1 Introduction and Preliminaries

Let \(\mathcal {A}\) and \(\mathcal {B}\) be complex Banach algebras and \(\varphi :\mathcal {A}\longrightarrow \mathcal {B}\) be a linear map. Then \(\varphi \) is called an n-homomorphism if for all \(a_1, a_2, ...a_n\in \mathcal {A}\),

$$\begin{aligned} \varphi (a_1a_2...a_n)=\varphi (a_1)\varphi (a_2)...\varphi (a_n). \end{aligned}$$

The concept of n-homomorphisms was studied for complex algebras by Hejazian \(et \ al.\) in [10]. One may refer to [5], for certain properties of 3-homomorphisms.

A linear map \(\varphi \) between Banach algebras \(\mathcal {A}\) and \(\mathcal {B}\) is called an n-Jordan homomorphism if \(\varphi (a^n)=\varphi (a)^n\), for all \(a\in \mathcal {A}\). This notion was introduced by Herstein in [11]. For \(n = 2\), this concepts coincides the classical definitions of homomorphism and Jordan homomorphism, respectively. Moreover, Jordan homomorphism is equivalent by

$$\begin{aligned} \varphi (a\circ b)=\varphi (a)\circ \varphi (b), \ \ \ a,b\in \mathcal {A}, \end{aligned}$$

where \(a\circ b=ab+ba\).

It is obvious that each n-homomorphism is an n-Jordan homomorphism, but the converse is false, in general. In fact, the converse is true under certain conditions. For example, it is shown in [8] that each n-Jordan homomorphism between two commutative algebras is an n-homomorphism for \(n\in \{3,4\}\), and this result extended to \(n<8\), in [3]. Note that for \(n = 2\), the proof is clear.

The following more general result is due to Gselmann.

Theorem 1.1

[9, Theorem 2.1] Let \(n\in \mathbb {N}\), \(\mathcal {R}\) and \(\mathcal {R}'\) be two commutative rings such that char\((\mathcal {R}')>n\) and suppose that \(\varphi :\mathcal {R}\longrightarrow \mathcal {R}'\) is an n-Jordan homomorphism. Then \(\varphi \) is an n-homomorphism.

Moreover, if \(\mathcal {R}\) is unital, then \(\varphi (e)=\varphi (e)^{n}\) and the map \(\psi :\mathcal {R}\longrightarrow \mathcal {R}'\) defined by \(\psi (x)=\varphi (e)^{n-2}\varphi (x)\) is a homomorphism.

In 2018, Bodaghi and İnceboz proved that every additive n-Jordan homomorphism between two commutative algebras is an n-homomorphism [2]. However, their proof is different from that of Gselmann. We remark that since char\((\mathcal {B})>n\) for each algebra \(\mathcal {B}\), so Theorem 1.1 is stronger than the result of Bodaghi and İnceboz.

When the domain is not necessarily commutative, Żelazko in [15] proved the following theorem (see also [13]).

Theorem 1.2

Suppose that \(\mathcal {A}\) is a Banach algebra, which need not be commutative, and suppose that \(\mathcal {B}\) is a semisimple commutative Banach algebra. Then each Jordan homomorphism \(\varphi :\mathcal {A}\longrightarrow \mathcal {B}\) is a homomorphism.

This result has been proved by the author in [16] and [17] for 3-Jordan and 5-Jordan homomorphism with the additional hypothesis that the Banach algebra \(\mathcal {A}\) is unital. In other words, he presented the next theorem.

Theorem 1.3

Let \(n\in \lbrace 3,5\rbrace \) be fixed. Let \(\mathcal {A}\) be a unital Banach algebra, which need not be commutative, and \(\mathcal {B}\) be a semisimple commutative Banach algebra. Then each n-Jordan homomorphism \(\varphi :\mathcal {A}\longrightarrow \mathcal {B}\) is an n-homomorphism.

Later in 2017, An extended Theorem 1.3 and obtained the next result (for alternative proof see [20, Corollary 2.3]).

Theorem 1.4

[1, Corollary 2.5] Let \(\mathcal {A}\) be a unital Banach algebra and \(\mathcal {B}\) be a semisimple commutative Banach algebra. Then each n-Jordan homomorphism \(\varphi :\mathcal {A}\longrightarrow \mathcal {B}\) is an n-homomorphism. Moreover, if \(\varphi \) is surjective, then \(\varphi \) is automatically continuous.

Recall that a bounded approximate identity for \(\mathcal {A}\) is a bounded net \((e_{\alpha })_{\alpha \in I}\) in \(\mathcal {A}\) such that \(ae_{\alpha }\longrightarrow a\) and \(e_{\alpha }a\longrightarrow a\), for all \(a\in \mathcal {A}\). For example, it is known that the group algebra \(L^1(G)\), for a locally compact group G, and \(C^*\)-algebras have a bounded approximate identity bounded by one [7].

In this paper we prove that each n-Jordan homomorphism from Banach algebra \(\mathcal {A}\) into a semisimple commutative Banach algebra \(\mathcal {B}\) is automatically continuous, and generalize Theorem 1.4, for non unital Banach algebras which equipped a bounded approximate identity. In the other word, the results presented here are indeed extensions and generalizations of the above mentioned results.

2 Characterization of 3-Jordan homomorphisms

The next example provided that we cannot assert that n-Jordan homomorphisms of rings are always n-homomorphisms.

Example 2.1

Let \(\mathcal {A}=K[x,y]\) be the polynomial ring in two independent in determinates over a field K of characteristic not two, and let \(\mathcal {B}=K[X,Y,Z]\) be the polynomial ring in the three elements X, Y, Z that satisfy the relations

$$\begin{aligned} YX=XY+Z,\ \ \ \ \ XZ=ZX,\ \ \ \ YZ=ZY,\ \ \ \ \ Z^2=0. \end{aligned}$$

Then the linear mapping \(\varphi \) that sends \(x^my^n\) into \(\frac{1}{2}(X^m\circ Y^n)\), \(m,n=0,1,2,...\) is a Jordan homomorphism as is shown in [12, Example 1], hence it is an n-Jordan homomorphism by [12, Theorem 1], or [14, Lemma 6.3.2]. On the other hand, since

$$\begin{aligned} \big (\frac{1}{2}(X^{m_1}\circ Y^{n_1})\big )... \big (\frac{1}{2}(X^{m_k}\circ Y^{n_k})\big )\ne \frac{1}{2}\big (X^{m_1+...+m_k}\circ Y^{n_1+...+n_k}\big ), \end{aligned}$$

thus, \(\varphi \) is not n-homomorphism.

The commutativity of Banach algebra \(\mathcal {B}\) in Theorem 1.3 is essential. The following example illustrates this fact.

Example 2.2

Let

$$\begin{aligned} \mathcal {A}=\left\{ \begin{bmatrix} X &{}\quad 0\\ 0 &{}\quad Y \end{bmatrix}:\ \ \ X,Y\in M_2(\mathbb {C}) \right\} . \end{aligned}$$

Then under the usual matrix operations, \(\mathcal {A}\) is a unital and semisimple Banach algebra but it is not commutative. Define a continuous linear map \(\varphi :\mathcal {A}\longrightarrow \mathcal {A}\) by

$$\begin{aligned} \varphi \left( \begin{bmatrix} X &{}\quad 0 \\ 0 &{}\quad Y \end{bmatrix}\right) =\begin{bmatrix} X &{}\quad 0\\ 0 &{}\quad Y^T \end{bmatrix}, \end{aligned}$$

where \(Y^T\) denote the transpose of Y. Then, for all \(A\in \mathcal {A}\) and for each \(n\in \mathbb {N}\),

$$\begin{aligned} \varphi (A^n)=\varphi (A)^n. \end{aligned}$$

Thus, \(\varphi \) is an n-Jordan homomorphism, but it is not an n-homomorphism.

Note that in the above example \(\mathcal {A}\) is unital. Next we construct an example of n-Jordan homomorphism \(\varphi :\mathcal {A}\longrightarrow \mathcal {B}\), such that \(\mathcal {A}\) is not unital.

Example 2.3

Let

$$\begin{aligned} \mathcal {A}=\left\{ \begin{bmatrix} X &{}\quad Y\\ 0 &{}\quad 0 \end{bmatrix}:\ \ \ X,Y\in M_2(\mathbb {C}) \right\} , \end{aligned}$$

where X is the form \(\begin{bmatrix} a_{11} &{}\quad 0\\ a_{21} &{}\quad a_{22} \end{bmatrix} \), and let

$$\begin{aligned} \mathcal {B}=\left\{ \begin{bmatrix} z_{11} &{}\quad z_{12}\\ 0 &{}\quad z_{22} \end{bmatrix}:\ \ \ z_{ij}\in \mathbb {C} \right\} . \end{aligned}$$

Then \(\mathcal {A}\) is neither unital nor commutative and \(\mathcal {B}\) is a noncommutative semisimple Banach algebra. Define a continuous linear map \(\varphi :\mathcal {A}\longrightarrow \mathcal {B}\) by

$$\begin{aligned} \varphi \left( \begin{bmatrix} X &{}\quad Y \\ 0 &{}\quad 0 \end{bmatrix}\right) =X^T. \end{aligned}$$

Then,

$$\begin{aligned} \varphi (A^n)=\varphi (A)^n, \end{aligned}$$

for all \(A\in \mathcal {A}\). Consequently, \(\varphi \) is a n-Jordan homomorphism, but it is easy to see that \(\varphi \) is not an n-homomorphism.

To achieve our aim in this section, we need the following theorem.

Theorem 2.4

[19, Theorem 2.3] Suppose that \(\mathcal {A}\) is a Banach algebra. Then every 3-Jordan homomorphism \(\varphi :\mathcal {A}\longrightarrow \mathbb {C}\) is automatically continuous.

Next we generalize Theorem 1.3 for nonunital Banach algebras.

Theorem 2.5

Let \(\mathcal {A}\) be a Banach algebra with a bounded approximate identity. Then each 3-Jordan homomorphism \(\varphi :\mathcal {A}\longrightarrow \mathbb {C}\) is a 3-homomorphism.

Proof

Assume that \(\varphi \) is a 3-Jordan homomorphism, then \(\varphi (a^3)=\varphi (a)^3\), for all \(a\in \mathcal {A}\). Replacing a by \(a+b\), we get

$$\begin{aligned} \varphi (ab^2+b^2a+a^2b+ba^2+aba+bab)=3\varphi (a)^2\varphi (b)+3\varphi (a)\varphi (b)^2, \end{aligned}$$
(1)

and interchanging a by \(-a\) in (1), to obtain

$$\begin{aligned} \varphi (-ab^2-b^2a+a^2b+ba^2+aba-bab)=3\varphi (a)^2\varphi (b)-3\varphi (a)\varphi (b)^2. \end{aligned}$$
(2)

By (1) and (2),

$$\begin{aligned} \varphi (a^2b+ba^2+aba)=3\varphi (a)^2\varphi (b),\ \ \ \ (a,b\in \mathcal {A}). \end{aligned}$$
(3)

Suppose that \((e_{\alpha })_{\alpha \in I}\) is a bounded approximate identity for \(\mathcal {A}\), and let

$$\begin{aligned} E=\lbrace \varphi (e_{\alpha }):\ \ \alpha \in I\rbrace . \end{aligned}$$

Then we may suppose, by passing to a subnet, that \(\varphi (e_{\alpha })\longrightarrow \beta \in \mathbb {C}\). Replacing b by \(e_{\alpha }\) in (3) and using Theorem 2.4, we obtain

$$\begin{aligned} \varphi (a^2)=\beta \varphi (a)^2, \end{aligned}$$
(4)

for all \(a\in \mathcal {A}\). Replacing a by \(a+e_{\alpha }\) in (4), we arrive at

$$\begin{aligned} \varphi (a)=\beta \varphi (a)\lim _{\alpha }\varphi (e_{\alpha })=\beta ^2 \varphi (a), \end{aligned}$$
(5)

which proves that \(\beta ^2 =1\). Define \(\psi :\mathcal {A}\longrightarrow \mathbb {C}\) by \(\psi (x)=\beta \varphi (x)\), for all \(x\in \mathcal {A}\). Then \(\psi (x^2)=\beta \varphi (x^2)=\beta ^2\varphi (x)^2=\psi (x)^2\), therefore \(\psi \) is a Jordan homomorphism and hence by Theorem 1.2, \(\psi \) is a homomorphism. Thus,

$$\begin{aligned} \beta \varphi (abc)=\psi (abc)=\psi (a)\psi (b)\psi (c)=\beta ^3 \varphi (a)\varphi (b)\varphi (c), \end{aligned}$$

for all \(a,b,c\in \mathcal {A}\). Consequently, \(\varphi \) is a 3-homomorphism. \(\square \)

Corollary 2.6

Let \(\mathcal {A}\) be a Banach algebra with a bounded approximate identity and let \(\mathcal {B}\) be a semisimple commutative Banach algebra. Then each 3-Jordan homomorphism \(\varphi :\mathcal {A}\longrightarrow \mathcal {B}\) is a 3-homomorphism.

Proof

Let \(\mathfrak {M}(\mathcal {B})\) be the maximal ideal space of \(\mathcal {B}\). We associate with each \(f\in \mathfrak {M}(\mathcal {B})\) a function \(\varphi _f:\mathcal {A}\longrightarrow \mathbb {C}\) defined by

$$\begin{aligned} \varphi _f(a):=f(\varphi (a)),\ \ \ \ \ \ (a\in \mathcal {A}). \end{aligned}$$

Then \(\varphi _f\) is a 3-Jordan homomorphism, so by Theorem 2.5 it is a 3-homomorphism. Hence, by the definition of \(\varphi _f\) we have

$$\begin{aligned} f(\varphi (abc))=f(\varphi (a))f(\varphi (b))f(\varphi (c))=f(\varphi (a)\varphi (b)\varphi (c)). \end{aligned}$$

Since \(f\in \mathfrak {M}(\mathcal {B})\) was arbitrary and \(\mathcal {B}\) is assumed to be semisimple, we obtain

$$\begin{aligned} \varphi (abc)=\varphi (a)\varphi (b)\varphi (c), \end{aligned}$$

for all \(a,b,c\in \mathcal {A}\). This completes the proof. \(\square \)

For a Banach algebra \(\mathcal {A}\) without bounded approximate identity, the next result characterizes the 3-Jordan homomorphisms.

Theorem 2.7

Let \(\mathcal {A}\) be a Banach algebra and \(\varphi \) be a 3-Jordan homomorphism from \(\mathcal {A}\) into a commutative semisimple Banach algebra \(\mathcal {B}\) such that for all \(a,b,c\in \mathcal {A}\),

$$\begin{aligned} \varphi (abc-cba)=0. \end{aligned}$$

Then \(\varphi \) is a 3-homomorphism.

Proof

By a careful adaption of the methods of Theorem 1.1, the result follows. \(\square \)

3 Characterization of n-Jordan homomorphisms

It is shown in [19] that every n-Jordan homomorphism from unital Banach algebra \(\mathcal {A}\) into \(\mathbb {C}\) is automatically continuous, and without any extra condition asked the following: Is every n-Jordan homomorphism from \(\mathcal {A}\) into \(\mathbb {C}\) automatically continuous? ([19, Question 2.12]).

Next we answer this question in the affirmative. This result is the main key to characterize n-Jordan homomorphism. For the case \(n=2\), it is [18, Proposition 2.1], and for \(n=3\) it is Theorem 2.4.

Our main theorem in this section is the following.

Theorem 3.1

Every n-Jordan homomorphism \(\varphi \) from Banach algebra \(\mathcal {A}\) into \(\mathbb {C}\) is automatically continuous.

Proof

Let \(n\geqslant 4\) be fixed and let \(\varphi :\mathcal {A}\longrightarrow \mathbb {C}\) be an n-Jordan homomorphism. First we prove that for every \(a\in \mathcal {A}\) with \(\Vert a\Vert <1\), \(\varphi (a)\ne 1\). We argue by contradiction. Suppose that there exist \(a\in \mathcal {A}\) with \(\Vert a\Vert <1\) and \(\varphi (a)=1\). Thus, \(\varphi (a^n)=\varphi (a)^n=1\). Let \(\mathfrak {A}\) be a Banach subalgebra of \(\mathcal {A}\) generated by the above element a of norm \(\Vert a\Vert <1\). Define \(\psi :\mathfrak {A}\longrightarrow \mathbb {C}\) by \(\psi (x)=\varphi (x)\). Then \(\psi \) is an n-Jordan homomorphism, that is \(\psi (x^n)=\psi (x)^n\), for all \(x\in \mathfrak {A}\). Since \(\mathfrak {A}\) is commutative by Theorem 1.1, we have

$$\begin{aligned} \psi (x_1x_2...x_{n-1}x_n)=\psi (x_1)\psi (x_2)...\psi (x_{n-1})\psi (x_n), \end{aligned}$$
(6)

for all \(x_1,x_2,...,x_n\in \mathfrak {A}\). Replacing \(x_{n-1}\) by \(a^{n-1}\) and \(x_i\) by a for all \(i\geqslant 3\) with \(i\ne n-1\), in (6), gives

$$\begin{aligned} \psi (x_1x_2a^{2n-4})=\psi (x_1x_2a...a^{n-1}a)=\psi (x_1)\psi (x_2)\psi (a)...\psi (a^{n-1})\psi (a), \end{aligned}$$
(7)

for all \(x_1,x_2\in \mathfrak {A}\). Since \(\psi (a)=1\), by (7), we have

$$\begin{aligned} \psi (x_1x_2a^{2n-4})=\psi (x_1)\psi (x_2)\psi (a^{n-1}). \end{aligned}$$

Let \(\lambda =\psi (a^{n-1})\), then

$$\begin{aligned} \psi (x_1)\psi (a^{n-2}x_2)=\psi (x_1)\psi (a^{n-2}x_2)\psi (a)^{n-2}=\psi (x_1x_2a^{2n-4})=\lambda \psi (x_1)\psi (x_2), \end{aligned}$$

and since \(\psi \ne 0\) we obtain

$$\begin{aligned} \psi (a^{n-2}x_2)=\lambda \psi (x_2), \end{aligned}$$
(8)

for all \(x_2\in \mathfrak {A}\). From (6) and (8) we get

$$\begin{aligned} \psi (x_1)\psi (x_2)=\psi (x_1)\psi (x_2)\psi (a)^{n-2}=\psi (a^{n-2}x_1x_2)=\lambda \psi (x_1x_2), \end{aligned}$$

for all \(x_1,x_2\in \mathfrak {A}\). Continuing in this way, we conclude that

$$\begin{aligned} \psi (x_1x_2...x_{n-1}x_n)=\lambda ^{n-1} \psi (x_1)\psi (x_2)...\psi (x_{n-1})\psi (x_n). \end{aligned}$$

Therefore \(\lambda ^{n-1}=1\) and hence \(\vert \lambda \vert =1\). Now define \(f:\mathfrak {A}\longrightarrow \mathbb {C}\) by \(f(x)=\lambda ^{n-2}\psi (x)\). For all \(x,y\in \mathfrak {A}\),

$$\begin{aligned} f(x)f(y)= & {} \lambda ^{n-2}\psi (x)\lambda ^{n-2}\psi (y)\\= & {} \lambda ^{2n-4}\big (\lambda \psi (xy)\big )\\= & {} \lambda ^{n-1}\lambda ^{n-2}\psi (xy)\\= & {} \lambda ^{n-2}\psi (xy)\\= & {} f(xy). \end{aligned}$$

Thus, f is a multiplicative linear functional and hence by [4, Proposition 3, \(\S \ 16\)], it is continuous and \(\Vert f\Vert \leqslant 1\). This implies that \(\Vert \psi \Vert \leqslant 1\), and hence \(\psi \) is continuous. This is a contradiction with \(\Vert a\Vert <1\) and \(\psi (a)=1\). Consequently, for all \(a\in \mathcal {A}\) with \(\Vert a\Vert <1\), we have \(\varphi (a)\ne 1\).

Now it is easy to see that if \(x\in \mathcal {A}\) in such that \(\Vert x\Vert \leqslant 1\), then \(\vert \varphi (x)\vert \leqslant 1\). Therefore \(\varphi \) is norm decreasing and hence it is continuous. This finishes the proof. \(\square \)

As a consequence of preceding theorem, we get the next result.

Corollary 3.2

Suppose that \(\mathcal {A}\) and \(\mathcal {B}\) are two Banach algebras, where \(\mathcal {B}\) is semisimple and commutative. Then each n-Jordan homomorphism \(\varphi :\mathcal {A}\longrightarrow \mathcal {B}\) is continuous.

Corollary 3.3

Every n-homomorphism \(\varphi \) from a Banach algebra \(\mathcal {A}\) into a semisimple commutative Banach algebra \(\mathcal {B}\) is automatically continuous.

A well-known result due to \(\breve{S}\)ilov [7, Theorem 2.3.3] or [4, Theorem 8, \(\S \ 17\)], states that every homomorphism \(\varphi \) from Banach algebra \(\mathcal {A}\) into a semisimple commutative Banach algebra \(\mathcal {B}\) is automatically continuous. Corollary 3.3 is an extension of this result for all \(n\in \mathbb {N}\).

Lemma 3.4

Let \(\mathcal {A}\) be a Banach algebra with a bounded approximate identity \((e_\alpha )_{\alpha \in I}\), and \(\varphi :\mathcal {A}\longrightarrow \mathbb {C}\) be an n-Jordan homomorphism. Then for all \(a\in \mathcal {A}\),

  1. (i)

    \(\varphi (a)=\beta ^{n-1} \varphi (a)\), where \(\beta =\lim \limits _{\alpha }\varphi (e_\alpha )\), and

  2. (ii)

    \(\varphi (a^2)=\beta ^{n-2}\varphi (a)^2\).

Proof

The conclusion is proved for \(n=3\), as is done in the proof of Theorem 2.5. By applying Theorem 3.1 and the same method which has been used in [1, Theorem 2.4], we can prove the result for \(n>3\). \(\square \)

Theorem 3.5

Let \(\mathcal {A}\) be a Banach algebra with a bounded approximate identity. Then each n-Jordan homomorphism \(\varphi :\mathcal {A}\longrightarrow \mathbb {C}\) is an n-homomorphism.

Proof

Define a mapping \(\psi :\mathcal {A}\longrightarrow \mathbb {C}\) by

$$\begin{aligned} \psi (a)=\beta ^{n-2} \varphi (a), \end{aligned}$$

for all \(a\in \mathcal {A}\). It follows from Lemma 3.4, that \(\psi \) is a Jordan homomorphism and hence it is a homomorphism by Theorem 1.2. By the definition of \(\psi \) and Lemma 3.4, we have

$$\begin{aligned} \beta \psi (a)=\varphi (a). \end{aligned}$$
(9)

By Lemma 3.4 and (9), we have

$$\begin{aligned} \varphi (a_1a_2...a_n)= & {} \beta \psi (a_1a_2...a_n)\\= & {} \beta \psi (a_1)\psi (a_2)...\psi (a_n)\\= & {} \beta \big (\beta ^{n-2} \varphi (a_1)\big )\big (\beta ^{n-2} \varphi (a_2)\big )...\big (\beta ^{n-2} \varphi (a_n)\big ) \\= & {} \beta ^{(n-1)^2} \varphi (a_1)\varphi (a_2)...\varphi (a_n)\\= & {} \varphi (a_1)\varphi (a_2)...\varphi (a_n). \end{aligned}$$

Consequently, \(\varphi \) is an n-homomorphism. \(\square \)

From Theorem 3.5, we deduce the following result which generalize Corollary 2.5 of [1].

Corollary 3.6

Suppose that \(\mathcal {A}\) is a Banach algebra with a bounded approximate identity, and \(\mathcal {B}\) is a semisimple commutative Banach algebra. Then each n-Jordan homomorphism \(\varphi :\mathcal {A}\longrightarrow \mathcal {B}\) is an n-homomorphism.

As a consequence of Corollary 3.6, we have the following result.

Corollary 3.7

Let \(\mathcal {A}\) be an amenable Banach algebra or a \(C^*\)-algebra. Then each n-Jordan homomorphism \(\varphi \) from \(\mathcal {A}\) into a semisimple commutative Banach algebra \(\mathcal {B}\) is an n-homomorphism.