Abstract
In this paper, we look at the concept of multi-\(C^{*}\)-ternary algebras and consider some properties. As an application we approximate multi-\(C^{*}\)-ternary algebra homomorphisms and derivations in these spaces.
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1 Introduction and preliminaries
Ternary algebraic structures arise naturally in theoretical and mathematical physics, for example, the quark model inspired a particular brand of ternary algebraic system. We also refer the reader to ‘Nambu mechanics’ [1] (see also [2, 3] and [4]).
A \(C^{*}\)-ternary algebra is a complex Banach space A, equipped with a ternary product \((x, y, z) \mapsto[x, y, z]\) of \(A^{3}\) into A, which is C-linear in the outer variables, conjugate C-linear in the middle variable, and associative in the sense that \([x, y, [z, w, v]] = [x, [w, z, y], v] = [[x, y, z], w, v]\), and satisfies \(\|[x, y, z]\| \le\|x\| \cdot\|y\| \cdot \|z\|\) and \(\|[x, x, x]\| = \|x\|^{3}\) (see [4]).
If a \(C^{*}\)-ternary algebra \((A, [\cdot, \cdot, \cdot] )\) has the identity, i.e., the element \(e\in A\) such that \(x = [x, e, e] = [e, e, x]\) for all \(x\in A\), then it is routine to verify that A, endowed with \(x\circ y : = [x, e, y]\) and \(x^{*}:=[e, x, e]\), is a unital \(C^{*}\)-algebra. Conversely, if \((A, \circ)\) is a unital \(C^{*}\)-algebra, then \([x, y, z] : = x \circ y^{*} \circ z\) makes A into a \(C^{*}\)-ternary algebra.
A C-linear mapping \(H: A \rightarrow B\) is called a \(C^{*}\)-ternary algebra homomorphism if
for all \(x, y, z \in A\). A C-linear mapping \(\delta: A \rightarrow A\) is called a \(C^{*}\)-ternary derivation if
for all \(x, y, z \in A\) (see [5]).
Ternary structures and their generalization, the so-called n-ary structures, are important in view of their applications in physics (see [6]).
Let X be a set. A function \(d: X \times X \rightarrow[0, \infty]\) is called a generalized metric on X if d satisfies the following conditions:
-
(1)
\(d(x, y) = 0\) if and only if \(x=y\);
-
(2)
\(d(x, y) = d(y, x)\) for all \(x, y \in X\);
-
(3)
\(d(x, z) \le d(x, y) + d(y, z)\) for all \(x, y, z\in X\).
Theorem 1.1
([7])
Let \((X, d)\) be a complete generalized metric space and let \(J: X \rightarrow X\) be a strictly contractive mapping with Lipschitz constant \(L<1\). Then, for each \(x\in X\), either
for all non-negative integers n or there exists a positive integer \(n_{0}\) such that
-
(1)
\(d(J^{n} x, J^{n+1}x) <\infty\) for all \(n\ge n_{0}\);
-
(2)
the sequence \(\{J^{n} x\}\) converges to a fixed point \(y^{*}\) of J;
-
(3)
\(y^{*}\) is the unique fixed point of J in the set \(Y = \{y\in X \mid d(J^{n_{0}} x, y) <\infty\}\);
-
(4)
\(d(y, y^{*}) \le\frac{1}{1-L} d(y, Jy)\) for all \(y \in Y\).
2 Multi-normed spaces
The notion of a multi-normed space was introduced by Dales and Polyakov in [8] and many examples are given in [8–10].
Let \(( {\mathcal{E}},\|\cdot\|)\) be a complex normed space and let \(k\in\mathbf{N}\). We denote by \(\mathcal{E}^{k}\) the linear space \(\mathcal{E}\oplus\cdots\oplus\mathcal{E}\) consisting of k-tuples \((x_{1}, \ldots, x_{k})\), where \(x_{1}, \ldots, x_{k}\in\mathcal{E}\). 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. We denote by \(\mathbf{N}_{k}\) the set \(\{1, 2, \ldots ,k\}\) and by \(\Sigma_{k}\) the group of permutations on k symbols.
Definition 2.1
A multi-norm on \(\{ {\mathcal{E}}^{k}: k\in\mathbf{N}\}\) is a sequence
such that \(\|\cdot\|_{k}\) is a norm on \({\mathcal{E}}^{k}\) for each \(k\in\mathbf{N}\) with \(k\geq2\):
-
(A1)
\(\|(x_{\sigma(1)},\ldots,x_{\sigma(k)})\|_{k}=\|(x_{1},\ldots,x_{k})\|_{k}\) for any \(\sigma\in\Sigma_{k}\) and \(x_{1},\ldots,x_{k}\in\mathcal{E}\);
-
(A2)
\(\|(\alpha_{1}x_{1},\ldots,\alpha_{k}x_{k})\|_{k}\leq (\max_{i\in{\mathbf{N}}_{k}}|\alpha_{i}| ) \|(x_{1},\ldots,x_{k})\| _{k}\) for any \(\alpha_{1},\ldots,\alpha_{k} \in\mathbf{C}\) and \(x_{1}, \ldots, x_{k}\in\mathcal{E}\);
-
(A3)
\(\|(x_{1},\ldots,x_{k-1},0)\|_{k}=\|(x_{1},\ldots,x_{k-1})\|_{k-1}\) for any \(x_{1}, \ldots, x_{k-1}\in\mathcal{E}\);
-
(A4)
\(\|(x_{1},\ldots,x_{k-1},x_{k-1})\|_{k}=\|(x_{1},\ldots,x_{k-1})\| _{k-1}\) for any \(x_{1},\ldots, x_{k-1}\in\mathcal{E}\).
In this case, we say that \(((\mathcal{E}^{k},\|\cdot\|_{k}): k\in\mathbf{N})\) is a multi-normed space.
Lemma 2.2
([10])
Suppose that \(((\mathcal{E}^{k},\|\cdot\|_{k}): k\in\mathbf{N})\) is a multi-normed space and let \(k\in\mathbf{N}\). Then
-
(1)
\(\|(x,\ldots,x)\|_{k}=\|x\|\) for any \(x\in\mathcal{E}\);
-
(2)
\(\max_{i\in\mathbf{N}_{k}}\|x_{i}\|\leq \|x_{1},\ldots,x_{k}\|_{k}\leq\sum_{i=1}^{k}\|x_{i}\|\leq k \max_{i\in {\mathbf{N}}_{k}}\|x_{i}\|\) for any \(x_{1},\ldots, x_{k}\in\mathcal{E}\).
It follows from (2) that, if \(( \mathcal{E},\|\cdot\|)\) is a Banach space, then \(( \mathcal{E}^{k},\|\cdot\|_{k})\) is a Banach space for each \(k\in\mathbf{N}\). In this case, \(((\mathcal{E}^{k},\|\cdot\|_{k}): k\in\mathbf{N})\) is a multi-Banach space.
Now, we present two examples (see [8]).
Example 2.3
The sequence \((\|\cdot\|_{k}: k\in\mathbf{N})\) on \(\{\mathcal{E}^{k}: k\in\mathbf{N}\}\) defined by
for any \(x_{1}, \ldots, x_{k}\in\mathcal{E}\) is a multi-norm, which is called the minimum multi-norm.
Example 2.4
Let \(\{(\|\cdot\|_{k}^{\alpha}: k\in\mathbf{N}):\alpha\in A\}\) be the (non-empty) family of all multi-norms on \(\{\mathcal{E}^{k}:k\in\mathbf{N}\}\). For each \(k\in\mathbf{N} \), set
for any \(x_{1}, \ldots, x_{k}\in \mathcal{E}\). Then \(( \|\cdot\|_{k} : k\in\mathbf{N})\) is a multi-norm on \(\{\mathcal{E}^{k}: k\in\mathbf{N}\}\), which is called the maximum multi-norm.
Now, we need the following observation which can easily be deduced from Lemma 2.2(2) of multi-norms.
Lemma 2.5
Suppose that \(k\in\mathbf{N}\) and \((x_{1},\ldots, x_{k})\in \mathcal{E}^{k} \). For each \(j\in\{1,\ldots,k\}\), let \((x_{n}^{j})\) be a sequence in \(\mathcal{E} \) such that \(\lim_{n\to\infty}x_{n}^{j}=x_{j}\). Then, for each \((y_{1},\ldots,y_{k})\in\mathcal{E}^{k}\),
Definition 2.6
Let \(((\mathcal{E}^{k},\|\cdot\|_{k}): k\in\mathbf{N})\) be a multi-normed space. A sequence \((x_{n})\) in \(\mathcal{E}\) is a multi-null sequence if, for each \(\epsilon>0\), there exists \(n_{0}\in\mathbf{N}\) such that
for any \(n\geq n_{0}\). Let \(x\in\mathcal{E}\). We say that the sequence \((x_{n})\) is multi-convergent to \(x\in\mathcal{E}\) and write
if \((x_{n}-x)\) is a multi-null sequence.
Definition 2.7
Let \(({A},\|\cdot\|)\) be a normed algebra such that \((({A}^{k},\|\cdot\|_{k}): k\in\mathbf{N})\) is a multi-normed space. Then \((({A}^{k},\|\cdot\|_{k}): k\in\mathbf{N})\) is called a multi-normed algebra if
for all \(k\in\mathbf{N}\) and \(a_{1},\ldots,a_{k},b_{1},\ldots,b_{k}\in {A}\). Further, the multi-normed algebra \((({A}^{k},\|\cdot\|_{k}): k\in\mathbf{N})\) is a multi-Banach algebra if \((({A}^{k},\|\cdot\|_{k}): k\in\mathbf{N})\) is a multi-Banach space.
Example 2.8
Let p, q with \(1 \leq p \leq q < \infty\) and let \({A}=\ell^{p}\). The algebra A is a Banach sequence algebra with respect to a coordinate-wise multiplication of sequences (see [12]). Let \((\|\cdot\|_{k}: k\in\mathbf{N})\) be the standard \((p, q)\)-multi-norm on \(\{{A}^{k}: k\in\mathbf{N}\}\). Then \((({A}^{k},\|\cdot\|_{k}): k\in\mathbf{N})\) is a multi-Banach algebra.
Definition 2.9
Let \((({ A}^{k},\|\cdot\|_{k}): k\in\mathbf{N})\) be a multi-Banach algebra. A multi-\(C^{*}\)-algebra is a complex multi-Banach algebra \((({ A}^{k},\|\cdot\|_{k}): k\in\mathbf{N})\) with an involution ∗ satisfying
for all \(k\in\mathbf{N}\) and \(a_{1},\ldots,a_{k}\in {A}\).
Definition 2.10
Let \((({ A}^{k},\|\cdot\|_{k}): k\in\mathbf{N})\) be a multi-Banach space. A multi-\(C^{*}\)-ternary algebra is a complex multi-Banach space \((({ A}^{k},\|\cdot\|_{k}): k\in\mathbf{N})\) equipped with a ternary product.
3 Approximation of homomorphisms in multi-Banach algebras
Throughout this paper, assume that A, B are \(C^{*}\)-ternary algebras.
For a given mapping \(f: A \to B\), we define
for all \(\mu\in{\mathbf{T}}^{1}:=\{ \lambda\in\mathbf{C}: |\lambda|=1\}\) and \(x_{1},\ldots,x_{p},y_{1},\ldots,y_{d}\in A\).
One can easily show that a mapping \(f:A \rightarrow B\) satisfies
for all \(\mu\in {\mathbf{T}}^{1}\) and all \(x_{1},\ldots,x_{p},y_{1},\ldots,y_{d}\in A\) if and only if
for all \(\mu, \lambda\in {{\mathbf{T}}}^{1}\) and \(x, y \in A\).
Lemma 3.1
([13])
Let \(f:A \rightarrow B\) be an additive mapping such that \(f(\mu x) = \mu f(x)\) for all \(x\in A\) and \(\mu\in{\mathbf{T}}^{1}\). Then the mapping f is C-linear.
Lemma 3.2
Let \(\{x_{n}\}\), \(\{y_{n}\}\), and \(\{z_{n}\}\) be the convergent sequences in A. Then the sequence \(\{[x_{n},y_{n},z_{n}]\}\) is convergent in A.
Proof
Let \(x,y,z\in A\) be such that
Since
for all \(n\geq1\), we get
for all \(n\geq1\), and so
This completes the proof. □
Using Theorem 1.1, we approximate homomorphisms in multi-\(C^{*}\)-ternary algebras for the functional equation \(C_{\mu}f(x_{1},\ldots,x_{m}) =0\).
Theorem 3.3
Let \((( {B}^{k},\|\cdot\|_{k}): k\in\mathbf{N})\) be a multi-\(C^{*}\)-ternary algebra. Let \(f: A \rightarrow B\) be a mapping for which there are functions \(\varphi: A^{(p+d)k} \rightarrow[0, \infty)\) and \(\psi: A^{3k} \rightarrow[0, \infty)\) such that
for all \(\mu\in\mathbf{T}^{1}\) and \(x_{11},\ldots,x_{1p},y_{11},\ldots ,y_{1d},\ldots,x_{k1},\ldots,x_{kp},y_{k1},\ldots, y_{kd}, x_{1},\ldots,x_{k},y_{1}, \ldots,y_{k},z_{1}, \ldots,z_{k}\in A\), where \(\gamma=\frac{p+2d}{2}\). If there exists a constant \(L<1\) such that
for all \(x_{1},x_{2},\ldots,x_{k} \in A\), then there exists a unique homomorphism \(H : A \rightarrow B\) such that
for all \(x_{1},\ldots,x_{k} \in A\).
Proof
Let \(\mu= 1\) and \(x_{ij} = y_{ij} = x_{i}\) for \(1\leq i\leq k \) in (2). Then we get
for all \(x_{1},\ldots,x_{k} \in A\). Consider the set
and introduce the generalized metric on E:
It is easy to see that \((E, d)\) is complete (see also [9]).
First we show that d is metric on E. It is obvious \(d(g,g)=0\) for all \(g\in E\). If \(d(g,h)=0\), then, for every fixed \(x_{1},\ldots,x_{k} \in A\),
and therefore \(g=h\). If \(d(g,h)=a<\infty\) and \(d(h,l)=b<\infty\) for all \(g,h,l\in E\), then
So we have \(d(g,l)\leq d(g,h) + d(h,l)\).
Let \(\{g_{n}\}\) be a Cauchy sequence in \((E,d)\). Then for all \(\epsilon >0\) there exists N such that \(d(g_{n},g_{i}) < \epsilon\), if \(n,i \geq N\), Let \(n,i\geq N\). Since \(d(g_{n},g_{i}) < \epsilon\) there exists \(C\in [0,\epsilon)\) such that
for all \(x_{1},\ldots,x_{k} \in A\), so for each \(x_{1},\ldots,x_{k}\in A\), \(\{g_{n}(x_{1},\ldots,x_{k})\}\) is a Cauchy sequence in B. Since B is complete, there exists \(g(x_{1},\ldots,x_{k})\in B\) such that \(g_{n}(x_{1},\ldots,x_{k})\rightarrow g(x_{1},\ldots,x_{k})\) as \(n\rightarrow\infty\). Thus, we have \(g\in E\). Taking the limit as \(i\rightarrow\infty\) in (9) we obtain, for \(n\geq N\),
Therefore \(d(g_{n},g)\leq\epsilon\). Hence \(g_{n}\rightarrow g\) as \(n\rightarrow\infty\), so \((E,d)\) is complete. Now, we consider the linear mapping \(\Lambda: E \rightarrow E\) such that
for all \(x \in A\). From Theorem 3.1 of [14] (also see Lemma 3.2 of [9]),
for all \(g, h \in E\). Let \(g,h\in E\) and let \(C\in[0,\infty]\) be an arbitrary constant with \(d(g,h)\leq C\). From the definition of d, we have
for all \(x_{1},\ldots,x_{k}\in A\). From our assumption and the last inequality, we have
for all \(x_{1},\ldots,x_{k} \in A\) and so
for all \(x_{1},\ldots,x_{k} \in A\). Hence \(d(\Lambda f, f) \le \frac{1}{2\gamma}\). From Theorem 1.1, the sequence \(\{\Lambda^{n} f\}\) converges to a fixed point H of Λ, i.e., \(H:A\rightarrow B\) is a mapping defined by
and \(H(\gamma x)=\gamma H(x)\) for all \(x \in A\). Also, H is the unique fixed point of Λ in the set \(E'=\{ g \in E : d(f,g)< \infty\}\) and
i.e., the inequality (7) hold for all \(x_{1},\ldots ,x_{k} \in A\). Thus it follows from the definition of H, (1), and (2) that
for all \(\mu\in\mathbf{T}^{1}\) and \(x_{11},\ldots,x_{1p},y_{11},\ldots ,y_{1d},\ldots,x_{k1},\ldots,x_{kp},y_{k1},\ldots,y_{kd} \in A\). Hence we have
for all \(\mu\in\mathbf{T}^{1}\), \(x_{11},\ldots,x_{1p},y_{11},\ldots,y_{1d},\ldots,x_{k1},\ldots ,x_{kp},y_{k1},\ldots,y_{kd} \in A\) and \(1\leq i \leq k\) and so \(H(\lambda x + \mu y)=\lambda H(x) + \mu H(y)\) for all \(\lambda, \mu\in\mathbf{T}^{1}\) and \(x,y \in A\). Therefore, by Lemma 3.1, the mapping \(H : A \rightarrow B\) is C-linear.
Also it follows from (3) and (4) that
for all \(x_{1}, y_{1}, z_{1},\ldots,x_{k},y_{k},z_{k} \in A\). Thus we have
for all \(x, y, z \in A\). Thus \(H : A \rightarrow B\) is a homomorphism satisfying (7).
Now, let \(T : A \rightarrow B\) be another \(C^{*}\)-ternary-algebras homomorphism satisfying (7). Since \(d(f,T)\leq \frac{1}{(1-L)2\gamma}\) and T is C-linear, we get \(T\in E'\) and \((\Lambda T)(x)=\frac{1}{\gamma}(T\gamma x)=T(x)\) for all \(x\in A\), i.e., T is a fixed point of Λ. Since H is the unique fixed point of \(\Lambda\in E'\), we get \(H=T\). This completes the proof. □
Theorem 3.4
Let \((( {B}^{k},\|\cdot\|_{k}): k\in\mathbf{N})\) be a multi-\(C^{*}\)-ternary algebra. Let \(f: A \rightarrow B\) be a mapping for which there are the functions \(\varphi: A^{(p+d)k} \rightarrow[0, \infty)\) and \(\psi: A^{3k} \rightarrow[0, \infty)\) satisfying the inequalities (2) and (3) such that
for all \(\mu\in\mathbf{T}^{1}\) and \(x_{11},\ldots,x_{1p},y_{11},\ldots ,y_{1d},\ldots,x_{k1},\ldots,x_{kp},y_{k1},\ldots, y_{kd}, x_{1},\ldots,x_{k}, y_{1},\ldots,y_{k},z_{1}, \ldots,z_{k} \in A\), where \(\gamma=\frac{p+2d}{2}\). If the constant \(L<1\) exists such that
for all \(x_{1},x_{2},\ldots,x_{k} \in A\), then there exists a unique homomorphism \(H : A \rightarrow B\) such that
for all \(x_{1},\ldots,x_{k} \in A\).
Proof
If we replace \(x_{i}\) in (8) by \(\frac{x_{i}}{\gamma} \) for \(1\leq i\leq k \), then we get
for all \(x_{1},\ldots,x_{k} \in A\). Consider the set
and introduce the generalized metric on E:
It is easy to see that \((E, d)\) is complete (see [9]).
Now, we consider the linear mapping \(\Lambda: E \rightarrow E\) such that
for all \(x \in A\). From Theorem 3.1 of [14] (also see Lemma 3.2 of [9]),
for all \(g, h \in E\). Let \(g,h\in E\) and let \(C\in[0,\infty]\) be an arbitrary constant with \(d(g,h)\leq C\). From the definition of d, we have
for all \(x_{1},\ldots,x_{k}\in A\). From our assumption and the last inequality, we have
for all \(x_{1},\ldots,x_{k} \in A\) and so \(d(\Lambda g ,\Lambda h)\leq Ld(g,h)\) for any \(g,h \in E\). It follows from (16) that \(d(\Lambda f,f)\leq\frac{1}{2\gamma}\). Therefore, according to Theorem 1.1, the sequence \(\{\Lambda^{n} f\}\) converges to a fixed point H of Λ, i.e., \(H:A\rightarrow B\) is a mapping defined by
for all \(x \in A\).
The rest of the proof is similar to the proof of Theorem 3.3 and so we omit it. This completes the proof. □
Theorem 3.5
Let r and θ be non-negative real numbers such that \(r\notin[1,3]\) and let \((( {B}^{k},{\|\cdot\|_{k}}): k\in\mathbf{N})\) be a multi-\(C^{*}\)-ternary algebra. Let \(f : A \rightarrow B\) be a mapping such that
and
for all \(\mu\in\mathbf{T}^{1}\) and \(x_{11},\ldots,x_{1p},y_{11},\ldots,y_{1d},\ldots,x_{k1},\ldots ,x_{kp},y_{k1},\ldots,y_{kd},x_{1},\ldots,x_{k}, y_{1},\ldots,y_{k},z_{1}, \ldots,z_{k} \in A\). Then there exists a unique \(C^{*}\)-ternary algebra homomorphism \(H : A \rightarrow B\) such that
for all \(x_{1},\ldots,x_{k} \in A\).
Proof
The proof follows from Theorem 3.3 by taking
for all \(\mu\in\mathbf{T}^{1}\) and \(x_{11},\ldots,x_{1p},y_{11},\ldots ,y_{1d},\ldots,x_{k1},\ldots,x_{kp},y_{k1},\ldots, y_{kd},x_{1},\ldots,x_{k}, y_{1},\ldots,y_{k},z_{1}, \ldots,z_{k}\in A\). Then we can choose \(L=2^{1-r}(p+2d)^{r-1}\), when \(0< r<1\), and \(L=2-2^{1-r}(p+2d)^{r-1}\), when \(r>3\), and so we get the desired result. This completes the proof. □
Theorem 3.6
Let \((( {B}^{k},\|\cdot\|_{k}): k\mathbf{N})\) be a multi-\(C^{*}\)-ternary algebra. Let \(f: A \rightarrow B\) be a mapping for which there are functions \(\varphi: A^{(p+d)k} \rightarrow[0, \infty)\) and \(\psi: A^{3k} \rightarrow[0, \infty)\) such that
for all \(\mu\in\mathbf{T}^{1}\) and \(x_{11},\ldots,x_{1p},y_{11},\ldots ,y_{1d},\ldots,x_{k1},\ldots,x_{kp},y_{k1},\ldots,y_{kd}, x_{1},\ldots,x_{k}, y_{1},\ldots,y_{k},z_{1}, \ldots,z_{k}\in A\), where \(\gamma=\frac{p+2d}{2}\). If there exists the constant \(L<1\) such that
for all \(x_{1},x_{2},\ldots,x_{k} \in A\), then there exists a unique homomorphism \(H : A \rightarrow B\) such that
for all \(x_{1},\ldots,x_{k} \in A\).
Proof
Let \(\mu= 1\) and \(x_{ij} = 0 \), \(y_{ij} = x_{i}\) for \(1\leq i\leq k \) in (22). Then we get
for all \(x_{1},\ldots,x_{k} \in A\). Consider the set
and introduce the generalized metric on E:
It is easy to see that \((E, d)\) is complete (see [9]).
Now, we consider the linear mapping \(\Lambda: E \rightarrow E\) such that
for all \(x \in A\). From Theorem 3.1 of [14] (also see Lemma 3.2 of [9]),
for all \(g, h \in E\). Let \(g,h\in E\) and let \(C\in[0,\infty]\) be an arbitrary constant with \(d(g,h)\leq C\). From the definition of d, we have
for all \(x_{1},\ldots,x_{k}\in A\). From our assumption and the last inequality, we have
for all \(x_{1},\ldots,x_{k} \in A\). Thus we have
for all \(x_{1},\ldots,x_{k} \in A\). Hence \(d(\Lambda f, f) \le \frac{1}{2d}\). From Theorem 1.1, the sequence \(\{\Lambda^{n} f\}\) converges to a fixed point H of Λ, i.e., \(H:A\rightarrow B\) is a mapping defined by
and \(H(d x)=d H(x)\) for all \(x \in A\). Also, H is the unique fixed point of Λ in the set \(E'=\{ g \in E : d(f,g)< \infty\}\) and
i.e., the inequality (27) hold for all \(x_{1},\ldots ,x_{k} \in A\). It follows from the definition of H, (21), and (22) that
for all \(\mu\in\mathbf{T}^{1}\) and \(x_{11},\ldots,x_{1p},y_{11},\ldots,y_{1d},\ldots,x_{k1},\ldots ,x_{kp},y_{k1},\ldots,y_{kd} \in A\). Hence we have
for all \(\mu\in\mathbf{T}^{1}\), \(x_{11},\ldots,x_{1p},y_{11},\ldots,y_{1d},\ldots,x_{k1},\ldots ,x_{kp},y_{k1},\ldots,y_{kd} \in A\) and \(1\leq i \leq k\) and so \(H(\lambda x + \mu y)=\lambda H(x) + \mu H(y)\) for all \(\lambda, \mu\in\mathbf{T}^{1}\) and all \(x,y \in A\). Therefore, by Lemma 3.1, the mapping \(H : A \rightarrow B\) is C-linear.
Also it follows from (23) and (24) that
for all \(x_{1}, y_{1}, z_{1},\ldots,x_{k},y_{k},z_{k} \in A\). Thus
for all \(x, y, z \in A\).Thus \(H : A \rightarrow B\) is a homomorphism satisfying (26).
Now, let \(T : A \rightarrow B\) be another \(C^{*}\)-ternary algebras homomorphism satisfying (27). Since \(d(f,T)\leq \frac{1}{(1-L)2d}\) and T is C-linear, we get \(T\in E'\) and \((\Lambda T)(x)=\frac{1}{d}(T\gamma x)=T(x)\) for all \(x\in A\), i.e., T is a fixed point of Λ. Since H is the unique fixed point of \(\Lambda\in E'\), we get \(H=T\). This completes the proof. □
Theorem 3.7
Let r, s, and θ be non-negative real numbers such that \(0< r\neq1\), \(0< s\neq3\), and let \(d\ge2\). Suppose that \(f : A \rightarrow B\) is a mapping with \(f(0)=0\) satisfying (18) and
for all \(\mu\in\mathbf{T}^{1}\) and \(x_{1},\ldots,x_{k}, y_{1},\ldots,y_{k}, z_{1},\ldots,z_{k} \in A\). Then there exists a unique \(C^{*}\)-ternary algebra homomorphism \(H : A \rightarrow B\) such that
for all \(x_{1},\ldots,x_{k} \in A\).
Proof
We only prove the theorem when \(0< r<1\) and \(0< s<3\). One can prove the theorem for the other cases in a similar way. The proof follows from Theorem 3.6 by taking
for all \(\mu\in\mathbf{T}^{1}\) and \(x_{11},\ldots,x_{1p},y_{11},\ldots,y_{1d},\ldots,x_{k1},\ldots ,x_{kp},y_{k1},\ldots,y_{kd},x_{1},\ldots,x_{k},y_{1},\ldots,y_{k}, z_{1}, \ldots,z_{k} \in A \). Then we can choose \(L=d^{r-1}\), when \(0< r<1\) and \(0< s<3\), and \(L=2-d^{r-1}\), when \(r>1\) and \(s>3\), and so we get the desired result. □
Now, assume that A is a unital \(C^{*}\)-ternary algebra with norm \(\| \cdot\|\) and unit e and B is a unital \(C^{*}\)-ternary algebra with norm \(\| \cdot\|\) and unit \(e'\).
We investigate homomorphisms in \(C^{*}\)-ternary algebras associated with the functional equation \(C_{\mu}f(x_{1}, \ldots, x_{p}, y_{1}, \ldots, y_{d})=0\).
Theorem 3.8
([5])
Let \(r > 1\) (resp., \(r<1\)) and θ be non-negative real numbers and let \(f : A \rightarrow B\) be a bijective mapping satisfying (18) and
for all \(x, y, z \in A\). If \(\lim_{n\rightarrow\infty} \frac{(p+2d)^{n}}{2^{n}} f(\frac{2^{n}e}{(p+2d)^{n}}) = e'\) (resp., \(\lim_{n\rightarrow\infty} \frac{2^{n}}{(p+2d)^{n}} f(\frac{(p+2d)^{n}}{2^{n}} e) = e'\)), then the mapping \(f : A \rightarrow B\) is a \(C^{*}\)-ternary algebra isomorphism.
Theorem 3.9
Let \(r<1\) and θ be non-negative real numbers and let \(f : A \rightarrow B\) be a mapping satisfying (18) and (19). If there exist a real number \(\lambda>1\) (resp., \(0<\lambda<1\)) and an element \(x_{0}\in A\) such that \(\lim_{n\rightarrow\infty} \frac{1}{\lambda^{n}} f(\lambda^{n} x_{0}) = e'\) (resp., \(\lim_{n\rightarrow\infty} \lambda^{n} f(\frac{x_{0}}{\lambda^{n}}) = e'\)), then the mapping \(f : A \rightarrow B\) is a multi-\(C^{*}\)-ternary algebra homomorphism.
Proof
By using the proof of Theorem 3.5, there exists a unique multi-\(C^{*}\)-ternary algebra homomorphism \(H : A \rightarrow B\) satisfying (20). It follows from (20) that
for all \(x\in A\) and \(\lambda>1\) (\(0<\lambda<1\)). Therefore, from our assumption, we get \(H(x_{0})=e'\).
Let \(\lambda>1\) and \(\lim_{n\rightarrow\infty} \frac{1}{\lambda^{n}} f(\lambda^{n} x_{0}) = e'\). It follows from (19) that
for all \(x_{1},\ldots,x_{k}\in A\). Thus \([H(x),H(y),H(z)]=[H(x), H(y), f(z)]\) for all \(x,y,z\in A\). Letting \(x=y=x_{0}\) in the last equality, we get \(f(z)=H(z)\) for all \(z\in A\). Similarly, one can show that \(H(x)=f(x)\) for all \(x\in A\) when \(0<\lambda<1\) and \(\lim_{n\rightarrow\infty} \lambda^{n} f(\frac{x_{0}}{\lambda^{n}})=e'\).
Similarly, one can show the theorem for the case \(\lambda>1\). Therefore, the mapping \(f : A \rightarrow B\) is a multi-\(C^{*}\)-ternary algebra homomorphism. This completes the proof. □
Theorem 3.10
Let \(r>1\) and θ be non-negative real numbers and let \(f : A \rightarrow B\) be a mapping satisfying (18) and (19). If there exist a real number \(\lambda>1\) (resp., \(0<\lambda<1\)) and an element \(x_{0}\in A\) such that \(\lim_{n\rightarrow\infty} \frac{1}{\lambda^{n}} f(\lambda^{n} x_{0}) = e'\) (resp., \(\lim_{n\rightarrow\infty} \lambda^{n} f(\frac{x_{0}}{\lambda^{n}}) = e'\)), then the mapping \(f : A \rightarrow B\) is a multi-\(C^{*}\)-ternary algebra homomorphism.
Proof
The proof is similar to the proof of Theorem 3.9 and we omit it. □
4 Approximation of derivations on multi-\(C^{*}\)-ternary algebras
Throughout this section, assume that A is a \(C^{*}\)-ternary algebra with norm \(\| \cdot\|\).
Park [5] studied approximation of derivations on \(C^{*}\)-ternary algebras for the functional equation \(C_{\mu}f(x_{1}, \ldots, x_{p}, y_{1}, \ldots, y_{d}) =0\) (see also [5, 13, 15–59] and [60]).
For any mapping \(f : A \rightarrow A\), let
for all \(x, y, z \in A\).
Theorem 4.1
([13])
Let r and θ be non-negative real numbers such that \(r\notin[1,3]\) and let \(f:A \rightarrow A\) be a mapping satisfying (19) and
for all \(x, y, z \in A\). Then there exists a unique \(C^{*}\)-ternary derivation \(\delta:A\rightarrow A\) such that
for all \(x \in A\).
In the following theorem, we generalize and improve the result in Theorem 4.1.
Theorem 4.2
Let \((( {A}^{k},\|\cdot\|_{k}): k\in\mathbf{N})\) be a multi-\(C^{*}\)-ternary algebra. Let \(f:A\rightarrow A\) be a mapping for which there are the functions \(\varphi: A^{(p+d)k} \rightarrow[0, \infty)\) and \(\psi : A^{3k} \rightarrow[0, \infty)\) satisfying the inequalities (1), (2), and (4) such that
for all \(\mu\in\mathbf{T}^{1}\) and \(x_{11},\ldots,x_{1p},y_{11},\ldots ,y_{1d},\ldots,x_{k1},\ldots,x_{kp},y_{k1},\ldots,y_{kd}, x_{1},\ldots,x_{k}, y_{1},\ldots,y_{k},z_{1}, \ldots,z_{k}\in A\), where \(\gamma=\frac{p+2d}{2}\). If the constant \(L<1\) exists such that
for all \(x_{1},x_{2},\ldots,x_{k} \in A\), then there exists a unique \(C^{*}\)-ternary derivation \(\delta: A \rightarrow B\) such that
for all \(x_{1},\ldots,x_{k} \in A\).
Proof
The same reasoning as in the proof of Theorem 3.3, guarantees there exists a unique C-linear mapping \(\delta :A\rightarrow A\) satisfying (32). The mapping \(\delta:A\rightarrow A\) is given by
and \(\delta(\gamma x)=\gamma\delta(x)\) for all \(x \in A\). Also, H is the unique fixed point of Λ in the set \(E'=\{ g \in E : d(f,g)< \infty\}\) and
i.e., the inequality (6) holds for all \(x_{1},\ldots ,x_{k} \in A\). It follows from the definition of δ, (1) and (2), and (35) that
for all \(\mu\in\mathbf{T}^{1}\) and \(x_{11},\ldots,x_{1p},y_{11},\ldots,y_{1d},\ldots,x_{k1},\ldots ,x_{kp},y_{k1},\ldots,y_{kd} \in A\). Hence we have
for all \(\mu\in\mathbf{T}^{1}\), \(x_{11},\ldots,x_{1p},y_{11},\ldots,y_{1d},\ldots,x_{k1},\ldots ,x_{kp},y_{k1},\ldots,y_{kd} \in A\) and \(1\leq i \leq k\) and so \(\delta(\lambda x + \mu y)=\lambda \delta(x) + \mu \delta(y)\) for all \(\lambda, \mu\in\mathbf{T}^{1}\) and \(x,y \in A\). Therefore, by Lemma 3.1, the mapping \(\delta: A \rightarrow B\) is C-linear.
Also it follows from (4) and (32) that
for all \(x_{1}, y_{1}, z_{1},\ldots,x_{k},y_{k},z_{k} \in A\) and hence
for all \(x, y, z \in A\) and so the mapping \(\delta: A \rightarrow A\) is a \(C^{*}\)-ternary derivation. It follows from (32) and (4) that
for all \(x_{1},y_{1},z_{1},\ldots,x_{k},y_{k},z_{k}\in A\) and so we have
for all \(x,y,z\in A\). Hence it follows from (36) and (37) that
for all \(x,y,z\in A\). Letting \(x=y=f(z)-\delta(z)\) in (38), we get
for all \(z_{1},\ldots,z_{k} \in A\) and hence \(f(z)=\delta(z)\) for all \(z\in A\). Therefore, the mapping \(f:A\rightarrow A\) is a \(C^{*}\)-ternary derivation. This completes the proof. □
Corollary 4.3
Let \(r<1\), \(s<2\), and θ be non-negative real numbers and let \(f:A \rightarrow A\) be a mapping satisfying (18) and
for all \(x_{1}, y_{1}, z_{1},\ldots,x_{k},y_{k},z_{k} \in A\). Then the mapping \(f:A \rightarrow A\) is a \(C^{*}\)-ternary derivation.
Proof
Define
and
for all \(x_{1}, y_{1}, z_{1},\ldots,x_{k},y_{k},z_{k} , x_{11},\ldots,x_{1p},y_{11},\ldots,y_{1d},\ldots,x_{k1},\ldots ,x_{kp},y_{k1},\ldots,y_{kd}\in A\) and applying Theorem 4.2, we get the desired result. □
Theorem 4.4
Let \((( {A}^{k},\|\cdot\|_{k}): k\in\mathbf{N})\) be a multi-\(C^{*}\)-ternary algebra. Let \(f: A \rightarrow A\) be a mapping for which there are the functions \(\varphi: A^{(p+d)k} \rightarrow[0, \infty)\) and \(\psi: A^{3k} \rightarrow[0, \infty)\) satisfying the inequalities (2), (11), (12), and (32) for all \(\mu\in\mathbf{T}^{1}\) and \(x_{11},\ldots ,x_{1p},y_{11},\ldots, y_{1d},\ldots,x_{k1}, \ldots,x_{kp},y_{k1}, \ldots,y_{kd}, x_{1},\ldots,x_{k},y_{1},\ldots,y_{k},z_{1},\ldots,z_{k}\in A\), where \(\gamma=\frac{p+2d}{2}\). If there exists the constant \(L<1\) such that
for all \(x_{1},x_{2},\ldots,x_{k} \in A\), then there exists a unique homomorphism \(\delta: A \rightarrow A\) such that
for all \(x_{1},\ldots,x_{k} \in A\).
Proof
The same reasoning as in the proof of Theorem 3.4 guarantees there exists a unique C-linear mapping \(\delta :A\rightarrow A\) satisfying (32). The rest of the proof is similar to the proof of Theorem 4.2 and so we omit it. □
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Alsulami, H.H., Kenari, H.M., O’Regan, D. et al. Multi-\(C^{*}\)-ternary algebras and applications. J Inequal Appl 2015, 223 (2015). https://doi.org/10.1186/s13660-015-0746-9
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DOI: https://doi.org/10.1186/s13660-015-0746-9