Almost multiplicative maps with respect to almost associative operations on Banach algebras

Abstract

We prove an Ulam type stability result for a non-associative version of the multiplicativity equation, that is, $T(xy)=\Psi (T(x),T(y))$, where T is a unital bounded operator acting from an amenable Banach algebra to a dual Banach algebra.

1 Introduction

Let ${\mathcal {A}}$ and ${\mathcal {B}}$ be Banach algebras, and let ${\mathscr {L}}^n({\mathcal {A}},{\mathcal {B}})$ be the space of n-linear bounded operators from ${\mathcal {A}}^n$ to ${\mathcal {B}}$. In this note, we study an Ulam type stability problem for a non-associative version of the multiplicativity equation, namely,

$$\begin{aligned} T(xy)=\Psi (T(x),T(y))\qquad (x,y\in {\mathcal {A}}), \end{aligned}$$

(1.1)

where $T\in {\mathscr {L}}^1({\mathcal {A}},{\mathcal {B}})$ and $\Psi \in {\mathscr {L}}^2({\mathcal {B}},{\mathcal {B}})$. An expected stability effect should be quite different from the one for almost multiplicative maps. Indeed, $\Psi$ being merely ‘almost’ (but not exactly) associative on the range of T means that (1.1) cannot be satisfied accurately, since the left-hand side yields an associative operation on (x, y). Hence, the best accuracy with which Eq. (1.1) can be satisfied should be expressed in terms of the ‘defect’ of $\Psi$ concerning its associativity.

Our main result says, roughly speaking, that for certain Banach algebras, any operator T satisfying (1.1) with sufficient accuracy can be approximated to a prescribed accuracy by an operator for which the obvious obstacle of non-associativity of $\Psi$ is actually the only obstacle in the way of satisfying (1.1).

To be more precise, for any linear map $T:{\mathcal {A}}\rightarrow {\mathcal {B}}$ and any bilinear map $\Psi :{\mathcal {B}}\times {\mathcal {B}}\rightarrow {\mathcal {B}}$, consider $T^{\,\vee }:{\mathcal {A}}^2\rightarrow {\mathcal {B}}$ defined by

$$\begin{aligned} T^{\,\vee }(x,y)=T(xy)-\Psi (T(x),T(y)). \end{aligned}$$

We define the $\Psi$-multiplicative defect of T as $\Vert T^{\,\vee }\Vert$, that is,

$$\begin{aligned} {\textsf{m}}\text{- }\text {def}_\Psi (T)=\sup \big \{\Vert T(xy)-\Psi (T(x),T(y))\Vert :x,y\in {\mathcal {A}},\, \Vert x\Vert , \Vert y\Vert \le 1\big \}. \end{aligned}$$

Similarly, we define the associative defect of $\Psi$ by

$$\begin{aligned} {\textsf{a}}\text{- }\text {def}(\Psi )=\sup \big \{\Vert \Psi (u,\Psi (v,w))-\Psi (\Psi (u,v),w)\Vert :u,v,w\in {\mathcal {B}},\, \Vert u\Vert , \Vert v\Vert , \Vert w\Vert \le 1\big \}. \end{aligned}$$

Our main result then reads as follows.

Theorem 1

Let ${\mathcal {A}}$ be a unital amenable Banach algebra and ${\mathcal {B}}$ be a unital dual Banach algebra with an isometric predual. Then for arbitrary $K,L\ge 1$ and $\varepsilon ,\eta \in (0,1)$ there exists $\delta >0$ such that the following holds true: If $\Psi \in {\mathscr {L}}^2({\mathcal {B}},{\mathcal {B}})$ satisfies $\Vert \Psi \Vert \le L$ and $\Psi (1,u)=\Psi (u,1)=u$ for every $u\in {\mathcal {B}}$, and $T\in {\mathscr {L}}^1({\mathcal {A}},{\mathcal {B}})$ is a unital operator with $\Vert T\Vert \le K$ and ${\textsf{m}}\text{- }\text {def}_\Psi (T)\le \delta$, then there exists a unital $S\in {\mathscr {L}}^1({\mathcal {A}},{\mathcal {B}})$ such that

$$\begin{aligned} {\textsf{m}}\text{- }\text {def}_\Psi (S)<{\textsf{a}}\text{- }\text {def}(\Psi )^{1-\eta }\quad \text{ and } \quad \Vert T-S\Vert <\varepsilon . \end{aligned}$$

In order to put this result in a wider context, recall that the theory of almost multiplicative maps between Banach algebras was developed by B.E. Johnson in the series of papers [8,9,10,11], where he introduced the AMNM property. A pair $({\mathcal {A}},{\mathcal {B}})$ of Banach algebras has this property, provided that for any $K,\varepsilon >0$ there is $\delta >0$ such that for every $T\in {\mathscr {L}}^1({\mathcal {A}},{\mathcal {B}})$ satisfying $\Vert T\Vert \le K$ and $\Vert T(xy)-T(x)T(y)\Vert <\delta \Vert x\Vert \Vert y\Vert$ for all nonzero $x,y\in {\mathcal {A}}$, there exists a multiplicative operator $S\in {\mathscr {L}}^1({\mathcal {A}},{\mathcal {B}})$ with $\Vert T-S\Vert <\varepsilon$. One of the brilliant ideas of Johnson was to define a ‘Banach-algebraic’ analogue of the Newton–Raphson approximation procedure which allowed him to show that if ${\mathcal {A}}$ is amenable and ${\mathcal {B}}$ is a dual Banach algebra, then $({\mathcal {A}},{\mathcal {B}})$ has the AMNM property (see [11, Thm. 3.1]). His result was generalized in the PhD thesis of Horváth [7], and widely developed in the recent paper [4] by Choi, Horváth and Laustsen, where they exhibited a large collection of AMNM pairs of algebras of bounded operators on Banach spaces. Many other Ulam type stability problems on Banach algebras have been studied in recent years; see e.g. [1,2,3, 12, 13].

The proof of Theorem 1 closely follows the above-mentioned method of proof [11, Thm. 3.1] by using a Newton–Raphson-like algorithm defined in terms of an approximate diagonal. However, unlike in the case of Johnson’s method, unless $\Psi$ is associative, our approximation process must terminate at a certain point.

2 Terminology and preparatory lemmas

Throughout this section, we fix unital Banach algebras ${\mathcal {A}}$ and ${\mathcal {B}}$, as well as operators $T\in {\mathscr {L}}^1({\mathcal {A}},{\mathcal {B}})$ and $\Psi \in {\mathscr {L}}^2({\mathcal {B}},{\mathcal {B}})$.

One of the main ideas of approximating almost multiplicative maps (see [4, 11]) is based on a modification of standard coboundary operators in the Hochschild cohomology theory, as defined in [14, Ch. 2], by introducing as a parameter a map $\phi \in {\mathscr {L}}({\mathcal {A}},{\mathcal {B}})$ playing the role of ‘approximate action’ of ${\mathcal {A}}$ on ${\mathcal {B}}$. According to [4, Def. 5.2], for any $n\in {\mathbb {N}}_0$, the n-coboundary operator $\partial ^n_\phi :{\mathscr {L}}^n({\mathcal {A}},{\mathcal {B}})\rightarrow {\mathscr {L}}^{n+1}({\mathcal {A}},{\mathcal {B}})$ is defined by the formula

$$\begin{aligned} \begin{aligned} \partial ^n_\phi \psi (a_1,\ldots ,a_{n+1})&=\phi (a_1)\psi (a_2,\ldots ,a_{n+1}) +\sum _{j=1}^n (-1)^j \psi (a_1,\ldots ,a_ja_{j+1},\ldots ,a_{n+1})\\&\quad +(-1)^{n+1}\psi (a_1,\ldots ,a_n)\phi (a_{n+1}). \end{aligned} \end{aligned}$$

In fact, in the study of almost multiplicative maps only the 2-coboundary operator is relevant. For studying Eq. (1.1), we introduce the following modification of $\partial _\phi ^2$. Define $\delta _T^2:{\mathscr {L}}^2({\mathcal {A}},{\mathcal {B}})\rightarrow {\mathscr {L}}^3({\mathcal {A}},{\mathcal {B}})$ by the formula

$$\begin{aligned} \delta ^2_T\,\phi (x,y,z)=\Psi (T(x),\phi (y,z))-\phi (xy,z)+\phi (x,yz)-\Psi (\phi (x,y),T(z)). \end{aligned}$$

Observe that if $\Psi$ is ‘almost’ associative, then the operator $T^{\,\vee }$ ‘almost’ satisfies the relation $T^{\,\vee }\in \textrm{ker}\,\delta _T^2$ in the following sense.

Lemma 2

For any $T\in {\mathscr {L}}^1({\mathcal {A}},{\mathcal {B}})$ and $\Psi \in {\mathscr {L}}^2({\mathcal {B}},{\mathcal {B}})$, we have

$$\begin{aligned} \Vert \delta _T^2\, T^{\,\vee }\Vert \le {\textsf{a}}\text{- }\text {def}(\Psi )\cdot \Vert T\Vert ^3. \end{aligned}$$

Proof

Note that

$$\begin{aligned} \begin{aligned}{}[\delta _T^2\, T^{\,\vee }](x,y,z)&= \Psi (T(x),T^{\,\vee }(y,z))+\Psi (T(xy),T(z))\\&\hspace{64pt}-\Psi (T(x),T(yz))-\Psi (T^{\,\vee }(x,y),T(z))\\&=\Psi [\Psi (T(x),T(y)),T(z)]-\Psi [T(x),\Psi (T(y),T(z))], \end{aligned} \end{aligned}$$

from which the result follows readily. $\square$

Lemma 3

The Fréchet derivative of the map

$$\begin{aligned} {\mathscr {L}}^1({\mathcal {A}},{\mathcal {B}})\ni S\mapsto {}S^{\,\vee }\in {\mathscr {L}}^2({\mathcal {A}},{\mathcal {B}}) \end{aligned}$$

at the point T is given by the formula

$$\begin{aligned}{}[{\textsf{D}}H](x,y)=H(xy)-\Psi (T(x),H(y))-\Psi (H(x),T(y))\qquad (H\in {\mathscr {L}}^1({\mathcal {A}},{\mathcal {B}})) \end{aligned}$$

and, moreover,

$$\begin{aligned} \Vert (T+H)^{\,\vee }-T^{\,\vee }-{\textsf{D}}H\Vert \le \Vert \Psi \Vert \cdot \Vert H\Vert ^2\qquad (H\in {\mathscr {L}}^1({\mathcal {A}},{\mathcal {B}})). \end{aligned}$$

(2.1)

Proof

It follows by a straightforward calculation:

$$\begin{aligned} \begin{aligned}{}[(T+H)^{\vee }-T^{\,\vee }](x,y)&=H(xy)-\Psi (T(x)+H(x),T(y)+H(y))+\Psi (T(x),T(y))\\&=H(xy)-\Psi (T(x),H(y))-\Psi (H(x),T(y))-r(H)(x,y), \end{aligned} \end{aligned}$$

where $r(H)(x,y)=\Psi (H(x),H(y))$ and hence $\Vert r(H)\Vert \le \Vert \Psi \Vert \cdot \Vert H\Vert ^2$. $\square$

In what follows, we use standard notation and facts concerning projective tensor products of Banach spaces, as described e.g. in [16, Ch. 2]. By ${\mathcal {A}}{\hat{\otimes }}{\mathcal {A}}$ we denote the completion of ${\mathcal {A}}\otimes {\mathcal {A}}$ in the projective norm. Recall that for every $\Phi \in {\mathscr {L}}^2({\mathcal {A}},{\mathcal {B}})$ there exists a unique operator $\Theta \in {\mathscr {L}}^1({\mathcal {A}}{\hat{\otimes }}{\mathcal {A}},{\mathcal {B}})$ such that $\Theta (a\otimes b)=\Phi (a,b)$ for all $a,b\in {\mathcal {A}}$ and, moreover, $\Vert \Phi \Vert =\Vert \Theta \Vert$. From now on, we adopt the convention that for any $\Phi \in {\mathscr {L}}^2({\mathcal {A}},{\mathcal {B}})$, we write $\widetilde{\Phi }$ for the corresponding linear operator from ${\mathcal {A}}{\hat{\otimes }}{\mathcal {A}}$ to ${\mathcal {B}}$. Let also $\pi _{\mathcal {A}}:{\mathcal {A}}{\hat{\otimes }}{\mathcal {A}}\rightarrow {\mathcal {A}}$ stand for the unique bounded linear map such that $\pi _{\mathcal {A}}(a\otimes b)=ab$ for all $a,b\in {\mathcal {A}}$.

Let us now recall some elementary facts concerning amenable Banach algebras; for a nice exposition of this theory, see [14, 15]. A net $(\Delta _\alpha )_{\alpha \in A}\subset {\mathcal {A}}{\hat{\otimes }}{\mathcal {A}}$ is called an approximate diagonal, provided that

$$\begin{aligned} \lim _\alpha (a\cdot \Delta _\alpha -\Delta _\alpha \cdot a)=0\quad \text{ and } \quad \lim _\alpha a\pi _{\mathcal {A}}(\Delta _\alpha )=a\quad \,\,(a\in {\mathcal {A}}). \end{aligned}$$

Note that in the case ${\mathcal {A}}$ is unital the latter condition means that $\lim _\alpha \pi _{\mathcal {A}}(\Delta _\alpha )=1$ in norm. The approximate diagonal $(\Delta _\alpha )_{\alpha \in A}$ is called bounded if $\sup _\alpha \Vert \Delta \Vert _\alpha <\infty$. Notice that, in our convention, $\pi _{\mathcal {A}}=\widetilde{\Lambda }$, where $\Lambda (a,b)=ab$. Therefore, $\Vert \pi _{\mathcal {A}}\Vert =\Vert \Lambda \Vert =1$ and since $\lim _\alpha \pi _{\mathcal {A}}(\Delta _\alpha )=1$, we must have $\sup _\alpha \Vert \Delta _\alpha \Vert \ge 1$. A Banach algebra ${\mathcal {A}}$ is called amenable if there exists a bounded approximate diagonal $(\Delta _\alpha )_{\alpha \in A}$ for ${\mathcal {A}}$.

Concerning the codomain algebra in our stability problem, we need the following definition: A Banach algebra ${\mathcal {B}}$ is called a dual Banach algebra with isometric predual X, provided that X is a Banach space such that ${\mathcal {B}}$ and $X^*$ are isometrically isomorphic and multiplication in ${\mathcal {B}}$ is separately $\sigma ({\mathcal {B}},X)$-continuous. In particular, ${\mathcal {B}}$ is equipped with the weak$^*$ topology and the condition of the continuity of multiplication is equivalent to saying that $i^*\circ \kappa (X)$ is a sub-${\mathcal {B}}$-bimodule of ${\mathcal {B}}^*$, where $i:{\mathcal {B}}\rightarrow X^*$ is the said linear isometry and $\kappa :X\rightarrow X^{**}$ is the canonical embedding (see [6, §2]).

It should be remarked that the original assumption made by Johnson was that the codomain algebra ${\mathcal {B}}$ is isomorphic, as a Banach ${\mathcal {B}}$-bimodule, to the dual $({\mathcal {B}}_*)^*$ of some Banach ${\mathcal {B}}$-bimodule ${\mathcal {B}}_*$. Our choice follows the definition proposed in [4] which was influenced by [5, 6].

3 Proof of the main result

From now on, we fix Banach algebras ${\mathcal {A}}$ and ${\mathcal {B}}$ satisfying the assumptions of Theorem 1, as well as an isometric predual ${\mathcal {B}}_*$ of ${\mathcal {B}}$. Let $(\Delta _\alpha )_{\alpha \in A}\subset {\mathcal {A}}{\hat{\otimes }}{\mathcal {A}}$ be a bounded approximate diagonal for ${\mathcal {A}}$ with

$$\begin{aligned} \Delta _\alpha =\sum _j a_{\alpha ,j}\otimes b_{\alpha ,j}\qquad (\alpha \in A). \end{aligned}$$

All these are finite sums yet we do not need to indicate the sets over which we sum. The following observation is due to Johnson (see also [4, Lemma 5.11]); we include a short proof for the sake of completeness.

Lemma 4

For any $R\in {\mathscr {L}}^2({\mathcal {A}},{\mathcal {B}})$, the limit

$$\begin{aligned} \lim _\alpha \sum _j R(a_{\alpha ,j},b_{\alpha ,j}) \end{aligned}$$

exists in the weak$^*$ topology $\sigma ({\mathcal {B}},{\mathcal {B}}_*)$ on ${\mathcal {B}}$, and it is bounded by $\sup _\alpha \Vert \Delta _\alpha \Vert \Vert R\Vert$.

Proof

Since $\sup _\alpha \Vert \Delta _\alpha \Vert <\infty$, using the Banach–Alaoglu theorem and passing to a subnet we may assume that $\kappa _{{\mathcal {A}}{\hat{\otimes }}{\mathcal {A}}}(\Delta _\alpha )$ converges in the weak$^*$ topology

to some ${\textsf{M}}\in ({\mathcal {A}}{\hat{\otimes }}{\mathcal {A}})^{**}$, and then $\Vert {\textsf{M}}\Vert \le \sup _\alpha \Vert \Delta _\alpha \Vert$. Thus, by the weak$^*$-continuity of $\kappa _{{\mathcal {B}}_*}^*$ and $(\widetilde{R})^{**}$, the net

$$\begin{aligned} \begin{aligned} \sum _j R(a_{\alpha ,j},b_{\alpha ,j})&=\sum _j \widetilde{R}(a_{\alpha ,j}\otimes b_{\alpha ,j})=\widetilde{R}(\Delta _\alpha )\\&=\kappa _{{\mathcal {B}}_*}^*(\widetilde{R})^{**}\kappa _{{\mathcal {A}}{\hat{\otimes }}{\mathcal {A}}}(\Delta _\alpha ) \end{aligned} \end{aligned}$$

converges in the weak$^*$ topology to $\kappa _{{\mathcal {B}}_*}^*(\widetilde{R})^{**}({\textsf{M}})$, and is bounded by $\sup _\alpha \Vert \Delta _\alpha \Vert \Vert (\widetilde{R})^{**}\Vert =\sup _\alpha \Vert \Delta _\alpha \Vert \Vert R\Vert$. Note also that for any weak$^*$ convergent net $(y_i)_{i\in I}\subset {\mathcal {B}}$ we have $\Vert \!\lim _i y_i\Vert \le \limsup _i\Vert y_i\Vert$, hence the upper estimate by $\sup _\alpha \Vert \Delta _\alpha \Vert \Vert R\Vert$ holds true for our weak$^*$ limit. $\square$

Similarly as in Johnson’s method (see the proof of [11, Thm. 3.1]), we define an operator ${\textsf{J}}:{\mathscr {L}}^2({\mathcal {A}},{\mathcal {B}})\rightarrow {\mathscr {L}}^1({\mathcal {A}},{\mathcal {B}})$ by

$$\begin{aligned} {\textsf{J}}R(x)=\lim _\alpha \sum _j \Psi (T(a_{\alpha ,j}),R(b_{\alpha ,j},x)). \end{aligned}$$

(3.1)

This definition is correct in view of Lemma 4, and we have $\Vert {\textsf{J}}R\Vert \le \sup _\alpha \Vert \Delta _\alpha \Vert \Vert T\Vert \Vert R\Vert \Vert \Psi \Vert$. We will use this observation several times in the sequel. Obviously, ${\textsf{J}}R$ depends on the operator T which we will not indicate explicitely, as it should be obvious from the context.

We shall show that formula (3.1) yields an approximate right inverse of the derivative ${\textsf{D}}$ given in Lemma 3 on the set of operators ‘almost’ belonging to $\textrm{ker}\,\delta _T^2$. In order to simplify writing let us denote $u\circ v=\Psi (u,v)$.

Proposition 5

Assume that T is unital and $\Psi (1,u)=u$ for each $u\in {\mathcal {B}}$. For every $R\in {\mathscr {L}}^2({\mathcal {A}},{\mathcal {B}})$, we have

$$\begin{aligned} \begin{aligned} \Vert R+{\textsf{D}}{\textsf{J}}R\Vert \le \big (&2\!\cdot \!{\textsf{m}}\text{- }\text {def}_\Psi (T)\cdot \!\Vert \Psi \Vert \!\cdot \!\Vert R\Vert \\&+\Vert \psi \Vert \!\cdot \!\Vert T\Vert \!\cdot \!\Vert \delta _T^2 R\Vert +3\!\cdot \!{\textsf{a}}\text{- }\text {def}(\Psi )\!\cdot \!\Vert T\Vert ^2\!\cdot \!\Vert R\Vert \big )\!\cdot \!\sup _\alpha \Vert \Delta _\alpha \Vert . \end{aligned}\nonumber \\ \end{aligned}$$

(3.2)

Proof

First, observe that since $\lim _\alpha \pi _{\mathcal {A}}(\Delta _\alpha )=1$, we have $\lim _\alpha \sum _j a_{\alpha ,j}b_{\alpha ,j}=1$ in norm and hence $\lim _\alpha \sum _j T(a_{\alpha ,j}b_{\alpha ,j})=1$. For any $x,y\in {\mathcal {A}}$, using the identity $\Psi (1,u)=u$, Lemma 3, and applying Lemma 4 to the bilinear maps $(a,b)\mapsto T(x)\circ (T(a)\circ R(b,y))$ and $(a,b)\mapsto (T(a)\circ R(b,x))\circ T(y)$, we thus obtain

$$\begin{aligned} \begin{aligned}{}[R+{\textsf{D}}{\textsf{J}}R](x,y)&=R(x,y)+[{\textsf{J}}R](xy)-T(x)\circ [{\textsf{J}}R](y)-[{\textsf{J}}R](x)\circ T(y)\\&=\lim _\alpha \sum _j\Big \{T(a_{\alpha ,j}b_{\alpha ,j})\circ R(x,y)+T(a_{\alpha ,j})\circ R(b_{\alpha ,j},xy)\\&\quad -T(x)\circ \big (T(a_{\alpha ,j})\circ R(b_{\alpha ,j},y)\big )\\&\quad -\big (T(a_{\alpha ,j})\circ R(b_{\alpha ,j},x)\big )\circ T(y)\Big \}. \end{aligned} \end{aligned}$$

(3.3)

Since

$$\begin{aligned} R(b_{\alpha ,j},xy)=\delta _T^2\,R(b_{\alpha ,j},x,y)-T(b_{\alpha ,j})\circ R(x,y)+R(b_{\alpha ,j}x,y)+R(b_{\alpha ,j},x)\circ T(y), \end{aligned}$$

we have

$$\begin{aligned} \begin{aligned}&\Big \Vert \lim _\alpha \sum _j\Big \{T( a_{\alpha ,j})\circ R(b_{\alpha ,j},xy)-T(a_{\alpha ,j})\circ R(b_{\alpha ,j}x,y)\\&\qquad +T(a_{\alpha ,j})\circ \big (T(b_{\alpha ,j})\circ R(x,y)\big )-T(a_{\alpha ,j})\circ \big (R(b_{\alpha ,j},x)\circ T(y)\big )\Big \}\Big \Vert \\&\quad \le \Vert \Psi \Vert \!\cdot \!\Vert T\Vert \!\cdot \!\Vert \delta _T^2 R\Vert \!\cdot \!\Vert x\Vert \Vert y\Vert \sup _\alpha \Vert \Delta _\alpha \Vert . \end{aligned} \end{aligned}$$

This estimate follows by considering the bilinear operator $\Phi _1:{\mathcal {A}}\times {\mathcal {A}}\rightarrow {\mathcal {B}}$ given by

$$\begin{aligned} \Phi _1(a,b)=T(a)\circ \delta _T^2 R(b,x,y) \end{aligned}$$

for which $\Vert \Phi _1\Vert \le \Vert \Psi \Vert \!\cdot \!\Vert T\Vert \!\cdot \!\Vert \delta _T^2 R\Vert \!\cdot \!\Vert x\Vert \Vert y\Vert$, thus

$$\begin{aligned}{} & {} \Big \Vert \lim _\alpha \sum _j T(a_{\alpha ,j})\circ \delta _T^2 R(b_{\alpha ,j},x,y)\Big \Vert \\{} & {} \quad \le \Vert \Psi \Vert \!\cdot \!\Vert T\Vert \!\cdot \!\Vert \delta _T^2 R\Vert \!\cdot \!\Vert x\Vert \Vert y\Vert \sup _\alpha \Vert \Delta _\alpha \Vert =:\!\eta _1(x,y). \end{aligned}$$

Coming back to (3.3) we see that $[R+{\textsf{D}}{\textsf{J}}R](x,y)$ is at distance at most $\eta _1(x,y)$ from

$$\begin{aligned} \begin{aligned}&S(x,y){:}{=}\lim _\alpha \sum _j \Big \{T( a_{\alpha ,j}b_{\alpha ,j})\circ R(x,y)-T(a_{\alpha ,j})\circ \big (T(b_{\alpha ,j})\circ R(x,y)\big )\\&\qquad +T(a_{\alpha ,j})\circ \big (R(b_{\alpha ,j},x)\circ T(y)\big )-\big (T(a_{\alpha ,j})\circ R(b_{\alpha ,j},x)\big )\circ T(y)\\&\qquad +T(a_{\alpha ,j})\circ R(b_{\alpha ,j}x,y)-T(x)\circ \big (T(a_{\alpha ,j})\circ R(b_{\alpha ,j},y)\big )\Big \}. \end{aligned} \end{aligned}$$

Considering the bilinear operator $\Phi _2:{\mathcal {A}}\times {\mathcal {A}}\rightarrow {\mathcal {B}}$ given by

$$\begin{aligned} \Phi _2(a,b)=T(a)\circ (T(b)\circ R(x,y))-(T(a)\circ T(b))\circ R(x,y), \end{aligned}$$

we have $\Vert \Phi _2\Vert \le {\textsf{a}}\text{- }\text {def}(\Psi )\!\cdot \!\Vert T\Vert ^2\!\cdot \!\Vert R\Vert \!\cdot \!\Vert x\Vert \Vert y\Vert$. Hence, the difference in the first line of the definition of S(x, y) can be estimated as follows:

$$\begin{aligned} \begin{aligned}&\Big \Vert \lim _\alpha \sum _j \Big \{ T(a_{\alpha ,j}b_{\alpha ,j})\circ R(x,y)-T(a_{\alpha ,j})\circ \big (T(b_{\alpha ,j})\circ R(x,y)\big )\Big \}\Big \Vert \\&\quad \le \Big \Vert \lim _\alpha \sum _j \Big \{\big (T(a_{\alpha ,j}b_{\alpha ,j})-T(a_{\alpha ,j})\circ T(b_{\alpha ,j})\big )\circ R(x,y)\Big \}\Big \Vert +\eta _2(x,y), \end{aligned} \end{aligned}$$

(3.4)

where

$$\begin{aligned} \eta _2(x,y){:}{=}{\textsf{a}}\text{- }\text {def}(\Psi )\!\cdot \!\Vert T\Vert ^2\!\cdot \!\Vert R\Vert \!\cdot \!\Vert x\Vert \Vert y\Vert \sup _\alpha \Vert \Delta _\alpha \Vert . \end{aligned}$$

Similarly, for the difference in the second line, we have

$$\begin{aligned}{} & {} \Big \Vert \lim _\alpha \sum _j \Big \{T(a_{\alpha ,j})\circ \big (R(b_{\alpha ,j},x)\circ T(y)\big )-\big (T(a_{\alpha ,j})\circ R(b_{\alpha ,j},x)\big )\circ T(y)\Big \}\Big \Vert \nonumber \\{} & {} \quad \le \eta _2(x,y). \end{aligned}$$

(3.5)

Finally, in order to estimate the difference in the third line of the definition of S(x, y), consider the bilinear map $\Phi _3:{\mathcal {A}}\times {\mathcal {A}}\rightarrow {\mathcal {B}}$ defined by $\Phi _3(a,b)=T(a)\circ R(b,y)$. By the diagonal property, $\lim _\alpha (x\cdot \Delta _\alpha -\Delta _\alpha \cdot x)=0$ in norm, therefore

$$\begin{aligned} 0=\lim _\alpha \widetilde{\Phi }_3(x\cdot \Delta _\alpha -\Delta _\alpha \cdot x)=\lim _\alpha \Big (\sum _j\Phi _3(xa_{\alpha ,j},b_{\alpha ,j})-\sum _j\Phi _3(a_{\alpha ,j},b_{\alpha ,j}x)\Big ) \end{aligned}$$

in norm, so also in the weak$^*$ topology. By Lemma 4 applied to the bilinear maps $(a,b)\mapsto \Phi _3(xa,b)$ and $(a,b)\mapsto \Phi _3(a,bx)$, we infer that both $\lim _\alpha \sum _j\Phi _3(xa_{\alpha ,j},b_{\alpha ,j})$ and $\lim _\alpha \sum _j\Phi _3(a_{\alpha ,j},b_{\alpha ,j}x)$ exist in the weak$^*$ sense, thus

$$\begin{aligned} \lim _\alpha \sum _j\Phi _3(xa_{\alpha ,j},b_{\alpha ,j})=\lim _\alpha \sum _j\Phi _3(a_{\alpha ,j},b_{\alpha ,j}x) \end{aligned}$$

in the weak$^*$ sense. This means that the norm of the difference in the third line defining S(x, y) equals

$$\begin{aligned} \begin{aligned}&\Big \Vert \lim _\alpha \sum _j \Big \{T(xa_{\alpha ,j})\circ R(b_{\alpha ,j},y)-T(x)\circ \big (T(a_{\alpha ,j})\circ R(b_{\alpha ,j},y)\big )\Big \}\Big \Vert \\&\quad \le \Big \Vert \lim _\alpha \sum _j \Big \{T(xa_{\alpha ,j})\circ R(b_{\alpha ,j},y)-\big (T(x)\circ T(a_{\alpha ,j})\big )\circ R(b_{\alpha ,j},y)\Big \}\Big \Vert \\&\qquad +\eta _2(x,y). \end{aligned} \end{aligned}$$

(3.6)

Combining (3.4), (3.5) and (3.6) we obtain

$$\begin{aligned} \begin{aligned} \Vert [R+{\textsf{D}}{\textsf{J}}R](x,y)\Vert&\le \Big \Vert \lim _\alpha \sum _j \Big \{\big (T(a_{\alpha ,j}b_{\alpha ,j})-T(a_{\alpha ,j})\circ T(b_{\alpha ,j})\big )\circ R(x,y)\Big \}\Big \Vert \\&\qquad +\Big \Vert \lim _\alpha \sum _j \Big \{T(xa_{\alpha ,j})\circ R(b_{\alpha ,j},y)-\big (T(x)\circ T(a_{\alpha ,j})\big )\circ R(b_{\alpha ,j},y)\Big \}\Big \Vert \\&\qquad +\eta _1(x,y)+3\eta _2(x,y)\\&\quad \le 2\!\cdot \!{\textsf{m}}\text{- }\text {def}_\Psi (T)\cdot \!\Vert \Psi \Vert \!\cdot \!\Vert R\Vert \!\cdot \!\Vert x\Vert \Vert y\Vert \sup _\alpha \Vert \Delta _\alpha \Vert +\eta _1(x,y)+3\eta _2(x,y), \end{aligned} \end{aligned}$$

which gives the desired estimate (3.2). $\square$

Corollary 6

Assume that T is unital and $\Psi (1,u)=u$ for each $u\in {\mathcal {B}}$. Let also $(\Delta _\alpha )_{\alpha \in A}$ be a bounded approximate diagonal in ${\mathcal {A}}{\hat{\otimes }}{\mathcal {A}}$ with $M=\sup _\alpha \Vert \Delta _\alpha \Vert <\infty$. Then, the operator ${\textsf{J}}:{\mathscr {L}}^2({\mathcal {A}},{\mathcal {B}})\rightarrow {\mathscr {L}}^1({\mathcal {A}},{\mathcal {B}})$ defined by (3.1) satisfies

$$\begin{aligned} \begin{aligned}&\Vert T^{\,\vee }+{\textsf{D}}{\textsf{J}}\,T^{\,\vee }\Vert \le \big ( 2\!\cdot \!({\textsf{m}}\text{- }\text {def}_\Psi (T))^2\Vert \Psi \Vert \\&\quad +3\!\cdot \!{\textsf{a}}\text{- }\text {def}(\Psi )\!\cdot \!{\textsf{m}}\text{- }\text {def}_\Psi (T)\!\cdot \!\Vert T\Vert ^2+{\textsf{a}}\text{- }\text {def}(\Psi )\!\cdot \!\Vert \psi \Vert \!\cdot \!\Vert T\Vert ^4\big ) M. \end{aligned} \end{aligned}$$

(3.7)

Proof

We simply apply Proposition 5 to $R=T^{\,\vee }$ and appeal to Lemma 2.

$\square$

Following Johnson’s idea, for any fixed bounded approximate diagonal $(\Delta _\alpha )_{\alpha \in A}\subset {\mathcal {A}}{\hat{\otimes }}{\mathcal {A}}$, we define the ‘improving operator’ ${\mathcal {F}}:{\mathscr {L}}^1({\mathcal {A}},{\mathcal {B}})\rightarrow {\mathscr {L}}^1({\mathcal {A}},{\mathcal {B}})$ by

$$\begin{aligned} {\mathcal {F}}\,T=T+{\textsf{J}}T^{\,\vee }, \end{aligned}$$

(3.8)

where ${\textsf{J}}$ is given by formula (3.1).

Proposition 7

Under the same assumptions as in Corollary 6, there exist constants $C_i=C_i(\Vert \Psi \Vert ,\Vert T\Vert ,M)\ge 0$, for $i=1,2,3$, such that

$$\begin{aligned} \begin{aligned} {\textsf{m}}\text{- }\text {def}_\Psi ({\mathcal {F}}T)\le C_1(&{\textsf{m}}\text{- }\text {def}_\Psi (T))^2\\&+C_2\!\cdot \!{\textsf{a}}\text{- }\text {def}(\Psi )\!\cdot \!{\textsf{m}}\text{- }\text {def}_\Psi (T)+C_3\!\cdot \!{\textsf{a}}\text{- }\text {def}(\Psi ). \end{aligned} \end{aligned}$$

Moreover, these constant are given by the formulas

$$\begin{aligned} \left\{ \begin{array}{rcl} C_1(\Vert \Psi \Vert ,\Vert T\Vert ,M) &{} = &{} \Vert \Psi \Vert ^3\!\cdot \!\Vert T\Vert ^2\!\cdot \! M^2+2\Vert \Psi \Vert \!\cdot \! M\\ C_2(\Vert \Psi \Vert ,\Vert T\Vert ,M) &{} = &{} 3\Vert T\Vert ^2\!\cdot \! M\\ C_3(\Vert \Psi \Vert ,\Vert T\Vert ,M) &{} = &{} \Vert \Psi \Vert \!\cdot \!\Vert T\Vert ^4\!\cdot \! M. \end{array}\right. \end{aligned}$$

(3.9)

Proof

First, observe that considering the bilinear map $\Phi :{\mathcal {A}}\times {\mathcal {A}}\rightarrow {\mathcal {B}}$ given by $\Phi (a,b)=T(a)\circ T^{\,\vee }(b,x)$, for any fixed $x\in {\mathcal {A}}$, by Lemma 4, we have

$$\begin{aligned} \Vert \Phi \Vert \le {\textsf{m}}\text{- }\text {def}_\Psi (T)\!\cdot \! \Vert \Psi \Vert \!\cdot \!\Vert T\Vert \Vert x\Vert , \end{aligned}$$

hence

$$\begin{aligned} \Vert {\textsf{J}}T^{\,\vee }(x)\Vert =\lim _\alpha \Vert \widetilde{\Phi }(\Delta _\alpha )\Vert \le {\textsf{m}}\text{- }\text {def}_\Psi (T)\!\cdot \! \Vert \Psi \Vert \!\cdot \!\Vert T\Vert \!\cdot \! M\Vert x\Vert . \end{aligned}$$

(3.10)

Using (3.10) in combination with Lemma 3, in particular estimate (2.1), as well as Corollary 6, we get

$$\begin{aligned} \begin{aligned} {\textsf{m}}\text{- }\text {def}_\Psi ({\mathcal {F}}T)&=\Vert (T+{\textsf{J}}T^{\,\vee })^{\,\vee }\Vert \\&\le \Vert (T+{\textsf{J}}T^{\,\vee })^{\,\vee }-T^{\,\vee }-{\textsf{D}}{\textsf{J}}T^{\,\vee }\Vert +\Vert T^{\,\vee }+{\textsf{D}}{\textsf{J}}T^{\,\vee }\Vert \\&\le \Vert \Psi \Vert \!\cdot \!\Vert {\textsf{J}}T^{\,\vee }\Vert ^2+\Vert T^{\,\vee }+{\textsf{D}}{\textsf{J}}T^{\,\vee }\Vert \\&\le C_1 ({\textsf{m}}\text{- }\text {def}_\Psi (T))^2+C_2\!\cdot \!{\textsf{a}}\text{- }\text {def}(\Psi )\!\cdot \!{\textsf{m}}\text{- }\text {def}_\Psi (T)+C_3\!\cdot \!{\textsf{a}}\text{- }\text {def}(\Psi ), \end{aligned} \end{aligned}$$

where $C_1$, $C_2$ and $C_3$ are given as in (3.9). $\square$

The proof of our main result relies on an approximation procedure in which a given operator $T_n$ is replaced by its ‘improved’ version $T_n+{\mathcal {F}}T_n$. We iterate this process as long as the obtained multiplicative defects are roughly larger than the associative defect of $\Psi$. This terminates at some point, unless we can continue and obtain an exact solution of equation (1.1).

Proof of Theorem 1

Fix any $K,L\ge 1$, $\varepsilon , \theta \in (0,1)$, pick a bounded approximate diagonal $(\Delta _\alpha )_{\alpha \in A}$ in ${\mathcal {A}}{\hat{\otimes }}{\mathcal {A}}$ with $M=\sup _\alpha \Vert \Delta _\alpha \Vert <\infty$. Define $\delta \in (0,1)$ by the formula

$$\begin{aligned} \delta =\Big [2\big (2LM+e^{4LM}(L^3M^2+3M)K^2+e^{8LM}LMK^4\big )\Big ]^{-1/\theta }\mathbf {\varepsilon }. \end{aligned}$$

(3.11)

Fix any unital operator $T\in {\mathscr {L}}^1({\mathcal {A}},{\mathcal {B}})$ with $\Vert T\Vert \le K$ and ${\textsf{m}}\text{- }\text {def}_\Psi (T)\le \delta$. Assume also that $\Vert \Psi \Vert \le L$ and define $\alpha ={\textsf{a}}\text{- }\text {def}(\Psi )$. Let ${\mathcal {F}}:{\mathscr {L}}^1({\mathcal {A}},{\mathcal {B}})\rightarrow {\mathscr {L}}^1({\mathcal {A}},{\mathcal {B}})$ be the ‘improving operator’ defined by (3.8). We define a sequence $(T_n)_{n=0}^\infty \subset {\mathscr {L}}^1({\mathcal {A}},{\mathcal {B}})$ recursively by

$$\begin{aligned} T_0=T\quad \,\,\text{ and }\,\,\quad T_n={\mathcal {F}}T_{n-1}\qquad \text{ for } n\in {\mathbb {N}}. \end{aligned}$$

Suppose that $\alpha \le ({\textsf{m}}\text{- }\text {def}_\Psi (T))^{1+\theta }$. We define $N\in {\mathbb {N}}\cup \{\infty \}$ as the least natural number n for which

$$\begin{aligned} \alpha >({\textsf{m}}\text{- }\text {def}_\Psi (T_n))^{1+\theta }, \end{aligned}$$

provided that such an n exists, and we set $N=\infty$ otherwise. Define $(\omega _n)_{n=0}^\infty$ recursively by $\omega _0=0$ and $\omega _n=1+(1+\theta )\omega _{n-1}$ for $n\in {\mathbb {N}}$. For any $n\in {\mathbb {N}}_0$, we also define

$$\begin{aligned} \delta _n=2^{-\omega _n}\delta \quad \text{ and } \quad \beta _n=\prod _{j=0}^{n-1}(1+LM\delta _j). \end{aligned}$$

Observe that since $\sum _{j=0}^\infty \delta _j<\infty$, the product $\prod _{j=0}^\infty (1+LM\delta _j)$ converges, hence we have $\beta _n\nearrow B$ for some $B<\infty$. In fact, since $\omega _n\ge n$ for $n\in {\mathbb {N}}_0$, we have

$$\begin{aligned} B\le \prod _{j=0}^\infty (1+2^{-j}LM\delta )\le \exp \Big \{\sum _{j=0}^\infty 2^{-j}LM\Big \}=\exp (2LM). \end{aligned}$$

(3.12)

Set $K_n=\beta _n K$, so that we have $K_n<BK$ for each $n\in {\mathbb {N}}_0$.

$\underline{{Claim.}}$ For each integer $0\le n<N$, the following conditions hold true:

• (i) $T_n$ is unital;

• (ii) ${\textsf{m}}\text{- }\text {def}_\Psi (T_n)\le \delta _n$;

• (iii) $\Vert T_n\Vert \le K_n$.

We proceed by induction. In order to show (i) notice that the identity $\Psi (u,1)=u$ implies $T^{\,\vee }(b,1)=0$ for every $b\in {\mathcal {B}}$. Hence, ${\textsf{J}} T^{\,\vee }(1)=0$ which gives $T_1(1)={\mathcal {F}}T(1)=T(1)=1$ and, by induction, $T_n(1)=1$ for every $n\in {\mathbb {N}}$.

Assertions (ii) and (iii) are plainly true for $n=0$ as $\delta _0=\delta$ is not smaller than the defect of $T=T_0$, and $K_0=\beta _0K=K$. Fix $n\in {\mathbb {N}}_0$ with $n+1<N$ and suppose that estimates (ii) and (iii) hold true. Using Proposition 7, formulas (3.9), and the fact that $\alpha \le ({\textsf{m}}\text{- }\text {def}_\Psi (T_n))^{1+\theta }$, we obtain

$$\begin{aligned} \begin{aligned} {\textsf{m}}\text{- }\text {def}_\Psi (T_{n+1})&={\textsf{m}}\text{- }\text {def}_\Psi ({\mathcal {F}}T_n)\\&\quad \le C_1(\Vert \Psi \Vert ,\Vert T_n\Vert ,M)\delta _n^2+C_2(\Vert \Psi \Vert ,\Vert T_n\Vert ,M)\alpha \delta _n\\&\quad +C_3(\Vert \Psi \Vert ,\Vert T_n\Vert ,M)\alpha \\&\quad \le \big (C_1(\Vert \Psi \Vert ,\Vert T_n\Vert ,M)+C_2(\Vert \Psi \Vert ,\Vert T_n\Vert ,M)\\&\quad +C_3(\Vert \Psi \Vert ,\Vert T_n\Vert ,M)\big )\delta _n^{1+\theta }\\&\quad \le \big (2LM+K_n^2L^3M^2+3K_n^2M+K_n^4LM\big )\delta _n^{1+\theta }\\&\quad \le \big (2LM+(L^3M^2+3M)B^2K^2+B^4K^4LM\big )\delta _n^{1+\theta }. \end{aligned} \end{aligned}$$

Therefore, appealing to the definition of $\delta$, that is, formula (3.11), and inequality (3.12), we obtain

$$\begin{aligned} {\textsf{m}}\text{- }\text {def}_\Psi (T_{n+1})\le \frac{1}{2}\delta ^{-\theta }\delta _n^{1+\theta }=\frac{1}{2}\!\cdot \!2^{-(1+\theta )\omega _n}\delta =2^{-\omega _{n+1}}\delta =\delta _{n+1}. \end{aligned}$$

By inequality (3.10), we also obtain

$$\begin{aligned} \begin{aligned} \Vert T_{n+1}\Vert&=\Vert {\mathcal {F}}T_n\Vert \le \Vert T_n\Vert +\Vert {\textsf{J}}T_n^{\,\vee }\Vert \\&\le K_n+\delta _n K_nLM\\&=\beta _n(1+\delta _n LM)K=\beta _{n+1}K=K_{n+1}, \end{aligned} \end{aligned}$$

which completes the induction.

Now, suppose that the inequality $({\textsf{m}}\text{- }\text {def}_\Psi (T_n))^{1+\theta }<\alpha$ is never satisfied, which means that our process does not terminate and we have $N=\infty$. Then, since for each $n\in {\mathbb {N}}$ we have

$$\begin{aligned} \Vert {\textsf{J}}T_n^{\,\vee }\Vert \le \delta _n K_nLM<\delta _n BKLM\le 2^{-n}\delta BKLM, \end{aligned}$$

the sequence $(T_n)_{n=1}^\infty$ is a Cauchy sequence, thus the norm limit $S=\lim _{n\rightarrow \infty }T_n$ exists. In view of (ii), we have $\lim _{n\rightarrow \infty }{\textsf{m}}\text{- }\text {def}_\Psi (T_n)=0$ which yields ${\textsf{m}}\text{- }\text {def}_\Psi (S)=0$ because for any $x,y\in {\mathcal {A}}$ with $\Vert x\Vert ,\Vert y\Vert \le 1$, we have

$$\begin{aligned} \Vert S(xy)-\Psi (S(x),S(y))\Vert= & {} \lim _{n\rightarrow \infty }\Vert T_n(xy)-\Psi (T_n(x),T_n(y))\Vert \\\le & {} \lim _{n\rightarrow \infty }{\textsf{m}}\text{- }\text {def}_\Psi (T_n)=0. \end{aligned}$$

Moreover, $S-T=\sum _{n=0}^\infty {\textsf{J}}T_n^{\,\vee }$ and hence

$$\begin{aligned} \Vert S-T\Vert \le \sum _{n=0}^\infty 2^{-n}\delta BKLM\le 2\delta e^{2LM}KLM<\varepsilon , \end{aligned}$$

which means that in this case we have produced an exact solution of the equation $S(xy)=S(x)\circ S(y)$ lying at distance smaller than $\varepsilon$ from T.

If our process terminates, i.e. $N<\infty$, then we have

$$\begin{aligned} {\textsf{m}}\text{- }\text {def}_\Psi (T_N)<\alpha ^{\frac{1}{1+\theta }} \end{aligned}$$

and, likewise in the previous case,

$$\begin{aligned} \Vert T_N-T\Vert \le \sum _{n=0}^{N-1}\Vert {\textsf{J}}T_n^{\,\vee }\Vert <\varepsilon . \end{aligned}$$

This completes the proof since for any $\eta \in (0,1)$ we can pick $\theta \in (0,1)$ so that $(1+\theta )^{-1}>1-\eta$ and then the parameter $\delta$ defined by (3.11) does the job in the sense that whenever T is as above, then the approximation $T_N$ satisfies ${\textsf{m}}\text{- }\text {def}_\Psi (T_N)<{\textsf{a}}\text{- }\text {def}(\Psi )^{1-\eta }$.

Remark

It may be of some interest to ask if the power function $\alpha \mapsto \alpha ^{1-\eta }$ appearing in Theorem 1 is optimal and, in general, how close to ${\textsf{a}}\text{- }\text {def}(\Psi )$ one can get with the multiplicative defect ${\textsf{m}}\text{- }\text {def}_\Psi (S)$.