Approximation of a generalized additive mapping in multi-Banach modules and isomorphisms in multi-$C^{\ast }$-algebras: a fixed-point approach

Abstract

Let $\mathcal{X}$, $\mathcal{Y}$ be vector spaces. It is shown that if an odd mapping $f:\mathcal{X}\to \mathcal{Y}$ satisfies the functional equation

(0.1)

then the odd mapping $f:\mathcal{X}\to \mathcal{Y}$ is additive, and we use a fixed-point method to prove the Hyers-Ulam stability of the functional equation (0.1) in multi-Banach modules over a unital multi-$C^{\ast }$-algebra. As an application, we show that every almost linear bijection $h:\mathcal{A}\to \mathcal{B}$ of a unital multi-$C^{\ast }$-algebra $\mathcal{A}$ onto a unital multi-$C^{\ast }$-algebra $\mathcal{B}$ is a $C^{\ast }$-algebra isomorphism when $h(\frac{2^n}{r^n}uy)=h(\frac{2^n}{r^n}u)h(y)$ for all unitaries $u\in U(\mathcal{A})$, all $y\in \mathcal{A}$, and $n=0,1,2,\dots$ .

MSC:39B52, 46L05, 47H10, 47B48.

1 Introduction

Throughout this paper we assume that r is a positive rational number and d, l are integers with $1<l<\frac{d}{2}$.

Let X and Y be Banach spaces. Consider a mapping $f:X\to Y$ such that $f(tx)$ is continuous in $t\in \mathbb{R}$ for each fixed $x\in X$, and assume that there exist constants $\theta \ge 0$ and $p\in [0,1)$ with

$$\parallel f(x+y)-f(x)-f(y)\parallel \le \theta ({\parallel x\parallel }^p+{\parallel y\parallel }^p),x,y\in X.$$

Rassias [1] showed that there exists a unique $\mathbb{R}$-linear mapping $T:X\to Y$ such that

$$\parallel f(x)-T(x)\parallel \le \frac{2\theta }{2-2^p}{\parallel x\parallel }^p,x\in X.$$

Găvruta [2] extended the above theorem as follows: let G be an Abelian group, Y be a Banach space and put

$$\tilde{\phi }(x,y)=\sum _{j=0}^{\mathrm{\infty }}\frac{1}{2^j}\phi (2^{j}x,2^{j}y)<\mathrm{\infty },x,y\in G.$$

If $f:G\to Y$ is a mapping satisfying

$$\parallel f(x+y)-f(x)-f(y)\parallel \le \phi (x,y),x,y\in G,$$

then there exists a unique additive mapping $T:G\to Y$ such that

$$\parallel f(x)-T(x)\parallel \le \frac{1}{2}\tilde{\phi }(x,x),x\in G.$$

Park [3] applied Găvruta’s result to linear functional equations in Banach modules over a $C^{\ast }$-algebra. Several functional equations have been investigated in [4, 5] and [6]. In 2006 Baak, Boo and Rassias [7] solved the following functional equation:

(1.1)

(any solution of (1.1) will be called a generalized additive mapping) and proved its Hyers-Ulam stability in Banach modules over a unital $C^{\ast }$-algebra via the direct method. These results were applied to investigate $C^{\ast }$-algebra isomorphisms in unital $C^{\ast }$-algebras.

In this paper, we prove the Hyers-Ulam stability of the functional equation (1.1) in multi-Banach modules over a unital multi-$C^{\ast }$-algebra via the fixed-point method. These results are applied to investigate $C^{\ast }$-algebra isomorphisms in unital multi-$C^{\ast }$-algebras.

2 Fixed-point theorems

We recall two fundamental results in the fixed-point theory.

Theorem 2.1 [8, 9]

Let $(X,d)$ be a complete metric space and let $J:X\to X$ be strictly contractive, i.e.,

$$d(Jx,Jy)\le Ld(x,y),x,y\in X$$

for a Lipschitz constant $L<1$. Then

• (1) the mapping J has a unique fixed point $x^{\ast }\in X$,

• (2) the fixed point $x^{\ast }$ is globally attractive, i.e.,

  $$\underset{n\to \mathrm{\infty }}{lim}{J}^{n}x=x^{\ast },x\in X,$$

• (3) the following inequalities hold:

  $$\begin{array}{c}d(J^{n}x,x^{\ast })\le L^{n}d(x,x^{\ast }),\hfill \\ d(J^{n}x,x^{\ast })\le \frac{1}{1-L}d(J^{n}x,J^{n+1}x),\hfill \\ d(x,x^{\ast })\le \frac{1}{1-L}d(x,Jx)\hfill \end{array}$$

for all $x\in X$ and nonnegative integers n.

Let X be a non-empty set. A function $d:X\times X\to [0,\mathrm{\infty }]$ is called a generalized metric on X if for any $x,y,z\in X$, we have:

• (1) $d(x,y)=0$ if and only if $x=y$,

• (2) $d(x,y)=d(y,x)$,

• (3) $d(x,z)\le d(x,y)+d(y,z)$.

Theorem 2.2 [8, 10]

Let $(X,d)$ be a complete generalized metric space and let $J:X\to X$ be a strictly contractive mapping with a Lipschitz constant $L<1$. Then, for each $x\in X$, either

$$d(J^{n}x,J^{n+1}x)=\mathrm{\infty },n\ge 0$$

or there exists a positive integer $n_0$ such that:

• (1) $d(J^{n}x,J^{n+1}x)<\mathrm{\infty }$, $n\ge n_0$,

• (2) the sequence $(J^{n}x)$ converges to a fixed point $y^{\ast }$ of J,

• (3) $y^{\ast }$ is the unique fixed point of J in the set $Y=\{y\in X:d(J^{n_0}x,y)<\mathrm{\infty }\}$,

• (4) $d(y,y^{\ast })\le \frac{1}{1-L}d(y,Jy)$, $y\in Y$.

3 Multi-normed spaces

The notion of a multi-normed space was introduced by Dales and Polyakov [11]. This concept is somewhat similar to an operator sequence space and has some connections with operator spaces and Banach lattices. Motivations for the study of multi-normed spaces and many examples are given in [11–13].

Let $(\mathcal{E},\parallel \cdot \parallel )$ be a complex normed space and $k\in \mathbb{N}$. We denote by ${\mathcal{E}}^k$ the linear space $\mathcal{E}\oplus \cdots \oplus \mathcal{E}$ consisting of k-tuples $(x_1,\dots ,x_k)$, where $x_1,\dots ,x_k\in \mathcal{E}$. The linear operations on ${\mathcal{E}}^k$ are defined coordinate-wise. The zero element of either $\mathcal{E}$ or ${\mathcal{E}}^k$ is denoted by 0. Finally, we denote by ${\mathbb{N}}_k$ the set $\{1,\dots ,k\}$ and by ${\mathrm{\Sigma }}_k$ the group of permutations on k symbols.

Definition 3.1 [11, 14] A multi-norm on $\{{\mathcal{E}}^k:k\in \mathbb{N}\}$ is a sequence

$$\left({\parallel \cdot \parallel }_k\right)=({\parallel \cdot \parallel }_k:k\in \mathbb{N})$$

such that ${\parallel \cdot \parallel }_k$ is a norm on ${\mathcal{E}}^k$ for $k\in \mathbb{N}$, ${\parallel \cdot \parallel }_1=\parallel \cdot \parallel$, and for any integer $k\ge 2$, we have

(A1) ${\parallel (x_{\sigma (1)},\dots ,x_{\sigma (k)})\parallel }_k={\parallel (x_1,\dots ,x_k)\parallel }_k$, $\sigma \in {\mathrm{\Sigma }}_k$, $x_1,\dots ,x_k\in \mathcal{E}$,

(A2) ${\parallel ({\alpha }_1{x}_1,\dots ,{\alpha }_k{x}_k)\parallel }_k\le ({max}_{i\in {\mathbb{N}}_k}|{\alpha }_i|){\parallel (x_1,\dots ,x_k)\parallel }_k$, ${\alpha }_1,\dots ,{\alpha }_k\in \mathbb{C}$, $x_1,\dots ,x_k\in \mathcal{E}$,

(A3) ${\parallel (x_1,\dots ,x_{k-1},0)\parallel }_k={\parallel (x_1,\dots ,x_{k-1})\parallel }_{k-1}$, $x_1,\dots ,x_{k-1}\in \mathcal{E}$,

(A4) ${\parallel (x_1,\dots ,x_{k-1},x_{k-1})\parallel }_k={\parallel (x_1,\dots ,x_{k-1})\parallel }_{k-1}$, $x_1,\dots ,x_{k-1}\in \mathcal{E}$.

A sequence $(({\mathcal{E}}^k,{\parallel \cdot \parallel }_k):k\in \mathbb{N})$ is then said to be a multi-normed space.

Lemma 3.2 [11, 13]

Suppose that $(({\mathcal{E}}^k,{\parallel \cdot \parallel }_k):k\in \mathbb{N})$ is a multi-normed space. Then for any $k\in \mathbb{N}$, we have

• (a) ${\parallel (x,\dots ,x)\parallel }_k=\parallel x\parallel$, $x\in \mathcal{E}$,

• (b) ${max}_{i\in {\mathbb{N}}_k}\parallel x_i\parallel \le {\parallel (x_1,\dots ,x_k)\parallel }_k\le {\sum }_{i=1}^k\parallel x_i\parallel \le k{max}_{i\in {\mathbb{N}}_k}\parallel x_i\parallel$, $x_1,\dots ,x_k\in \mathcal{E}$.

From Lemma 3.2(b), it follows that if $(\mathcal{E},\parallel \cdot \parallel )$ is a Banach space, then $({\mathcal{E}}^k,{\parallel \cdot \parallel }_k)$ is a Banach space for each $k\in \mathbb{N}$ (in this case we say that $(({\mathcal{E}}^k,{\parallel \cdot \parallel }_k):k\in \mathbb{N})$ is a multi-Banach space).

Now, we recall two important examples of multi-norms (see [11, 12]).

Example 3.3 The sequence $({\parallel \cdot \parallel }_k:k\in \mathbb{N})$ on $\{{\mathcal{E}}^k:k\in \mathbb{N}\}$ defined by

$${\parallel (x_1,\dots ,x_k)\parallel }_k:=\underset{i\in {\mathbb{N}}_k}{max}\parallel x_i\parallel ,x_1,\dots ,x_k\in \mathcal{E}$$

is a multi-norm called the minimum multi-norm. The terminology ‘minimum’ is justified by property (b) from Lemma 3.2.

Example 3.4 Let $\{({\parallel \cdot \parallel }_k^{\alpha }:k\in \mathbb{N}):\alpha \in A\}$ be a (non-empty) family of all multi-norms on $\{{\mathcal{E}}^k:k\in \mathbb{N}\}$. For $k\in \mathbb{N}$, set

$$⦀(x_1,\dots ,x_k){⦀}_k:=\underset{\alpha \in A}{sup}{\parallel (x_1,\dots ,x_k)\parallel }_k^{\alpha },x_1,\dots ,x_k\in \mathcal{E}.$$

Then $(⦀\cdot {⦀}_k:k\in \mathbb{N})$ is a multi-norm on $\{{\mathcal{E}}^k:k\in \mathbb{N}\}$ called the maximum multi-norm.

Lemma 3.5 [14]

Suppose that $k\in \mathbb{N}$ and $(x_1,\dots ,x_k)\in {\mathcal{E}}^k$. For each $j\in \{1,\dots ,k\}$, let ${(x_n^j)}_{n\in \mathbb{N}}$ be a sequence in $\mathcal{E}$ such that ${lim}_{n\to \mathrm{\infty }}{x}_n^j=x_j$. Then for each $(y_1,\dots ,y_k)\in {\mathcal{E}}^k$ we have

$$\underset{n\to \mathrm{\infty }}{lim}(x_n^1-y_1,\dots ,x_n^k-y_k)=(x_1-y_1,\dots ,x_k-y_k).$$

Definition 3.6 [12, 14]

Let $(({\mathcal{E}}^k,{\parallel \cdot \parallel }_k):k\in \mathbb{N})$ be a multi-normed space. A sequence ${(x_n)}_{n\in \mathbb{N}}$ in $\mathcal{E}$ is said to be a multi-null sequence if for each $\epsilon >0$, there exists an $n_0\in \mathbb{N}$ such that

$$\underset{k\in \mathbb{N}}{sup}{\parallel (x_n,\dots ,x_{n+k-1})\parallel }_k<\epsilon ,n\ge n_0.$$

We say that the sequence ${(x_n)}_{n\in \mathbb{N}}$ is multi-convergent to $x\in \mathcal{E}$ and write ${lim}_{n\to \mathrm{\infty }}{x}_n=x$ if ${(x_n-x)}_{n\in \mathbb{N}}$ is a multi-null sequence.

Definition 3.7 [11, 14]

Let $(\mathcal{A},\parallel \cdot \parallel )$ be a normed algebra such that $(({\mathcal{A}}^k,{\parallel \cdot \parallel }_k):k\in \mathbb{N})$ is a multi-normed space. Then $(({\mathcal{A}}^k,{\parallel \cdot \parallel }_k):k\in \mathbb{N})$ is called a multi-normed algebra if

$${\parallel (a_1{b}_1,\dots ,a_k{b}_k)\parallel }_k\le {\parallel (a_1,\dots ,a_k)\parallel }_k\cdot {\parallel (b_1,\dots ,b_k)\parallel }_k,k\in \mathbb{N},a_1,\dots ,a_k,b_1,\dots ,b_k\in \mathcal{A}.$$

The multi-normed algebra $(({\mathcal{A}}^k,{\parallel \cdot \parallel }_k):k\in \mathbb{N})$ is said to be a multi-Banach algebra if $(({\mathcal{A}}^k,{\parallel \cdot \parallel }_k):k\in \mathbb{N})$ is a multi-Banach space.

Example 3.8 Let p, q be such that $1\le p\le q<\mathrm{\infty }$ and $\mathcal{A}={\ell }^p$. The algebra $\mathcal{A}$ is a Banach sequence algebra with respect to coordinate-wise multiplication of sequences (see Example 4.1.42 of [15]). Let $({\parallel \cdot \parallel }_k:k\in \mathbb{N})$ be the standard $(p,q)$-multi-norm on $\{{\mathcal{A}}^k:k\in \mathbb{N}\}$ (see [11]). Then $(({\mathcal{A}}^k,{\parallel \cdot \parallel }_k):k\in \mathbb{N})$ is a multi-Banach algebra.

Definition 3.9 Let $(({\mathcal{A}}^k,{\parallel \cdot \parallel }_k):k\in \mathbb{N})$ be a multi-Banach algebra and assume that $\mathcal{A}$ is a (unital) $C^{\ast }$-algebra. If the involution ∗ satisfies

$${\parallel (a_1^{\ast }{a}_1,\dots ,a_k^{\ast }{a}_k)\parallel }_k={\parallel (a_1,\dots ,a_k)\parallel }_k^2,k\in \mathbb{N},a_1,\dots ,a_k\in \mathcal{A},$$

then $(({\mathcal{A}}^k,{\parallel \cdot \parallel }_k):k\in \mathbb{N})$ is called a (unital) multi-$C^{\ast }$-algebra.

Definition 3.10 Let $(({\mathcal{A}}^k,{\parallel \cdot \parallel }_k):k\in \mathbb{N})$ be a multi-Banach algebra and $(({\mathcal{X}}^k,{\parallel \cdot \parallel }_k):k\in \mathbb{N})$ be a multi-Banach space. Assume also that $\mathcal{X}$ is a Banach left module over $\mathcal{A}$. We say that $(({\mathcal{X}}^k,{\parallel \cdot \parallel }_k):k\in \mathbb{N})$ is a multi-Banach left module over $(({\mathcal{A}}^k,{\parallel \cdot \parallel }_k):k\in \mathbb{N})$ if there is an $M\ge 0$ such that

$${\parallel (a_1{x}_1,\dots ,a_k{x}_k)\parallel }_k\le M{\parallel (a_1,\dots ,a_k)\parallel }_k\cdot {\parallel (x_1,\dots ,x_k)\parallel }_k$$

for all $k\in \mathbb{N}$, $a_1,\dots ,a_k\in \mathcal{A}$, $x_1,\dots ,x_k\in \mathcal{X}$.

4 Stability of an odd functional equation in multi-Banach modules over a multi-$C^{\ast }$-algebra

Throughout this section, we assume that $(({\mathcal{A}}^k,{\parallel \cdot \parallel }_k):k\in \mathbb{N})$ is a unital multi-$C^{\ast }$-algebra, and $(({\mathcal{X}}^k,{\parallel \cdot \parallel }_k):k\in \mathbb{N})$ and $(({\mathcal{Y}}^k,{\parallel \cdot \parallel }_k):k\in \mathbb{N})$ are multi-Banach left modules over $(({\mathcal{A}}^k,{\parallel \cdot \parallel }_k):k\in \mathbb{N})$. Moreover, by $U(\mathcal{A})$ we denote the unitary group of $\mathcal{A}$.

Lemma 4.1 [7]

Let X and Y be vector spaces. An odd mapping $f:X\to Y$ satisfies (1.1) for all $x_1,\dots ,x_d\in X$ if and only if f is additive.

Corollary 4.2 [7]

Let X and Y be vector spaces. An odd mapping $f:X\to Y$ satisfies

$$rf\left(\frac{x+y}{r}\right)=f(x)+f(y),x,y\in X$$

if and only if f is additive.

Given a mapping $f:\mathcal{X}\to \mathcal{Y}$, we set

$$\begin{array}{rcl}{D}_{u}f(x_1,\dots ,x_d)& :=& rf\left(\frac{{\sum }_{j=1}^{d}u{x}_j}{r}\right)+\underset{{\sum }_{j=1}^d\iota (j)=l}{\sum _{\iota (j)=0,1}}rf\left(\frac{{\sum }_{j=1}^d{(-1)}^{\iota (j)}u{x}_j}{r}\right)\\ -{(}_{d-1}{C}_l{-}_{d-1}{C}_{l-1}+1)\sum _{j=1}^{d}uf(x_j)\end{array}$$

for all $u\in U(\mathcal{A})$ and $x_1,\dots ,x_d\in \mathcal{X}$.

Theorem 4.3 Let $r\ne 2$ and $f:\mathcal{X}\to \mathcal{Y}$ be an odd mapping such that for every $k\in \mathbb{N}$ there is a function ${\phi }_k:{\mathcal{X}}^{kd}\to [0,\mathrm{\infty })$ with

(4.1)

(4.2)

for all $u\in U(\mathcal{A})$ and $x_{11},\dots ,x_{1d},\dots ,x_{k1},\dots ,x_{kd}\in \mathcal{X}$. If there exists an $L<1$ such that

$$\begin{array}{c}{\phi }_k(\stackrel{d}{\overbrace{\frac{2}{r}{x}_{11},\frac{2}{r}{x}_{11},\dots ,0}},\dots ,\stackrel{d}{\overbrace{\frac{2}{r}{x}_{k1},\frac{2}{r}{x}_{k1},\dots ,0}})\hfill \\ \le \frac{2}{r}L{\phi }_k(\stackrel{d}{\overbrace{x_{11},x_{11},\dots ,0}},\dots ,\stackrel{d}{\overbrace{x_{k1},x_{k1},\dots ,0}})\hfill \end{array}$$

for all $k\in \mathbb{N}$ and $x_{11},\dots ,x_{k1}\in \mathcal{X}$, then there is a unique $\mathcal{A}$-linear generalized additive mapping $\mathcal{L}:\mathcal{X}\to \mathcal{Y}$ with

(4.3)

for all $k\in \mathbb{N}$ and $x_1,\dots ,x_k\in \mathcal{X}$.

Proof Put

$$X:=\{\mathcal{L}:\mathcal{X}\to \mathcal{Y}\}$$

and

$$\begin{array}{rcl}d(\mathcal{L},h)& =& inf\{C\in {\mathbb{R}}_{+}:{\parallel (\mathcal{L}(x_1)-h(x_1),\dots ,\mathcal{L}(x_k)-h(x_k))\parallel }_k\\ \le & C{\phi }_k(\stackrel{d}{\overbrace{x_1,x_1,0,\dots ,0}},\dots ,\stackrel{d}{\overbrace{x_k,x_k,0,\dots ,0}}),k\in \mathbb{N},x_1,\dots ,x_k\in \mathcal{X}\}\end{array}$$

for all $\mathcal{L},h\in X$. It is easy to show that $(X,d)$ is a complete generalized metric space.

Define a mapping $J:X\to X$ by

$$J\mathcal{L}(x):=\frac{r}{2}\mathcal{L}\left(\frac{2}{r}x\right),\mathcal{L}\in X,x\in \mathcal{X}.$$

Analysis similar to that in the proof of Theorem 3.1 in [8] (see also the proof of Lemma 3.2 in [12]) shows that

$$d(J\mathcal{L},Jh)\le Ld(\mathcal{L},h),\mathcal{L},h\in X.$$

Fix $k\in \mathbb{N}$. Putting $u=1\in U(\mathcal{A})$, $x_{i1}=x_{i2}=x_1$, and $x_{i3}=\cdots =x_{id}=0$ for $i\in \{1,\dots ,k\}$ in (4.2), we have

$$\begin{array}{c}{\parallel (rf\left(\frac{2}{r}{x}_1\right)-2f(x_1),\dots ,rf\left(\frac{2}{r}{x}_k\right)-2f(x_k))\parallel }_k\hfill \\ \le \frac{1}{{}_{d-2}C_l{-}_{d-2}{C}_{l-2}+1}{\phi }_k(x_1,x_1,\underset{d-2\mathrm{times}}{\underset{⏞}{0,\dots ,0}},\dots ,x_k,x_k,\underset{d-2\mathrm{times}}{\underset{⏞}{0,\dots ,0}}),\hfill \end{array}$$

because f is odd and $t:{=}_{d-2}{C}_l{-}_{d-2}{C}_{l-2}+1{=}_{d-1}{C}_l{-}_{d-1}{C}_{l-1}+1$. We thus get

$$\begin{array}{c}{\parallel (f(x_1)-\frac{r}{2}f\left(\frac{2}{r}{x}_1\right),\dots ,f(x_k)-\frac{r}{2}f\left(\frac{2}{r}{x}_k\right))\parallel }_k\hfill \\ \le \frac{1}{2t}{\phi }_k(x_1,x_1,\underset{d-2\mathrm{times}}{\underset{⏞}{0,\dots ,0}},\dots ,x_k,x_k,\underset{d-2\mathrm{times}}{\underset{⏞}{0,\dots ,0}}),x_1,\dots ,x_k\in \mathcal{X},\hfill \end{array}$$

and therefore,

$$d(f,Jf)\le \frac{1}{2t}.$$

(4.4)

Consequently, by Theorem 2.2, there exists a mapping $\mathcal{L}:\mathcal{X}\to \mathcal{Y}$ such that

• (1) $\mathcal{L}$ is a fixed point of J, i.e.,

  $$\mathcal{L}\left(\frac{2}{r}x\right)=\frac{2}{r}\mathcal{L}(x),x\in \mathcal{X},$$

  (4.5)

and $\mathcal{L}$ is unique in the set

$$Y=\{\mathcal{L}\in X:d(f,\mathcal{L})<\mathrm{\infty }\}.$$

This means that $\mathcal{L}$ is a unique mapping satisfying (4.5) such that there exists a $C\in (0,\mathrm{\infty })$ with

$${\parallel (\mathcal{L}(x_1)-f(x_1),\dots ,\mathcal{L}(x_k)-f(x_k))\parallel }_k\le C{\phi }_k(x_1,x_1,\underset{d-2\mathrm{times}}{\underset{⏞}{0,\dots ,0}},\dots ,x_k,x_k,\underset{d-2\mathrm{times}}{\underset{⏞}{0,\dots ,0}})$$

for all $k\in \mathbb{N}$ and $x_1,\dots ,x_k\in \mathcal{X}$.

• (2) $d(J^{n}f,\mathcal{L})\to 0$ as $n\to \mathrm{\infty }$. This implies the equality

  $$\underset{n\to \mathrm{\infty }}{lim}\frac{r^n}{2^n}f\left(\frac{2^n}{r^n}x\right)=\mathcal{L}(x)x\in \mathcal{X}.$$

  (4.6)

• (3) $d(f,\mathcal{L})\le \frac{1}{1-L}d(f,Jf)$, which together with (4.4) gives

  $$d(f,\mathcal{L})\le \frac{1}{2t-2tL},$$

and therefore, inequality (4.3) holds for all $x_1,\dots ,x_k\in \mathcal{X}$.

Next, note that the fact that the mapping f is odd and (4.6) imply that $\mathcal{L}$ is odd. Moreover, by (4.1) and (4.2), we get

$$\begin{array}{c}{\parallel (D_1\mathcal{L}(x_{11},\dots ,x_{1d}),\dots ,D_1\mathcal{L}(x_{k1},\dots ,x_{kd}))\parallel }_k\hfill \\ =\underset{n\to \mathrm{\infty }}{lim}\frac{r^n}{2^n}{\parallel (D_{1}f(\frac{2^n}{r^n}{x}_{11},\dots ,\frac{2^n}{r^n}{x}_{1d}),\dots ,D_{1}f(\frac{2^n}{r^n}{x}_{k1},\dots ,\frac{2^n}{r^n}{x}_{kd}))\parallel }_k\hfill \\ \le \underset{n\to \mathrm{\infty }}{lim}\frac{r^n}{2^n}{\phi }_k(\frac{2^n}{r^n}{x}_{11},\dots ,\frac{2^n}{r^n}{x}_{1d},\dots ,\frac{2^n}{r^n}{x}_{k1},\dots ,\frac{2^n}{r^n}{x}_{kd})=0\hfill \end{array}$$

for all $k\in \mathbb{N}$ and $x_{11},\dots ,x_{1d},\dots ,x_{k1},\dots ,x_{kd}\in \mathcal{X}$, and therefore, $\mathcal{L}$ is a generalized additive mapping.

Fix $u\in U(\mathcal{A})$ and $x\in \mathcal{X}$. Using (4.1) and (4.2), we have

$$\begin{array}{c}{\parallel (D_u\mathcal{L}(x,\underset{d-1\mathrm{times}}{\underset{⏞}{0,\dots ,0}}),\dots ,D_u\mathcal{L}(x,\underset{d-1\mathrm{times}}{\underset{⏞}{0,\dots ,0}})\parallel }_k\hfill \\ =\underset{n\to \mathrm{\infty }}{lim}\frac{r^n}{2^n}{\parallel (D_{u}f(\frac{2^n}{r^n}x,\underset{d-1\mathrm{times}}{\underset{⏞}{0,\dots ,0}}),\dots ,D_{u}f(\frac{2^n}{r^n}x,\underset{d-1\mathrm{times}}{\underset{⏞}{0,\dots ,0}}))\parallel }_k\hfill \\ \le \underset{n\to \mathrm{\infty }}{lim}\frac{r^n}{2^n}{\phi }_k(\frac{2^n}{r^n}x,\underset{d-1\mathrm{times}}{\underset{⏞}{0,\dots ,0}},\dots ,\frac{2^n}{r^n}x,\underset{d-1\mathrm{times}}{\underset{⏞}{0,\dots ,0}})=0,\hfill \end{array}$$

and consequently,

$${(}_{d-1}{C}_l{-}_{d-1}{C}_{l-1}+1)r\mathcal{L}\left(\frac{ux}{r}\right)={(}_{d-1}{C}_l{-}_{d-1}{C}_{l-1}+1)u\mathcal{L}(x).$$

Since $\mathcal{L}$ is a generalized additive mapping, from Lemma 4.1 it follows that $\mathcal{L}$ is additive, and therefore,

$$\mathcal{L}(ux)=r\mathcal{L}\left(\frac{ux}{r}\right)=u\mathcal{L}(x),u\in U(\mathcal{A}),x\in \mathcal{X}.$$

As in the proof of Theorem 3.1, in [7] one can now show that $\mathcal{L}$ is an $\mathcal{A}$-linear mapping.  □

Corollary 4.4 Let $r\ne 2$ and $\theta ,p\in (0,\mathrm{\infty })$. Assume also that $p>1$ for $r>2$, and $p<1$ for $r<2$. If $f:\mathcal{X}\to \mathcal{Y}$ is an odd mapping such that

$${\parallel (D_{u}f(x_{11},\dots ,x_{1d}),\dots ,D_{u}f(x_{k1},\dots ,x_{kd}))\parallel }_k\le \theta (\sum _{j=1}^d{\parallel x_{1j}\parallel }^p+\cdots +\sum _{j=1}^d{\parallel x_{kj}\parallel }^p)$$

for all $u\in U(\mathcal{A})$, $k\in \mathbb{N}$, and $x_{11},\dots ,x_{1d},\dots ,x_{k1},\dots ,x_{kd}\in \mathcal{X}$, then there exists a unique $\mathcal{A}$-linear generalized additive mapping $\mathcal{L}:\mathcal{X}\to \mathcal{Y}$ with

$$\begin{array}{c}{\parallel (\mathcal{L}(x_1)-f(x_1),\dots ,\mathcal{L}(x_k)-f(x_k))\parallel }_k\hfill \\ \le \frac{r^{p-1}\theta }{(r^{p-1}-2^{p-1}){(}_{d-2}{C}_l{-}_{d-2}{C}_{l-2}+1)}({\parallel x_1\parallel }^p+\cdots +{\parallel x_k\parallel }^p)\hfill \end{array}$$

for all $k\in \mathbb{N}$ and $x_1,\dots ,x_k\in \mathcal{X}$.

Proof Putting $L=\frac{2^{p-1}}{r^{p-1}}$ and

$$\begin{array}{c}{\phi }_k(x_{11},\dots ,x_{1d},\dots ,x_{k1},\dots ,x_{kd})\hfill \\ =\theta (\sum _{j=1}^d{\parallel x_{1j}\parallel }^p+\cdots +\sum _{j=1}^d{\parallel x_{kj}\parallel }^p),\hfill \end{array}$$

for all $k\in \mathbb{N}$ and $x_{11},\dots ,x_{1d},\dots ,x_{k1},\dots ,x_{kd}\in \mathcal{X}$, in Theorem 4.3, we get the desired assertion. □

Theorem 4.5 Let $r\ne 2$. Let $f:\mathcal{X}\to \mathcal{Y}$ be an odd mapping for which there is a function $\phi :{\mathcal{X}}^{kd}\to [0,\mathrm{\infty })$ such that

(4.7)

for all $u\in U(\mathcal{A})$ and all $x_{11},\dots ,x_{1d},\dots ,x_{k1},\dots ,x_{kd}\in \mathcal{X}$. If there exists an $L<1$ such that

$$\begin{array}{c}\phi (\stackrel{d}{\overbrace{\frac{r}{2}{x}_{11},\frac{r}{2}{x}_{11},\dots ,0}},\stackrel{d}{\overbrace{\frac{r}{2}{x}_{21},\frac{r}{2}{x}_{21},\dots ,0}},\dots ,\stackrel{d}{\overbrace{\frac{r}{2}{x}_{k1},\frac{r}{2}{x}_{k1},\dots ,0}})\hfill \\ \le \frac{r}{2}L\phi (\stackrel{d}{\overbrace{x_{11},x_{11},\dots ,0}},\stackrel{d}{\overbrace{x_{21},x_{21},\dots ,0}},\dots ,\stackrel{d}{\overbrace{x_{k1},x_{k1},\dots ,0}})\hfill \end{array}$$

for all $x_{11},x_{21},\dots ,x_{k1}\in \mathcal{X}$. Then there exists a unique $\mathcal{A}$-linear generalized additive mapping $\mathcal{L}:\mathcal{X}\to \mathcal{Y}$ such that

$$\begin{array}{c}\underset{k\in \mathbb{N}}{sup}{\parallel (\mathcal{L}(x_1)-f(x_1),\dots ,\mathcal{L}(x_k)-f(x_k))\parallel }_k\hfill \\ \le \underset{k\in \mathbb{N}}{sup}\frac{L}{2{(}_{d-2}{C}_l{-}_{d-2}{C}_{l-2}+1)(1-L)}\phi (x_1,x_1,\underset{d-2\mathrm{times}}{\underset{⏞}{0,\dots ,0}},\dots ,x_k,x_k,\underset{d-2\mathrm{times}}{\underset{⏞}{0,\dots ,0}})\hfill \end{array}$$

for all $x_1,\dots ,x_k\in \mathcal{X}$.

Proof Note that $f(0)=0$ and $f(-x)=-f(x)$ for all $x\in \mathcal{X}$ since f is an odd mapping. Let $u=1\in U(\mathcal{A})$. Putting $x_{i1}=x_{i2}=x_1$ and $x_{i3}=\cdots =x_{im}=0$, $1\le i\le k$ in (4.7), we have

$$\begin{array}{c}{\parallel (rf\left(\frac{2}{r}{x}_1\right)-2f(x_1),\dots ,rf\left(\frac{2}{r}{x}_k\right)-2f(x_k))\parallel }_k\hfill \\ \le \frac{1}{{}_{d-2}C_l{-}_{d-2}{C}_{l-2}+1}\phi (x_1,x_1,\underset{d-2\mathrm{times}}{\underset{⏞}{0,\dots ,0}},\dots ,x_k,x_k,\underset{d-2\mathrm{times}}{\underset{⏞}{0,\dots ,0}}).\hfill \end{array}$$

Letting $t:{=}_{d-2}{C}_l{-}_{d-2}{C}_{l-2}+1$, we get

$$\begin{array}{c}{\parallel (f(x_1)-\frac{2}{r}f\left(\frac{r}{2}{x}_1\right),\dots ,f(x_k)-\frac{2}{r}f\left(\frac{r}{2}{x}_k\right))\parallel }_k\hfill \\ \le \frac{1}{rt}\phi (\frac{r}{2}{x}_1,\frac{r}{2}{x}_1,\underset{d-2\mathrm{times}}{\underset{⏞}{0,\dots ,0}},\dots ,\frac{r}{2}{x}_k,\frac{r}{2}{x}_k,\underset{d-2\mathrm{times}}{\underset{⏞}{0,\dots ,0}})\hfill \\ \le \frac{L}{2t}\phi (x_1,x_1,\underset{d-2\mathrm{times}}{\underset{⏞}{0,\dots ,0}},\dots ,x_k,x_k,\underset{d-2\mathrm{times}}{\underset{⏞}{0,\dots ,0}})\hfill \end{array}$$

for all $x_1,\dots ,x_k\in \mathcal{X}$.

The rest of the proof is similar to the proof of Theorem 4.3. □

Corollary 4.6 Let $r<2$, and let θ and $p>1$ be positive real numbers, or let $r>2$, and let θ and $p<1$ be positive real numbers. Let $f:\mathcal{X}\to \mathcal{Y}$ be an odd mapping such that

$${\parallel (D_{u}f(x_{11},\dots ,x_{1d}),\dots ,D_{u}f(x_{k1},\dots ,x_{kd}))\parallel }_k\le \theta (\sum _{j=1}^d{\parallel x_{1j}\parallel }^p+\cdots +\sum _{j=1}^d{\parallel x_{kj}\parallel }^p)$$

for all $u\in U(\mathcal{A})$ and all $x_{11},\dots ,x_{1d},\dots ,x_{k1},\dots ,x_{kd}\in \mathcal{X}$. Then there exists a unique $\mathcal{A}$-linear generalized additive mapping $\mathcal{L}:\mathcal{X}\to \mathcal{Y}$ such that

$$\begin{array}{c}\underset{k\in \mathbb{N}}{sup}{\parallel (\mathcal{L}(x_1)-f(x_1),\dots ,\mathcal{L}(x_k)-f(x_k))\parallel }_k\hfill \\ \le \underset{k\in \mathbb{N}}{sup}\frac{r^{p-1}\theta }{(2^{p-1}-r^{p-1}){(}_{d-2}{C}_l{-}_{d-2}{C}_{l-2}+1)}({\parallel x_1\parallel }^p+\cdots +{\parallel x_k\parallel }^p)\hfill \end{array}$$

for all $x\in X$.

Proof Define

$$\phi (x_{11},\dots ,x_{1d},\dots ,x_{k1},\dots ,x_{kd})=\theta (\sum _{j=1}^d{\parallel x_{1j}\parallel }^p+\cdots +\sum _{j=1}^d{\parallel x_{kj}\parallel }^p).$$

Putting $L=\frac{r^{p-1}}{2^{p-1}}$ in Theorem 4.5, we get the desired result. □

Now we investigate the Hyers-Ulam stability of linear mappings for the case $d=2$.

Theorem 4.7 Let $r\ne 2$. Let $f:\mathcal{X}\to \mathcal{Y}$ be an odd mapping for which there is a function $\phi :{\mathcal{X}}^{2k}\to [0,\mathrm{\infty })$ such that

(4.8)

for all $u\in U(\mathcal{A})$ and all $x_1,\dots x_k,y_1\dots ,y_k\in \mathcal{X}$. If there exists an $L<1$ such that

$$\phi (\frac{2}{r}{x}_1,\frac{2}{r}{x}_1,\frac{2}{r}{x}_2,\frac{2}{r}{x}_2,\dots ,\frac{2}{r}{x}_k,\frac{2}{r}{x}_k)\le \frac{2}{r}L\phi (x_1,x_1,x_2,x_2,\dots ,x_k,x_k)$$

for all $x_1,\dots ,x_k\in \mathcal{X}$. Then there exists a unique $\mathcal{A}$-linear generalized additive mapping $\mathcal{L}:\mathcal{X}\to \mathcal{Y}$ such that

$$\begin{array}{c}\underset{k\in \mathbb{N}}{sup}{\parallel (\mathcal{L}(x_1)-f(x_1),\dots ,\mathcal{L}(x_k)-f(x_k))\parallel }_k\hfill \\ \le \underset{k\in \mathbb{N}}{sup}\frac{L}{2(1-L)}\phi (x_1,x_1,\dots ,x_k,x_k)\hfill \end{array}$$

for all $x_1,\dots ,x_k\in \mathcal{X}$.

Proof Let $u=1\in U(\mathcal{A})$. Putting $x=y$ in (4.8), we have

$${\parallel (rf\left(\frac{2}{r}{x}_1\right)-2f(x_1),\dots ,rf\left(\frac{2}{r}{x}_k\right)-2f(x_k))\parallel }_k\le \phi (x_1,x_1,\dots ,x_k,x_k)$$

for all $x\in X$. So

$${\parallel (f(x_1)-\frac{r}{2}f\left(\frac{2}{r}{x}_1\right),\dots ,f(x_k)-\frac{r}{2}f\left(\frac{2}{r}{x}_k\right))\parallel }_k\le \frac{1}{2}\phi (x_1,x_1,\dots ,x_k,x_k)$$

for all $x\in X$.

The rest of the proof is the same as in the proof of Theorem 4.3. □

Corollary 4.8 Let $r>2$, and let θ and $p>1$ be positive real numbers, or let $r<2$, and let θ and $p<1$ be positive real numbers. Let $f:X\to Y$ be an odd mapping such that

$$\begin{array}{c}{\parallel (rf\left(\frac{u{x}_1+u{y}_1}{r}\right)-uf(x_1)-uf(y_1),\dots ,rf\left(\frac{u{x}_k+u{y}_k}{r}\right)-uf(x_k)-uf(y_k))\parallel }_k\hfill \\ \le \theta \sum _{j=1}^k({\parallel x_j\parallel }^p+{\parallel y_j\parallel }^p)\hfill \end{array}$$

for all $u\in U(\mathcal{A})$ and for all $x_1,\dots ,x_k\in \mathcal{X}$. Then there exists a unique $\mathcal{A}$-linear generalized additive mapping $\mathcal{L}:\mathcal{X}\to \mathcal{Y}$ such that

$$\underset{k\in \mathbb{N}}{sup}{\parallel (\mathcal{L}(x_1)-f(x_1),\dots ,\mathcal{L}(x_k)-f(x_k))\parallel }_k\le \underset{k\in \mathbb{N}}{sup}\frac{r^{p-1}\theta }{r^{p-1}-2^{p-1}}\sum _{j=1}^k{\parallel x_j\parallel }^p$$

for all $x_1,\dots ,x_k\in \mathcal{X}$.

Proof Define $\phi (x_1,y_1,\dots ,x_k,y_k)=\theta {\sum }_{j=1}^k({\parallel x_j\parallel }^p+{\parallel y_j\parallel }^p)$, and apply Theorem 4.7. Then we get the desired result. □

Theorem 4.9 Let $r\ne 2$. Let $f:\mathcal{X}\to \mathcal{Y}$ be an odd mapping for which there is a function $\phi :{\mathcal{X}}^{2k}\to [0,\mathrm{\infty })$ such that

(4.9)

for all $u\in U(\mathcal{A})$ and all $x_1,\dots ,x_k,y_1,\dots ,y_k\in \mathcal{X}$. If there exists an $L<1$ such that

$$\phi (\frac{r}{2}{x}_1,\frac{r}{2}{x}_1,\frac{r}{2}{x}_2,\frac{r}{2}{x}_2,\dots ,\frac{r}{2}{x}_k,\frac{r}{2}{x}_k)\le \frac{r}{2}L\phi (x_1,x_1,x_2,x_2,\dots ,x_k,x_k)$$

for all $x_1,\dots ,x_k\in \mathcal{X}$. Then there exists a unique $\mathcal{A}$-linear generalized additive mapping $\mathcal{L}:\mathcal{X}\to \mathcal{Y}$ such that

$$\underset{k\in \mathbb{N}}{sup}{\parallel (\mathcal{L}(x_1)-f(x_1),\dots ,\mathcal{L}(x_k)-f(x_k))\parallel }_k\le \underset{k\in \mathbb{N}}{sup}\frac{1}{2(1-L)}\phi (x_1,x_1,\dots ,x_k,x_k)$$

for all $x_1,\dots ,x_k\in \mathcal{X}$.

Proof Let $u=1\in U(\mathcal{A})$. Putting $x=y$ in (4.9), we have

$${\parallel (rf\left(\frac{2}{r}{x}_1\right)-2f(x_1),\dots ,rf\left(\frac{2}{r}{x}_k\right)-2f(x_k))\parallel }_k\le \phi (x_1,x_1,\dots ,x_k,x_k)$$

for all $x_1,\dots ,x_k\in \mathcal{X}$. So

$$\begin{array}{rcl}{\parallel (f(x_1)-\frac{2}{r}f\left(\frac{r}{2}{x}_1\right),\dots ,f(x_k)-\frac{2}{r}f\left(\frac{r}{2}{x}_k\right))\parallel }_k& \le & \frac{1}{r}\phi (\frac{r}{2}{x}_1,\frac{r}{2}{x}_1,\dots ,\frac{r}{2}{x}_k,\frac{r}{2}{x}_k)\\ \le & \frac{1}{2}L\phi (x_1,x_1,\dots ,x_k,x_k)\end{array}$$

for all $x_1,\dots ,x_k\in \mathcal{X}$.

The rest of the proof is similar to the proof of Theorem 4.3. □

Corollary 4.10 Let $r>2$, and let θ and $p>1$ be positive real numbers. Or let $r<2$, and let θ and $p<1$ be positive real numbers. Let $f:X\to Y$ be an odd mapping such that

$$\begin{array}{c}{\parallel (rf\left(\frac{u{x}_1+u{y}_1}{r}\right)-uf(x_1)-uf(y_1),\dots ,rf\left(\frac{u{x}_k+u{y}_k}{r}\right)-uf(x_k)-uf(y_k))\parallel }_k\hfill \\ \le \theta \sum _{j=1}^k({\parallel x_j\parallel }^p+{\parallel y_j\parallel }^p)\hfill \end{array}$$

for all $u\in U(\mathcal{A})$ and all $x_1,\dots ,x_k\in \mathcal{X}$. Then there exists a unique $\mathcal{A}$-linear generalized additive mapping $\mathcal{L}:\mathcal{X}\to \mathcal{Y}$ such that

$$\underset{k\in \mathbb{N}}{sup}{\parallel (\mathcal{L}(x_1)-f(x_1),\dots ,\mathcal{L}(x_k)-f(x_k))\parallel }_k\le \underset{k\in \mathbb{N}}{sup}\frac{r^{p-1}\theta }{2^{p-1}-r^{p-1}}\sum _{j=1}^k{\parallel x_j\parallel }^p$$

for all $x_1,\dots ,x_k\in \mathcal{X}$.

Proof Define $\phi (x_1,y_1,\dots ,x_k,y_k)=\theta {\sum }_{j=1}^k({\parallel x_j\parallel }^p+{\parallel y_j\parallel }^p)$, and apply Theorem 4.9. Then we get the desired result. □

5 Isomorphisms in unital multi-$C^{\ast }$-algebras

Throughout this section, assume that $\mathcal{A}$ and $\mathcal{B}$ are unital multi-$C^{\ast }$-algebras with unit e. Let $U(\mathcal{A})$ be the set of unitary elements in $\mathcal{A}$.

We investigate $C^{\ast }$-algebra isomorphisms in unital multi-$C^{\ast }$-algebras.

Theorem 5.1 Let $r\ne 2$. Let $h:\mathcal{A}\to \mathcal{B}$ be an odd bijective mapping satisfying $h(\frac{2^n}{r^n}uy)=h(\frac{2^n}{r^n}u)h(y)$ for all $u\in U(\mathcal{A})$, all $y\in \mathcal{A}$, and $n=0,1,2,\dots$, for which there exists a function $\phi :{\mathcal{A}}^{kd}\to [0,\mathrm{\infty })$ such that

$$\begin{array}{c}\underset{j\to \mathrm{\infty }}{lim}\frac{r^j}{2^j}\phi (\frac{2^j}{r^j}{x}_{11},\dots ,\frac{2^j}{r^j}{x}_{1d},\dots ,\frac{2^j}{r^j}{x}_{k1},\dots ,\frac{2^j}{r^j}{x}_{kd})=0,\hfill \\ {\parallel (D_{\mu }h(x_{11},\dots ,x_{1d}),\dots ,D_{\mu }h(x_{k1},\dots ,x_{kd}))\parallel }_k\hfill \\ \le \phi (x_{11},\dots ,x_{1d},\dots ,x_{k1},\dots ,x_{kd}),\hfill \\ {\parallel (h\left(\frac{2^n}{r^n}{u}_1^{\ast }\right)-h{\left(\frac{2^n}{r^n}{u}_1\right)}^{\ast },\dots ,h\left(\frac{2^n}{r^n}{u}_k^{\ast }\right)-h{\left(\frac{2^n}{r^n}{u}_k\right)}^{\ast })\parallel }_k\hfill \\ \le \phi (\underset{d\mathrm{times}}{\underset{⏞}{\frac{2^n}{r^n}{u}_1,\dots ,\frac{2^n}{r^n}{u}_1}},\dots ,\underset{d\mathrm{times}}{\underset{⏞}{\frac{2^n}{r^n}{u}_k,\dots ,\frac{2^n}{r^n}{u}_k}})\hfill \end{array}$$

for all $\mu \in S^1:=\{\lambda \in \mathbb{C}\mid |\lambda |=1\}$, all $u_1,\dots ,u_k\in U(\mathcal{A})$, $n=0,1,2,\dots$, and all $x_{11},\dots ,x_{kd}\in A$. Assume that ${lim}_{n\to \mathrm{\infty }}\frac{r^n}{2^n}h(\frac{2^n}{r^n}e)$ is invertible. Then the odd bijective mapping $h:\mathcal{A}\to \mathcal{B}$ is a $C^{\ast }$-algebra isomorphism.

Proof Consider the multi-$C^{\ast }$-algebras $\mathcal{A}$ and $\mathcal{B}$ as left Banach modules over the unital multi-$C^{\ast }$-algebra $\mathbb{C}$. By Theorem 4.3, there exists a unique $\mathbb{C}$-linear generalized additive mapping $\mathcal{H}:\mathcal{A}\to \mathcal{B}$ such that

$$\begin{array}{c}\underset{k\in \mathbb{N}}{sup}{\parallel (h(x_1)-\mathcal{H}(x_1),\dots ,h(x_k)-\mathcal{H}(x_k)\parallel }_k\hfill \\ \le \underset{k\in \mathbb{N}}{sup}\frac{1}{2{(}_{d-2}{C}_l{-}_{d-2}{C}_{l-2}+1)}\phi (x_1,x_1,\underset{d-2\mathrm{times}}{\underset{⏞}{0,\dots ,0}},\dots ,x_k,x_k,\underset{d-2\mathrm{times}}{\underset{⏞}{0,\dots ,0}})\hfill \end{array}$$

for all $x_1,\dots ,x_k\in \mathcal{A}$ in which $\mathcal{H}:\mathcal{A}\to \mathcal{B}$ is given by

$$\mathcal{H}(x)=\underset{n\to \mathrm{\infty }}{lim}\frac{r^n}{2^n}h\left(\frac{2^n}{r^n}x\right)$$

for all $x\in \mathcal{A}$.

The rest of the proof is similar to the proof of Theorem 4.1 of [7]. □

Corollary 5.2 Let $r>2$, and let θ and $p>1$ be positive real numbers. Or let $r<2$, and let θ and $p<1$ be positive real numbers. Let $h:\mathcal{A}\to \mathcal{B}$ be an odd bijective mapping satisfying $h(\frac{2^n}{r^n}uy)=h(\frac{2^n}{r^n}u)h(y)$ for all $u\in U(\mathcal{A})$, all $y\in \mathcal{A}$, and all $n=0,1,2,\dots$, such that

$$\begin{array}{c}{\parallel (D_{\mu }h(x_{11},\dots ,x_{1d}),\dots ,D_{\mu }h(x_{k1},\dots ,x_{kd}))\parallel }_k\le \theta \sum _{j=1}^d({\parallel x_{1j}\parallel }^p+\cdots +{\parallel x_{kj}\parallel }^p),\hfill \\ {\parallel (h\left(\frac{2^n}{r^n}{u}_1^{\ast }\right)-h{\left(\frac{2^n}{r^n}{u}_1\right)}^{\ast },\dots ,h\left(\frac{2^n}{r^n}{u}_k^{\ast }\right)-h{\left(\frac{2^n}{r^n}{u}_k\right)}^{\ast })\parallel }_k\le kd\frac{2^{pn}}{r^{pn}}\theta \hfill \end{array}$$

for all $\mu \in S^1$, all $u\in U(\mathcal{A})$, $n=0,1,2,\dots$, and all $x_{11},\dots ,x_{kd}\in \mathcal{A}$. Assume that ${lim}_{n\to \mathrm{\infty }}\frac{r^n}{2^n}h(\frac{2^n}{r^n}e)$ is invertible. Then the odd bijective mapping $h:\mathcal{A}\to \mathcal{B}$ is a $C^{\ast }$-algebra isomorphism.

Proof Define $\phi (x_{11},\dots ,x_{1d},\dots ,x_{k1},\dots ,x_{kd})=\theta {\sum }_{j=1}^d({\parallel x_{1j}\parallel }^p+\cdots +{\parallel x_{kj}\parallel }^p)$, and apply Theorem 5.1. Then we get the desired result. □