Some Inequalities for Continuous Algebra-Multiplications on a Banach Space

TL;DR

This paper investigates inequalities related to algebraic properties of continuous algebra-multiplications on Banach spaces and explores the structure of the space of all such multiplications.

Contribution

It introduces inequalities comparing algebraic properties of different multiplications and analyzes the structure of the space of all continuous algebra-multiplications.

Findings

Derived inequalities for algebraic properties of multiplications

Basic observations on the structure of the space of all continuous algebra-multiplications

Abstract

In this short note, we first consider some inequalities for comparison of some algebraic properties of two continuous algebra-multiplications on an arbitrary Banach space and then, as an application, we consider some very basic observations on the space of all continuous algebra-multiplications on a Banach space.

Equations

$$\Vert b-a\Vert \le r(\Vert a^{-1}\Vert |\ast {|}^2{)}^{-1}.$$

$$\Vert b^{-1}-a^{-1}\Vert \le (\Vert e\Vert |\ast |+r(1-r{)}^{-1})|\ast {|}^2\Vert a^{-1}{\Vert }^2\Vert b-a\Vert .$$

$$|\ast |\sum _{n=1}^{\infty }\Vert c^n\Vert \le \sum _{n=1}^{\infty }(|\ast |\Vert c\Vert {)}^n\le \sum _{n=1}^{\infty }{r}^n=r(1-r{)}^{-1}.$$

$$\begin{array}{rl}\Vert b^{-1}-a^{-1}\Vert & =\Vert \sum _{n=1}^{\infty }{c}^n{a}^{-1}\Vert \\ & \le |\ast {|}^2\Vert a^{-1}\Vert \Vert c\Vert \Vert \sum _{n=0}^{\infty }{c}^n\Vert \\ & \le |\ast {|}^2\Vert a^{-1}\Vert \Vert c\Vert (\Vert e\Vert +\sum _{n=1}^{\infty }\Vert c^n\Vert )\\ & \le (\Vert e\Vert |\ast |+r(1-r{)}^{-1})|\ast {|}^2\Vert a^{-1}{\Vert }^2\Vert b-a\Vert .\end{array}$$

$$\Vert x\Vert \Vert e_{\ast }{\Vert }^{-1}\le |{\mathfrak{l}}_{\ast ,x}|,|{\mathfrak{r}}_{\ast ,x}|\le |\ast |\Vert x\Vert .$$

$$\Vert e_{\diamond }-e_{\ast }\Vert \le s|\diamond -\ast |\Vert e_{\ast }{\Vert }^2\le rs\Vert e_{\ast }\Vert .$$

$$|{\mathfrak{l}}_{\diamond ,e_{\ast }}-\mathrm{id}|=|{\mathfrak{l}}_{\diamond ,e_{\ast }}-{\mathfrak{l}}_{\ast ,e_{\ast }}|\le |\diamond -\ast |\Vert e_{\ast }\Vert \le r.$$

$$x\diamond e^{\prime }=(x^{\prime }\diamond e_{\ast })\diamond e^{\prime }=x^{\prime }\diamond (e_{\ast }\diamond e^{\prime })=x^{\prime }\diamond e_{\ast }=x.$$

$$|{\mathfrak{l}}_{\diamond ,e_{\ast }}^{-1}-\mathrm{id}|\le s|\diamond -\ast |\Vert e_{\ast }\Vert \le rs.$$

$$\Vert e_{\diamond }-e_{\ast }\Vert \le |\diamond -\ast |\Vert e_{\diamond }\Vert \Vert e_{\ast }\Vert .$$

$$|\ast |,\Vert e_{\ast }\Vert ,\Vert a\Vert ,\Vert a_{\ast }^{-1}\Vert \le M\text{and}|\diamond -\ast |\le r{M}^{-3}$$

$$\Vert a_{\diamond }^{-1}-a_{\ast }^{-1}\Vert \le s^2{C}_M|\diamond -\ast |,$$

$$|{\mathfrak{l}}_{\diamond ,a}-{\mathfrak{l}}_{\ast ,a}|\le \Vert a\Vert |\diamond -\ast |<(\Vert a_{\ast }^{-1}\Vert |\ast |{)}^{-1}\le |{\mathfrak{l}}_{\ast ,a}^{-1}{|}^{-1}.$$

$$|{\mathfrak{l}}_{\diamond ,a}-{\mathfrak{l}}_{\ast ,a}|\le r|{\mathfrak{l}}_{\ast ,a}^{-1}{|}^{-1}.$$

$$|{\mathfrak{l}}_{\diamond ,a}^{-1}-{\mathfrak{l}}_{\ast ,a}^{-1}|\le s|\ast {|}^2\Vert a_{\ast }^{-1}{\Vert }^2\Vert a\Vert |\diamond -\ast |\le s{M}^5|\diamond -\ast |$$

$$\begin{array}{rl}\Vert a_{\diamond }^{-1}-a_{\ast }^{-1}\Vert & =\Vert {\mathfrak{l}}_{\diamond ,a}^{-1}(e_{\diamond })-{\mathfrak{l}}_{\ast ,a}^{-1}(e_{\ast })\Vert \\ & \le \Vert {\mathfrak{l}}_{\diamond ,a}^{-1}(e_{\diamond })-{\mathfrak{l}}_{\ast ,a}^{-1}(e_{\diamond })\Vert +\Vert {\mathfrak{l}}_{\ast ,a}^{-1}(e_{\diamond })-{\mathfrak{l}}_{\ast ,a}^{-1}(e_{\ast })\Vert \\ & \le |{\mathfrak{l}}_{\diamond ,a}^{-1}-{\mathfrak{l}}_{\ast ,a}^{-1}|\Vert e_{\diamond }\Vert +|{\mathfrak{l}}_{\ast ,a}^{-1}|\Vert e_{\diamond }-e_{\ast }\Vert .\end{array}$$

$$\Vert e_{\diamond }-e_{\ast }\Vert \le s\Vert e_{\ast }{\Vert }^2|\diamond -\ast |\le s{M}^2|\diamond -\ast |$$

$$\Vert e_{\diamond }\Vert \le \Vert e_{\diamond }-e_{\ast }\Vert +\Vert e_{\ast }\Vert \text{and}|{\mathfrak{l}}_{\ast ,a}^{-1}|\le |\ast |\Vert a_{\ast }^{-1}\Vert \le M^2.$$

$${\rho }_{\star }(a)\subseteq \mathrm{liminf}{\rho }_{{\star }_n}(a):={\cup }_n{\cap }_{k\ge n}{\rho }_{{\star }_k}(a),$$

$${\sigma }_{\star }(a)\supseteq \mathrm{limsup}{\sigma }_{{\star }_n}(a):={\cap }_n{\cup }_{k\ge n}{\sigma }_{{\star }_k}(a).$$

$$\Vert a_n{\star }_n{b}_n-a\star b\Vert \le |{\star }_n-\star |\Vert a_n\Vert \Vert b_n\Vert +|\star |\Vert a_n\Vert \Vert b_n-b\Vert +|\star |\Vert b\Vert \Vert a_n-a\Vert .$$

$$(a\star b)\star c=(\underset{n}{\lim }a{\star }_{n}b)\star c=\underset{n}{\lim }(a{\star }_{n}b){\star }_{n}c=\underset{n}{\lim }a{\star }_n(b{\star }_{n}c)=a\star (\underset{n}{\lim }b{\star }_{n}c)=a\star (b\star c).$$

$$\sum _{\ell }{\lambda }_{ij\ell }{\lambda }_{\ell kp}-\sum _q{\lambda }_{iqp}{\lambda }_{jkq}(1\le i,j,k,p\le n).$$

$$(x_1,x_2)\star (x_1^{\prime },x_2^{\prime }):=(x_1{\star }_1{x}_1^{\prime },x_2{\star }_2{x}_2^{\prime }).$$

$$\begin{array}{rl}|{\diamond }_s-{\ast }_t|& =|{\Lambda }_{s^{-1}}{\Gamma }_{s,s}(\diamond )-{\Lambda }_{t^{-1}}{\Gamma }_{t,t}(\ast )|\\ & \le |{\Lambda }_{s^{-1}}{\Gamma }_{s,s}(\diamond )-{\Lambda }_{t^{-1}}{\Gamma }_{s,s}(\diamond )|+|{\Lambda }_{t^{-1}}{\Gamma }_{s,s}(\diamond )-{\Lambda }_{t^{-1}}{\Gamma }_{t,t}(\ast )|\\ & \le |{\Lambda }_{s^{-1}-t^{-1}}||{\Gamma }_{s,s}(\diamond )|+|{\Lambda }_{t^{-1}}||{\Gamma }_{s,s}(\diamond )-{\Gamma }_{t,t}(\ast )|\\ & \le |\diamond ||s{|}^2|s^{-1}-t^{-1}|+|t^{-1}||{\Gamma }_{s,s}(\diamond )-{\Gamma }_{t,t}(\ast )|.\end{array}$$

$$\begin{array}{rl}& |{\Gamma }_{s,s}(\diamond )-{\Gamma }_{t,t}(\ast )|\\ \le & |{\Gamma }_{s,s}(\diamond )-{\Gamma }_{s,t}(\diamond )|+|{\Gamma }_{s,t}(\diamond )-{\Gamma }_{t,t}(\diamond )|+|{\Gamma }_{t,t}(\diamond )-{\Gamma }_{t,t}(\ast )|\\ \le & |\diamond |(|s|+|t|)|s-t|+|t{|}^2|\diamond -\ast |.\end{array}$$

$$(f{\star }_{\mu }g)(x,y):={\int }_{X}f(x,z)g(z,y)\mathrm{d}\mu (z)(f,g\in {\mathbf{M}}_X,x,y\in X).$$

$$(\mu {\star }_s\nu )(B):=\mu \times \nu (S^{-1}(B))(\mu ,\nu \in \mathcal{M}(X),B\subseteq X).$$

$$(a,b){\star }_T(a^{\prime },b^{\prime }):=(a{a}^{\prime }+aT(b^{\prime })+T(b)a,b{b}^{\prime })((a,b),(a^{\prime },b^{\prime })\in A\oplus B).$$

Full text

MSC 2010. Primary 46H05; Secondary 46H20.

Keywords. Banach space; Banach algebra; continuous algebra-multiplication; moduli space of associative multiplications.

1 Introduction

One of the fundamental and hardest questions in the theory of Banach algebras is the following.

Question 1.1**.**

Let $A$ be a Banach space and let $\star$ and $\diamond$ be two Banach algebra multiplications on $A$. How are algebraic properties of the Banach algebras $(A,\star)$ and $(A,\diamond)$ related if $\star$ and $\diamond$ be sufficiently close?

This question which is the central subject of perturbation theory of Banach algebras, has been considered by many authors. See, for instance, [7, 9] and references therein.

In this short note, we first consider some inequalities for comparison of some algebraic properties of two continuous algebra-multiplications on an arbitrary Banach space and then, as an application, we consider some very basic observations on the space of all continuous algebra-multiplications on a Banach space.

Conventions & Notations. Operator norm of bounded linear and bilinear operators, all are denoted by $|\cdot|$. Throughout, $E$ denotes a non-zero Banach space. The Banach space of bounded linear operators on $E$ is denoted by $\mathrm{B}(E)$, and the Banach space of bounded bilinear operators from $E\times E$ into $E$ is denoted by $\mathrm{B}_{2}(E)$. Suppose that $*\in\mathrm{B}_{2}(E)$ is associative as a binary operation on $E$. Then $*$ is called continuous multiplication on $E$. We denote by $\mathrm{Mltp}(E)\subset\mathrm{B}_{2}(E)$ the set of all continuous multiplications on $E$. For $*\in\mathrm{Mltp}(E)$, the pair $(E,*)$ is called complete normed algebra. If $|*|\leq 1$, then $*$ is called Banach algebra multiplication and $(E,*)$ is called Banach algebra. A multiplication $*\in\mathrm{Mltp}(E)$ is called unital (resp. commutative) if the algebra $(E,*)$ is unital (resp. commutative). We denote by $\mathrm{Mltp}_{\mathrm{u}}(E)\subset\mathrm{Mltp}(E)$ (resp. $\mathrm{Mltp}_{\mathrm{c}}(E)\subset\mathrm{Mltp}(E)$) the set of unital (resp. commutative) continuous multiplications on $E$. For $*\in\mathrm{Mltp}_{\mathrm{u}}(E)$ we denote by $e_{*}$ the unite element of $(E,*)$.

For a complete normed algebra $(A,*)$ the products of elements of $A$, if there is no danger of misunderstanding about the multiplication $*$, is denoted by juxtaposition. In the case that $(A,*)$ is unital, the inverse of an invertible element $a\in A$ is denoted by $a^{-1}$ or by $a^{-1}_{*}$ in order to indicate that the inverse is with respect to $*$. Also, the set of invertible elements of $(A,*)$ is denoted by $\mathrm{Inv}(A,*)$, and for any $a\in A$, $\sigma_{*}(a)$ and $\rho_{*}(a)$ denote respectively the spectrum and resolvent of $a$.

2 Some Inequalities

We begin by a simple generalization of a famous lemma in Banach Algebra Theory ([1, Lemma 1.2.5],[3, Theorem 2.1.29],[12, Theorem 10.11]).

Theorem 2.1**.**

Let $(A,*)$ be a unital complete normed algebra with unit $e=e_{*}$. Let $a\in A$ be invertible, $r$ be a numerical constant with $0\leq r<1$, and let $b\in A$ be such that

[TABLE]

Then, $b$ is invertible. Moreover,

[TABLE]

Proof.

Let $c:=e-a^{-1}b=a^{-1}(a-b)$. We have $\|c\|\leq\|b-a\|\|a^{-1}\||*|\leq r|*|^{-1}$. Thus,

[TABLE]

This shows that $(\sum_{n=0}^{\infty}c^{n})\in A$ is the inverse of $a^{-1}b=e-c$. It follows that $b\in\mathrm{Inv}(A,*)$. Moreover, $b^{-1}=\sum_{n=0}^{\infty}c^{n}a^{-1}$, and we have,

[TABLE]

∎

We introduce some notations: Let $(E,*)$ be a complete normed algebra. For any $x\in E$, let $\mathfrak{l}_{*,x}\in\mathrm{B}(E)$ denote the left multiplication operator by $x$, i.e. $\mathfrak{l}_{*,x}(y)=x*y$. Similarly, we let $\mathfrak{r}_{*,x}\in\mathrm{B}(E)$ denote the right multiplication operator by $x$. If $(E,*)$ is unital then we have,

[TABLE]

Also, it is easily checked that $x\in\mathrm{Inv}(E,*)$ iff both of $\mathfrak{l}_{*,x}$ and $\mathfrak{r}_{*,x}$ are invertible in the Banach algebra $\mathrm{B}(E)$.

Theorem 2.2**.**

Let $*,\diamond\in\mathrm{Mltp}(E)$. Suppose that $*$ is unital and for a numerical constant $r$ with $0\leq r<1$ we have $|\diamond-*|\leq r\|e_{*}\|^{-1}$. Then, $\diamond$ is also unital. Moreover, letting $s=(1+r(1-r)^{-1})$, we have

[TABLE]

Proof.

We have

[TABLE]

Thus, by Theorem 2.1, $\mathfrak{l}_{\diamond,e_{*}}$ is invertible in $\mathrm{B}(E)$. Similarly, it is proved that $\mathfrak{r}_{\diamond,e_{*}}$ is invertible in $\mathrm{B}(E)$. By invertibility of $\mathfrak{l}_{\diamond,e_{*}}$, there is $e^{\prime}\in E$ such that $e_{*}\diamond e^{\prime}=e_{*}$. By invertibility of $\mathfrak{r}_{\diamond,e_{*}}$, for every $x\in E$ there is $x^{\prime}$ such that $x=x^{\prime}\diamond e_{*}$. Thus, we have

[TABLE]

This shows that $e^{\prime}$ is a right unit of $(E,\diamond)$. Similarly, it is proved that $(E,\diamond)$ has a left unit. Thus, $(E,\diamond)$ is unital and $e_{\diamond}=e^{\prime}$. It follows from (2) and Theorem 2.1 that

[TABLE]

Thus, $\|e_{\diamond}-e_{*}\|=\|\mathfrak{l}_{\diamond,e_{*}}^{-1}(e_{*})-\mathrm{id}(e_{*})\|\leq s|\diamond-*|\|e_{*}\|^{2}\leq rs\|e_{*}\|$. ∎

We have the following easy estimate of distance between unit elements.

Theorem 2.3**.**

Let $*,\diamond\in\mathrm{Mltp}_{\mathrm{u}}(E)$. Then,

[TABLE]

Proof.

We have, $\|e_{\diamond}-e_{*}\|=\|e_{\diamond}*e_{*}-e_{\diamond}\diamond e_{*}\|\leq|\diamond-*|\|e_{\diamond}\|\|e_{*}\|$. ∎

Theorem 2.4**.**

Let $*\in\mathrm{Mltp}_{\mathrm{u}}(E)$, and let $a\in\mathrm{Inv}(E,*)$. Suppose that $\diamond\in\mathrm{Mltp}(E)$ is such that $|\diamond-*|<(\|a\|\|a^{-1}_{*}\||*|)^{-1}$. Then $\diamond$ is unital and $a\in\mathrm{Inv}(E,\diamond)$. Moreover, for every $M>0$ there exists a universal constant $C_{M}>0$ such that if

[TABLE]

for some constant $r$ with $0\leq r<1$, then $\diamond$ is unital, $a\in\mathrm{Inv}(E,\diamond)$, and

[TABLE]

where $s=1+r(1-r)^{-1}$.

Proof.

Suppose that $|\diamond-*|<(\|a\|\|a^{-1}_{*}\||*|)^{-1}$. Since $\|e_{*}\|\leq\|a\|\|a^{-1}_{*}\||*|$, we have $|\diamond-*|<\|e_{*}\|^{-1}$. Thus, by Theorem 2.2, $\diamond$ is unital. We have,

[TABLE]

Note that $\mathfrak{l}_{*,a_{*}^{-1}}=\mathfrak{l}^{-1}_{*,a}$. From (4) and Theorem 2.1 it is concluded that $\mathfrak{l}_{\diamond,a}$ is invertible in $\mathrm{B}(E)$. Similarly, it is proved that $\mathfrak{r}_{\diamond,a}$ is invertible in $\mathrm{B}(E)$. Thus, $a\in\mathrm{Inv}(E,\diamond)$.

Suppose that $M>0$ be such that (3) is satisfied. We have $|\diamond-*|\leq rM^{-3}<(\|a\|\|a^{-1}_{*}\||*|)^{-1}$. Thus, by the first part, $\diamond$ is unital and $a\in\mathrm{Inv}(E,\diamond)$. Analogous to (4), we have

[TABLE]

Thus, by Theorem 2.1, we have $|\mathfrak{l}^{-1}_{\diamond,a}-\mathfrak{l}^{-1}_{*,a}|\leq s|\mathfrak{l}^{-1}_{*,a}|^{2}|\mathfrak{l}_{\diamond,a}-\mathfrak{l}_{*,a}|$. It follows that

[TABLE]

We have,

[TABLE]

By Theorem 2.2, we have

[TABLE]

Also, we have,

[TABLE]

Now, the second part of the theorem follows from (5),(6),(7), and (8). ∎

The following is a direct corollary of Theorem 2.4.

Corollary 2.5**.**

Let $(\star_{n})_{n}$ be a sequence in $\mathrm{Mltp}_{\mathrm{u}}(E)$ converging to $\star\in\mathrm{Mltp}_{\mathrm{u}}(E)$. Then, for every $a\in E$ we have

[TABLE]

or equivalently,

[TABLE]

3 Elementary aspects of the geometry of $\mathrm{Mltp}(E)$

In this section, we consider some elementary observations on the geometry of $\mathrm{Mltp}(E)$ as a subset of the normed space $\mathrm{B}_{2}(E)$.

Lemma 3.1**.**

Let $(\star_{n})_{n}$ be a sequence in $\mathrm{B}_{2}(E)$ such that it converges to $\star\in\mathrm{B}_{2}(E)$. Let $(a_{n})_{n},(b_{n})_{n}$ be two convergent sequence in $E$ such that $a_{n}\rightarrow a$ and $b_{n}\rightarrow b$. Then, $a_{n}\star_{n}b_{n}\rightarrow a\star b$.

Proof.

It follows from the following simple inequality:

[TABLE]

∎

Theorem 3.2**.**

$\mathrm{Mltp}(E)$* is a closed subset of $\mathrm{B}_{2}(E)$.*

Proof.

Let $(\star_{n})_{n}$ be a sequence in $\mathrm{Mltp}(E)$ such that it converges to $\star\in\mathrm{B}_{2}(E)$. We must show that $\star\in\mathrm{Mltp}(E)$. Let $a,b,c\in E$. By permanent application of Lemma 3.1, we have

[TABLE]

Thus, $\star$ is associative and the proof is complete. ∎

Theorem 3.3**.**

$\mathrm{Mltp}_{\mathrm{u}}(E)$* and $\mathrm{Mltp}_{\mathrm{c}}(E)$ are, respectively, open and closed in $\mathrm{Mltp}(E)$.*

Proof.

It follows from Theorem 2.2 that $\mathrm{Mltp}_{\mathrm{u}}(E)$ is open in $\mathrm{Mltp}(E)$. By Lemma 3.1, it is easily verified that any limit point of $\mathrm{Mltp}_{\mathrm{c}}(E)$ is a commutative multiplication. ∎

Theorem 3.4**.**

The map $\star\mapsto e_{\star}$ from $\mathrm{Mltp}_{\mathrm{u}}(E)$ to $E$ is continuous. Moreover, if $\star$ converges to a point in the boundary of $\mathrm{Mltp}_{\mathrm{u}}(E)$ in $\mathrm{Mltp}(E)$, then $\|e_{\star}\|$ converges to $\infty$.

Proof.

The first part follows directly from Inequality (1). For the second part, let $(\star_{n})_{n}$ be a sequence in $\mathrm{Mltp}_{\mathrm{u}}(E)$ converging to $\star$ such that $\star$ is in the boundary of $\mathrm{Mltp}_{\mathrm{u}}(E)$ in $\mathrm{Mltp}(E)$. Since $\mathrm{Mltp}_{\mathrm{u}}(E)$ is open in $\mathrm{Mltp}(E)$, $\star$ is not unital. If the sequence $\|e_{\star_{n}}\|$ is bounded, then Theorem 2.2 shows that $\star$ must be unital. The proof is complete. ∎

Theorem 3.5**.**

Let $a$ be a nonzero element of $E$. Let $U_{a}:=\{\star\in\mathrm{Mltp}(E):a\in\mathrm{Inv}(E,\star)\}$. Then $U_{a}$ is open in $\mathrm{Mltp}_{\mathrm{u}}(E)$ and the map $\star\mapsto a_{\star}^{-1}$ from $U_{a}$ to $E$ is continuous.

Proof.

It follows easily from Theorem 2.4. ∎

Let us mention that for finite dimensional $E$, $\mathrm{Mltp}(E)$ can be considered as an affine algebraic variety: For any natural number $n$, let $\mathbb{C}^{n}$ denote the direct sum of $n$-copy of complex field $\mathbb{C}$. Let $\alpha_{1},\ldots,\alpha_{n}$ denote the standard basis of $\mathbb{C}^{n}$. Then any bilinear operator $\star\in\mathrm{B}_{2}(\mathbb{C}^{n})$ is exactly determined by a $n^{3}$-tuple $(\lambda_{ijk})_{1\leq i,j,k\leq n}$ of complex numbers such that $\alpha_{i}\star\alpha_{j}=\sum_{k=1}^{n}\lambda_{ijk}\alpha_{k}$. It is easily checked that $\star$ belongs to $\mathrm{Mltp}(\mathbb{C}^{n})$ iff the associated $n^{3}$-tuple $(\lambda_{ijk})$ is in the zero locus of the following class of quadratic polynomials:

[TABLE]

In other word, $\mathrm{Mltp}(\mathbb{C}^{n})$ is identified with a Zariski-closed algebraic subset of affine space $\mathbb{C}^{n^{3}}$. Similar to our result (Theorem 3.3), it has been shown in [6] that $\mathrm{Mltp}_{\mathrm{u}}(\mathbb{C}^{n})$ is Zariski-open in $\mathrm{Mltp}(\mathbb{C}^{n})$. (Note that the the $|\cdot|$-topology of $\mathrm{B}_{2}(\mathbb{C}^{n})$ coincides with the Euclidean topology of $\mathbb{C}^{n^{3}}$, and the Zariski topology is strictly coarser than Euclidean topology.)

Let us observe that $\mathrm{Mltp}(E)$ is a ‘large’ subset of $\mathrm{B}_{2}(E)$: First of all, note that if $F$ be a $n$-dimensional vector space, then any semigroup $S$ with $n$ elements together with any vector basis $(\alpha_{s})_{s\in S}$ of $F$, indexed by elements of $S$, define a multiplication $\star\in\mathrm{Mltp}(F)$ by $\alpha_{s}\star\alpha_{s^{\prime}}=\alpha_{ss^{\prime}}$. Now, suppose that $E_{1},E_{2}$ are Banach spaces and let $\star_{1}\in\mathrm{Mltp}(E_{1})$ and $\star_{2}\in\mathrm{Mltp}(E_{2})$. Then one can define a multiplication $\star\in\mathrm{Mltp}(E_{1}\oplus E_{2})$ by

[TABLE]

Thus, $(\star_{1},\star_{2})\mapsto\star$ defines an embedding $\mathrm{Mltp}(E_{1})\times\mathrm{Mltp}(E_{2})\rightarrow\mathrm{Mltp}(E_{1}\oplus E_{2})$. These observations show that for an arbitrary Banach space $E$, one can define an element of $\mathrm{Mltp}(E)$, simply by choosing a $n$-dimensional subspace $E_{1}\subset E$, any topological complement $E_{2}$ of $E_{1}$ in $E$, any semigroup with $n$ elements, and any vector basis of $E_{1}$.

In the following, we shall introduce a quotient $\mathbb{M}(E)$ of $\mathrm{Mltp}(E)$ which is much smaller than $\mathrm{Mltp}(E)$ and is more reasonable for study.

Let $\mathrm{Hom}(E)\subset\mathrm{B}(E)$ be the topological group of linear homeomorphisms from $E$ onto $E$, with $|\cdot|$-topology. $\mathrm{Hom}(E)$ acts on $\mathrm{Mltp}(E)$ from left in a canonical way: Let $t\in\mathrm{Hom}(E)$ and $*\in\mathrm{Mltp}(E)$. Then $t\cdot*=*_{t}$, the action of $t$ on $*$, is defined to be the continuous multiplication on $E$ given by $a*_{t}b:=t^{-1}(t(a)*t(b))$. Note that the linear map $t^{-1}$ becomes a homeomorphic algebra isomorphism from $(E,*)$ onto $(E,*_{t})$.

We show that the described group action is continuous: For any $s,t\in\mathrm{B}(E)$ let $\Lambda_{t},\Gamma_{s,t}:\mathrm{B}_{2}(E)\rightarrow\mathrm{B}_{2}(E)$ be defined respectively by $\varphi\mapsto t\varphi,\varphi\mapsto\varphi(s,t)$. Then these are bounded linear operators on $\mathrm{B}_{2}(E)$, and it is not hard to see that $|\Lambda_{t}|=|t|$ and $|\Gamma_{s,t}|=|s||t|$. For $s,t\in\mathrm{Hom}(E)$ and $\diamond,*\in\mathrm{Mltp}(E)$ we have,

[TABLE]

We also have,

[TABLE]

It follows from (9) and (10) that the canonical action $\mathrm{Hom}(E)\times\mathrm{Mltp}(E)\rightarrow\mathrm{Mltp}(E)$ is continuous. We denote the orbit space $\mathrm{Mltp}(E)/\mathrm{Hom}(E)$ by $\mathbb{M}(E)$. It is appropriate to call $\mathbb{M}(E)$ moduli space of multiplications on $E$.

A natural problem related to geometry of Banach spaces arises:

Problem 3.6**.**

Describe the ‘geometry’ of $\mathbb{M}(E)$.

It is clear that if two Banach spaces $E,F$ are homeomorphic linear isomorphic then $\mathrm{Mltp}(E)$ and $\mathbb{M}(E)$ are respectively homeomorphic with $\mathrm{Mltp}(F)$ and $\mathbb{M}(F)$. Thus, $\mathrm{Mltp}(E)$ and $\mathbb{M}(E)$ (and their invariants) can be considered as invariants of $E$ in category of topological vector spaces. However, the study of $\mathbb{M}(E)$ is a hard work even if $E$ is finite dimensional.

In the following we remark some aspects of the geometry of $\mathbb{M}(E)$. (For a multiplication $*\in\mathrm{Mltp}(E)$, its orbit $\{t\cdot*:t\in\mathrm{Hom}(E)\}$, mutually as an element in $\mathbb{M}(E)$ and as a subset of $\mathrm{Mltp}(E)$, is denoted by $\bar{*}$.)

Remark 3.7**.**

**

1. (i)

The only orbit $\bar{*}$ which is closed in $\mathrm{Mltp}(E)$ is the orbit of the zero multiplication [math]: We have $\bar{0}=\{0\}$ and thus $\bar{0}$ is closed. Let $*\neq 0$ be in $\mathrm{Mltp}(E)$. It is clear that [math] does not belong to $\bar{*}$. For every $n\geq 1$, let $t_{n}:=n^{-1}\mathrm{id}\in\mathrm{Hom}(E)$. Then $|t_{n}\cdot*|\rightarrow 0$. This shows that [math] is a limit point of $\bar{*}$. Therefore $\bar{*}$ is not closed in $\mathrm{Mltp}(E)$ if $*\neq 0$. Note that since [math] is in the closure of any orbit, $\bar{0}$ belongs to any nonempty closed subset of $\mathbb{M}(E)$. 2. (ii)

A multiplication $*\in\mathrm{Mltp}(E)$ is called rigid [7, Definition 3.13] if there exist $\epsilon>0$ such that for every $\diamond\in\mathrm{Mltp}(E)$ with $|\diamond-*|<\epsilon$ there exists $t\in\mathrm{Hom}(E)$ with $\diamond=t\cdot*$. Thus, $*$ is rigid iff $\bar{*}$ is an open subset of $\mathrm{Mltp}(E)$, or equivalently, $\bar{*}$ is an isolated point of $\mathbb{M}(E)$. Johnson has shown [9, Theorem 2.1] that for $*\in\mathrm{Mltp}(E)$ if the second and the third Hochschild cohomology groups ([8]) of $(E,*)$ with values in $E$ vanish, then $*$ is rigid. 3. (iii)

The algebraic geometry of $\mathbb{M}(\mathbb{C}^{5})$, has been considered by G. Mazzola in [10]. He has shown that $\mathbb{M}_{\mathrm{u}}(\mathbb{C}^{5})$, the moduli space of unital algebras on $\mathbb{C}^{5}$, has exactly $59$ members. Algebraic geometry of $\mathbb{M}(\mathbb{C}^{n})$, for $n=2,3,4$ has been considered respectively in [2, 4, 5]. 4. (iv)

Finding numerical functions (with some continuity properties) on $\mathrm{Mltp}(E)$ and $\mathbb{M}(E)$ may be useful. By Theorem 3.4, the function $\star\mapsto\|e_{\star}\|$ is continuous on $\mathrm{Mltp}_{\mathrm{u}}(E)$. It is easily seen that for a $\star\in\mathrm{Mltp}(E)$, the Hochschild cohomology groups ([8]) of $(E,\star)$ with values in $E$ and its topological dual, is only depends on $\bar{\star}$. Thus, for instance, the dimensions of these groups can be considered as a function on $\mathbb{M}(E)$ with discrete values.

We end this note by some examples of families of continuous multiplications which the above theorems and inequalities may be effectively applied to them.

Example 3.8**.**

Let $X$ be a compact metric space and $\mathbf{M}_{X}$ be the Banach space of complex-valued continuous functions on $X\times X$, with $\sup$-norm. Any Borel complex measure $\mu$ on $X$, induces a $\star_{\mu}\in\mathrm{Mltp}(\mathbf{M}_{X})$ defined by

[TABLE]

In the case that $\mu$ is a probability measure, $(\mathbf{M}_{X},\star_{\mu})$ is a Banach algebra. Some algebraic aspects of this class of Banach algebras has been considered in [13]. It is easily seen that the family $\{\star_{\mu}\}_{\mu}$ is continuously depends on $\mu$ as $\mu$ varies in the Banach space $\mathcal{M}(X)$ of Borel complex measures, with total variation norm. Another multiplication on $\mathbf{M}_{X}$, is the point-wise multiplication which makes $\mathbf{M}_{X}$ to a commutative C-algebra. Note that in the case that $X:=\{1,\ldots,n\}$ and $\mu$ is the counting measure, $(\mathbf{M}_{X},\star_{\mu})$ is isomorphic to the algebra of $n\times n$ matrixes.

---

Example 3.9**.**

Let $X$ be a locally compact space and $\mathcal{M}(X)$ be as in Example 3.8. Any continuous semigroup multiplication $s:X\times X\rightarrow X$ defines a $\star_{s}\in\mathrm{Mltp}(\mathcal{M}(X))$ as follows.

[TABLE]

Example 3.10**.**

Let $A$ and $B$ be Banach algebras. Consider the Banach space $A\oplus B$ with an $\ell_{p}$-norm ($1\leq p\leq\infty$). Any continuous algebra homomorphism $T:B\rightarrow A$ induces a $\star_{T}\in\mathrm{Mltp}(A\oplus B)$ defined by

[TABLE]

*It is easily verified that the family $\{\star_{T}\}_{T}$ is continuously depends on $T$ when $T$ varies in the space of continuous algebra homomorphisms as a subspace of the Banach space of bounded linear operators from $B$ to $A$. Some properties of algebras of the type $(A\oplus B,\star_{T})$ has been considered by some authors. See [11] and references therein.

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