Abstract
In this paper, we prove that each n-Jordan homomorphism \(\varphi \) from Banach algebra \(\mathcal {A}\) into a semisimple commutative Banach algebra \(\mathcal {B}\) is automatically continuous. Some useful results about characterization of n-Jordan homomorphisms and interesting examples of them on Banach algebras are given as well.
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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 \(a_1, a_2, ...a_n\in \mathcal {A}\),
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
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
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
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
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
where \(Y^T\) denote the transpose of Y. Then, for all \(A\in \mathcal {A}\) and for each \(n\in \mathbb {N}\),
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
where X is the form \(\begin{bmatrix} a_{11} &{}\quad 0\\ a_{21} &{}\quad a_{22} \end{bmatrix} \), and let
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
Then,
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
and interchanging a by \(-a\) in (1), to obtain
Suppose that \((e_{\alpha })_{\alpha \in I}\) is a bounded approximate identity for \(\mathcal {A}\), and let
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
for all \(a\in \mathcal {A}\). Replacing a by \(a+e_{\alpha }\) in (4), we arrive at
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,
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
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
Since \(f\in \mathfrak {M}(\mathcal {B})\) was arbitrary and \(\mathcal {B}\) is assumed to be semisimple, we obtain
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 \(a,b,c\in \mathcal {A}\),
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
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
for all \(x_1,x_2\in \mathfrak {A}\). Since \(\psi (a)=1\), by (7), we have
Let \(\lambda =\psi (a^{n-1})\), then
and since \(\psi \ne 0\) we obtain
for all \(x_2\in \mathfrak {A}\). From (6) and (8) we get
for all \(x_1,x_2\in \mathfrak {A}\). Continuing in this way, we conclude that
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 \(x,y\in \mathfrak {A}\),
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 \(a\in \mathcal {A}\),
-
(i)
\(\varphi (a)=\beta ^{n-1} \varphi (a)\), where \(\beta =\lim \limits _{\alpha }\varphi (e_\alpha )\), and
-
(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
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
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.
References
An, G.: Characterization of \(n\)-Jordan homomorphism. Linear Multi. Alge. 66(4), 671–680 (2018)
Bodaghi, A., Inceboz, H.: \(n\)-Jordan homomorphisms on commutative algebras. Acta. Math. Univ. Comenianae. 87(1), 141–146 (2018)
Bodaghi, A., Shojaee, B.: \(n\)-Jordan homomorphisms on \(C^*\)-algebras. J. Linear Topological Alge. 1, 1–7 (2012)
Bonsall, F.F., Duncan, J.: Complete normed algebra. Springer-Verlag, New York (1973)
Brac̆ic̆, J., Moslehian, M.S.: On automatic continuity of \(3\)-homomorphisms on Banach algebras. Bull. Malaysian Math. Sci. Soc. 30(2), 195–200 (2007)
Bres̆ar, M.: Characterizing homomorphisms; derivations and multipliers in rings with idempotents. Proc. Roy. Soc. Edinburgh Sect., A 137, 9–21 (2007)
Dales, H.G.: Banach Algebras and Automatic Continuity, LMS Monographs 24. Clarenden Press, Oxford (2000)
Eshaghi Gordji, M.: \(n\)-Jordan homomorphisms. Bull. Aust. Math. Soc. 80(1), 159–164 (2009)
Gselmann, E.: On approximate \(n\)-Jordan homomorphisms. Ann. Math. Sil. 28, 47–58 (2014)
Hejazian, Sh., Mirzavaziri, M., Moslehian, M.S.: \(n\)-homomorphisms. Bull. Iranian Math. Soc. 31(1), 13–23 (2005)
Herstein, I.N.: Jordan homomorphisms. Trans. Amer. Math. Soc. 81(1), 331–341 (1956)
Jacobson, N., Rickart, C.E.: Jordan homomorphisms of rings. Trans. Amer. Math. Soc. 69(3), 479–502 (1950)
Miura, T., Takahasi, S.E., Hirasawa, G.: Hyers-Ulam-Rassias stability of Jordan homomorphisms on Banach algebras. J. Ineq. Appl. 2005(4), 435–441 (2005)
Palmer, T.: Banach algebras and the general theory \(C^*\)-algebras, vol. I. Univ Press, Cambridge (1994)
Zelazko, W.: A characterization of multiplicative linear functionals in complex Banach algebras. Studia Math. 30, 83–85 (1968)
Zivari-kazempour, A.: A characterization of \(3\)-Jordan homomorphisms on Banach algebras. Bull. Aust. Math. Soc. 93(2), 301–306 (2016)
Zivari-kazempour, A.: A characterization of Jordan and \(5\)-Jordan homomorphisms between Banach algebras. Asian Eur. J. Math. 11(2), 1–10 (2018)
Zivari-Kazempour, A.: Automatic continuity of \(n\)-Jordan homomorphisms on Banach algebras. Commun. Korean Math. Soc. 33(1), 165–170 (2018)
Zivari-Kazempour, A.: Characterization of \(n\)-Jordan homomorphisms and automatic continuity of \(3\)-Jordan homomorphisms on Banach algebras. Iran. J. Sci. Technol. Trans. Sci. 44(1), 213–218 (2020)
Zivari-Kazempour, A., Valaei, M.: Characterization of \(n\)-Jordan homomorphisms on rings. Tamkang J. Math. 53(1), 89–96 (2022)
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Zivari-Kazempour, A. Characterization of n-Jordan homomorphisms and their automatic continuity on banach algebras. Ann Univ Ferrara 69, 263–271 (2023). https://doi.org/10.1007/s11565-022-00425-6
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DOI: https://doi.org/10.1007/s11565-022-00425-6