each $n\in\mathbb{N}$, and each $\varepsilon>0$ there exists $M\in(\frac{\mathcal{A}}{I}\widehat{\otimes}\frac{\mathcal{A}}{I},\hat{r}_{n})$ such that

[TABLE]

[TABLE]

Proof.

First suppose that $(m_{\alpha})_{\alpha}\subseteq\frac{\mathcal{A}}{I}\widehat{\otimes}\frac{\mathcal{A}}{I}$ is a bounded approximate diagonal modulo $I$ for $\mathcal{A}$. So for each $n\in\mathbb{N}$, and each $\varepsilon>0$ there exists $\alpha_{0}$ such that for each $\alpha\geq\alpha_{0}$ we have

[TABLE]

[TABLE]

Now set $B=(m_{\alpha})_{\alpha}\subseteq\frac{\mathcal{A}}{I}\widehat{\otimes}\frac{\mathcal{A}}{I}$. Let $n\in\mathbb{N}$, $\varepsilon>0$ and $F\subseteq\mathcal{A}\setminus I$, be a finite set. So there exists $M=m_{\alpha}$ which ${\alpha\geq\alpha_{0}}$ such that

[TABLE]

[TABLE]

Conversely, suppose that there exists a bounded set $B\subseteq\frac{\mathcal{A}}{I}\widehat{\otimes}\frac{\mathcal{A}}{I}$ such that for each finite set $F\subseteq\mathcal{A}\setminus I$, each $n\in\mathbb{N}$, and each $\varepsilon>0$ there exists $M\in(\frac{\mathcal{A}}{I}\widehat{\otimes}\frac{\mathcal{A}}{I},\hat{r}_{n})$ such that

[TABLE]

[TABLE]

Now take

[TABLE]

So $S$ is a directed set as follows:

[TABLE]

Therefore there exists a bounded net $(m_{\alpha})\subseteq B\subseteq\frac{\mathcal{A}}{I}\widehat{\otimes}\frac{\mathcal{A}}{I}$. Furthermore let $k\in\mathbb{N}$ and $\varepsilon>0$ and the finite set $F\subseteq\mathcal{A}\setminus I$ be arbitrary. Therefore for each $(n,\delta,C)\geq(k,\varepsilon,F)$ we have

[TABLE]

[TABLE]

So $\mathcal{A}$ has a bounded approximate diagonal modulo $I$. ∎

An immediate consequence of Definition 2.2 and Proposition 2.4 is the following.

Remark 2.5**.**

For a Banach algebra the notions of locally bounded and bounded approximate diagonal modulo an ideal are equivalent.

Let $(\mathcal{A},p_{n})$ be a Fr$\acute{e}$chet algebra and $I$ be a closed ideal of $\mathcal{A}$. As we have already mentioned $(\frac{\mathcal{A}}{I},\hat{p}_{n})$ and $(\frac{\mathcal{A}}{I}\widehat{\otimes}\frac{\mathcal{A}}{I},\hat{r}_{n})$ are Fr$\acute{e}$chet algebras and for each $n\in\mathbb{N}$, $(\mathcal{A}_{n},\|\cdot\|_{n})$ is a Banach algebra where $\|\varphi_{n}(a){\|}_{n}=\|a+\ker p_{n}\|_{n}=p_{n}(a)$. Also $(\frac{\mathcal{A}_{n}}{\overline{I_{n}}},\|\cdot\hat{\|}_{n})$ is a Banach algebra, where

[TABLE]

and $(\frac{\mathcal{A}_{n}}{\overline{I_{n}}}\widehat{\otimes}\frac{\mathcal{A}_{n}}{\overline{I_{n}}},|\|\cdot|\|_{n})$ is a Banach algebra, where $|\|\cdot|\|_{n}$, is the projective tensor norm of Banach algebras $(\frac{\mathcal{A}_{n}}{\overline{I_{n}}},\|\cdot\hat{\|}_{n})$. On the other hand

[TABLE]

Also if $z\in\overline{I_{n}}$, then there is a sequence $(z_{m,n})_{m\in\mathbb{N}}\in I_{n}$ such that $z=\lim_{m}z_{m,n}$. Now we have

[TABLE]

Therefore $\hat{p}_{n}(a+I)\leq\|\varphi_{n}(a)+\overline{I_{n}}\hat{\|}_{n}$, and so $\hat{p}_{n}(a+I)=\|\varphi_{n}(a)+\overline{I_{n}}\hat{\|}_{n}$.

Also

[TABLE]

Where $\varphi_{n}$ stands for the canonical map from $\mathcal{A}$ into $\mathcal{A}_{n}$ and $\bar{\varphi}_{n}$ is defined from $\frac{\mathcal{A}}{I}$ into $\frac{\mathcal{A}_{n}}{\overline{I_{n}}}$.

Proposition 2.6**.**

Let $(\mathcal{A},p_{n})$ be a Fr$\acute{e}$chet algebra and $I$ be a closed ideal of $\mathcal{A}$. Then the following conditions are equivalent:

1. (i)

$\mathcal{A}$* has a locally bounded approximate diagonal modulo $I$.* 2. (ii)

For each Banach algebra $\mathcal{B}$ and each continuous homomorphism $\varphi:\mathcal{A}\to\mathcal{B}$ with dense range, the Banach algebra $\mathcal{B}$ has a bounded approximate diagonal modulo $J$, where $J$ is a closed ideal of $\mathcal{B}$ such that $\varphi(I)\subseteq J$.

Proof.

$(i)\Rightarrow(ii)$. It is straightforward from Theorem 2.3 and Remark 2.5.

$(ii)\Rightarrow(i)$. Let $n\in\mathbb{N}$ be arbitrary and $\mathcal{A}=\underleftarrow{\lim}\mathcal{A}_{n}$ be an Arens-Michael decomposition of $\mathcal{A}$ and $I=\underleftarrow{\lim}\overline{I_{n}}$ be an Arens-Michael decomposition of $I$. Since $\varphi_{n}:\mathcal{A}\to\mathcal{A}_{n}$ is a continuous homomorphism with dense range, by assumption the Banach algebra $\mathcal{A}_{n}$ has a bounded approximate diagonal modulo $\overline{I_{n}}$. Therefore by Proposition 2.4, there exists $C_{n}>0$ such that for each finite set $F_{n}\subseteq{\mathcal{A}_{n}}\setminus{\overline{I_{n}}}$ and each $\varepsilon>0$ there exists $M^{{}^{\prime}}\in\frac{\mathcal{A}_{n}}{\overline{I_{n}}}\widehat{\otimes}\frac{\mathcal{A}_{n}}{\overline{I_{n}}}$ for which

[TABLE]

and

[TABLE]

and

[TABLE]

Suppose that $q_{\mathcal{A}_{n}}:\mathcal{A}_{n}\to{\frac{\mathcal{A}_{n}}{\overline{I_{n}}}}$ be the quotient map. Then $q_{\mathcal{A}_{n}}\circ\varphi_{n}:\mathcal{A}\to{\frac{\mathcal{A}_{n}}{\overline{I_{n}}}}$ defined by $q_{\mathcal{A}_{n}}\circ\varphi_{n}(a)=\varphi_{n}(a)+{\overline{I_{n}}}$ is a continuous homomorphism with dense range and $I\subseteq\ker(q_{\mathcal{A}_{n}}\circ\varphi_{n})$. Therefore we can define a continuous homomorphism with dense range $\bar{\varphi}_{n}:\frac{\mathcal{A}}{I}\to\frac{\mathcal{A}_{n}}{\overline{I_{n}}}$ such that $\bar{\varphi}_{n}(a+I)={\varphi}_{n}(a)+\overline{I_{n}}$ and so $\bar{\varphi}_{n}\otimes\bar{\varphi}_{n}:\frac{\mathcal{A}}{I}\widehat{\otimes}\frac{\mathcal{A}}{I}\to\frac{\mathcal{A}_{n}}{\overline{I_{n}}}\widehat{\otimes}\frac{\mathcal{A}_{n}}{\overline{I_{n}}}$ is a continuous homomorphism with dense range. If $F=\{a_{1},a_{2},...,a_{m}\}\subseteq\mathcal{A}\setminus I$ is a finite set, then there exists $n_{0}\in\mathbb{N}$, such that $a_{n}^{i}\notin\overline{I_{n}}$ for each $n\geq n_{0}$, where $a_{i}=(a_{n}^{i})$ and $1\leq i\leq m$. In fact if $a_{i}\notin I$, then there exists $n_{i}\in\mathbb{N}$, such that $a_{n_{i}}^{i}\notin\overline{I_{n_{i}}}$. Also the maps $\varphi_{mn}:\mathcal{A}_{m}\to\mathcal{A}_{n}$ defined by $\varphi_{mn}(a_{m})=a_{n}$, for each $m\geq n$. So $a_{n}^{i}\notin\overline{I_{n}}$, for each $n\geq n_{i}$. Now put

[TABLE]

Thus $a_{n}\notin\overline{I_{n}}$, for each $n\geq n_{0}$. In the sequel take a finite set $F\subseteq\mathcal{A}\setminus I$, $\varepsilon>0$ and $k\in\mathbb{N}$. Also set

[TABLE]

Because of the above argument we can assume that $F^{\prime}=\{\varphi_{n}(a):\,\,a\in F\}\subseteq\mathcal{A}_{n}\setminus\overline{I_{n}}$ is a finite set. Put

[TABLE]

Since $\bar{\varphi}_{n}\otimes\bar{\varphi}_{n}$ is a continuous homomorphism with dense range, so there exists $M\in\frac{\mathcal{A}}{I}\hat{\otimes}\frac{\mathcal{A}}{I}$ such that

[TABLE]

Hence

[TABLE]

Furthermore

[TABLE]

Which in view of Definition 2.2, this completes the proof. ∎

Corollary 2.7**.**

Let $(\mathcal{A},p_{n})$ be a Fr$\acute{e}$chet algebra and $I$ be a closed ideal of $\mathcal{A}$. Suppose that $\mathcal{A}=\underleftarrow{\lim}\mathcal{A}_{n}$ is an Arens-Michael decomposition of $\mathcal{A}$ and $I=\underleftarrow{\lim}\overline{I_{n}}$ is an Arens-Michael decomposition of $I$. Then $\mathcal{A}$ has a locally bounded approximate diagonal modulo $I$ if and only if each $\mathcal{A}_{n}$ has a bounded approximate diagonal modulo $\overline{I_{n}}$.

Proof.

Since $\varphi_{n}:\mathcal{A}\to\mathcal{A}_{n}$ is a continuous homomorphism with dense range, Theorem 2.3 and Remark 2.5, imply that if $\mathcal{A}$ has a locally bounded approximate diagonal modulo $I$, then $\mathcal{A}_{n}$ has a bounded approximate diagonal modulo $\overline{I_{n}}$, for each $n\in\mathbb{N}$.

Conversely, suppose that each $\mathcal{A}_{n}$ has a bounded approximate diagonal modulo $\overline{I_{n}}$ and $\Phi:\mathcal{A}\to\mathcal{B}$ is a continuous homomorphism for some Banach algebra $\mathcal{B}$, with dense range. Since $\mathcal{B}$ is a Banach algebra and $\Phi$ is a continuous homomorphism, there exists a continuous homomorphism with dense range $\varPhi_{n_{0}}:\mathcal{A}_{n_{0}}\to\mathcal{B}$ for some $n_{0}\in\mathbb{N}$. On the other hand if $J$ is a closed ideal of $\mathcal{B}$ such that $\varPhi(I)\subseteq J$, then $\varPhi_{n_{0}}(I_{n_{0}})=\varPhi(I)\subseteq J$. Therefore by Theorem 2.3, $\mathcal{B}$ has a bounded approximate diagonal modulo $J$ such that $\Phi(I)\subseteq J$. Thus by using Proposition 2.6, $\mathcal{A}$ has a locally bounded approximate diagonal modulo $I$. ∎

We now state the main result of this section which is an extention of corresponding result for Banach algebras.

Theorem 2.8**.**

Let $(\mathcal{A},p_{n})$ be a Fr$\acute{e}$chet algebra and $I$ be a closed ideal of $\mathcal{A}$. Then the following statements are equivalent;

1. (i)

$\mathcal{A}$* is amenable modulo $I$.* 2. (ii)

$\mathcal{A}$* has a locally bounded approximate diagonal modulo $I$.*

Proof.

$(i)\Rightarrow(ii)$. Suppose that $\mathcal{A}=\underleftarrow{\lim}\mathcal{A}_{n}$ is an Arens-Michael decomposition of $\mathcal{A}$ and $I=\underleftarrow{\lim}\overline{I_{n}}$ is an Arens-Michael decomposition of $I$. Since $\mathcal{A}$ is amenable modulo $I$, by [11, Theorem 2.3] each $\mathcal{A}_{n}$ is amenable modulo $\overline{I_{n}}$. So $\mathcal{A}_{n}$ has a bounded approximate diagonal modulo $\overline{I_{n}}$ for each $n\in\mathbb{N}$ by [12, Theorem 7]. Then $\mathcal{A}$ has a locally bounded approximate diagonal modulo $I$, by Corollary 2.7.

$(ii)\Rightarrow(i)$. Let $\mathcal{A}$ has a locally bounded approximate diagonal modulo $I$ and $\mathcal{A}=\underleftarrow{\lim}\mathcal{A}_{n}$ be an Arens-Michael decomposition of $\mathcal{A}$ and $I=\underleftarrow{\lim}\overline{I_{n}}$ be an Arens-Michael decomposition of $I$. Therefore by Corollary 2.7, $\mathcal{A}_{n}$ has a bounded approximate diagonal modulo $\overline{I_{n}}$ for each $n\in\mathbb{N}$. So each $\mathcal{A}_{n}$ is amenable modulo $\overline{I_{n}}$, by [12, Theorem 7]. Thus $\mathcal{A}$ is amenable modulo $I$, by [11, Theorem 2.3] . ∎

Corollary 2.9**.**

Let $(\mathcal{A},p_{n})$ be a Fr$\acute{e}$chet algebra and $I$ be a closed ideal of $\mathcal{A}$. If $\mathcal{A}$ has a locally bounded approximate diagonal modulo $I$, then $\frac{\mathcal{A}}{I}$ has a locally bounded approximate diagonal.

Proof.

Since $q_{\mathcal{A}}:\mathcal{A}\to\frac{\mathcal{A}}{I}$ is a continuous homomorphism with dense range by Theorem 2.3, $\frac{\mathcal{A}}{I}$ has a locally bounded approximate diagonal modulo $I=\{0_{\frac{\mathcal{A}}{I}}\}$. So it is clear that $\frac{\mathcal{A}}{I}$ is amenable. By applying [10, Theorem 9.7], $\frac{\mathcal{A}}{I}$ has a locally bounded approximate diagonal. ∎

In the next propositions we give two hereditary properties of amenability modulo an ideal for Fr$\acute{e}$chet algebras.

Proposition 2.10**.**

Let $(\mathcal{A},p_{n})$ be a Fr$\acute{e}$chet algebra and $I$ be a closed ideal of $\mathcal{A}$. Then $\mathcal{A}$ is amenable modulo $I$ if and only if $\mathcal{A}_{e}$ is amenable modulo $I$.

Proof.

Suppose that $\mathcal{A}=\underleftarrow{\lim}\mathcal{A}_{n}$ be an Arens-Michael decomposition of $\mathcal{A}$ and $I=\underleftarrow{\lim}\overline{I_{n}}$ is an Arens-Michael decomposition of $I$. By [11, Theorem 2.3], $\mathcal{A}$ is amenable modulo $I$ if and only if for each $n\in\mathbb{N}$, $\mathcal{A}_{n}$ is amenamble modulo $\overline{I_{n}}$. This is equivalent to $(\mathcal{A}_{n})_{e}$ is amenamble modulo $\overline{I_{n}}$, for each $n\in\mathbb{N}$ by using [13, Theorem 3.2]. On the other hand by [2, Proposition 3.11], $\mathcal{A}_{e}=\underleftarrow{\lim}(\mathcal{A}_{n})_{e}$. So $\mathcal{A}$ is amenable modulo $I$ if and only if ${\mathcal{A}}_{e}$ is amenable modulo $I$. ∎

Proposition 2.11**.**

Let $(\mathcal{A},p_{n})$ be a Fr$\acute{e}$chet algebra and $I$ be a closed ideal of $\mathcal{A}$ and $J$ be a closed ideal of $I$. If $\mathcal{A}$ is amenable modulo $I$, then $\frac{\mathcal{A}}{J}$ is amenable modulo $\frac{I}{J}$.

Proof.

Let $\mathcal{A}=\underleftarrow{\lim}\mathcal{A}_{n}$ be an Arens-Michael decomposition of $\mathcal{A}$ and $I=\underleftarrow{\lim}\overline{I_{n}}$ be an Arens-Michael decomposition of $I$ and $J=\underleftarrow{\lim}\overline{J_{n}}$ be an Arens-Michael decomposition of $J$. Since $\mathcal{A}$ is amenable modulo $I$, by Theorem [11, Theorem 2.3], each $\mathcal{A}_{n}$ is amenable modulo $\overline{I_{n}}$. Therefore $\frac{\mathcal{A}_{n}}{\overline{J_{n}}}$, is amenable modulo $\frac{\overline{I_{n}}}{\overline{J_{n}}}$ for each $n\in\mathbb{N}$, by [13, Theorem 3.8]. So $\frac{\mathcal{A}}{J}$ is amenable modulo $\frac{I}{J}$, by [11, Theorem 2.3]. ∎

Acknowledgment. This work was supported by Iran national science foundation and we would like to acknowledge it with much appreciation. Also we are thankful to university of Isfahan in developing the project.