Characterizing Jordan embeddings between block upper-triangular subalgebras via preserving properties (2024)

Ilja Gogić, Tatjana Petek, Mateo TomaševićI.Gogić, Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatiailja@math.hrT.Petek, Faculty of Electrical Engineering and Computer Science, University of Maribor, Koroška cesta 46, SI-2000 Maribor, Slovenia;
Institute of Mathematics, Physics and Mechanics, Jadranska ulica 19, SI-1000 Ljubljana, Slovenia tatjana.petek@um.siM.Tomašević, Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatiamateo.tomasevic@math.hr

(Date: June 14, 2024)

Abstract.

We consider arbitrary block upper-triangular subalgebras $\mathcal{A}\subseteq M_{n}$ (i.e.subalgebras of $M_{n}$ which contain the algebra of upper-triangular matrices) and their Jordan embeddings. We first describe Jordan embeddings $\phi:\mathcal{A}\to M_{n}$ as maps of the form

\phi(X)=TXT^{-1}\qquad\mbox{or}\qquad\phi(X)=TX^{t}T^{-1},

where $T\in M_{n}$ is an invertible matrix, and then we obtain a simple criteria of when one block upper-triangular subalgebra Jordan-embeds into another (and in that case we describe the form of such embeddings). As a main result, we characterize Jordan embeddings $\phi:\mathcal{A}\to M_{n}$ (when $n\geq 3$ )as continuous injective maps which preserve commutativity and spectrum. We show by counterexamples that all these assumptions are indispensable (unless $\mathcal{A}=M_{n}$ when injectivity is superfluous).

Key words and phrases:

Jordan hom*omorphisms, spectrum preserver, commutativity preserver, upper-triangular matrices, matrix algebras, block upper-triangular algebras

2020 Mathematics Subject Classification:

47B49, 15A27, 16S50, 16W20

1. Introduction

Jordan algebras were first introduced by Pascual Jordan in 1933 in the context of quantum mechanics [21]. The majority of the practically relevant Jordan algebras naturally arise as subalgebras of an associative algebra $\mathcal{A}$ under a symmetric product given by

x\circ y=xy+yx.

This gave rise to the study of Jordan hom*omorphisms in the context of associative rings and algebras. Namely, recall that an additive (linear) map $\phi:\mathcal{A}\to\mathcal{B}$ between rings (algebras) $\mathcal{A}$ and $\mathcal{B}$ is a Jordan hom*omorphism if

\phi(a\circ b)=\phi(a)\circ\phi(b),\qquad\text{ for all }a,b\in\mathcal{A}.

When the rings (algebras) are $2$ -torsion-free, this is equivalent to

\phi(a^{2})=\phi(a)^{2},\qquad\text{ for all }a\in\mathcal{A}.

Clearly, multiplicative and antimultiplicative maps are immediate examples of such maps. One of the main problems in the context of Jordan hom*omorphisms is under which assumptions on rings (algebras) $\mathcal{A}$ and $\mathcal{B}$ , usually without $2$ -torsion, can we conclude that every Jordan hom*omorphism $\phi:\mathcal{A}\to\mathcal{B}$ (possibly satisfying some extra conditions such as surjectivity) is either multiplicative or antimultiplicative. More generally, the question is whether one can express all such Jordan hom*omorphisms as a suitable combination of ring (algebra) hom*omorphisms and antihom*omorphisms. This question goes a long way back. Namely, in 1950 Jacobson and Rickart [20] proved that a Jordan hom*omorphism from an arbitrary ring into an integral domain is either a hom*omorphism or an antihom*omorphism. This paper is particularly relevant for our discussion as it also proves that a Jordan hom*omorphism from the ring of $n\times n$ matrices, $n\geq 2$ , over an arbitrary unital ring is the sum of a hom*omorphism and an antihom*omorphism. In the same vein, Herstein [17] concludes that a Jordan hom*omorphism onto a prime ring is either a hom*omorphism or an antihom*omorphism. The same result was later refined by Smiley [32].

Let $M_{n}$ be the algebra of $n\times n$ matrices over the field of complex numbers. By combining the aforementioned result of Herstein with the well-known fact that all automorphisms of $M_{n}$ are inner (for a short and elegant proof see [31]), one obtains that all nonzero Jordan endomorphisms $\phi$ of $M_{n}$ are precisely maps of the form

(1.1)

\phi(X)=TXT^{-1}\qquad\text{or}\qquad\phi(X)=TX^{t}T^{-1}

(globally) for some invertible matrix $T\in M_{n}$ .

There have been many attempts to characterize Jordan hom*omorphisms, particularly on matrix algebras, using preserver properties. These attempts date back at least to 1970 and Kaplansky’s famous problem [22] which asks under which conditions on unital (complex) Banach algebras $A$ and $B$ is a linear unital map $\phi:A\to B$ which preserves invertibility necessarily a Jordan hom*omorphism. This problem received a lot of attention and progress was made in some special cases, but it is still widely open, even for $C^{*}$ -algebras (see [7], page 270). For other interesting types of linear preserver problems resulting in more general kinds of maps, we refer to the survey paper [23] and references within. We would also like to distinguish the following nonlinear preserver problem which elegantly characterizes Jordan automorphisms of $M_{n}$ :

Theorem 1.1 (Šemrl).

Let $\phi:M_{n}\to M_{n},n\geq 3$ be a continuous map which preserves commutativity and spectrum. Then there exists an invertible matrix $T\in M_{n}$ such that $\phi$ is of the form (1.1).

A precursor to this result was first formulated in [29] and it assumed its current form a decade later in [30]. It also serves as the main motivation for our investigation. Namely, we are interested in the following general problem:

Problem 1.2.

Find necessary and sufficient conditions on a (Jordan) subalgebra $\mathcal{A}$ of $M_{n}$ such that each Jordan automorphism of $\mathcal{A}$ , or more generally a Jordan embedding (monomorphism) $\phi:\mathcal{A}\to M_{n}$ , extends to a Jordan automorphism of $M_{n}$ . Additionally, for such $\mathcal{A}$ , characterize all such mappings $\phi$ via suitable preserving properties, similarly as in Theorem 1.1.

The first natural example to consider in the context of Problem 1.2 is the algebra $\mathcal{T}_{n}$ of $n\times n$ upper-triangular complex matrices. First of all, it is well-known that all Jordan automorphisms of $\mathcal{T}_{n}$ are of the form (1.1) for suitable invertible matrix $T\in M_{n}$ (see e.g.[27, Corollary 4]). The same holds true for all Jordan embeddings $\mathcal{T}_{n}\to M_{n}$ (a special case of our first result, Theorem 1.3). Also, Jordan automorphisms of $\mathcal{T}_{n}$ (as well as more general type of maps on $\mathcal{T}_{n}$ ) were characterized via both linear and nonlinear preserving properties by several authors (see e.g.[11, 12, 13, 14, 19, 24, 28, 37]). In particular, following Theorem 1.1, the second-named author [28] described all Jordan automorphisms of $\mathcal{T}_{n}$ as continuous spectrum and commutativity preserving surjective mappings $\mathcal{T}_{n}\to\mathcal{T}_{n}$ . Analyzing the main result of [28] it is easy to verify that the same holds true if instead of surjectivity one assumes injectivity, thus providing a positive solution for (both parts of) Problem 1.2 when $\mathcal{A}=\mathcal{T}_{n}$ .

Continuing in this vein, the next class of algebras we consider are subalgebras $\mathcal{A}$ of $M_{n}$ which contain $\mathcal{T}_{n}$ . As it turns out, these algebras are precisely the block upper-triangular subalgebras of $M_{n}$ (see [34]). More specifically, any such algebra is of the form

(1.2)

\mathcal{A}_{k_{1},\ldots,k_{r}}:=\begin{bmatrix}M_{k_{1},k_{1}}&M_{k_{1},k_{2%}}&\cdots&M_{k_{1},k_{r}}\\0&M_{k_{2},k_{2}}&\cdots&M_{k_{2},k_{r}}\\\vdots&\vdots&\ddots&\vdots\\0&0&\cdots&M_{k_{r},k_{r}}\end{bmatrix}

for some $r,k_{1},\ldots,k_{r}\in{\mathbb{N}}$ such that $k_{1}+\cdots+k_{r}=n$ . These algebras also appear in the literature as parabolic algebras (see e.g.[1, 34]). Note that the block upper-triangular algebras $\mathcal{A}_{1,n-1}$ and $\mathcal{A}_{n-1,1}$ are exactly (up to similarity) the unital strict subalgebras of $M_{n}$ of maximal dimension (see [1]). Our first introductory result verifies that block upper-triangular algebras indeed satisfy the desired extension property, providing an affirmative answer for the first part of Problem 1.2:

Theorem 1.3.

Let $\mathcal{A}\subseteq M_{n}$ be a block upper-triangular subalgebra and let $\phi:\mathcal{A}\to M_{n}$ be a Jordan embedding. Then there exists an invertible matrix $T\in M_{n}$ such that $\phi$ is of the form (1.1).

After providing the short proof of Theorem 1.3 (in Section §3) we also present simple criteria of when one block upper-triangular algebra (Jordan-)embeds into another (see Corollary 3.1 and 3.3). To put Theorem 1.3 in a wider context, there is a number of related (and somewhat more general) results concerning Jordan hom*omorphisms on certain classes of (block) upper-triangular rings and algebras. For example, an influential result by Beidar, Brešar and Chebotar states that every Jordan isomorphism of the algebra of upper-triangular matrices $\mathcal{T}_{n}(\mathcal{C}),n\geq 2$ over a $2$ -torsion-free commutative unital ring $\mathcal{C}$ without nontrivial idempotents onto an arbitrary $\mathcal{C}$ -algebra is necessarily multiplicative or antimultiplicative [4]. A generalization was given in [25] by removing the assumption that $\mathcal{C}$ has no nontrivial idempotents. These results were further developed by Benkovič in [5]; any Jordan hom*omorphism from $\mathcal{T}_{n}(\mathcal{C}),n\geq 2$ into an algebra $\mathcal{B}$ is a (so-called) near-sum of a hom*omorphism and an antihom*omorphism. We also mention papers [6, 15, 35, 36] which treat similar problems for Jordan hom*omorphisms between more general types of (block) triangular matrix rings.

Concerning block upper-triangular subalgebras in particular, the papers [3, 8, 9] (all sequels of the aforementioned paper [5]) describe Jordan hom*omorphisms of certain classes of more general algebras. As block upper-triangular algebras are (up to isomorphism) precisely finite-dimensional instances of nest algebras, papers [10, 26] are also relevant. Note that many of the mentioned results above assume that the image of Jordan hom*omorphisms are rings or algebras. In Theorem 1.3 we make no such assumption but this is obviously compensated by the fact that the codomain is restricted to $M_{n}$ , which incidentally also enables us to state the explicit form (1.1) of such maps.

After verifying that block upper-triangular algebras $\mathcal{A}$ satisfy the first part of Problem 1.2, the next step is to characterize Jordan embeddings $\phi:\mathcal{A}\to M_{n}$ via suitable preserver properties. Building upon both Theorem 1.1 and [28, Corollary 3], we arrived at the following theorem, which is also the main result of our paper:

Theorem 1.4.

Let $\mathcal{A}\subseteq M_{n},n\geq 3$ , be a block upper-triangular subalgebra and let $\phi:\mathcal{A}\to M_{n}$ be a continuous injective map which preserves commutativity and spectrum. Then $\phi$ is a Jordan embedding and hence of the form (1.1) for some invertible matrix $T\in M_{n}$ .

Moreover, if we additionally assume that the image of $\phi$ is contained in $\mathcal{A}$ , so that $\phi:\mathcal{A}\to\mathcal{A}$ , using the invariance of domain theorem we show that the spectrum preserving assumption can be further relaxed to spectrum shrinking (Corollary 4.4).

This paper is organized as follows. We begin by providing terminology and notation in Section §2 along with some preliminary technical results related to block upper-triangular algebras and Jordan hom*omorphisms. Section §3 contains the proof of Theorem 1.3 and its consequences regarding (Jordan) embeddings between two block upper-triangular subalgebras of $M_{n}$ . Section §4 is the core of the paper. It contains the proof of Theorem 1.4, which is nontrivial and contains most of the actual work. The proof is carried out by induction on $n$ . After proving Theorem 1.4 a few direct consequences are stated. Finally, in Section §5, we demonstrate the necessity of all assumptions of Theorem 1.4 via counterexamples.

2. Preliminaries

We begin this section by introducing some notation and terminology. First of all, for a unital algebra $\mathcal{A}$ by $\mathcal{A}^{\times}$ we denote the set of all invertible elements in $\mathcal{A}$ .

Let $n\in{\mathbb{N}}$ .

•
As usual, by $M_{n}:=M_{n}({\mathbb{C}})$ we denote the set of all $n\times n$ complex matrices and by $M_{m,n}$ ( $m\in{\mathbb{N}}$ ) the set of all $m\times n$ complex matrices.
•
For $A,B\in M_{n}$ we denote by $A\leftrightarrow B$ the fact that $A$ and $B$ commute, i.e. $AB=BA$ .
•
We say that the matrices $A,B\in M_{n}$ are orthogonal, and write $A\perp B$ , if $AB=BA=0$ . Furthermore, by $A^{\perp}$ we denote the set of all matrices in $M_{n}$ orthogonal to $A$ .
•
For $A,B\in M_{n}$ we write $A\circ B=AB+BA$ .
•
For $A\in M_{n}$ we denote the characteristic polynomial of $A$ by $p_{A}(x)=\det(A-xI)$ , by $R(A)$ the image of $A$ and by $N(A)$ the nullspace of $A$ .
•
By $\mathcal{T}_{n}$ and $\mathcal{D}_{n}$ we denote the sets of all upper-triangular and diagonal matrices of $M_{n}$ , respectively.
•
By $\mathcal{A}_{k_{1},\ldots,k_{r}}$ we denote the block upper-triangular algebra (1.2) for some $r,k_{1},\ldots,k_{r}\in{\mathbb{N}}$ such that $k_{1}+\cdots+k_{r}=n$ .
•
By $\Delta$ we denote the diagonal matrix $\operatorname{diag}(1,\ldots,n)$ .

As usual, we will frequently identify vectors $x=(x_{1},\ldots,x_{n})\in{\mathbb{C}}^{n}$ as column-matrices

x=\begin{bmatrix}x_{1}\\\vdots\\x_{n}\end{bmatrix}

and $x^{t}=\begin{bmatrix}x_{1}&\cdots&x_{n}\end{bmatrix}$ as row-matrices.

As any matrix $A=(A_{ij})_{ij}\in M_{n}$ can be understood as a map $\{1,\ldots,n\}^{2}\to{\mathbb{C}},(i,j)\mapsto A_{ij}$ , we consider its support $\operatorname{supp}A$ as the set of all indices $(i,j)\in\{1,\ldots,n\}^{2}$ such that $A_{ij}\neq 0$ . We also say that $A$ is supported in a set $S\subseteq\{1,\ldots,n\}^{2}$ if $\operatorname{supp}A\subseteq S$ .

Lemma 2.1.

Let

J:=\begin{bmatrix}0&0&\cdots&0&1\\0&0&\cdots&1&0\\\vdots&\vdots&\ddots&\vdots&\vdots\\0&1&\cdots&0&0\\1&0&\cdots&0&0\\\end{bmatrix}\in M_{n}.

Then the map $\mathcal{A}_{k_{1},\dots,k_{r}}\to\mathcal{A}_{k_{r},\dots,k_{1}},X\mapsto JX^%{t}J$ is an algebra antiisomorphism.

Proof.

From $J^{2}=I$ and the fact that transposition is antimultiplicative we conclude that the (obviously injective and additive) map $\mathcal{A}_{k_{1},\dots,k_{r}}\to M_{n}$ is an algebra anti-hom*omorphism.It remains to show that $JX^{t}J\in\mathcal{A}_{k_{r},\ldots,k_{1}}$ for all $X\in\mathcal{A}_{k_{1},\ldots,k_{r}}$ . Indeed, for each $0\leq s\leq r-1$ , $1\leq i\leq k_{1}+\cdots+k_{s+1}$ and $k_{1}+\cdots+k_{s}+1\leq j\leq n$ we have

JE_{ij}^{t}J=E_{n+1-j,n+1-i}\in\mathcal{A}_{k_{r},\ldots,k_{1}}

since $k_{r}+\cdots+k_{s}+1\leq n+1-i\leq n$ and $1\leq n+1-j\leq k_{r}+\cdots+k_{s+1}$ .∎

This map is so ubiquitous that we will introduce a notation

X^{\odot}:=JX^{t}J.

Note that it actually corresponds to mirroring the matrix $X$ along its secondary diagonal. Also, for a block upper-triangular algebra $\mathcal{A}\subseteq M_{n}$ we denote by $\mathcal{A}^{\odot}\subseteq M_{n}$ the image of $\mathcal{A}$ under the map $X\mapsto X^{\odot}$ . Then $\mathcal{A}^{\odot}$ is the block upper-triangular algebra obtained from $\mathcal{A}$ by reversing the sizes of the diagonal blocks.

Lemma 2.2.

Let $A\in\mathcal{A}$ be a matrix with $n$ distinct eigenvalues. Then $A$ is diagonalizable within the algebra $\mathcal{A}$ . Furthermore, if $A\leftrightarrow E_{ss}$ for some $1\leq s\leq n$ , then the similarity matrix $T=[t_{ij}]\in\mathcal{A}^{\times}$ can be chosen to satisfy $T\leftrightarrow E_{ss}$ and $t_{ss}=1$ .

Proof.

If $\mathcal{A}=M_{n}$ , the first statement is well-known. So let $\mathcal{A}$ be a proper block upper-triangular subalgebra of $M_{n}$ . We prove the lemma by induction on $n$ . For $n=1$ the statement is clear, so suppose that the first statement of the lemma is true for any $1\leq m<n$ . The algebra $\mathcal{A}$ can be split so that any $A\in\mathcal{A}$ can be written

A=\begin{bmatrix}A_{11}&A_{12}\\0&A_{22}\end{bmatrix},

where $A_{11}\in M_{k_{1}}$ , $A_{12}\in M_{k_{1},n-k_{1}}$ and $A_{22}$ is a member of a block-upper-triangular subalgebra $\mathcal{A}_{2}$ of $M_{n-k_{1}}$ . By the induction hypothesis, there exist invertible matrices $S_{1}\in M_{k_{1}}$ and $S_{2}\in\mathcal{A}_{2}$ , diagonal matrices $D_{1}\in M_{k_{1}}$ , $D_{2}\in\mathcal{A}_{2}$ such that $A_{11}=S_{1}D_{1}S_{1}^{-1}$ and $A_{22}=S_{2}D_{2}S_{2}^{-1}$ and, we obtain that

A=\begin{bmatrix}S_{1}&0\\0&S_{2}\end{bmatrix}\begin{bmatrix}D_{1}&S_{1}^{-1}A_{12}S_{2}\\0&D_{2}\end{bmatrix}\begin{bmatrix}S_{1}^{-1}&0\\0&S_{2}^{-1}.\end{bmatrix}

Further we diagonalize the matrix in the middle. We aim to find the matrix $X$ such that

\begin{bmatrix}D_{1}&S_{1}^{-1}A_{12}S_{2}\\0&D_{2}\end{bmatrix}=\begin{bmatrix}I_{k_{1}}&X\\0&I_{n-k_{1}}\end{bmatrix}\begin{bmatrix}D_{1}&0\\0&D_{2}\end{bmatrix}\begin{bmatrix}I_{k_{1}}&X\\0&I_{n-k_{1}}\end{bmatrix}^{-1}=\begin{bmatrix}D_{1}&XD_{2}-D_{1}X\\0&D_{2}\end{bmatrix}.

By the Sylvester equation [18], since $\sigma(D_{1})\cap\sigma(D_{2})$ is empty, we have a unique solution $X$ satisfying $XD_{2}-D_{1}X=S_{1}^{-1}A_{12}S_{2}$ .

Remark 2.3.

Let $\phi:\mathcal{A}\to\mathcal{B}$ be a Jordan hom*omorphism between complex algebras $\mathcal{A}$ and $\mathcal{B}$ . We have:

(a)
$\phi(aba)=\phi(a)\phi(b)\phi(a)$ for all $a,b\in\mathcal{A}$ .
(b)
$\phi([[a,b],c])=[[\phi(a),\phi(b)],\phi(c)]$ for all $a,b,c\in\mathcal{A}$ .
(c)
$\phi([a,b]^{2})=[\phi(a),\phi(b)]^{2}$ for all $a,b\in\mathcal{A}$ .
(d)
Obviously $\phi$ preserves idempotents, i.e.for any idempotent $a\in\mathcal{A}$ , the element $\phi(a)$ is an idempotent in $\mathcal{B}$ .
(e)
If the image of $\phi$ has a trivial commutant in $\mathcal{B}$ (i.e.if $b\in\mathcal{B}$ commutes with every element of the image of $\phi$ , then $b$ is zero or a multiple of the unity in $\mathcal{B}$ , if it exists), then by (b) and (c) one easily sees that $\phi$ preserves commutativity.

(f)

Suppose $\mathcal{A}$ and $\mathcal{B}$ are algebras with unities $1_{\mathcal{A}}$ and $1_{\mathcal{B}}$ , respectively. Let $\phi:\mathcal{A}\to\mathcal{B}$ be a Jordan hom*omorphism such that $1_{\mathcal{B}}$ is in the image of $\phi$ . Then $\phi$ is unital and preserves inverses in the sense that for every $a\in\mathcal{A}^{\times}$ we have $\phi(a)\in\mathcal{B}^{\times}$ and $\phi(a^{-1})=\phi(a)^{-1}$ ([16, Thm 2.5], in Slovenian). Indeed, suppose that $a_{0}\in\mathcal{A}$ satisfies $\phi(a_{0})=1_{\mathcal{B}}$ . Then

2\cdot 1_{\mathcal{B}}=\phi(a_{0}+a_{0})=\phi(1_{\mathcal{A}}\circ a_{0})=\phi%(1_{\mathcal{A}})\circ\phi(a_{0})=\phi(1_{\mathcal{A}})\circ 1_{\mathcal{B}}=2%\phi(1_{\mathcal{A}})

which implies $\phi(1_{\mathcal{A}})=1_{\mathcal{B}}$ so $\phi$ is unital.Let $a\in\mathcal{A}^{\times}$ be arbitrary. Then (a) gives

\phi(a)=\phi(aa^{-1}a)=\phi(a)\phi(a^{-1})\phi(a).

If we set $p_{1}:=\phi(a)\phi(a^{-1})$ and $p_{2}:=\phi(a^{-1})\phi(a)$ , it is immediate that $p_{1}$ and $p_{2}$ are idempotents in $\mathcal{B}$ . Furthermore, we have

p_{1}+p_{2}=\phi(a)\circ\phi(a^{-1})=2\phi(a\circ a^{-1})=2\phi(1_{\mathcal{A}%})=2\cdot 1_{\mathcal{B}}

and hence $2\cdot 1_{\mathcal{B}}-p_{1}=p_{2}$ is idempotent as well. It follows that $2(p_{1}-1_{\mathcal{B}})=0$ so we conclude $p_{1}=1_{\mathcal{B}}$ and then $p_{2}=1_{\mathcal{B}}$ as well. Therefore, $\phi(a)$ is invertible in $\mathcal{B}$ , with the inverse being equal to $\phi(a^{-1})$ . In particular, all linear unital Jordan hom*omorphisms between unital algebras are spectrum preserving.

3. Block upper-triangular subalgebras and their Jordan embeddings

We start this section by proving Theorem 1.3.

Proof of Theorem 1.3.

We prove the theorem by induction on $n$ . For $n=1$ the statement is clear. So suppose it holds for all $k$ , $1\leq k<n$ , and let $\mathcal{A}\subseteq M_{n}$ be a block upper-triangular algebra. As $\phi$ is Jordan hom*omorphism, it preserves idempotents. For every distinct $i,j\in\{1,2,\ldots,n\}$ we have $0=\phi(E_{ii}\circ E_{jj})=\phi(E_{ii})\circ\phi(E_{jj})$ and so, $\phi(E_{ii})$ , $i=1,2,\dots,n$ , is a family of $n$ pairwise orthogonal idempotents. This follows from a general fact: if $P$ , $Q\in M_{n}$ are idempotents such that $P\circ Q=0$ , then $P\perp Q$ . Indeed, we have $PQ+QP=0$ , so by multiplying this by $P$ from both sides, one gets $PQP=0$ . As $QP=-PQ$ , this implies $PQ=-PQP=0$ . Then also $PQ=-QP=0$ . It follows that there exists an invertible matrix $S$ such that $\phi(E_{ii})=SE_{ii}S^{-1}$ for all $1\leq i\leq n$ . With no loss of generality we assume that $\phi(E_{11})=E_{11}$ and in turn, $SE_{11}=E_{11}S$ . For every matrix $A\in\mathrm{span}\{E_{ij}:2\leq i,j\leq n\}\cap\mathcal{A}$ we have that $A\circ E_{11}=0$ and hence $\phi(A)\circ E_{11}=0$ , which implies $\phi(A)\perp E_{11}$ . So, we can define a Jordan embedding $\psi:\mathcal{A}_{n-1}\to M_{n-1}$ , where $\mathcal{A}_{n-1}\subseteq M_{n-1}$ is a block upper-triangular subalgebra, by

\phi\left(\begin{bmatrix}0&0\\0&A\end{bmatrix}\right)=\begin{bmatrix}0&0\\0&\psi(A)\end{bmatrix}.

Now we apply the induction hypothesis to get that there exists an invertible $T\in M_{n-1}$ such that either $\psi(A)=TAT^{-1}$ for all $A\in\mathcal{A}_{n-1}$ or, $\psi(A)=TA^{t}T^{-1}$ for all $A\in\mathcal{A}_{n-1}$ . By passing to the map $\operatorname{diag}(1,T)^{-1}\phi(\,\cdot\,)\operatorname{diag}(1,T)$ , we can further take $T$ to be identity matrix in $M_{n-1}$ and therefore, $\phi$ fixes all diagonal matrices. In particular, $\phi(E_{ii})=E_{ii}$ , for all $1\leq i\leq n$ . By linearity, it suffices to check that there is an invertible (diagonal in fact by our last assumption) matrix $T\in M_{n}$ such that $\phi(E_{ij})=TE_{ij}T^{-1}$ for all $1\leq i,j\leq n$ or, $\phi(E_{ij})=TE_{ji}T^{-1}$ for all $1\leq i,j\leq n$ .

Suppose that $E_{ij}\in\mathcal{A}$ , $i\neq j$ . We claim that $\phi(E_{ij})=\alpha_{ij}E_{ij}+\beta_{ij}E_{ji}$ , where exactly one of $\alpha_{ij}$ and $\beta_{ij}$ is non-zero. When $n>2$ , we first use that for every $k\notin\{i,j\}$ , $0=\phi(E_{ij}\circ E_{kk})=\phi(E_{ij})\circ E_{kk}$ . So, for $n\geq 2$ we have $\phi(E_{ij})\in\mathrm{span}\{E_{ii},E_{ij},E_{ji},E_{jj}\}$ . From $0=\phi(E_{ij}\circ E_{ij})=2\phi(E_{ij})^{2}$ we see that $\phi(E_{ij})$ is a rank-one nilpotent as $\phi$ is injective.Now, the claim follows from

\phi(E_{ij})=\phi(E_{ij}\circ E_{ii})=\phi(E_{ij})\circ E_{ii},

since this means that $\phi(E_{ij})$ is supported in the $i$ -th row and column. Therefore, $\phi(E_{ij})_{jj}=0$ and hence the other eigenvalue $\phi(E_{ij})_{ii}$ also has to be zero.

So, further suppose $n\geq 3$ . If $\psi(A)=A$ for every $A\in\mathcal{A}_{n-1}$ then we claim that $\phi(E_{1j})=\alpha_{1j}E_{1j}$ for all $1<j\leq n$ . If not, first suppose that $\phi(E_{1k})=\beta_{1k}E_{k1}$ for some $1<k<n$ . Then $\phi(E_{1k}\circ E_{kn})=\phi(E_{1n})$ but $\phi(E_{1k})\circ\phi(E_{kn})=\beta_{1k}E_{k1}\circ E_{kn}=0$ , a contradiction. Secondly, the case $\phi(E_{1n})=\beta_{1n}E_{n1}$ can be eliminated by comparing $\phi(E_{1n}\circ E_{1,n-1})=0$ and $\phi(E_{1n})\circ\phi(E_{1,n-1})=\beta_{1n}E_{n1}\circ E_{1,n-1}=\beta_{1n}E_{%n,n-1}$ . By a diagonal similarity implemented by $\operatorname{diag}(1,\alpha_{12},\ldots,\alpha_{1n})\in\mathcal{D}_{n}^{\times}$ , we can achieve that $\phi(E_{1j})=E_{1j}$ for every $j=2,\dots,n$ .

If $\mathcal{A}=\mathcal{T}_{n}$ , we are done, else, for every $E_{j1}\in\mathcal{A}$ with $2\leq j\leq n$ , we have that $\phi(E_{1j}\circ E_{j1})=E_{11}+E_{jj}$ , so $E_{1j}\circ(\alpha_{j1}E_{j1}+\beta_{j1}E_{1j})=\alpha_{j1}(E_{11}+E_{jj})$ gives $\alpha_{j1}=1$ and hence $\beta_{j1}=0$ .

The anti-multiplicative case of $\psi$ can be considered by replacing the map $\phi$ with the map $\phi(\cdot)^{t}$ . Finally, the linearity of $\phi$ closes the proof.∎

Corollary 3.1.

Let $\mathcal{A}$ and $\mathcal{B}$ be block-upper-triangular subalgebras of $M_{n}$ . Suppose that $\phi:\mathcal{A}\to\mathcal{B}$ is a Jordan embedding. Then one of the following is true:

(a)
$\mathcal{A}\subseteq\mathcal{B}$ and there exists $T\in\mathcal{B}^{\times}$ such that $\phi(X)=TXT^{-1}$ for all $X\in\mathcal{A}$ ,
(b)
$\mathcal{A}^{\odot}\subseteq\mathcal{B}$ and there exists $T\in\mathcal{B}^{\times}$ such that $\phi(X)=TX^{\odot}T^{-1}$ for all $X\in\mathcal{A}$ .

Proof.

We will show that $E_{ij}\in\mathcal{A}$ implies $E_{ij}\in\mathcal{B}$ for all $E_{ij}\in\mathcal{A}$ or, $E_{ij}\in\mathcal{A}^{\odot}$ implies $E_{ij}\in\mathcal{B}$ for all $E_{ij}\in\mathcal{A}^{\odot}$ . We use mathematical induction again. There is nothing to do when $n=1$ , so for the induction step, assume $n\geq 2$ and that the claim already holds for $n-1$ . Let $\mathcal{A}=\mathcal{A}_{k_{1},\dots,k_{p}}$ and $\mathcal{B}=\mathcal{A}_{l_{1},\dots,l_{q}}$ .

Assume that $\phi$ is of the form $\phi(A)=TAT^{-1}$ for some fixed invertible matrix $T\in M_{n}$ . It is not difficult to see, that $\phi(E_{ii})=SE_{\pi(i)\pi(i)}S^{-1}$ for some invertible matrix $S\in\mathcal{B}$ and a permutation $\pi$ . Indeed, since $\phi(\Delta)=T\Delta T^{-1}$ , by Lemma 2.2, there exists an $S\in\mathcal{B}^{\times}$ such that $T\Delta T^{-1}=S\Delta S^{-1}$ . For every $1\leq i\leq n$ , $E_{ii}$ commutes with $\Delta$ and so, $\phi(E_{ii})=SE_{\pi(i)\pi(i)}S^{-1}$ for some permutation $\pi$ on the set $\{1,2,\dots,n\}$ .We may and we do suppose that $S=I$ . For each fixed $i$ , $1\leq i\leq k_{1}$ , $\operatorname{span}\{E_{ij}:j=1,\dots,n\}$ is an $n$ -dimensional vector space of rank-one matrices living in only one row, which is mapped into $\operatorname{span}\{E_{\pi(i),s}:s=1,\dots,n\}$ . As $\phi$ is one-to-one, $\pi(i)\in\{1,2,\dots,l_{1}\}$ where $l_{1}\geq k_{1}$ . Therefore, by a similarity induced by a permutation matrix in $\mathcal{B}$ we can achieve that all matrices supported only in the first row are mapped by $\phi$ to the matrices of the same kind, i.e. $\pi(1)=1$ with no loss of generality. For all $E_{i1}\in\mathcal{A}$ we have that $\phi(E_{i1})=E_{\pi(i)1}\in\mathcal{B}$ for every $1\leq i\leq k_{1}$ .

Now we can define a Jordan embedding $\psi:\mathcal{A}^{\prime}\to\mathcal{B}^{\prime}$ , where $\mathcal{A}^{\prime},\,\mathcal{B}^{\prime}\subseteq M_{n-1}$ are block upper-triangular algebras, by

\phi\left(\begin{bmatrix}0&0\\0&A\end{bmatrix}\right)=\begin{bmatrix}0&0\\0&\psi(A)\end{bmatrix}\in\mathcal{B},\ \ \ A\in\mathcal{A}^{\prime},

which, by induction hypothesis, gives that $\mathcal{A}^{\prime}\subseteq\mathcal{B}^{\prime}$ so, for every $E_{ij}\in\mathcal{A}$ we have that $E_{ij}\in\mathcal{B}$ for all $i,j$ such that $i,j\neq 1$ . By linearity and the fact that $\phi$ preserves also the matrices supported in the first row and those supported in the first column, we deduce $\mathcal{A}\subseteq\mathcal{B}$ . That $T\in\mathcal{B}^{\times}$ is now almost obvious.

For the remaining case, when $\phi$ is antimultiplicative, we may instead consider the multiplicative linear map $A\mapsto\phi(A^{\odot})$ , $A\in\mathcal{A}^{\odot}$ , which ends the proof by applying the above consideration.∎

The next simple example shows that the image of a Jordan embedding $\phi:\mathcal{A}\to M_{n}$ , where $\mathcal{A}$ is a block-upper-triangular subalgebra of $M_{n}$ , needs not to be a block-upper-triangular subalgebra of $M_{n}$ .

Example 3.2.

Consider the algebra monomorphism $\phi:\mathcal{T}_{3}\to M_{3}$ given by $\phi(X)=JXJ^{-1}$ where $J\in M_{3}^{\times}$ is from Lemma 2.1.The image of $\phi$ is precisely the algebra of $3\times 3$ lower-triangular matrices, hence not equal to $\mathcal{T}_{3}$ nor $\mathcal{T}_{3}^{\odot}=\mathcal{T}_{3}$ .

Corollary 3.3.

Let $\mathcal{A}_{k_{1},\ldots,k_{p}}$ and $\mathcal{A}_{l_{1},\ldots,l_{q}}$ be block upper-triangular subalgebras of $M_{n}$ .

(a)
$\mathcal{A}_{k_{1},\ldots,k_{p}}$ and $\mathcal{A}_{l_{1},\ldots,l_{q}}$ are algebra-isomorphic if and only if $(k_{1},\ldots,k_{p})=(l_{1},\ldots,l_{q})$ .
(b)
$\mathcal{A}_{k_{1},\ldots,k_{p}}$ and $\mathcal{A}_{l_{1},\ldots,l_{q}}$ are algebra-antiisomorphic if and only if $(k_{1},\ldots,k_{p})=(l_{q},\ldots,l_{1})$ .
(c)
$\mathcal{A}_{k_{1},\ldots,k_{p}}$ and $\mathcal{A}_{l_{1},\ldots,l_{q}}$ are Jordan-isomorphic if and only if $(k_{1},\ldots,k_{p})=(l_{1},\ldots,l_{q})$ or $(k_{1},\ldots,k_{p})=(l_{q},\ldots,l_{1})$ .

Proof.

This follows directly from Corollary 3.1.∎

4. Proof of the main result

We now proceed with the proof of our main result, namely Theorem 1.4. We begin with the following auxiliary fact:

Lemma 4.1.

Let $\mathcal{A}$ be a block upper-triangular subalgebra of $M_{n}$ and $\phi:\mathcal{A}\to M_{n}$ a continuous map satisfying $\sigma(A)\subseteq\sigma(\phi(A))$ for every matrix $A$ in some open (relative to $\mathcal{A}$ ) set $\,\mathcal{U}\subseteq\mathcal{A}$ . Then $p_{A}=p_{\phi(A)}$ for all $A\in\mathcal{U}$ , thus $\phi$ is spectrum preserving on $\,\mathcal{U}$ and in turn, the algebraic multiplicities of all eigenvalues are also preserved by $\phi$ .

Proof.

The assertion is clearly true on every set $\mathcal{E}\subset\mathcal{U}$ of all matrices with $n$ distinct eigenvalues.By applying Schur’s triangularization on each diagonal block, every matrix $A\in\mathcal{U}$ can be represented as $A=S(\Lambda+N)S^{-1}$ for some diagonal matrix $\Lambda$ , an invertible matrix $S\in\mathcal{A}$ and a strictly upper-triangular matrix $N$ . Define a family of matrices $A_{v}=S(\Lambda+E(v)+N)S^{-1}$ where $E(v)=\mathrm{diag}(v_{1},\dots,v_{n})$ , and the diagonal entries of $\Lambda+E(v)=\mathrm{diag}(\lambda_{1}+v_{1},\dots,\lambda_{n}+v_{n})$ are all distinct whenever ${\left\|v\right\|}={\left\|(v_{1},\dots,v_{n})\right\|}$ is sufficiently small.Clearly, $A=\lim_{v\to 0}A_{v}$ and by continuity of $\phi$ and continuity of determinant

	$\displaystyle p_{A}(x)$	$\displaystyle=\mathrm{det}(A-xI)=\lim_{v\to 0}\mathrm{det}(A_{v}-xI)$
		$\displaystyle=\lim_{v\to 0}\mathrm{det}(\phi(A_{v})-xI)=\mathrm{det}(\phi(A)-%xI)=p_{\phi(A)}(x)$

confirming the assertion of the Lemma.∎

We will need the following technical Lemma.

Lemma 4.2.

Every upper-triangular rank-one idempotent matrix $R\in\mathcal{T}_{n}$ can be written as $R=(e_{k}+x)(e_{k}+y)^{t}$ , where $x=\sum_{i=1}^{k-1}x_{i}e_{i}$ , $y=\sum_{i=k+1}^{n}y_{i}e_{i}$ for some $1\leq k\leq n$ . It will be convenient to represent $R$ as

R=T_{k}(x)S_{k}(y)^{-1}E_{kk}S_{k}(y)T_{k}(x)^{-1}

where

S_{k}(y)=I+e_{k}y^{t},\qquad\text{ and }T_{k}(x)=I+xe_{k}^{t}

are upper-triangular matrices satisfying $S_{k}(y)^{-1}=S_{k}(-y)$ and $T_{k}(x)^{-1}=T_{k}(-x)$ . Note that $T_{k}(x)e_{i}=e_{i}$ and $e_{i}^{t}S_{k}(y)=e_{i}^{t}$ when $i\neq k$ . Moreover,

(4.1)		$\displaystyle S_{k}(y)^{-1}E_{ii}S_{k}(y)$	$\displaystyle=\left\{\begin{matrix}E_{kk}+e_{k}y^{t},&\text{if}&i=k,\\E_{ii}-y_{i}E_{ki},&\text{if}&i>k,\\E_{ii},&\text{if}&i<k,\end{matrix}\right.$
(4.2)		$\displaystyle T_{k}(x)E_{ii}T_{k}(x)^{-1}$	$\displaystyle=\left\{\begin{matrix}E_{kk}+xe_{k}^{t},&\text{if}&i=k,\\E_{ii}-x_{i}E_{ik},&\text{if}&i<k,\\E_{ii},&\text{if}&i>k.\end{matrix}\right.$

Proof.

By direct computation and is therefore left to the reader.∎

Proof of Theorem 1.4.

Throughout the proof, let $\mathcal{A}:=\mathcal{A}_{k_{1},k_{2},\dots,k_{r}}$ , $r\geq 2$ , be fixed.Let $\phi:\mathcal{A}\to M_{n}$ be a continuous injective map which preserves commutativity and spectrum. By Lemma 4.1, we first conclude that $\phi$ also preserves the algebraic multiplicities of the eigenvalues.

For easier comprehension, the proof will now be divided into several steps.

Step 1.

There exists an invertible matrix $S\in\mathcal{A}$ such that $\phi(D)=SDS^{-1}$ , for all diagonal matrices $D\in\mathcal{A}$ .

Proof.

Since the matrix $\phi(\Delta)\in M_{n}$ is diagonalizable with eigenvalues $1,\ldots,n$ , there exists an $S\in M_{n}^{\times}$ such that $\phi(\Delta)=S\Delta S^{-1}$ . For any $D\in\mathcal{D}_{n}$ we have $\phi(D)\leftrightarrow\phi(\Delta)=S\Delta S^{-1}$ , since $D\leftrightarrow\Delta$ . So, $\phi(D)=S\widetilde{D}S^{-1}$ for some diagonal $\widetilde{D}\in M_{n}$ . By continuity of $\phi$ , the argument from [30, Lemma 2.1] gives that $\widetilde{D}=D$ , which confirms the claim.∎

Without loss of generality, we can assume further that $\phi$ is the identity on all diagonal matrices.

Step 2.

$\phi$ is a hom*ogeneous map. Moreover, if $A$ and $B$ are diagonalizable matrices in $\mathcal{A}$ , with the similarity matrix in $\mathcal{A}^{\times}$ , satisfying $AB=BA=0$ , then $\phi(A)\phi(B)=\phi(B)\phi(A)=0$ .

Proof.

For any $S\in\mathcal{A}^{\times}$ there exists a matrix $T_{S}\in M_{n}^{\times}$ such that $\phi(S\Delta S^{-1})=T_{S}\Delta T_{S}^{-1}$ . Applying Step 1 for the map $X\mapsto T_{S}^{-1}\phi(SXS^{-1})T_{S}$ provides that $\phi(SDS^{-1})=T_{S}DT_{S}^{-1}$ for every diagonal matrix $D$ . Replacing $D$ by $\alpha D$ , $\alpha\in\mathbb{C}$ gives that $\phi$ is hom*ogeneous on the set of all diagonalizable matrices in $\mathcal{A}$ . By density of such matrices (a consequence of Lemma 2.2) we conclude that $\phi$ is a hom*ogeneous map on $\mathcal{A}$ .

To prove the second assertion, we first show that any pair of matrices $A\perp B$ is simultaneously diagonalizable in $\mathcal{A}$ . Assume that $A=S_{1}D_{1}S_{1}^{-1}$ for some $S_{1}\in\mathcal{A}^{\times}$ and $D_{1}=\operatorname{diag}(d_{1},d_{2},\dots,d_{n})$ . If $A=0$ or $A$ is invertible, we are done. So, we can assume that the index set $\mathcal{J}=\{j;\ d_{j}\neq 0\}$ is a nonempty proper subset of $\{1,2,\dots,n\}$ . We observe that $S_{1}^{-1}BS_{1}\in\left(\cap_{j\in\mathcal{J}}E_{jj}^{\perp}\right)\cap%\mathcal{A}$ and, since it is diagonalizable, there exists a $S_{2}\in\mathcal{A}^{\times}$ commuting with all $E_{jj}$ , $j\in\mathcal{J}$ , such that $S_{2}^{-1}S_{1}^{-1}BS_{1}S_{2}=:D_{2}$ is diagonal and $S_{2}^{-1}S_{1}^{-1}AS_{1}S_{2}=D_{1}$ . Now, for $S:=S_{1}S_{2}\in\mathcal{A}^{\times}$ we have $A=SD_{1}S^{-1}$ and $B=SD_{2}S^{-1}$ .

To close the proof of this step, there exists an invertible matrix $T$ such that $\phi(S\Delta S^{-1})=T\Delta T^{-1}$ . It follows that $\phi(SD_{j}S^{-1})=TD_{j}T^{-1}$ , $j=1,2$ . Hence $\phi(A)\phi(B)=\phi(B)\phi(A)=0$ .∎

Step 3.

Fix a $k\in\{1,2,\dots,n\}$ . Let $S\in\mathcal{A}$ and $T\in M_{n}$ be invertible matrices such that $\phi(SE_{ii}S^{-1})=TE_{ii}T^{-1}$ for all $i\neq k$ . Then $\phi(SE_{kk}S^{-1})=TE_{kk}T^{-1}$ .

Proof.

We know that $\phi(S\Delta S^{-1})=P\Delta P^{-1}$ for some invertible $P\in M_{n}$ which implies that $\phi(SDS^{-1})=PDP^{-1}$ for every diagonal matrix $D$ . It follows that $TE_{ii}T^{-1}=PE_{ii}P^{-1}$ , $i\neq k$ and so,

\phi(SE_{kk}S^{-1})=PE_{kk}P^{-1}=I-\sum_{i\neq k}PE_{ii}P^{-1}=I-\sum_{i\neq k%}TE_{ii}T^{-1}=TE_{kk}T^{-1}.

∎

Step 4 (Base of induction, $n=3$ ).

Now we prove Theorem 1.4 completely for $n=3$ .

It suffices to prove the theorem for $\mathcal{A}=\mathcal{A}_{1,1,1}$ or $\mathcal{A}=\mathcal{A}_{1,2}$ . Indeed, the $M_{3}$ case is covered by Theorem 1.1, while if $\phi:\mathcal{A}_{2,1}\to M_{3}$ is a continuous injective commutativity and spectrum preserving map, then

X\mapsto\phi(X^{\odot}):\mathcal{A}_{1,2}\to M_{3}

is also such a map and thus the theorem follows from the $\mathcal{A}_{1,2}$ case.

Step 4.1 (Matrix units).

We show that we can, without loss of generality, assume that there exist constants $c_{ij}\in{\mathbb{C}}^{\times}$ such that

\phi(E_{ij})=c_{ij}E_{ij},\qquad\text{ for all matrix units }E_{ij}\in\mathcal%{A}.

We have $E_{12}\leftrightarrow E_{33}$ so by our assumption that $\phi$ fixes all diagonal matrices, we have $\phi(E_{12})\leftrightarrow\phi(E_{33})=E_{33}$ . By the fact that $\phi$ is injective and preserves spectrum, the matrix $\phi(E_{12})$ is a nonzero nilpotent so we conclude

\phi(E_{12})=\begin{bmatrix}ac&a^{2}&0\\-c^{2}&-ac&0\\0&0&0\end{bmatrix}

for some $a,c\in{\mathbb{C}}$ not both equal to $0$ . Analogously we get

\phi(E_{13})=\begin{bmatrix}bd&0&b^{2}\\0&0&0\\-d^{2}&0&-bd\end{bmatrix}

for some $b,d\in{\mathbb{C}}$ not both equal to $0$ . Since $E_{12}\leftrightarrow E_{13}$ we have

\begin{bmatrix}abcd&0&ab^{2}c\\-bc^{2}d&0&-b^{2}c^{2}\\0&0&0\\\end{bmatrix}=\phi(E_{12})\phi(E_{13})=\phi(E_{13})\phi(E_{12})=\begin{bmatrix%}abcd&a^{2}bd&0\\0&0&0\\-acd^{2}&-a^{2}d^{2}&0\\\end{bmatrix}.

If $a\neq 0$ , then $-a^{2}d^{2}=0$ implies $d=0$ so $b\neq 0$ . Now $ab^{2}c=0$ implies $c=0$ and therefore

\phi(E_{12})=a^{2}E_{12},\quad\phi(E_{13})=b^{2}E_{13}.

If however $c\neq 0$ , then $-b^{2}c^{2}=0$ implies $b=0$ so $d\neq 0$ . Now $a^{2}bd=0$ implies $a=0$ and therefore

\phi(E_{12})=-c^{2}E_{21},\quad\phi(E_{13})=-d^{2}E_{31}.

Without loss of generality, we can suppose that we are in the first case, as otherwise we can pass to the map $\phi(\cdot)^{t}$ .

As above, we obtain

\phi(E_{23})=\begin{bmatrix}0&0&0\\0&ef&e^{2}\\0&-f^{2}&-ef\end{bmatrix}

for some $e,f\in{\mathbb{C}}$ not both equal to $0$ .By using $E_{23}\leftrightarrow E_{13}$ it follows that $f=0$ and therefore $\phi(E_{23})=e^{2}E_{23}$ .

This settles the matter for $\mathcal{A}=\mathcal{T}_{3}$ . In the case of $\mathcal{A}=\mathcal{A}_{1,2}$ , similarly as above using $E_{32}\leftrightarrow E_{11},E_{12}$ we obtain $\phi(E_{32})=c_{32}E_{32}$ for some $c_{32}\in{\mathbb{C}}^{\times}$ .

Notice that

\operatorname{diag}(1,c_{12},c_{13})\phi(E_{1j})\operatorname{diag}(1,c_{12},c%_{13})^{-1}=E_{1j},\quad 2\leq j\leq 3.

Therefore, by passing to the map

\operatorname{diag}(1,c_{12},c_{13})\phi(\cdot)\operatorname{diag}(1,c_{12},c_%{13})^{-1}

without loss of generality we can further assume that $c_{12}=c_{13}=1$ .

Step 4.2 (Rank-one matrices).

We show that $\phi(A)=A$ for all non-nilpotent rank-one matrices $A\in\mathcal{A}\subset M_{3}$ .

When we refer to orthogonality in $\mathbb{C}^{3}$ we mean orthogonality with respect to the standard inner product, i.e. $x\perp y$ if and only if $y^{\ast}x=0\ (=x^{\ast}y)$ .

Every rank-one matrix in $\mathcal{A}$ is either of the form

(a)\,\begin{bmatrix}\ast&\ast&\ast\\0&0&0\\0&0&0\end{bmatrix}\qquad\text{or}\qquad(b)\,\begin{bmatrix}0&\ast&\ast\\0&\ast&\ast\\0&\ast&\ast\end{bmatrix}.

We start with case (a) and introduce

R(\alpha,x,y):=\begin{bmatrix}\alpha&x&y\\0&0&0\\0&0&0\end{bmatrix}.

After careful consideration of [29, Lemma 5], we note that the same arguments can be conducted entirely within the algebra $\mathcal{A}$ in order to obtain continuous functions $a_{12},a_{13}:{\mathbb{C}}^{2}\to{\mathbb{C}}$ such that

\phi(R(\alpha,x,y))=\begin{bmatrix}\alpha&a_{12}(\alpha,x)&a_{13}(\alpha,y)\\0&0&0\\0&0&0\end{bmatrix},\qquad\text{ for all $\alpha,x,y\in{\mathbb{C}}$},

with the property $a_{12}(0,x)=x$ and $a_{13}(0,y)=y$ for all $x,y\in{\mathbb{C}}$ . In particular, for all $x,y\in{\mathbb{C}}$ holds

	$\displaystyle\phi(R(0,x,y))$	$\displaystyle=\lim_{\alpha\to 0}\phi(R(\alpha,x,y))=\lim_{\alpha\to 0}R(\alpha%,a_{12}(\alpha,x),a_{13}(\alpha,y))$
		$\displaystyle=R(0,a_{12}(0,x),a_{13}(0,y))=R(0,x,y).$

However, the rest of the argument from [29, Lemma 5] is not applicable as it leaves the algebra $\mathcal{A}$ . Denote $R=R(1,x,y)$ . From Lemma 4.2 we can write $R=S(x,y)^{-1}E_{11}S(x,y)$ for $S(x,y)=I+e_{1}\left[0\ x\ y\right]$ . Then

S(x,y)^{-1}E_{22}S(x,y)=E_{22}-xE_{12}\ \text{ and }\ S(x,y)^{-1}E_{33}S(x,y)=%E_{33}-yE_{13}.

We see that $E_{22}-xE_{12}\perp E_{33}$ and it commutes with $E_{13}$ . Applying also the spectrum preserving property, we get that $\phi(E_{22}-xE_{12})=E_{22}-f_{12}(x)E_{12}$ for some complex-valued continuous function $f_{12}:{\mathbb{C}}^{\times}\to{\mathbb{C}}$ . Similarly, by applying orthogonality with $E_{22}$ and commuting with $E_{12}$ one gets that $\phi(E_{33}-yE_{13})=E_{33}-f_{13}(x)E_{13}$ for some continuous function $f_{13}$ . Then we have

\phi(E_{22}-xE_{12})=S(f_{12}(x),f_{13}(y))^{-1}E_{22}S(f_{12}(x),f_{13}(y))

and

\phi(E_{33}-yE_{13})=S(f_{12}(x),f_{13}(y))^{-1}E_{33}S(f_{12}(x),f_{13}(y)).

By Step 3 we get that

\phi(R)=S(f_{12}(x),f_{13}(y))^{-1}E_{11}S(f_{12}(x),f_{13}(y))=R(1,f_{12}(x),%f_{13}(y)).

Let us next consider the matrix $E_{33}-zE_{23}$ , being orthogonal to $E_{11}$ and commuting with $E_{12}+zE_{13}=\phi(E_{12}+zE_{13})$ . The matrix $E_{33}-zE_{23}$ commutes with $E_{21}$ which is not applicable. However, it commutes with $E_{12}+zE_{13}$ . So, there exist functions $p(z)$ and $q(z)$ , not both equal to zero at any non-zero $z$ , such that

\phi(E_{33}-zE_{23})=\begin{bmatrix}0&0&0\\0&-p(z)z&-q(z)z\\0&p(z)&q(z)\end{bmatrix}.

We use this observation with the relation $R(1,x,y)\perp E_{33}-y/xE_{23}$ and thus obtain that $\left[1\ f_{12}(x)\ f_{13}(y)\right]\left[0\ -y/x\ 1\right]^{t}=0$ giving $f_{12}(x)/x=f_{13}(y)/y=c$ for some nonzero constant $c$ . By continuity and hom*ogeneity of $\phi$ we have

c_{12}E_{12}=\phi(E_{12})=\lim_{\varepsilon\to 0}\phi(\varepsilon E_{11}+E_{12%})=\lim_{\varepsilon\to 0}\varepsilon\phi(R(1,1/\varepsilon,0))=\lim_{%\varepsilon\to 0}R(\varepsilon,c,0)=cE_{12}.

By our reduction above $c_{12}=c_{13}=1$ and hence $c=1$ which proves the case (a).

Let us proceed with the case (b). We can write any such rank-one (possibly nilpotent) matrix as $ab^{\ast}$ where $a$ , $b\in\mathbb{C}^{3}$ and $b\perp e_{1}$ . Suppose $a\notin\mathrm{span}\{e_{1}\}$ (otherwise, we are in the case (a)). We claim that $\phi(ab^{\ast})=uv^{\ast}\in M_{3}$ for some nonzero vectors $u$ , $v$ . Indeed, if $ab^{\ast}$ is non-nilpotent, then it is diagonalizable. We then apply that the multiplicities of all eigenvalues are preserved. If $ab^{\ast}$ is nilpotent, it can be considered as a limit of a rank-one non-nilpotent matrix. By lower semicontinuity, the rank can only decrease. Obviously, $ab^{\ast}$ commutes with $e_{1}(a^{\perp})^{\ast}$ for any vector $a^{\perp}$ orthogonal to $a$ . From (a) we already know that $\phi(e_{1}(a^{\perp})^{\ast})=e_{1}(a^{\perp})^{\ast}$ . So,

(4.3)

(v^{*}e_{1})u(a^{\perp})^{\ast}=uv^{\ast}e_{1}(a^{\perp})^{\ast}=e_{1}(a^{%\perp})^{\ast}uv^{\ast}=((a^{\perp})^{\ast}u)e_{1}v^{\ast}.

We claim that $v\perp e_{1}$ and $u\perp a^{\perp}$ . Suppose the contrary. Then, since both $u(a^{\perp})^{\ast}$ and $e_{1}v^{\ast}$ are nonzero, (4.3) shows that both $v^{*}e_{1}$ and $(a^{\perp})^{\ast}u$ are zero or both nonzero. If they are nonzero, we have $u\in\mathrm{span}\{e_{1}\}$ , which conflicts injectivity. Therefore, $v\perp e_{1}$ and $u\perp a^{\perp}$ and so, $u\in(\{a\}^{\perp})^{\perp}=\mathbb{C}a$ . Without loss of any generality, we can assume $u=a$ , and the scalar factor can be absorbed in $v$ . So far we have obtained that $\phi(ab^{\ast})=av^{\ast}\in\mathcal{A}$ for some vector $v\perp e_{1}$ which depends on $a$ and $b$ . On the other hand, $ab^{\ast}$ commutes with $(b^{\perp})c^{\ast}$ (which is possibly nilpotent), where $b^{\perp}\perp b$ and $c\in\{e_{1},a\}^{\perp}$ . From the previous argument, we know that $\phi(b^{\perp}c^{\ast})=b^{\perp}w^{\ast}$ for some $w\neq 0$ . Then $\phi(ab^{\ast})=av^{\ast}\leftrightarrow b^{\perp}w^{\ast}=\phi(b^{\perp}c^{%\ast})$ . This gives

(v^{\ast}b^{\perp})aw^{\ast}=av^{*}b^{\perp}w^{*}=b^{\perp}w^{*}av^{*}=(w^{%\ast}a)b^{\perp}v^{\ast}.

At this point assume that $ab^{\ast}$ is not nilpotent, i.e. $b$ is not orthogonal to $a$ . Then $b^{\perp}$ (defined up to a scalar factor) and $a$ are linearly independent, hence $v^{\ast}b^{\perp}=0$ . This implies that $v=\beta b$ for some scalar $\beta\neq 0$ . From the spectrum preserving property, comparing the traces, we get that $\beta=1$ , and we are done.

Step 4.3 (Conclusion of the base step).

$\phi(A)=A$ for all matrices $A\in\mathcal{A}\subset M_{3}$ .

By density, it suffices to prove that $\phi$ is the identity map on the set of matrices in $\mathcal{A}$ with $3$ distinct eigenvalues. By Lemma 2.2 for every matrix $A$ of this kind there exists a matrix $S\in\mathcal{A}^{\times}$ and a diagonal matrix $D$ such that $A=SDS^{-1}$ . Applying Step 1 for the map $X\mapsto\phi(SXS^{-1})$ we get that there exists a $T\in M_{3}^{\times}$ such that

\phi(SDS^{-1})=TDT^{-1},\quad\text{ for all }D\in\mathcal{D}_{3}.

By Step 4.2 we also have

SE_{jj}S^{-1}=\phi(SE_{jj}S^{-1})=TE_{jj}T^{-1},\qquad 1\leq j\leq 3,

so by linearity $TDT^{-1}=SDS^{-1}$ and consequently,

\phi(SDS^{-1})=TDT^{-1}=SDS^{-1}

for all diagonal matrices $D$ . We conclude that $\phi$ is the identity map.

Step 5 (Induction step).

By way of induction, suppose that $n\geq 4$ and that Theorem 1.4 holds for $n-1$ .

By Step 1, we can assume without loss of any generality, that $\phi(D)=D$ for any diagonal matrix $D$ and, in particular, $\phi(E_{ii})=E_{ii}$ for all $1\leq i\leq n$ .

Step 5.1.

There exists a diagonal matrix $T$ such that $\phi(A)=TAT^{-1}$ for all $A\in\mathcal{A}\cap E_{ss}^{\perp}$ , $1\leq s\leq n$ or, $\phi(A)=TA^{t}T^{-1}$ for all $A\in\mathcal{A}\cap E_{ss}^{\perp}$ , $1\leq s\leq n$ .

Proof.

By continuity of $\phi$ it suffices to consider only diagonalizable matrices. By Step 2, for any diagonalizable matrix $A\in\mathcal{A}\cap E_{ss}^{\perp}$ we have that $\phi(A)\in E_{ss}^{\perp}$ . By the induction hypothesis, for each $1\leq s\leq n$ there exists an invertible matrix $T_{s}\in M_{n}^{\times}$ commuting with $E_{ss}$ such that the map $\phi$ satisfies

(4.4)

\phi(A)=T_{s}AT_{s}^{-1},\ \ \ A\in\mathcal{A}\cap E_{ss}^{\perp},

\phi(A)=T_{s}A^{t}T_{s}^{-1},\ \ \ A\in\mathcal{A}\cap E_{ss}^{\perp}.

The matrix $T_{s}$ is diagonal by our assumption preceding this step, and so, let $T_{s}=\operatorname{diag}(t^{(s)}_{1},\dots,t^{(s)}_{n})$ . We can set the $s$ -th diagonal entry of $T_{s}$ to be equal to $1$ for every $s$ . We may and we do assume that $\phi$ is of the form (4.4) when $s=1$ .Replacing $\phi$ by a map $X\mapsto T_{1}^{-1}XT_{1}$ we can assume that $T_{1}=I$ . Let $s\neq 1$ . Applying that for every $A\in E_{11}^{\perp}\cap E_{ss}^{\perp}\cap\mathcal{A}$ holds $\phi(A)=T_{1}AT_{1}^{-1}=A$ and that $E_{11}^{\perp}\cap E_{ss}^{\perp}\cap\mathcal{A}$ contains at least $\mathrm{span}\{E_{jj},E_{kk},E_{jk}\}$ for some $1<j<k\leq n$ as $n\geq 4$ , we conclude that $\phi$ is of the form (4.4) on $E_{ss}^{\perp}\cap\mathcal{A}$ since otherwise the restriction of $\phi$ to $E_{11}^{\perp}\cap E_{ss}^{\perp}\cap\mathcal{A}$ would be multiplicative and anti-multiplicative simultaneously. Comparing the action of $\phi$ on different subspaces $E_{11}^{\perp}\cap E_{ss}^{\perp}\cap\mathcal{A}$ one observes that all $T_{s}$ , $s\neq 1$ , possibly differ from $T_{1}=I$ only in the first term. Additionally, for every $1<k,s\leq n$ , $k\neq s$ , we have

\phi(E_{1k})=T_{s}E_{1k}T_{s}^{-1}=\frac{t^{(s)}_{1}}{t^{(s)}_{k}}E_{1k}=t^{(s%)}_{1}E_{1k}

so, $t^{(s)}_{1}$ is in fact independent of $s$ . Hence $T_{2}=T_{3}=\dots=T_{n}=:T_{0}$ . Replacing $\phi$ by $X\mapsto T_{0}^{-1}\phi(X)T_{0}$ does not affect the action of $\phi$ on $E_{11}^{\perp}$ , so we have obtained that $\phi(E_{ij})=E_{ij}$ for every $E_{ij}\in\mathcal{A}$ . ∎

By the previous Step we assume that $\phi(A)=A$ for every matrix $A\in E_{ss}^{\perp}\cap\mathcal{A}$ , $s=1,2,\dots,n$ .

Step 5.2.

$\phi(R)=R$ for every rank-one matrix $R\in\mathcal{A}\subsetneq M_{n}$ .

Proof.

It suffices to consider only idempotent matrices $R$ due to the continuity and hom*ogeneity of $\phi$ .

We first show that $\phi(R)=R$ for every matrix $R$ supported in the first row. By Lemma 4.2, such a matrix can be written as $R=S_{1}(y)^{-1}E_{11}S_{1}(y)$ for some $y=\sum_{i=1}^{n}y_{i}e_{i}$ . Applying Step 3 we observe that for every $i\neq 1$ and $R_{i}:=S_{1}(y)^{-1}E_{ii}S_{1}(y)=E_{ii}-y_{i}E_{1i}\perp E_{nn}$ when $i\neq n$ and $R_{n}\perp E_{22}$ . Therefore, $\phi(R_{i})=R_{i}$ for every $i=2,\dots,n$ and thus, by Step 3, $\phi(R)=R$ . A very similar argument works in the case that $R$ is supported only in the last column as it can then be represented as $R=T_{n}(x)E_{nn}T_{n}(x)^{-1}$ for some $x=\sum_{i=1}^{n}x_{i}e_{i}$ .

Let further $R$ be supported in the first block-row but not in the first row. Then, for a block diagonal unitary matrix $U\in\mathcal{A}$ , satisfying $Ue_{n}=e_{n}=U^{\ast}e_{n}$ , we have $R=U^{\ast}S_{1}(y)^{-1}E_{11}S_{1}(y)U$ . Indeed, by Schur’s triangularization, there exists a unitary matrix $V\in M_{k_{1}}$ which upper-triangularizes the first block of $R$ such that the diagonal is exactly $1,0,\ldots,0$ . Then we can set $U:=\operatorname{diag}(V,I)\in\mathcal{A}$ . For $1<i\leq n$ consider

R_{i}:=U^{\ast}S_{1}(y)^{-1}E_{ii}S_{1}(y)U=U^{\ast}(E_{ii}-y_{i}E_{1i})U.

It is not difficult to see that for every $1<i<n$ , we have $R_{i}\perp E_{nn}$ , and so, $\phi(R_{i})=R_{i}$ . On the other hand, for $i=n$ , observe that $R_{n}$ is supported in the last column and hence, by the argument in the previous paragraph, $\phi(R_{n})=R_{n}$ . Then Step 3 closes this part. In the same spirit we handle the case when $R$ is supported in the last block-column. We are done if $\mathcal{A}=\mathcal{A}_{k_{1},k_{2}}$ .

Further, suppose that the number of diagonal blocks in $\mathcal{A}$ is greater than $2$ and that $R$ is supported in $k$ -th block-row or block-column for some $1<k<n$ . By unitary triangularization, we have that $R=U^{*}T_{k}S_{k}^{-1}E_{kk}S_{k}T_{k}^{-1}U$ for some $1<k<n$ , a unitary matrix block-diagonal matrix $U\in\mathcal{A}$ satisfying $Ue_{1}=e_{1}=U^{\ast}e_{1}$ and $Ue_{n}=e_{n}=U^{\ast}e_{n}$ , $S_{k}=S_{k}(y)$ and $T_{k}=T_{k}(x)$ are as in Lemma 4.2. Similarly as before, for $i\neq k$ consider

R_{i}:=U^{\ast}T_{k}S_{k}^{-1}E_{ii}S_{k}T_{k}^{-1}U.

For every $i<k$ , by a direct computation we obtain that $R_{i}\perp E_{nn}$ and so, $\phi(R_{i})=R_{i}$ . If $i>k$ , then $R_{i}\perp E_{11}$ and hence, $\phi(R_{i})=R_{i}$ . Finally Step 3 gives $\phi(R)=R$ as desired.∎

Step 6 (Conclusion of the inductive step).

$\phi(A)=A$ for all matrices $A\in\mathcal{A}$ .

Proof.

The verification of this step is essentially the same as in Step 6 for the $3\times 3$ case.∎

∎

Theorem 4.3.

Let $\mathcal{A},\mathcal{B}\subseteq M_{n}$ be two block upper-triangular algebras and let $\phi:\mathcal{A}\to\mathcal{B}$ be a continuous injective spectrum and commutativity preserving map. Then one of the following is true:

•
$\mathcal{A}\subseteq\mathcal{B}$ and there exists $T\in\mathcal{B}^{\times}$ such that $\phi(X)=TXT^{-1}$ for all $X\in\mathcal{A}$ .
•
$\mathcal{A}^{\odot}\subseteq\mathcal{B}$ and there exists $T\in\mathcal{B}^{\times}$ such that $\phi(X)=TX^{\odot}T^{-1}$ for all $X\in\mathcal{A}$ .

Proof.

Theorem 1.4 implies that $\phi$ is a Jordan embedding. The rest of the result follows from Corollary 3.1.∎

When we further assume that $\mathcal{B}=\mathcal{A}$ , so that $\phi:\mathcal{A}\to\mathcal{A}$ , we can relax the spectrum preserving assumption to spectrum shrinking ( $\sigma(\phi(X))\subseteq\sigma(X)$ for all $X\in\mathcal{A}$ ). More precisely, we obtain the following result (similarly as in [33], the proof relies on the invariance of domain theorem).

Corollary 4.4.

Let $\mathcal{A}\subseteq M_{n}$ be a block upper-triangular algebra and let $\phi:\mathcal{A}\to\mathcal{A}$ be a continuous injective commutativity preserving spectrum shrinking map. Then one of the following is true:

•
There exists $T\in\mathcal{A}^{\times}$ such that $\phi(X)=TXT^{-1}$ for all $X\in\mathcal{A}$ .
•
$\mathcal{A}^{\odot}=\mathcal{A}$ and there exists $T\in\mathcal{A}^{\times}$ such that $\phi(X)=TX^{\odot}T^{-1}$ for all $X\in\mathcal{A}$ .

Proof.

We shall prove that $\phi$ actually preserves the spectrum, and so Theorem 4.3 applies. By the invariance of domain theorem, the image $\mathcal{R}=\phi(\mathcal{A})$ is an open set in $\mathcal{A}$ and $\phi$ is a homeomorphism onto $\mathcal{R}$ . Then $\phi^{-1}:\mathcal{R}\to\mathcal{A}$ enlarges the spectrum: $\sigma(A)\subseteq\sigma(\phi^{-1}(A))$ . By Lemma 4.1 the characteristic polynomial is being preserved and hence, $\phi$ preserves the spectrum.∎

5. Counterexamples

We show the optimality of Theorem 1.4 via counterexamples. In short, all assumptions except injectivity are indispensable for all block upper-triangular algebras except $M_{n}$ , while injectivity is superfluous in the $M_{n}$ case and necessary in all other cases. The necessity of $n\geq 3$ is shown by examples from [29, Example 7] and [28, Theorem 4]. Therefore we assume $n\geq 3$ and let $\mathcal{A}\subseteq M_{n}$ be an arbitrary block upper-triangular algebra not equal to $M_{n}$ .

Example 5.1 (Injectivity, continuity and commutativity preserving is not enough).

Let $D$ be the open unit disk in ${\mathbb{C}}$ . Define a map $g:D\to D$ by

g(z)=\frac{1-3z}{3-z}.

It is easy to check that $g$ is a holomorphic bijection (actually, it is an involution). Consider the map $\phi:\mathcal{A}\to M_{n}$ given by

\phi(X)=g\left(\frac{X}{1+{\left\|X\right\|}}\right),

where ${\left\|\cdot\right\|}$ denotes the spectral norm. The map $\phi$ is well-defined, as for each $X\in\mathcal{A}$ the matrix $\frac{X}{1+{\left\|X\right\|}}$ has norm $<1$ and hence its spectrum is contained in $D$ at which point we can apply $g$ using the holomorphic functional calculus. Using the properties of the holomorphic functional calculus, we conclude that $\phi$ is continuous and preserves commutativity. Moreover, since the map $X\mapsto\frac{X}{1+{\left\|X\right\|}}$ is injective, via the application of $g^{-1}=g$ we conclude that $\phi$ is injective. However, $\phi$ is clearly not linear as

\phi(0)=g(0)=\frac{1}{3}I.

Example 5.2 (Commutativity preserving is necessary).

Consider the map $f:M_{n}\to M_{n}^{\times}$ given by

f(X)=\operatorname{diag}(e^{\det X},1,\ldots,1).

Then the map $\phi:\mathcal{A}\to M_{n}$ given by

\phi(X)=f(X)Xf(X)^{-1}

is clearly continuous and preserves the spectrum. Moreover, since $f$ is a similarity invariant and $\phi(X)$ is similar to $X$ , we can see that $\phi$ is bijective with the inverse $Y\mapsto f(Y)^{-1}Yf(Y)$ . However, $\phi$ is not linear:

\phi(I+E_{12})=I+eE_{12}\neq I+E_{12}=\phi(I)+\phi(E_{12}).

Example 5.3 (Continuity is necessary).

As in [29], consider the map $\phi:\mathcal{A}\to M_{n}$ given by

\phi(X)=\begin{cases}\operatorname{diag}(\lambda_{2},\lambda_{1},\ldots,%\lambda_{n}),\quad&\text{ if }X=\operatorname{diag}(\lambda_{1},\ldots,\lambda%_{n})\text{ and all $\lambda_{i}$ are distinct,}\\X,&\text{ otherwise}\end{cases}

which is bijective, spectrum and commutativity preserving but clearly not continuous.

Example 5.4 (Injectivity is necessary).

Consider the map $\phi:\mathcal{A}\to M_{n}$ given by

\phi\left(\begin{bmatrix}X_{11}&X_{12}&\cdots&X_{1n}\\0&X_{22}&\cdots&X_{2n}\\\vdots&\vdots&\ddots&\vdots\\0&0&\cdots&X_{rr}\end{bmatrix}\right)=\begin{bmatrix}X_{11}&0&\cdots&0\\0&X_{22}&\cdots&0\\\vdots&\vdots&\ddots&\vdots\\0&0&\cdots&X_{rr}\end{bmatrix}.

Then $\phi$ is clearly a unital Jordan hom*omorphism (and hence satisfies all assumptions of Theorem 1.4), but is not injective.

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Characterizing Jordan embeddings between block upper-triangular subalgebras via preserving properties (2024)

Abstract.

Key words and phrases:

2020 Mathematics Subject Classification:

1. Introduction

Theorem 1.1 (Šemrl).

Problem 1.2.

Theorem 1.3.

Theorem 1.4.

2. Preliminaries

Lemma 2.1.

Proof.

Lemma 2.2.

Proof.

Remark 2.3.

3. Block upper-triangular subalgebras and their Jordan embeddings

Proof of Theorem 1.3.

Corollary 3.1.

Proof.

Example 3.2.

Corollary 3.3.

Proof.

4. Proof of the main result

Lemma 4.1.

Proof.

Lemma 4.2.

Proof.

Proof of Theorem 1.4.

Step 1.

Proof.

Step 2.

Proof.

Step 3.

Proof.

Step 4 (Base of induction, n=3𝑛3n=3italic_n = 3).

Step 4.1 (Matrix units).

Step 4.2 (Rank-one matrices).

Step 4.3 (Conclusion of the base step).

Step 5 (Induction step).

Step 5.1.

Proof.

Step 5.2.

Proof.

Step 6 (Conclusion of the inductive step).

Proof.

Theorem 4.3.

Proof.

Corollary 4.4.

Proof.

5. Counterexamples

Example 5.1 (Injectivity, continuity and commutativity preserving is not enough).

Example 5.2 (Commutativity preserving is necessary).

Example 5.3 (Continuity is necessary).

Example 5.4 (Injectivity is necessary).

References

Step 4 (Base of induction, $n=3$ ).