#  $*$-Lie-type maps on alternative $*$-algebras

Aline Jaqueline de Oliveira Andrade  Aline Jaqueline de Oliveira Andrade,
Federal University of ABC, dos Estados Avenue, 5001, 09210-580, Santo André,
Brazil
E-mail<EMAIL_ADDRESS>,  Elisabete Barreiro  Elisabete
Barreiro, University of Coimbra, CMUC, Department of Mathematics, Apartado
3008 EC Santa Cruz 3001 – 501 Coimbra, Portugal
E-mail<EMAIL_ADDRESS>and  Bruno Leonardo Macedo Ferreira  Bruno Leonardo
Macedo Ferreira, Federal University of Technology, Avenida Professora Laura
Pacheco Bastos, 800, 85053-510, Guarapuava, Brazil
E-mail<EMAIL_ADDRESS>
###### Abstract.

Let ${\mathfrak{A}}\,$ and ${\mathfrak{A}}\,^{\prime}$ be two alternative
$*$-algebras with identities $1_{{\mathfrak{A}}\,}$ and
$1_{{\mathfrak{A}}\,^{\prime}}$, respectively, and $e_{1}$ and
$e_{2}=1_{{\mathfrak{A}}\,}-e_{1}$ nontrivial symmetric idempotents in
${\mathfrak{A}}\,$. In this paper we study the characterization of
multiplicative $*$-Lie-type maps. As application, we get a result on
alternative $W^{*}$-algebras.

Keywords: alternative $*$-algebra, alternative $W^{*}$-algebras.

2020 MSC: 17D05; 16W20.

The first author was supported by the Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior - Brasil (CAPES)-Finance 001. The second author was
supported by the Centre for Mathematics of the University of Coimbra -
UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES

##  1\. Introduction and Preliminaries

The study of additivity of maps have received a fair amount of attention of
mathematicians. The first quite surprising result is due to Martindale who
established a condition on a ring such that multiplicative bijective maps are
all additive [17]. Besides, over the years several works have been published
considering different types of associative and non-associative algebras among
them we can mention [3, 8, 9, 10, 11, 12, 13]. In order to add new ingredients
to the study of additivity of maps, many researches have devoted themselves to
the investigation of two new products, presented by Bre$\check{s}$ar and
Fo$\check{s}$ner in [2, 14], where the definition is as follows: for $a,b\in
R$, where $R$ is a $*-$ring, we denote by $\\{a,b\\}_{*}=ab+ba^{*}$ and
$[a,b]_{*}=ab-ba^{*}$ the $*$-Jordan product and the $*$-Lie product,
respectively. In [5], the authors proved that a map $\varphi$ between two
factor von Newmann algebras is a $*$-ring isomorphism if and only if
$\varphi(\\{a,b\\}_{*})=\\{\varphi(a),\varphi(b)\\}_{*}$. In [7], Ferreira and
Costa extended these new products and defined two other types of applications,
named multiplicative $*$-Jordan n-map and multiplicative $*$-Lie n-map and
used it to impose condition such that a map between $C^{*}$-algebras is a
$*$-ring isomorphism.

With this picture in mind, in this article we will discuss when a
multiplicative $*$-Lie $n$-map is a $*$-isomorphism in the case of alternative
$*$-algebras and, just as it was done in [6], we provide an application on
alternative $W^{*}$-algebras. Throughout the paper, the ground field is
assumed to be the field of complex numbers.

Let ${\mathfrak{A}}\,$ and ${\mathfrak{A}}\,^{\prime}$ be two algebras with
identities $1_{{\mathfrak{A}}\,}$ and $1_{{\mathfrak{A}}\,^{\prime}},$
respectively, and
$\varphi:{\mathfrak{A}}\,\rightarrow{\mathfrak{A}}\,^{\prime}$ a map. We have
the following concepts:

  * i.

$\varphi$ preserves product if $\varphi(ab)=\varphi(a)\varphi(b)$, for all
elements $a,b\in{\mathfrak{A}}\,$;

  * ii.

$\varphi$ preserves Lie product if $\varphi(ab-
ba)=\varphi(a)\varphi(b)-\varphi(b)\varphi(a)$, for any
$a,b\in{\mathfrak{A}}\,$;

  * iii.

$\varphi$ is additive if $\varphi(a+b)=\varphi(a)+\varphi(b)$, for any
$a,b\in{\mathfrak{A}}\,$;

  * iv.

$\varphi$ is isomorphism if $\varphi$ is a bijection additive that preserves
products and scalar multiplication;

  * v.

$\varphi$ is unital if
$\varphi(1_{{\mathfrak{A}}\,})=1_{{\mathfrak{A}}\,^{\prime}}.$

An algebra ${\mathfrak{A}}\,$ is called $*$-algebra if ${\mathfrak{A}}\,$ is
endowed with a involution. By involution, we mean a mapping
$*:{\mathfrak{A}}\,\rightarrow{\mathfrak{A}}\,$ such that
$(x+y)^{*}=x^{*}+y^{*}$, $(x^{*})^{*}=x$ and $(xy)^{*}=y^{*}x^{*}$ for all
$x,y\in{\mathfrak{A}}\,$. An element $s\in{\mathfrak{A}}\,$ satisfying
$s^{*}=s$ is called symmetric element of ${\mathfrak{A}}\,$.

Let ${\mathfrak{A}}\,$ and ${\mathfrak{A}}\,^{\prime}$ be two $*$-algebras and
$\varphi:{\mathfrak{A}}\,\rightarrow{\mathfrak{A}}\,^{\prime}$ a map. We have
the following definitions:

  * i.

$\varphi$ preserves involution if $\varphi(a^{*})=\varphi(a)^{*}$, for all
elements $a\in{\mathfrak{A}}\,$;

  * ii.

$\varphi$ is $*$-isomorphism if $\varphi$ is an isomorphism that preserves
involution;

  * iii.

$\varphi$ is $*$-additive if it preserves involution and it is additive.

######  Definition 1.1 (To see [7]).

Consider a $*$-algebra ${\mathfrak{A}}\,,$ we denote
$[x_{1},x_{2}]_{*}=x_{1}x_{2}-x_{2}x_{1}^{*},$ for all $x_{1},$
$x_{2}\in{\mathfrak{A}}\,$ and the sequence of polynomials,

$p_{1_{*}}(x)=x\,\,\text{and}\,\,p_{n_{*}}(x_{1},x_{2},\ldots,x_{n})=\left[p_{(n-1)_{*}}(x_{1},x_{2},\ldots,x_{n-1}),x_{n}\right]_{*},$

for all integers $n\geq 2$ and $x_{1},\ldots,x_{n}\in{\mathfrak{A}}\,.$

Thus,
$p_{2_{*}}(x_{1},x_{2})=\left[x_{1},x_{2}\right]_{*}=x_{1}x_{2}-x_{2}x_{1}^{*},$
for all $x_{1},$ $x_{2}\in{\mathfrak{A}}\,,$
$p_{3_{*}}(x_{1},x_{2},x_{3})=\left[\left[x_{1},x_{2}\right]_{*},x_{3}\right]_{*},$
for all $x_{1},$ $x_{2},$ $x_{3}\in{\mathfrak{A}}\,,$ etc. Note that
$p_{2_{*}}$ is the product introduced by Bre$\check{s}$ar and Fo$\check{s}$ner
[2, 14]. Then, using the nomenclature introduced in [7] we have a new class of
maps (not necessarily additive).

######  Definition 1.2.

Consider two $*$-algebras ${\mathfrak{A}}\,$ and ${\mathfrak{A}}\,^{\prime}$.
A map $\varphi:{\mathfrak{A}}\,\longrightarrow{\mathfrak{A}}\,^{\prime}$ is
multiplicative $*$-Lie $n$-map if

$\displaystyle\varphi(p_{n_{*}}(x_{1},x_{2},\ldots,x_{j},\ldots,x_{n}))=p_{n_{*}}(\varphi(x_{1}),\varphi(x_{2}),\ldots,\varphi(x_{j}),\ldots,\varphi(x_{n})),$

for all $x_{1},x_{2},\ldots,x_{n}\in{\mathfrak{A}}\,,$ where $n\geq 2$ is an
integer. Multiplicative $*$-Lie $2$-map, $*$-Lie $3$-map and $*$-Lie $n$-map
are collectively referred to as multiplicative $*$-Lie-type maps.

An algebra ${\mathfrak{A}}\,$ (not necessarily associative or commutative) is
called alternative algebra if it satisfies the identities $a^{2}b=a(ab)$ and
$ba^{2}=(ba)a$, for all elements $a,b\in{\mathfrak{A}}\,$. One easily sees
that any associative algebra is an alternative algebra. An alternative algebra
${\mathfrak{A}}\,$ is called prime if for any elements
$a,b\in{\mathfrak{A}}\,$ satisfying the condition $a{\mathfrak{A}}\,b=0$, then
either $a=0$ or $b=0$.

We consider an alternative algebra $\mathfrak{A}$ with identity
$1_{{\mathfrak{A}}\,}.$ Fix a nontrivial idempotent element
$e_{1}\in\mathfrak{A}$ and denote $e_{2}=1_{{\mathfrak{A}}\,}-e_{1}.$ It is
easy to see that $(e_{k}a)e_{j}=e_{k}(ae_{j})~{}(k,j=1,2)$ for all
$a\in\mathfrak{A}$. Then $\mathfrak{A}$ has a Peirce decomposition

$\mathfrak{A}=\mathfrak{A}_{11}\oplus\mathfrak{A}_{12}\oplus\mathfrak{A}_{21}\oplus\mathfrak{A}_{22},$

where $\mathfrak{A}_{kj}:=e_{k}\mathfrak{A}e_{j}$ $(k,j=1,2)$ (see [15]),
satisfying the following multiplicative relations:

  1. (i)

$\mathfrak{A}_{kj}\mathfrak{A}_{jl}\subseteq\mathfrak{A}_{kl}\ (k,j,l=1,2);$

  2. (ii)

$\mathfrak{A}_{kj}\mathfrak{A}_{kj}\subseteq\mathfrak{A}_{jk}\ (k,j=1,2);$

  3. (iii)

$\mathfrak{A}_{kj}\mathfrak{A}_{ml}=\\{0\\},$ if $j\neq m$ and
$(k,j)\neq(m,l),\ (k,j,m,l=1,2);$

  4. (iv)

$x_{kj}^{2}=0,$ for all $x_{kj}\in\mathfrak{A}_{kj}\ (k,j=1,2;~{}k\neq j).$

##  2\. Main theorem

In the following we shall prove a part of the main result of this paper.

######  Theorem 2.1.

Let ${\mathfrak{A}}\,$ and ${\mathfrak{A}}\,^{\prime}$ be two alternative
$*$-algebras with identities $1_{{\mathfrak{A}}\,}$ and
$1_{{\mathfrak{A}}\,^{\prime}},$ respectively, and $e_{1}$ and
$e_{2}=1_{{\mathfrak{A}}\,}-e_{1}$ nontrivial symmetric idempotents in
${\mathfrak{A}}\,$. Suppose that ${\mathfrak{A}}\,$ satisfies

(1) $(e_{j}{\mathfrak{A}}\,)x=\left\\{0\right\\}\mbox{ for any }j\in\\{1,2\\}\
\ \ \mbox{ implies }\ \ \ x=0$

Suppose also that
$\varphi:{\mathfrak{A}}\,\rightarrow{\mathfrak{A}}\,^{\prime}$ is a
multiplication bijective unital map which satisfies

(2)
$\varphi(p_{n_{*}}(a,b,\xi,\ldots,\xi))=p_{n_{*}}(\varphi(a),\varphi(b),\varphi(\xi),\ldots,\varphi(\xi)),\
$

for all $a,b\in{\mathfrak{A}}\,$ and
$\xi\in\left\\{e_{1},e_{2},1_{{\mathfrak{A}}\,}\right\\}$. Then $\varphi$ is
$*$-additive.

The following claims and lemmas have the same hypotheses as the Theorem 2.1
and we need them to prove the $*$-additivity of $\varphi$.

######  Claim 2.1.

$*({\mathfrak{A}}\,_{kj})\subset{\mathfrak{A}}\,_{jk}$, for $j,k\in\\{1,2\\}$.

###### Proof.

If $a_{kj}\in{\mathfrak{A}}\,_{kj}$ then

$a^{*}_{kj}=(e_{k}a_{kj}e_{j})^{*}=(e_{j})^{*}(a_{kj})^{*}(e_{k})^{*}=e_{j}(a_{kj})^{*}e_{k}\in{\mathfrak{A}}\,_{jk}.$

∎

It is easy to check the following result (see [6]).

######  Claim 2.2.

Let $x,y,h$ in ${\mathfrak{A}}\,$ such that
$\varphi(h)=\varphi(x)+\varphi(y)$. Then, given $z\in{\mathfrak{A}}\,$,

$\varphi(p_{n_{*}}(h,z,\xi,\ldots,\xi))=\varphi(p_{n_{*}}(x,z,\xi,\ldots,\xi))+\varphi(p_{n_{*}}(y,z,\xi,\ldots,\xi))$

and

$\varphi(p_{n_{*}}(z,h,\xi,\ldots,\xi))=\varphi(p_{n_{*}}(z,x,\xi,\ldots,\xi))+\varphi(p_{n_{*}}(z,y,\xi,\ldots\xi))$

for $\xi\in\left\\{e_{1},e_{2},1_{{\mathfrak{A}}\,}\right\\}$.

######  Claim 2.3.

$\varphi(0)=0$.

###### Proof.

Since $\varphi$ is surjective, there exists $x\in{\mathfrak{A}}\,$ such that
$\varphi(x)=0$. Then,

$\displaystyle\varphi(0)$
$\displaystyle=\varphi(p_{n_{*}}(0,x,1_{{\mathfrak{A}}\,},\ldots,1_{{\mathfrak{A}}\,}))=p_{n_{*}}(\varphi(0),\varphi(x),\varphi(1_{{\mathfrak{A}}\,}),\ldots,\varphi(1_{{\mathfrak{A}}\,}))$
$\displaystyle=p_{n_{*}}(\varphi(0),0,\varphi(1_{{\mathfrak{A}}\,}),\ldots,\varphi(1_{{\mathfrak{A}}\,}))=0.$

∎

The next results aim to show the additivity of $\varphi$.

######  Lemma 2.1.

For any $a_{11}\in{\mathfrak{A}}\,_{11}$ and $b_{22}\in{\mathfrak{A}}\,_{22}$,
we have

$\varphi(a_{11}+b_{22})=\varphi(a_{11})+\varphi(b_{22}).$

###### Proof.

Since $\varphi$ is surjective, given
$\varphi(a_{11})+\varphi(b_{22})\in{\mathfrak{A}}\,^{\prime}$ there exists
$h\in{\mathfrak{A}}\,$ such that $\varphi(h)=\varphi(a_{11})+\varphi(b_{22})$.
We may write $h=h_{11}+h_{12}+h_{21}+h_{22}$, with
$h_{jk}\in\mathfrak{A}_{jk}\ (k,j=1,2)$. Besides, by Claims 2.2 and 2.3

$\varphi(p_{n_{*}}(e_{1},h,e_{1},\ldots,e_{1}))=\varphi(p_{n_{*}}(e_{1},a_{11},e_{1},\ldots,e_{1}))+\varphi(p_{n_{*}}(e_{1},b_{22},e_{1},\ldots,e_{1})),$

that is,

$\varphi(-h_{21}+h_{21}^{*})=\varphi(0)+\varphi(0)=0.$

Then, by injectivity of $\varphi$, $-h_{21}+h_{21}^{*}=0$. Thus $h_{21}=0$.
Moreover,

$\varphi(p_{n_{*}}(e_{2},h,e_{2},\ldots,e_{2}))=\varphi(p_{n_{*}}(e_{2},a_{11},e_{2},\ldots,e_{2}))+\varphi(p_{n_{*}}(e_{2},b_{22},e_{2},\ldots,e_{2})),$

that is,

$\varphi(-h_{12}+h_{12}^{*})=0.$

Again, by injectivity of $\varphi$ we conclude that $h_{12}=0$.

Furthermore, given $d_{21}\in{\mathfrak{A}}\,_{21}$,

$\varphi(p_{n_{*}}(d_{21},h,e_{1},\ldots,e_{1}))=\varphi(p_{n_{*}}(d_{21},a_{11},e_{1},\ldots,e_{1}))+\varphi(p_{n_{*}}(d_{21},b_{22},e_{1},\ldots,e_{1})),$

that is,

$\varphi(d_{21}h_{11}-(d_{21}h_{11})^{*})=\varphi(d_{21}a_{11}-(d_{21}a_{11})^{*}).$

Then we conclude, by injectivity of $\varphi$, that
$d_{21}h_{11}-(d_{21}h_{11})^{*}=d_{21}a_{11}-(d_{21}a_{11})^{*}$, that is,
$d_{21}(h_{11}-a_{11})=0$. Even more,
$(e_{2}{\mathfrak{A}}\,)(h_{11}-a_{11})=0$, which implies that $h_{11}=a_{11}$
by Condition (1) of Theorem 2.1.

Finally, given $d_{12}\in{\mathfrak{A}}\,_{12}$, a similar calculation gives
us $h_{22}=b_{22}$. Therefore $h=a_{11}+b_{22}$. ∎

######  Lemma 2.2.

For any $a_{12}\in{\mathfrak{A}}\,_{12}$ and $b_{21}\in{\mathfrak{A}}\,_{21}$,
we have $\varphi(a_{12}+b_{21})=\varphi(a_{12})+\varphi(b_{21})$.

###### Proof.

Since $\varphi$ is surjective, given
$\varphi(a_{12})+\varphi(b_{21})\in{\mathfrak{A}}\,^{\prime}$ there exists
$h\in{\mathfrak{A}}\,$ such that $\varphi(h)=\varphi(a_{12})+\varphi(b_{21})$.
We may write $h=h_{11}+h_{12}+h_{21}+h_{22}$, with
$h_{jk}\in\mathfrak{A}_{jk}\ (k,j=1,2)$. Now, by Claims 2.2 and 2.3

$\varphi(p_{n_{*}}(e_{1},h,e_{1},\ldots,e_{1}))=\varphi(p_{n_{*}}(e_{1},a_{12},e_{1},\ldots,e_{1}))+\varphi(p_{n_{*}}(e_{1},b_{21},e_{1},\ldots,e_{1})),$

that is,

$\varphi(-h_{21}+h_{21}^{*})=\varphi(-b_{21}+b_{21}^{*}).$

Then, by injectivity of $\varphi$, $-h_{21}+h_{21}^{*}=-b_{21}+b_{21}^{*}$.
Thus $h_{21}=b_{21}$. Moreover,

$\varphi(p_{n_{*}}(e_{2},h,e_{2},\ldots,e_{2}))=\varphi(p_{n_{*}}(e_{2},a_{12},e_{2},\ldots,e_{2}))+\varphi(p_{n_{*}}(e_{2},b_{21},e_{2},\ldots,e_{2})),$

that is,

$\varphi(-h_{12}+h_{12}^{*})=\varphi(-a_{12}+a_{12}^{*}).$

Again, by injectivity of $\varphi$ we conclude that $h_{12}=a_{12}$.

Furthermore, given $d_{21}\in{\mathfrak{A}}\,_{21}$,

$\displaystyle\varphi(d_{21}h_{11}-(d_{21}h_{11})^{*})$
$\displaystyle=\varphi(p_{n_{*}}(d_{21},h,e_{1},\ldots,e_{1}))$
$\displaystyle=\varphi(p_{n_{*}}(d_{21},a_{12},e_{1},\ldots,e_{1}))+\varphi(p_{n_{*}}(d_{21},b_{21},e_{1},\ldots,e_{1}))=0.$

Then we conclude, by injectivity of $\varphi$, that
$d_{21}h_{11}-(d_{21}h_{11})^{*}=0$, that is, $d_{21}h_{11}=0$. Even more,
$(e_{2}{\mathfrak{A}}\,)h_{11}=0$, which implies that $h_{11}=0$ by Condition
(1) of Theorem 2.1.

Finally, given $d_{12}\in{\mathfrak{A}}\,_{12}$, a similar calculation gives
us $h_{22}=0$. Therefore, we conclude that $h=a_{12}+b_{21}$. ∎

######  Lemma 2.3.

For any $a_{11}\in{\mathfrak{A}}\,_{11}$, $b_{12}\in{\mathfrak{A}}\,_{12}$,
$c_{21}\in{\mathfrak{A}}\,_{21}$ and $d_{22}\in{\mathfrak{A}}\,_{22}$ we have

$\varphi(a_{11}+b_{12}+c_{21}+d_{22})=\varphi(a_{11})+\varphi(b_{12})+\varphi(c_{21})+\varphi(d_{22}).$

###### Proof.

Since $\varphi$ is surjective, given
$\varphi(a_{11})+\varphi(b_{12})+\varphi(c_{21})+\varphi(d_{22})\in{\mathfrak{A}}\,^{\prime}$
there exists $h\in{\mathfrak{A}}\,$ such that
$\varphi(h)=\varphi(a_{11})+\varphi(b_{12})+\varphi(c_{21})+\varphi(d_{22})$.
We may write $h=h_{11}+h_{12}+h_{21}+h_{22}$, with
$h_{jk}\in\mathfrak{A}_{jk}\ (k,j=1,2)$. Applying Lemmas 2.1 and 2.2 we have

$\varphi(h)=\varphi(a_{11})+\varphi(b_{12})+\varphi(c_{21})+\varphi(d_{22})=\varphi(a_{11}+d_{22})+\varphi(b_{12}+c_{21}).$

Now, observing that
$p_{n_{*}}(e_{1},a_{11}+d_{22},e_{1},\ldots,e_{1})=0=p_{n_{*}}(e_{1},b_{12},e_{1},\ldots,e_{1})$
and by Claims 2.2 and 2.3 we obtain

$\displaystyle\varphi(p_{n_{*}}(e_{1},h,e_{1},\ldots,e_{1}))$
$\displaystyle=\varphi(p_{n_{*}}(e_{1},a_{11}+d_{22},e_{1},\ldots,e_{1}))+\varphi(p_{n_{*}}(e_{1},b_{12}+c_{21},e_{1},\ldots,e_{1}))$
$\displaystyle=\varphi(p_{n_{*}}(e_{1},c_{21},e_{1},\ldots,e_{1})),$

that is,

$\varphi(-h_{21}+h_{21}^{*})=\varphi(-c_{21}+c_{21}^{*}).$

Then, by injectivity of $\varphi$, $-h_{21}+h_{21}^{*}=-c_{21}+c_{21}^{*}$.
Thus $h_{21}=c_{21}$.

In a similar way, using $e_{2}$ rather than $e_{1}$ in the previous
calculation, we conclude that $h_{12}=b_{12}$. Also, given
$x_{21}\in{\mathfrak{A}}\,_{21}$,

$\displaystyle\varphi(p_{n_{*}}(x_{21},h,e_{1},\ldots,e_{1}))$
$\displaystyle=\varphi(p_{n_{*}}(x_{21},a_{11}+d_{22},\ldots,e_{1}))+\varphi(p_{n_{*}}(x_{21},b_{12}+c_{21},e_{1},\ldots,e_{1}))$
$\displaystyle=\varphi(p_{n_{*}}(x_{21},a_{11},e_{1},\ldots,e_{1})),$

since
$p_{n_{*}}(x_{21},b_{12}+c_{21},e_{1},\ldots,e_{1})=0=p_{n_{*}}(x_{21},d_{22},\ldots,e_{1})$.
Again, by injectivity of $\varphi$ we conclude, by following the same strategy
as in the proof of Lemma 2.1, that $h_{11}=a_{11}$. Now, using $e_{2}$ rather
than $e_{1}$ and $x_{12}$ rather than $x_{21}$ in the previous calculation we
obtain $h_{22}=d_{22}$. Therefore, $h=a_{11}+b_{12}+c_{21}+d_{22}$. ∎

######  Lemma 2.4.

For any $a_{jk},b_{jk}\in{\mathfrak{A}}\,_{jk}$, with $j\neq k$, we have
$\varphi(a_{jk}+b_{jk})=\varphi(a_{jk})+\varphi(b_{jk})$.

###### Proof.

We shall prove the case $j=1$ and $k=2$. The other case is done in a similar
way. Since $\varphi$ is surjective, given
$\varphi(a_{12})+\varphi(b_{12})\in{\mathfrak{A}}\,^{\prime}$ and
$\varphi(-a_{12}^{*})+\varphi(-b_{12}^{*})$ there exist $h\in{\mathfrak{A}}\,$
and $t\in{\mathfrak{A}}\,$ such that
$\varphi(h)=\varphi(a_{12})+\varphi(b_{12})$ and
$\varphi(t)=\varphi(-a_{12}^{*})+\varphi(-b_{12}^{*})$. We may write
$h=h_{11}+h_{12}+h_{21}+h_{22}$ and $t=t_{11}+t_{12}+t_{21}+t_{22}$, with
$h_{jk},t_{jk}\in\mathfrak{A}_{jk}\ (k,j=1,2)$.

First we show that $h\in{\mathfrak{A}}\,_{12}$. By Claim 2.2 we get

$\displaystyle\varphi(-h_{21}+h_{21}^{*})$
$\displaystyle=\varphi(p_{n_{*}}(e_{1},h,e_{1},\ldots,e_{1}))$
$\displaystyle=\varphi(p_{n_{*}}(e_{1},a_{12},e_{1},\ldots,e_{1}))+\varphi(p_{n_{*}}(e_{1},b_{12},e_{1},\ldots,e_{1}))=0.$

Then, by injectivity of $\varphi$ we obtain $h_{21}=0$. Also, given
$d_{12}\in{\mathfrak{A}}\,_{12}$,

$\displaystyle\varphi(d_{12}h_{22}-(d_{12}h_{22})^{*})$
$\displaystyle=\varphi(p_{n_{*}}(d_{12},h,e_{2},\ldots,e_{2}))$
$\displaystyle=\varphi(p_{n_{*}}(d_{12},a_{12},e_{2},\ldots,e_{2}))+\varphi(p_{n_{*}}(d_{12},b_{12},e_{2},\ldots,e_{2}))=0,$

that is, $d_{12}h_{22}=0$, which implies that $h_{22}=0$ by Condition (1) of
Theorem 2.1. Now, using $d_{21}\in{\mathfrak{A}}\,_{21}$ rather than $d_{12}$
in the previous calculation, we conclude that $h_{11}=0$. Therefore,
$h=h_{12}\in{\mathfrak{A}}\,_{12}$.

In a similar way, we obtain $t=t_{21}\in{\mathfrak{A}}\,_{21}$. Finally, by
Lemma 2.3

$\displaystyle\varphi(a_{12}+b_{12}-a_{12}^{*}-b_{12}^{*})$
$\displaystyle=\varphi(p_{n_{*}}(e_{1}+a_{12},e_{2}+b_{12},e_{2},\ldots,e_{2}))$
$\displaystyle=p_{n_{*}}(\varphi(e_{1}+a_{12}),\varphi(e_{2}+b_{12}),\varphi(e_{2}),\ldots,\varphi(e_{2}))$
$\displaystyle=p_{n_{*}}(\varphi(e_{1}),\varphi(e_{2}),\varphi(e_{2}),\ldots,\varphi(e_{2}))$
$\displaystyle+p_{n_{*}}(\varphi(e_{1}),\varphi(b_{12}),\varphi(e_{2}),\ldots,\varphi(e_{2}))$
$\displaystyle+p_{n_{*}}(\varphi(a_{12}),\varphi(e_{2}),\varphi(e_{2}),\ldots,\varphi(e_{2}))$
$\displaystyle+p_{n_{*}}(\varphi(a_{12}),\varphi(b_{12}),\varphi(e_{2}),\ldots,\varphi(e_{2}))$
$\displaystyle=\varphi(p_{n_{*}}(e_{1},e_{2},e_{2},\ldots,e_{2}))$
$\displaystyle+\varphi(p_{n_{*}}(e_{1},b_{12},e_{2},\ldots,e_{2}))$
$\displaystyle+\varphi(p_{n_{*}}(a_{12},e_{2},e_{2},\ldots,e_{2}))$
$\displaystyle+\varphi(p_{n_{*}}(a_{12},b_{12},e_{2},\ldots,e_{2}))$
$\displaystyle=\varphi(a_{12}-a_{12}^{*})+\varphi(b_{12}-b_{12}^{*})$
$\displaystyle=\varphi(a_{12})+\varphi(b_{12})+\varphi(-a_{12}^{*})+\varphi(-b_{12}^{*})$
$\displaystyle=\varphi(h_{12})+\varphi(t_{21})=\varphi(h_{12}+t_{21}).$

Since $\varphi$ is injective, we have
$a_{12}+b_{12}-a_{12}^{*}-b_{12}^{*}=h_{12}+t_{21}$, this is,
$h=h_{12}=a_{12}+b_{12}$. ∎

######  Lemma 2.5.

For any $a_{jj},b_{jj}\in{\mathfrak{A}}\,_{jj}$, with
$j\in\left\\{1,2\right\\}$, we have
$\varphi(a_{jj}+b_{jj})=\varphi(a_{jj})+\varphi(b_{jj})$.

###### Proof.

We shall prove the case $j=1$, since the other case is done in a similar way.
Since $\varphi$ is surjective, given
$\varphi(a_{11})+\varphi(b_{11})\in{\mathfrak{A}}\,^{\prime}$ there exists
$h\in{\mathfrak{A}}\,$ such that $\varphi(h)=\varphi(a_{11})+\varphi(b_{11})$.
We may write $h=h_{11}+h_{12}+h_{21}+h_{22}$, with
$h_{jk}\in\mathfrak{A}_{jk}\ (k,j=1,2)$. Now, by Claim 2.2

$\displaystyle\varphi(-h_{21}+h_{21}^{*})$
$\displaystyle=\varphi(p_{n_{*}}(e_{1},h,e_{1},\ldots,e_{1}))$
$\displaystyle=\varphi(p_{n_{*}}(e_{1},a_{11},e_{1},\ldots,e_{1}))+\varphi(p_{n_{*}}(e_{1},b_{11},e_{1},\ldots,e_{1}))=0.$

Then, by injectivity of $\varphi$ we obtain $h_{21}=0$. Also,

$\displaystyle\varphi(-h_{12}+h_{12}^{*})$
$\displaystyle=\varphi(p_{n_{*}}(e_{2},h,e_{2},\ldots,e_{2}))$
$\displaystyle=\varphi(p_{n_{*}}(e_{2},a_{11},e_{2},\ldots,e_{2}))+\varphi(p_{n_{*}}(e_{2},b_{11},e_{2},\ldots,e_{2}))=0,$

that is, $h_{12}=0$ by injectivity of $\varphi$. Moreover, given
$d_{12}\in{\mathfrak{A}}\,_{12}$,

$\displaystyle\varphi(d_{12}h_{22}-(d_{12}h_{22})^{*})$
$\displaystyle=\varphi(p_{n_{*}}(d_{12},h,e_{2},\ldots,e_{2}))$
$\displaystyle=\varphi(p_{n_{*}}(d_{12},a_{11},e_{2},\ldots,e_{2}))+\varphi(p_{n_{*}}(d_{12},b_{11},e_{2},\ldots,e_{2}))$
$\displaystyle=0.$

Then, by injectivity of $\varphi$, $d_{12}h_{22}=0$, which implies that
$h_{22}=0$ by Condition (1) of Theorem 2.1. Finally, given
$d_{21}\in{\mathfrak{A}}\,_{21}$, by Lemmas 2.3 and 2.4 we have

$\displaystyle\varphi(d_{21}h_{11}-(d_{21}h_{11})^{*})$
$\displaystyle=\varphi(p_{n_{*}}(d_{21},h,e_{1},\ldots,e_{1}))$
$\displaystyle=\varphi(p_{n_{*}}(d_{21},a_{11},e_{1},\ldots,e_{1}))+\varphi(p_{n_{*}}(d_{21},b_{11},e_{1},\ldots,e_{1}))$
$\displaystyle=\varphi(d_{21}a_{11}-(d_{21}a_{11})^{*})+\varphi(d_{21}b_{11}-(d_{21}b_{11})^{*})$
$\displaystyle=\varphi(d_{21}a_{11})+\varphi(-(d_{21}a_{11})^{*})+\varphi(d_{21}b_{11})+\varphi(-(d_{21}b_{11})^{*})$
$\displaystyle=\varphi(d_{21}a_{11}+d_{21}b_{11})+\varphi(-(d_{21}a_{11})^{*}-(d_{21}b_{11})^{*})$
$\displaystyle=\varphi(d_{21}(a_{11}+b_{11})-(a_{11}^{*}+b_{11}^{*})d_{21}^{*}),$

that is,
$d_{21}h_{11}-(d_{21}h_{11})^{*}=d_{21}(a_{11}+b_{11})-(a_{11}^{*}+b_{11}^{*})d_{21}^{*}$,
by injectivity of $\varphi$. Thus, $d_{21}(h_{11}-(a_{11}+b_{11}))=0$, which
implies that $h_{11}=a_{11}+b_{11}$ by Condition (1) of Theorem 2.1. ∎

Proof of Theorem 2.1. Now using Lemmas 2.3, 2.4 and 2.5 is easy see that
$\varphi$ is additive. Besides, using additivity of $\varphi$ and since
$\varphi$ is unital, we have for $a\in{\mathfrak{A}}\,$,

$\displaystyle 2^{n-2}(\varphi(a)-\varphi(a)^{*})$
$\displaystyle=p_{n_{*}}(\varphi(a),1_{{\mathfrak{A}}\,^{\prime}},\ldots,1_{{\mathfrak{A}}\,^{\prime}})=p_{n_{*}}(\varphi(a),\varphi(1_{{\mathfrak{A}}\,}),\ldots,\varphi(1_{{\mathfrak{A}}\,}))$
$\displaystyle=\varphi(p_{n_{*}}(a,1_{{\mathfrak{A}}\,},\ldots,1_{{\mathfrak{A}}\,}))=\varphi(2^{n-2}(a-a^{*}))$
$\displaystyle=2^{n-2}\varphi(a-a^{*})=2^{n-2}(\varphi(a)-\varphi(a^{*})),$

then $\varphi(a^{*})=\varphi(a)^{*}$ and we conclude that $\varphi$ preserves
involution. $\square$

######  Remark 2.1.

Observe that the Theorem 2.1 holds for any field of characteristic different
of $2.$ In the proof the Theorem 2.1 we established the additivity of
$\varphi$ without using the unital assumption of $\varphi$.

######  Theorem 2.2.

Let ${\mathfrak{A}}\,$ and ${\mathfrak{A}}\,^{\prime}$ be two alternative
$*$-algebras with identities $1_{{\mathfrak{A}}\,}$ and
$1_{{\mathfrak{A}}\,^{\prime}}$, respectively, and $e_{1}$ and
$e_{2}=1_{{\mathfrak{A}}\,}-e_{1}$ nontrivial symmetric idempotents in
${\mathfrak{A}}\,$. Let
$\varphi:{\mathfrak{A}}\,\rightarrow{\mathfrak{A}}\,^{\prime}$ be a complex
scalar multiplication bijective unital map. Suppose that ${\mathfrak{A}}\,$
satisfies the conditions of the Theorem 2.1, namely,

$(e_{j}{\mathfrak{A}}\,)x=\left\\{0\right\\}\mbox{ for any }j\in\\{1,2\\}\ \ \
\mbox{ implies }\ \ \ x=0,$
$\varphi(p_{n_{*}}(a,b,\xi,\ldots,\xi))=p_{n_{*}}(\varphi(a),\varphi(b),\varphi(\xi),\ldots,\varphi(\xi)),$

for all $a,b\in{\mathfrak{A}}\,$ and
$\xi\in\left\\{e_{1},e_{2},1_{{\mathfrak{A}}\,}\right\\}$.

Even more, if ${\mathfrak{A}}\,^{\prime}$ satisfies the condition

(3) $(\varphi(e_{j}){\mathfrak{A}}\,^{\prime})y=\left\\{0\right\\}\mbox{ for
any }j\in\\{1,2\\}\ \ \ \mbox{implies}\ \ \ y=0,$

then $\varphi$ is $*$-isomorphism.

With this hypothesis and Theorem 2.1 we have already proved that $\varphi$ is
$*$-additive. It remains for us to show that $\varphi$ preserves product. In
order to do that we will prove some more lemmas. Firstly, we observe that,

######  Claim 2.4.

$q_{j}=\varphi(e_{j})$ is an idempotent in ${\mathfrak{A}}\,^{\prime}$, for
$j\in\\{1,2\\}$.

###### Proof.

Since $\varphi$ is a complex scalar multiplication, it follows that

$\displaystyle 2^{n-1}i\,q_{j}$
$\displaystyle=2^{n-1}i\varphi(e_{j})=\varphi(2^{n-1}ie_{j})=\varphi(p_{n_{*}}(ie_{j},e_{j},1_{{\mathfrak{A}}\,},\ldots,1_{{\mathfrak{A}}\,}))$
$\displaystyle=p_{n_{*}}(i\varphi(e_{j}),\varphi(e_{j}),\varphi(1_{{\mathfrak{A}}\,}),\ldots,\varphi(1_{{\mathfrak{A}}\,}))$
$\displaystyle=p_{n_{*}}(i\varphi(e_{j}),\varphi(e_{j}),1_{{\mathfrak{A}}\,^{\prime}},\ldots,1_{{\mathfrak{A}}\,^{\prime}}))=2^{n-1}i\varphi(e_{j})^{2}=2^{n-1}i\,{q_{j}}^{2}.$

Then we can conclude that $q_{j}={q_{j}}^{2}$. Moreover, since $e_{j}$ is a
idempotent in ${\mathfrak{A}}\,$ we have that
$p_{n_{*}}(e_{j},1_{{\mathfrak{A}}\,},\ldots,1_{{\mathfrak{A}}\,})=0$.
Besides,

$0=\varphi(0)=\varphi(p_{n_{*}}(e_{j},1_{{\mathfrak{A}}\,},\ldots,1_{{\mathfrak{A}}\,}))=p_{n_{*}}(q_{j},1_{{\mathfrak{A}}\,^{\prime}},\ldots,1_{{\mathfrak{A}}\,^{\prime}}).$

Thus, $q_{j}-{q_{j}}^{*}=0$, that is, $q_{j}={q_{j}}^{*}$. ∎

######  Lemma 2.6.

For any $a\in{\mathfrak{A}}\,$, $\varphi(e_{j}a)=\varphi(e_{j})\varphi(a)$ and
$\varphi(ae_{j})=\varphi(a)\varphi(e_{j})$, with $j\in\\{1,2\\}$.

###### Proof.

Firstly, observe that

$p_{n_{*}}(ia,e_{j},1_{{\mathfrak{A}}\,},\ldots,1_{{\mathfrak{A}}\,})=2^{n-2}i(ae_{j}+e_{j}a^{*})$

and

$p_{n_{*}}(a,e_{j},1_{{\mathfrak{A}}\,},\ldots,1_{{\mathfrak{A}}\,})=2^{n-2}(ae_{j}-e_{j}a^{*}).$

Still, by Condition $(\ref{bullet})$ of Theorem 2.1 and $*$-additivity of
$\varphi$,

$\displaystyle\varphi(2^{n-2}i(ae_{j}+e_{j}a^{*}))$
$\displaystyle=\varphi(p_{n_{*}}(ia,e_{j},1_{{\mathfrak{A}}\,},\ldots,1_{{\mathfrak{A}}\,}))=p_{n_{*}}(\varphi(ia),\varphi(e_{j}),1_{{\mathfrak{A}}\,^{\prime}},\ldots,1_{{\mathfrak{A}}\,^{\prime}})$
$\displaystyle=2^{n-2}i(\varphi(a)\varphi(e_{j})+\varphi(e_{j})\varphi(a)^{*})$

and

$\displaystyle\varphi(2^{n-2}(ae_{j}-e_{j}a^{*}))$
$\displaystyle=\varphi(p_{n_{*}}(a,e_{j},1_{{\mathfrak{A}}\,},\ldots,1_{{\mathfrak{A}}\,}))=p_{n_{*}}(\varphi(a),\varphi(e_{j}),1_{{\mathfrak{A}}\,^{\prime}},\ldots,1_{{\mathfrak{A}}\,^{\prime}})$
$\displaystyle=2^{n-2}(\varphi(a)\varphi(e_{j})-\varphi(e_{j})\varphi(a)^{*}).$

Now, since $\varphi$ is $*$-additive, multiplying the second equality by $i$
and adding these two equations we obtain
$\varphi(ae_{j})=\varphi(a)\varphi(e_{j})$. The second statement is obtained
in a similar way. ∎

Consider the Peirce decomposition of ${\mathfrak{A}}\,^{\prime}$ with respect
to idempotents $q_{j}=\varphi(e_{j})$ of ${\mathfrak{A}}\,^{\prime}$ (with
$j\in\\{1,2\\}$) given by
${\mathfrak{A}}\,^{\prime}={\mathfrak{A}}\,^{\prime}_{11}\oplus{\mathfrak{A}}\,^{\prime}_{12}\oplus{\mathfrak{A}}\,^{\prime}_{21}\oplus{\mathfrak{A}}\,^{\prime}_{22},$
where ${\mathfrak{A}}\,^{\prime}_{kj}:=q_{k}{\mathfrak{A}}\,^{\prime}q_{j}$
for $k,j\in\\{1,2\\}$.

######  Lemma 2.7.

$\varphi({\mathfrak{A}}\,_{jk})\subset{\mathfrak{A}}\,_{jk}^{\prime}$, for
$j,k\in\\{1,2\\}$.

###### Proof.

Given $x\in{\mathfrak{A}}\,_{jk}$, we have $x=e_{j}xe_{k}$ and then, by Lemma
2.6,

$\varphi(x)=\varphi(e_{j})\varphi(xe_{k})=\varphi(e_{j})\varphi(x)\varphi(e_{k})\in{\mathfrak{A}}\,_{jk}^{\prime}.$

∎

######  Lemma 2.8.

For $j\neq k$, we have:

  * •

If $a_{jk}\in{\mathfrak{A}}\,_{jk}$ and $b_{kk}\in{\mathfrak{A}}\,_{kk}$ then
$\varphi(a_{jk}b_{kk})=\varphi(a_{jk})\varphi(b_{kk})$;

  * •

If $a_{jk}\in{\mathfrak{A}}\,_{jk}$ and $b_{jk}\in{\mathfrak{A}}\,_{jk}$ then
$\varphi(a_{jk}b_{jk})=\varphi(a_{jk})\varphi(b_{jk})$;

  * •

If $a_{jj}\in{\mathfrak{A}}\,_{jj}$ and $b_{jk}\in{\mathfrak{A}}\,_{jk}$ then
$\varphi(a_{jj}b_{jk})=\varphi(a_{jj})\varphi(b_{jk})$;

  * •

If $a_{jk}\in{\mathfrak{A}}\,_{jk}$ and $b_{kj}\in{\mathfrak{A}}\,_{kj}$ then
$\varphi(a_{jk}b_{kj})=\varphi(a_{jk})\varphi(b_{kj})$.

###### Proof.

In order to prove the first statement, on the one hand, by Lemma 2.7

$\displaystyle\varphi(a_{jk}b_{kk})-\varphi(a_{jk}b_{kk})^{*}$
$\displaystyle=\varphi(a_{jk}b_{kk}-(a_{jk}b_{kk})^{*})=\varphi(p_{n_{*}}(a_{jk},b_{kk},e_{k},\ldots,e_{k}))$
$\displaystyle=p_{n_{*}}(\varphi(a_{jk}),\varphi(b_{kk}),q_{k},\ldots,q_{k})$
$\displaystyle=\varphi(a_{jk})\varphi(b_{kk})-(\varphi(a_{jk})\varphi(b_{kk}))^{*}$

and then $\varphi(a_{jk}b_{kk})=\varphi(a_{jk})\varphi(b_{kk})$.

Now to prove the second statement, we have

$\displaystyle\varphi(a_{jk}b_{jk})$
$\displaystyle-\varphi(a_{jk}b_{jk})^{*}-2^{n-3}\varphi(b_{jk}a_{jk})^{*}+2^{n-3}\varphi(b_{jk}a_{jk}^{*})^{*}$
$\displaystyle=\varphi(a_{jk}b_{jk})-(a_{jk}b_{jk})^{*}-2^{n-3}(b_{jk}a_{jk})^{*}+2^{n-3}(b_{jk}a_{jk}^{*})^{*})$
$\displaystyle=\varphi(p_{n_{*}}(a_{jk},b_{jk},e_{j},\ldots,e_{j}))=p_{n_{*}}(\varphi(a_{jk}),\varphi(b_{jk}),q_{j},\ldots,q_{j})$
$\displaystyle=\varphi(a_{jk})\varphi(b_{jk})-\varphi(a_{jk})^{*}\varphi(b_{jk})^{*}$
$\displaystyle\
-2^{n-3}\varphi(b_{jk})^{*}\varphi(a_{jk})^{*}+2^{n-3}\varphi(b_{jk})^{*}\varphi(a_{jk}^{*})^{*}$

and then $\varphi(a_{jk}b_{jk})=\varphi(a_{jk})\varphi(b_{jk})$.

The others statements are proved in a similar way. ∎

Since alternative algebras are flexible, we have

$(x_{kj},a_{jj},b_{jj})+(b_{jj},a_{jj},x_{kj})=0,$

for all $x_{kj}\in{\mathfrak{A}}\,_{kj},$
$a_{jj},b_{jj}\in{\mathfrak{A}}\,_{jj},$ for $k,j\in\\{1,2\\}.$

######  Lemma 2.9.

If $a_{jj},b_{jj}\in{\mathfrak{A}}\,_{jj}$, with $j\in\left\\{1,2\right\\}$,
then $\varphi(a_{jj}b_{jj})=\varphi(a_{jj})\varphi(b_{jj})$.

###### Proof.

Let $x_{kj}$ be an element of ${\mathfrak{A}}\,_{kj}$, with $j\neq k$. Using
Lemma 2.8 we obtain

$\displaystyle\varphi(x_{kj})\varphi(a_{jj}b_{jj})=$
$\displaystyle\varphi(x_{kj}a_{jj}b_{jj})=\varphi((x_{kj}a_{jj})b_{jj})$
$\displaystyle=$
$\displaystyle(\varphi(x_{kj})\varphi(a_{jj}))\varphi(b_{jj})=\varphi(x_{kj})(\varphi(a_{jj})\varphi(b_{jj}))$

that is,

$\varphi(x_{kj})(\varphi(a_{jj}b_{jj})-\varphi(a_{jj})\varphi(b_{jj}))=0.$

Now, by Lemma 2.7, $\varphi(x_{kj})\in{\mathfrak{A}}\,^{\prime}_{kj}$ as well
as $\varphi(a_{jj}b_{jj})$ and
$\varphi(a_{jj})\varphi(b_{jj})\in{\mathfrak{A}}\,^{\prime}_{jj}$. Then,
$(\varphi(e_{k}){\mathfrak{A}}\,^{\prime})(\varphi(a_{jj}b_{jj})-\varphi(a_{jj})\varphi(b_{jj}))=0$,
which implies that $\varphi(a_{jj}b_{jj})=\varphi(a_{jj})\varphi(b_{jj})$ by
Condition $(\ref{clubsuit})$ of Theorem 2.2. ∎

Proof of Theorem 2.2. By additivity of $\varphi$ and Lemmas 2.8 and 2.9, it
follows that $\varphi(ab)=\varphi(a)\varphi(b)$, for all $a$,
$b\in{\mathfrak{A}}\,$, this is, $\varphi$ preserves product as required.
$\square$

##  3\. Corollaries

Now we present some consequences of our main results.

######  Corollary 3.1.

Let ${\mathfrak{A}}\,$ and ${\mathfrak{A}}\,^{\prime}$ be two alternative
$*$-algebras with identities $1_{\mathfrak{A}}\,$ and
$1_{{\mathfrak{A}}\,^{\prime}}$, respectively, and $e_{1}$ and
$e_{2}=1_{{\mathfrak{A}}\,}-e_{1}$ nontrivial symmetric idempotents in
${\mathfrak{A}}\,$. Let
$\varphi:{\mathfrak{A}}\,\rightarrow{\mathfrak{A}}\,^{\prime}$ be a complex
scalar multiplication bijective unital map. Suppose that ${\mathfrak{A}}\,$
satisfies

$(e_{j}{\mathfrak{A}}\,)x=\left\\{0\right\\}\mbox{ for any }j\in\\{1,2\\}\ \ \
\mbox{implies}\ \ \ x=0.$

Even more, suppose that ${\mathfrak{A}}\,^{\prime}$ satisfies

$(\varphi(e_{j}){\mathfrak{A}}\,^{\prime})y=\left\\{0\right\\}\mbox{ for any
}j\in\\{1,2\\}\ \ \ \mbox{implies}\ \ \ y=0.$

In this conditions,
$\varphi:{\mathfrak{A}}\,\rightarrow{\mathfrak{A}}\,^{\prime}$ is a
multiplicative $*$-Lie $n$-map if and only if $\varphi$ is $*$-isomorphism.

It is easy to see that any prime alternative algebra satisfy Conditions (1)
and (3), so we have the following result:

######  Corollary 3.2.

Let ${\mathfrak{A}}\,$ and ${\mathfrak{A}}\,^{\prime}$ be two prime
alternative $*$-algebras with identities $1_{{\mathfrak{A}}\,}$ and
$1_{{\mathfrak{A}}\,^{\prime}}$, respectively, and $e_{1}$ and
$e_{2}=1_{{\mathfrak{A}}\,}-e_{1}$ nontrivial symmetric idempotents in
${\mathfrak{A}}\,$. In this condition, a complex scalar multiplication
$\varphi:{\mathfrak{A}}\,\rightarrow{\mathfrak{A}}\,^{\prime}$ is a bijective
unital multiplicative $*$-Lie $n$-map if and only if $\varphi$ is
$*$-isomorphism.

To finish we will give an application of the Corollary 3.2. A complete normed
alternative complex $*$-algebra $A$ is called an alternative $C^{*}$-algebra
if it satisfies the condition: $\left\|a^{*}a\right\|=\left\|a\right\|^{2}$,
for all elements $a\in A$. Alternative $C^{*}$-algebras are non-associative
generalizations of $C^{*}$-algebras and appear in various areas in Mathematics
(see more details in the references [18] and [19]). An alternative
$C^{*}$-algebra $A$ is called an alternative $W^{*}$-algebra if it is a dual
Banach space and a prime alternative $W^{*}$-algebra is called alternative
$W^{*}$-factor. It is well known that non-zero alternative $W^{*}$-algebras
are unital.

######  Corollary 3.3.

Let ${\mathfrak{A}}\,$ and ${\mathfrak{A}}\,^{\prime}$ be two alternative
$W^{*}$-factors. In this condition, a complex scalar multiplication
$\varphi:{\mathfrak{A}}\,\rightarrow{\mathfrak{A}}\,^{\prime}$ is a bijective
unital multiplicative $*$-Lie $n$-map if and only if $\varphi$ is
$*$-isomorphism.

## References

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# Operator inequalities via the triangle inequality

Shigeru Furuichi, Mohammad Sababheh, and Hamid Reza Moradi

###### Abstract.

This article improves the triangle inequality for complex numbers, using the
Hermite-Hadamard inequality for convex functions. Then, applications of the
obtained refinement are presented to include some operator inequalities. The
operator applications include numerical radius inequalities and operator mean
inequalities.

###### Key words and phrases:

Triangle inequality; numerical radius; convex function

###### 2010 Mathematics Subject Classification:

Primary 47A63, Secondary 46L05, 47A60

##  1\. Main Results

In the field of Mathematical inequalities, interest in refining existing
inequalities or sharpening them has been at the center of researchers’
attention; see [5] for example.

One of the most powerful tools in obtaining new inequalities or sharpening
existing ones is the use of convex functions. Recall that a function
$f:J\to\mathbb{R}$ is said to be convex on the interval $J$ if it satisfies
the basic inequality

(1.1) $\displaystyle f((1-t)a+tb)\leq(1-t)a+tb,$

for all $a,b\in J$ and $0\leq t\leq 1.$ This inequality was refined by the
form

(1.2) $\displaystyle
f((1-t)a+tb)+2r_{t}\left(\frac{f(a)+f(b)}{2}-f\left(\frac{a+b}{2}\right)\right)\leq(1-t)f(a)+tf(b),$

where $r_{t}=\min\\{t,1-t\\}.$ Applications of this inequality can be found in
[12, 15, 16].

One of the most useful inequalities in convex analysis is the so called
Hermite-Hadamard inequality, which states

(1.3) $\displaystyle
f\left(\frac{a+b}{2}\right)\leq\frac{1}{b-a}\int_{a}^{b}f(t)dt\leq\frac{f(a)+f(b)}{2};$

as a refinement of (1.1) when $t=\frac{1}{2}.$

In this paper we employ the Inequality (1.3) to refine the well known triangle
inequality

$|c+d|\leq|c|+|d|,c,d\in\mathbb{C}.$

More precisely, we show that

(1.4) $\left|c+d\right|\leq
2\int\limits_{0}^{1}{\left|sc+(1-s)d\right|ds}\leq\left|c\right|+\left|d\right|.$

This will enable us to present a refinement of the Cauchy-Schwartz inequality
for the inner product. Then applications that include refined forms for some
numerical radius inequalities and operator mean inequalities will be given.

For this, we need to recall some notions related to Hilbert space operators.
Let $\mathcal{H}$ be a complex Hilbert space with inner product
$\left<\cdot,\cdot\right>:\mathcal{H}\times\mathcal{H}\to\mathbb{C}$ and let
$\mathcal{B}(\mathcal{H})$ be the $C^{*}$-algebra of all bounded linear
operators on $\mathcal{H}$. For $A\in\mathcal{B}(\mathcal{H})$, the operator
norm and the numerical radius are defined respectively as

$\|A\|=\sup_{\|x\|=1}\|Ax\|\;{}\;w(A)=\sup_{\|x\|=1}\left|\left<Ax,x\right>\right|.$

It is well known that

$\displaystyle\frac{1}{2}\|A\|\leq w(A)\leq\|A\|.$

Refining these inequalities has occupied an adequate area of research in this
field. For example, in [10] Kittaneh showed that

(1.5) $\displaystyle w(A)\leq\frac{1}{2}\left\|\;|A|+|A^{*}|\;\right\|,$

where $A^{*}$ is adjoint operator of $A$. The fact that this refines the
inequality $w(A)\leq\|A\|$ is due to the triangle inequality and the
observation $\|\;|A^{*}|\;\|=\|\;|A|\;\|.$

Using (1.4) we will be able to present a refined form of (1.5), where we find
a scalar $\alpha$ such that $\frac{1}{2}\leq\alpha\leq 1$ and

$w(A)\leq\frac{\alpha}{2}\left\|\;|A|+|A^{*}|\;\right\|,$

for a certain class of operators. We should remark that finding better bounds
for the numerical radius has received a renowned interest in the last few
years, as one can see in [2, 3, 4, 6, 7, 8, 10, 11, 13, 14].

Using the same approach, we will be able to find an inequality that relates
the geometric mean of $|A|^{2v}$ and $|A|^{2(1-v)}$ with the numerical radius.
For this, we recall that the weighted geometric mean of the two positive
definite operators $A,B$ is [1]

$A\sharp_{t}B=A^{\frac{1}{2}}\left(A^{-\frac{1}{2}}BA^{-\frac{1}{2}}\right)^{t}A^{\frac{1}{2}};0\leq
t\leq 1.$

When $t=\frac{1}{2},$ we write $\sharp$ instead of $\sharp_{\frac{1}{2}}.$

Our results will make use of the angle $\angle_{x,y}$ between two vectors
$x,y\in\mathbb{C}^{n}$ or $x,y\in\mathcal{H}.$ For such $x,y$ the Cauchy-
Schwartz inequality states that $|\left<x,y\right>|\leq\|x\|\;\|y\|.$ From
this, the angle between $x,y$ can be defined by

$\angle_{x,y}=\cos^{-1}\left(\frac{|\left<x,y\right>|}{\|x\|\;\|y\|}\right).$

It is implicitly understood that $0\leq\angle_{x,y}\leq\pi.$

##  2\. Main results

In this section, we present our main results. We begin by showing the scalar
inequalities, leading to operator inequalities.

###  2.1. Scalar inequalities

Our main result in this section is the following refinement of the triangle
inequality.

######  Theorem 2.1.

Let $c$ and $d$ be two complex numbers. Then

$\left|\frac{c+d}{2}\right|\leq\int\limits_{0}^{1}{\left|sc+(1-s)d\right|ds}\leq\dfrac{\left|c\right|+\left|d\right|}{2}.$

###### Proof.

Let $a,b\in\mathbb{C}$ and define the function $f:\mathbb{R}\to[0,\infty)$ by
$f(t)=|a+tb|.$ We will show that $f$ is convex on $\mathbb{R}$. For this,
assume first that $\frac{a}{b}=\alpha\in\mathbb{R}$. Then $f(t)=|a|\;|1+\alpha
t|$, which is convex; being the absolute value function. Now, if
$\frac{a}{b}\not\in\mathbb{R}$, then $f$ is twice differentiable on
$\mathbb{R}$ and

$f^{\prime\prime}(t)=\frac{\left(\operatorname{Re}(a)\cdot\operatorname{Im}(b)-\operatorname{Im}(a)\cdot\operatorname{Re}(b)\right)^{2}}{\left\\{\left(\mathcal{\operatorname{Re}}(a)+t\cdot\operatorname{Re}(b)\right)^{2}+\left(\operatorname{Im}(a)+t\cdot\operatorname{Im}(b)\right)^{2}\right\\}^{3/2}}\geq
0.$

Thus, $f$ is convex on $\mathbb{R}$. By the Hermite-Hadamard inequality (1.3),
we have

$\left|a+\left(\frac{x+y}{2}\right)b\right|\leq\frac{1}{x-y}\int\limits_{y}^{x}{\left|a+tb\right|dt}\leq\frac{\left|a+xb\right|+\left|a+yb\right|}{2},$

for $x>y$. Putting $c=a+xb$ and $d=a+yb$, we have $a=\dfrac{dx-cy}{x-y}$ and
$b=\dfrac{c-d}{x-y}$. This implies

$\left|\frac{c+d}{2}\right|\leq\frac{1}{x-y}\int\limits_{y}^{x}{\left|\frac{dx-
cy+(c-d)t}{x-y}\right|dt}\leq\frac{\left|c\right|+\left|d\right|}{2}.$

Now, if we set $\dfrac{dx-cy+(c-d)t}{x-y}=sc+(1-s)d$, then we have
$\dfrac{1}{x-y}dt=ds$. So for complex numbers $c$ and $d$,

$\left|\frac{c+d}{2}\right|\leq\int\limits_{0}^{1}{\left|sc+(1-s)d\right|ds}\leq\dfrac{\left|c\right|+\left|d\right|}{2},$

as desired. ∎

As a direct consequence of Theorem 2.1, we can improve the celebrated Cauchy-
Schwartz inequality.

######  Corollary 2.1.

Let $x,y\in\mathcal{H}$. Then

$\left|\left\langle
x,y\right\rangle\right|\leq\int\limits_{0}^{1}{\left|t{{e}^{i\theta}}+\left(1-t\right){{e}^{-i\theta}}\right|dt}\left\|x\right\|\left\|y\right\|\leq\left\|x\right\|\left\|y\right\|,$

where $\theta={{\angle}_{x,y}}$.

###### Proof.

We have

(2.1) $\left|\left\langle
x,y\right\rangle\right|=\left|\cos\theta\right|\left\|x\right\|\left\|y\right\|.$

Since

$\cos\theta=\operatorname{\mathcal{R}}\left({{e}^{i\theta}}\right)=\frac{{{e}^{i\theta}}+{{e}^{-i\theta}}}{2},$

Theorem 2.1 applied for $c=e^{i\theta}$ and $d=e^{-i\theta}$ implies that

$\left|\left\langle
x,y\right\rangle\right|\leq\int\limits_{0}^{1}{\left|t{{e}^{i\theta}}+\left(1-t\right){{e}^{-i\theta}}\right|dt}\left\|x\right\|\left\|y\right\|\leq\left\|x\right\|\left\|y\right\|,$

as desired. ∎

######  Remark 2.1.

The inequalities in Corollary 2.1 mean

$|\cos\theta|\leq\int\limits_{0}^{1}{\left|t{{e}^{i\theta}}+\left(1-t\right){{e}^{-i\theta}}\right|dt}\leq
1.$

To calculate the constant that appears in Corollary 2.1, notice that for an
arbitrary $\theta\in\mathbb{R}$,

$\displaystyle\left|t{{e}^{i\theta}}+\left(1-t\right){{e}^{-i\theta}}\right|$
$\displaystyle=\left|t\left(\cos\theta+i\sin\theta\right)+\left(1-t\right)\left(\cos\theta-i\sin\theta\right)\right|$
$\displaystyle=\left|\cos\theta+i\left(2t-1\right)\sin\theta\right|$
$\displaystyle=\sqrt{{{\cos}^{2}}\theta+{{\left(2t-1\right)}^{2}}{{\sin}^{2}}\theta}.$

For the case $\sin\theta=0$, we have
$\int\limits_{0}^{1}\sqrt{\cos^{2}\theta}dt=|\cos\theta|$. If we assume
$\sin\theta\neq 0$, then

$\displaystyle\int_{0}^{1}{\sqrt{{{\cos}^{2}}\theta+{{\left(2t-1\right)}^{2}}{{\sin}^{2}}\theta}dt}$
$\displaystyle=\frac{|\sin\theta|}{2}\int_{-1}^{1}\sqrt{s^{2}+\cot^{2}\theta}ds$
$\displaystyle=\frac{|\sin\theta|}{4}\left[s\sqrt{s^{2}+\cot^{2}\theta}+\cot^{2}\theta\log\left|s+\sqrt{s^{2}+\cot^{2}\theta}\right|\right]_{-1}^{1}$
$\displaystyle=\frac{1}{2}+\frac{1}{4}|\sin\theta|\cot^{2}\theta\log\left|\frac{1+|\sin\theta|}{1-|\sin\theta|}\right|=:\mu(\theta).$

Since $\sin\theta>0$ for $n\pi<\theta<(n+1)\pi$ and $\sin\theta<0$ for
$(n+1)<\pi\theta<(n+2)\pi$, where $n=0,1,2,\cdots$,

$\displaystyle\mu(\theta)=\frac{1}{2}+\frac{1}{4}\sin\theta\cot\theta\log\left|\frac{1+\sin\theta}{1-\sin\theta}\right|,\quad
n\pi<\theta<(n+1)\pi,$
$\displaystyle\mu(\theta)=\frac{1}{2}-\frac{1}{4}\sin\theta\cot\theta\log\left|\frac{1-\sin\theta}{1+\sin\theta}\right|,\quad(n+1)\pi<\theta<(n+2)\pi.$

Thus we have

$\mu(\theta)=\frac{1}{4}\left(2+\cos\theta\cot\theta\log\frac{1+\sin\theta}{1-\sin\theta}\right),\quad\theta\neq
n\pi,\quad\text{where}\quad n=0,1,2,\cdots,$

since $1\pm\sin\theta>0$ for all $\theta\neq n\pi$, where $n=0,1,2,\cdots$.
Since we have $\mu(\theta)\to 1$ when $\theta\to n\pi$ and $|\cos\theta|=1$
for $\theta=n\pi$, where $n=0,1,2,\cdots$, we have

$\mu(\theta)=\frac{1}{4}\left(2+\cos\theta\cot\theta\log\frac{1+\sin\theta}{1-\sin\theta}\right),\quad\theta\geq
0.$

We study the properties of the function $\mu(\theta)$ for $\theta\geq 0$. It
is sufficient to consider $0\leq\theta<\pi$, since
$\mu(\theta+\pi)=\mu(\theta)$. For this purpose, we prepare the following
lemma.

######  Lemma 2.1.

If $0\leq x<1$, then

(2.2) $\dfrac{2x}{x^{2}+1}\leq\log\dfrac{1+x}{1-x}.$

If $-1<x\leq 0$, then the reversed inequality holds.

###### Proof.

We firstly prove the first statement. Putting $t:=\dfrac{1+x}{1-x}$ for $0\leq
x<1$, the inequality (2.2) is equivalent to the inequality
$\dfrac{t^{2}-1}{t^{2}+1}\leq\log t$ for $t\geq 1$. Letting $a:=t^{2}\geq 1$
and using the inequality $\dfrac{a-1}{\log a}\leq\dfrac{a+1}{2}$ for $a>0$, we
get the desired inequality. This proves the first inequality.

To prove the reversed inequality for $-1<x\leq 0,$ we set $s:=\dfrac{1}{t}$.
Then we have $\dfrac{s^{2}-1}{s^{2}+1}\geq\log s$ for $0<s\leq 1$. This
inequality is equivalent to the reversed inequality of (2.2) for $-1<x\leq 0$
setting $s:=\dfrac{1+x}{1-x}$. ∎

Now we present the monotonicity of the function $\mu(\theta).$ As we mentioned
earlier, this function has period $\pi,$ so we study it only on the interval
$[0,\pi].$

######  Proposition 2.1.

The function $\mu(\theta)$ is decreasing on the interval
$\left[0,\frac{\pi}{2}\right]$ and is increasing in
$\left[\frac{\pi}{2},\pi\right].$

###### Proof.

By elementary calculations, we have

$\mu^{\prime}(\theta)=\frac{\cos\theta}{8\sin^{2}\theta}\nu(\theta),\quad
where\;\nu(\theta):=4\sin\theta-2(\sin^{2}\theta+1)\log\frac{1+\sin\theta}{1-\sin\theta}.$

Since we have $\lim\limits_{\theta\to n\pi/2}\mu^{\prime}(\theta)=0$ for
$n=0,1,2$, we consider the values $\theta\neq n\pi/2$, where $n=0,1,2$.
Putting $x:=\sin\theta$ for $0<\theta<\pi$, we consider the function
$\hat{\nu}(x)=4x-2(x^{2}+1)\log\dfrac{1+x}{1-x}$ for $0<x<1$. We have
$\hat{\nu}(x)<0$ by Lemma 2.1. Therefore we have $\nu(\theta)\leq 0$ for
$0\leq\theta\leq\pi$.

Taking account that $\cos\theta$ is positive when $0<\theta<\dfrac{\pi}{2}$
and is negative when $\dfrac{\pi}{2}<\theta<\pi$, we have
$\mu^{\prime}(\theta)\leq 0$ when $0\leq\theta\leq\dfrac{\pi}{2}$ and
$\mu^{\prime}(\theta)\geq 0$ when $\dfrac{\pi}{2}\leq\theta\leq\pi$. This
completes the proof. ∎

######  Corollary 2.2.

The inequality $\dfrac{1}{2}\leq\mu(\theta)\leq 1$ holds for $\theta\geq 0.$

###### Proof.

It suffices to consider the values $0\leq\theta\leq\pi.$ We have
$\lim\limits_{\theta\to 0}\mu(\theta)=\lim\limits_{\theta\to\pi}\mu(\theta)=1$
and $\lim\limits_{\theta\to\pi/2}\mu(\theta)=\dfrac{1}{2}$. By Proposition
2.1, we infer that $\dfrac{1}{2}\leq\mu(\theta)\leq 1$. ∎

###  2.2. Hilbert space Operator inequalities

In the following theorem, we improve the mixed Schwarz inequality, which
states that [9]

$\left|\left\langle
Ax,y\right\rangle\right|\leq\sqrt{\left\langle{{\left|A\right|}^{2v}}x,x\right\rangle\left\langle{{\left|{{A}^{*}}\right|}^{2\left(1-v\right)}}y,y\right\rangle},x,y\in\mathcal{H},0\leq
v\leq 1.$

This inequality has been used extensively in the literature when dealing with
numerical radius inequalities, see [17] for example.

######  Theorem 2.2.

Let $A\in\mathcal{B}\left(\mathcal{H}\right)$ with the the polar decomposition
$A=U\left|A\right|$ and let $x,y\in\mathcal{H}$. Then for any $0\leq v\leq 1$,

$\left|\left\langle
Ax,y\right\rangle\right|\leq\mu\left(\theta\right)\sqrt{\left\langle{{\left|A\right|}^{2v}}x,x\right\rangle\left\langle{{\left|{{A}^{*}}\right|}^{2\left(1-v\right)}}y,y\right\rangle},$

where
$\theta={{\angle}_{{{\left|A\right|}^{v}}x,{{\left|A\right|}^{1-v}}{{U}^{*}}y}}$.

###### Proof.

According to the assumptions and by employing Corollary 2.1, we have

$\displaystyle\left|\left\langle Ax,y\right\rangle\right|$
$\displaystyle=\left|\left\langle U\left|A\right|x,y\right\rangle\right|$
$\displaystyle=\left|\left\langle
U{{\left|A\right|}^{1-v}}{{\left|A\right|}^{v}}x,y\right\rangle\right|$
$\displaystyle=\left|\left\langle{{\left|A\right|}^{v}}x,{{\left|A\right|}^{1-v}}{{U}^{*}}y\right\rangle\right|$
$\displaystyle\leq\mu\left(\theta\right)\left\|{{\left|A\right|}^{v}}x\right\|\left\|{{\left|A\right|}^{1-v}}{{U}^{*}}y\right\|$
$\displaystyle=\mu\left(\theta\right)\sqrt{\left\langle{{\left|A\right|}^{2v}}x,x\right\rangle\left\langle
U{{\left|A\right|}^{2\left(1-v\right)}}{{U}^{*}}y,y\right\rangle}$
$\displaystyle=\mu\left(\theta\right)\sqrt{\left\langle{{\left|A\right|}^{2v}}x,x\right\rangle\left\langle{{\left|{{A}^{*}}\right|}^{2\left(1-v\right)}}y,y\right\rangle}$

as desired. ∎

The following is a straightforward consequence from Proposition 2.1.

######  Corollary 2.3.

The following holds.

  * (i)

If $0\leq{{\theta}_{1}}<\theta<{{\theta}_{2}}\leq\frac{\pi}{2}$, then

$\mu\left(\theta\right)\leq\mu\left({{\theta}_{1}}\right).$

  * (ii)

If $\frac{\pi}{2}\leq{{\theta}_{1}}<\theta<{{\theta}_{2}}\leq\pi$, then

$\mu\left(\theta\right)\leq\mu\left({{\theta}_{2}}\right).$

Now we are ready to present a new bound for the numerical radius. This form
refines (1.5), when $v=\frac{1}{2}$ for a certain class of operators.

######  Corollary 2.4.

Let $A\in\mathcal{B}\left(\mathcal{H}\right)$ have the polar decomposition
$A=U\left|A\right|$, $0\leq v\leq 1$ and let
$\theta_{x}={{\angle}_{{{\left|A\right|}^{v}}x,{{\left|A\right|}^{1-v}}{{U}^{*}}x}}$
where $x\in\mathcal{H}$ with $\left\|x\right\|=1$. If

  * (i)

If $0\leq{{\theta}_{1}}<\theta_{x}<{{\theta}_{2}}\leq\frac{\pi}{2}$ for all
unit vectors $x\in\mathcal{H}$, then

$\omega\left(A\right)\leq\frac{\mu\left({{\theta}_{1}}\right)}{2}\left\|{{\left|A\right|}^{2v}}+{{\left|{{A}^{*}}\right|}^{2\left(1-v\right)}}\right\|.$

  * (ii)

If $\frac{\pi}{2}\leq{{\theta}_{1}}<\theta<{{\theta}_{2}}\leq\pi$ for all unit
vectors $x\in\mathcal{H}$, then

$\omega\left(A\right)\leq\frac{\mu\left({{\theta}_{2}}\right)}{2}\left\|{{\left|A\right|}^{2v}}+{{\left|{{A}^{*}}\right|}^{2\left(1-v\right)}}\right\|.$

###### Proof.

We prove the first inequality. Let $x\in\mathcal{H}$ be a unit vector. By
Theorem 2.2,

$\displaystyle\left|\left\langle Ax,x\right\rangle\right|$
$\displaystyle\leq\mu\left(\theta_{x}\right)\sqrt{\left\langle{{\left|A\right|}^{2v}}x,x\right\rangle\left\langle{{\left|{{A}^{*}}\right|}^{2\left(1-v\right)}}x,x\right\rangle}$
$\displaystyle\leq\mu\left({{\theta}_{1}}\right)\sqrt{\left\langle{{\left|A\right|}^{2v}}x,x\right\rangle\left\langle{{\left|{{A}^{*}}\right|}^{2\left(1-v\right)}}x,x\right\rangle}$
$\displaystyle\leq\mu\left({{\theta}_{1}}\right)\left(\frac{\left\langle{{\left|A\right|}^{2v}}x,x\right\rangle+\left\langle{{\left|{{A}^{*}}\right|}^{2\left(1-v\right)}}x,x\right\rangle}{2}\right)$
$\displaystyle=\frac{\mu\left({{\theta}_{1}}\right)}{2}\left\langle\left({{\left|A\right|}^{2v}}+{{\left|{{A}^{*}}\right|}^{2\left(1-v\right)}}\right)x,x\right\rangle$
$\displaystyle\leq\frac{\mu\left({{\theta}_{1}}\right)}{2}\left\|{{\left|A\right|}^{2v}}+{{\left|{{A}^{*}}\right|}^{2\left(1-v\right)}}\right\|,$

where the second inequality is obtained from Corollary 2.3, and the third
inequality follows from the arithmetic-geometric mean inequality. Therefore,

$\left|\left\langle
Ax,x\right\rangle\right|\leq\frac{\mu\left({{\theta}_{1}}\right)}{2}\left\|{{\left|A\right|}^{2v}}+{{\left|{{A}^{*}}\right|}^{2\left(1-v\right)}}\right\|.$

Now, we get the desired result by taking the supremum over all unit vector
$x\in\mathcal{H}$. ∎

######  Remark 2.2.

Put $v={1}/{2}\;$. If $\theta={\pi}/{2}\;$, then

$\omega\left(A\right)=\frac{1}{2}\left\|A\right\|,$

since

$\frac{1}{2}\left\|A\right\|\leq\omega\left(A\right)\leq\frac{1}{4}\left\|\left|A\right|+\left|{{A}^{*}}\right|\right\|\leq\frac{1}{2}\left\|A\right\|.$

In the following, we present a reverse of the triangle inequality.

######  Theorem 2.3.

Let $c$ and $d$ be two complex numbers. Then for any $0\leq t\leq 1$,

$\frac{\left|c\right|+\left|d\right|}{2}-\frac{1}{2r_{t}}\left(\left(1-t\right)\left|c\right|+t\left|d\right|-\left|\left(1-t\right)c+td\right|\right)\leq\left|\frac{c+d}{2}\right|,$

where $r_{t}=\min\left\\{t,1-t\right\\}$.

###### Proof.

We know that if $f$ is a convex function on $\mathbb{R}$, then for
$x,y\in\mathbb{R}$ and any $0\leq t\leq 1,$

$f\left(\left(1-t\right)x+ty\right)\leq\left(1-t\right)f\left(x\right)+tf\left(y\right)-2r_{t}\left(\frac{f\left(x\right)+f\left(y\right)}{2}-f\left(\frac{x+y}{2}\right)\right),$

where $r_{t}=\min\left\\{t,1-t\right\\}$, see (1.2). Since for
$a,b\in\mathbb{C}$, $f\left(t\right)=\left|a+tb\right|$ is convex on
$\mathbb{R}$, we get

$\left|a+\left(\left(1-t\right)x+ty\right)b\right|\leq\left(1-t\right)\left|a+xb\right|+t\left|a+yb\right|-2r_{t}\left(\frac{\left|a+xb\right|+\left|a+yb\right|}{2}-\left|a+\frac{x+y}{2}b\right|\right).$

Now, applying the same method as in Theorem 2.1, we infer

$\left|\left(1-t\right)c+td\right|\leq\left(1-t\right)\left|c\right|+t\left|d\right|-2r_{t}\left(\frac{\left|c\right|+\left|d\right|}{2}-\left|\frac{c+d}{2}\right|\right),$

as desired ∎

######  Remark 2.3.

Let $x,y\in\mathcal{H}$, and let $r_{t}=\min\left\\{t,1-t\right\\}$ with
$0\leq t\leq 1$. By Theorem 2.3,

$\displaystyle 0$ $\displaystyle\leq
1-\frac{1}{2r_{t}}\left(1-\left|t{{e}^{i\theta}}+\left(1-t\right){{e}^{-i\theta}}\right|\right)$
$\displaystyle\leq\left|\frac{{{e}^{i\theta}}+{{e}^{-i\theta}}}{2}\right|$
$\displaystyle=\left|\cos\theta\right|,$

where $\theta={{\angle}_{x,y}}$. Thus we have

(2.3)
$0\leq\gamma_{t}(\theta)\left\|x\right\|\left\|y\right\|\leq\left|\left\langle
x,y\right\rangle\right|,$

where

$\gamma_{t}(\theta):=1-\frac{1}{2r_{t}}\left(1-\left|t{{e}^{i\theta}}+\left(1-t\right){{e}^{-i\theta}}\right|\right)=1-\frac{1}{2r_{t}}\left(1-\sqrt{{{\cos}^{2}}\theta+\left(2t-1\right)^{2}{{\sin}^{2}}\theta}\right)$

for $\theta\in[0,\pi]$ and $r_{t}=\min\left\\{t,1-t\right\\}$ with $0\leq
t\leq 1$. This provides a reverse of the Cauchy-Schwarz inequality.

We notice that $\gamma_{t}(\theta)=\gamma_{1-t}(\theta)$ for $0\leq t\leq 1.$
Also, since we have the Cauchy-Schwartz inequality
$|\left<x,y\right>|\leq\|x\|\;\|y\|$, we can use (2.3) to obtain sufficient
conditions on the equality $|\left<x,y\right>|=\|x\|\;\|y\|$, as follows. We
notice that $\gamma_{t}(\theta)=1$ when $\theta=0,\pi.$ Thus, when
$\angle_{x,y}=0,\pi$, we have $|\left<x,y\right>|=\|x\|\;\|y\|$.
Further when $\theta=\frac{\pi}{2}$, $\gamma_{t}(\theta)=0.$

Also, we notice that $\max_{t}\gamma_{t}(\theta)=\cos\theta,$ which is evident
because $\left<x,y\right>=\|x\|\;\|y\|\cos\theta.$

Again, since $\gamma_{t}(\theta)$ is periodic in $\theta$ with period $\pi,$
it is sufficient to study it on the interval $[0,\pi].$

######  Proposition 2.2.

If $0\leq\theta\leq\dfrac{\pi}{2}$, then ${{\gamma}_{t}}\left(\theta\right)$
is decreasing. If $\dfrac{\pi}{2}\leq\theta\leq\pi$, then
${{\gamma}_{t}}\left(\theta\right)$ is increasing. In addition, we have
$0\leq\gamma_{t}(\theta)\leq 1$.

###### Proof.

Elementary calculations show that

$\dfrac{d\gamma_{t}(\theta)}{d\theta}=-\frac{t(1-t)\sin
2\theta}{r_{t}\sqrt{\cos^{2}\theta+(2t-1)^{2}\sin^{2}\theta}}.$

This shows the assertion of monotonicity. Since we have
$\lim\limits_{\theta\to n\pi/2}\gamma_{t}(\theta)=1$ for $n=0,2$ and
$\lim\limits_{\theta\to
n\pi/2}\gamma_{t}(\theta)=\dfrac{2r_{t}-1+\sqrt{(2t-1)^{2}}}{2r_{t}}=0$ for
$n=1$, we have $0\leq\gamma_{t}(\theta)\leq 1$ by monotonicity of
$\gamma_{t}(\theta).$ ∎

Now we present an inequality that gives a relation between the geometric mean
and the numerical radius of the operator $A$, noting that
$\max_{t}\gamma_{t}(\theta)=\cos\theta.$

######  Corollary 2.5.

Let $A\in\mathcal{B}\left(\mathcal{H}\right)$ have the polar decomposition
$A=U\left|A\right|$, $0\leq v\leq 1$ and let
$\theta_{x}={{\angle}_{{{\left|A\right|}^{v}}x,{{\left|A\right|}^{1-v}}{{U}^{*}}x}}$
where $x\in\mathcal{H}$ with $\left\|x\right\|=1$.

  * (i)

If $0\leq{{\theta}_{1}}<\theta_{x}<{{\theta}_{2}}\leq\frac{\pi}{2}$, then

$\cos\left({{\theta}_{2}}\right)\left\|{{\left|A\right|}^{2v}}\sharp{{\left|{{A}^{*}}\right|}^{2\left(1-v\right)}}\right\|\leq\omega\left(A\right).$

  * (ii)

If $\frac{\pi}{2}\leq{{\theta}_{1}}<\theta_{x}<{{\theta}_{2}}\leq\pi$, then

$\cos\left({{\theta}_{1}}\right)\left\|{{\left|A\right|}^{2v}}\sharp{{\left|{{A}^{*}}\right|}^{2\left(1-v\right)}}\right\|\leq\omega\left(A\right).$

###### Proof.

We prove case (i) since case (ii) can be proved similarly. In this case, by
Proposition 2.2, we have
${{\gamma}_{t}}\left({{\theta}_{2}}\right)\leq{{\gamma}_{t}}\left(\theta\right)$.
Putting $x={{\left|A\right|}^{1-t}}x$, $y={{\left|A\right|}^{1-v}}{{U}^{*}}x$,
where $x\in\mathcal{H}$ is a unit vector, in the inequality (2.3), we have

$\displaystyle\cos\left({{\theta}_{2}}\right)\left\langle{{\left|A\right|}^{2v}}\sharp{{\left|{{A}^{*}}\right|}^{2\left(1-v\right)}}x,x\right\rangle$
$\displaystyle\leq\cos\left({{\theta}_{2}}\right)\sqrt{\left\langle{{\left|A\right|}^{2v}}x,x\right\rangle\left\langle{{\left|{{A}^{*}}\right|}^{2\left(1-v\right)}}x,x\right\rangle}$
$\displaystyle=\cos\left({{\theta}_{2}}\right)\sqrt{\left\langle{{\left|A\right|}^{2v}}x,x\right\rangle\left\langle
U{{\left|A\right|}^{2\left(1-v\right)}}{{U}^{*}}x,x\right\rangle}$
$\displaystyle=\cos\left({{\theta}_{2}}\right)\left\|{{\left|A\right|}^{v}}x,x\right\|\left\|{{\left|A\right|}^{1-v}}{{U}^{*}}x,x\right\|$
$\displaystyle\leq\left|\left\langle{{\left|A\right|}^{v}}x,{{\left|A\right|}^{1-v}}{{U}^{*}}x\right\rangle\right|$
$\displaystyle=\left|\left\langle Ax,x\right\rangle\right|.$

Notice that the first inequality follows from the following fact for positive
operators $A,B$,

$\left\langle A\sharp Bx,x\right\rangle\leq\sqrt{\left\langle
Ax,x\right\rangle\left\langle Bx,x\right\rangle}.$

Thus,

$\cos\left({{\theta}_{2}}\right)\left\langle{{\left|A\right|}^{2v}}\sharp{{\left|{{A}^{*}}\right|}^{2\left(1-v\right)}}x,x\right\rangle\leq\left|\left\langle
Ax,x\right\rangle\right|,$

and this implies

$\cos\left({{\theta}_{2}}\right)\left\|{{\left|A\right|}^{2v}}\sharp{{\left|{{A}^{*}}\right|}^{2\left(1-v\right)}}\right\|\leq\omega\left(A\right),$

as desired. ∎

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(S. Furuichi) Department of Information Science, College of Humanities and
Sciences, Nihon University, Setagaya-ku, Tokyo, Japan

E-mail address<EMAIL_ADDRESS>
(M. Sababheh) Department of Basic Sciences, Princess Sumaya University for
Technology, Amman, Jordan

E-mail address<EMAIL_ADDRESS>
(H. R. Moradi) Department of Mathematics, Payame Noor University (PNU), P.O.
Box, 19395-4697, Tehran, Iran

E-mail address<EMAIL_ADDRESS>


# Avoided ferromagnetic quantum critical point in CeZn

Hisashi Kotegawa1, Toshiaki Uga1, Hideki Tou1, Eiichi Matsuoka1, Hitoshi
Sugawara1 1Department of Physics, Kobe University, Kobe 658-8530, Japan

###### Abstract

Cubic CeZn shows a structural phase transition under pressure, and it modifies
the ground state from an antiferromagnetic (AFM) state to a ferromagnetic (FM)
state. To investigate how the FM state terminates at a quantum phase
transition, we measured the electrical resistivity under pressure for a single
crystal CeZn. The transition temperature into the FM state decreases
monotonously with increasing pressure, accompanied by the pronounced Kondo
effect, but a drastic change in the field response occurs before the ordered
phase terminates. This result suggests that the FM quantum critical point is
avoided by the appearance of an AFM-like state.

††preprint: APS/123-QED

In condensed matter physics, a combination of elements and crystal structure
brings diversity in the electronic state of materials. Therefore, universality
present in various materials is important. The ”absence of ferromagnetic (FM)
quantum critical point (QCP)” is an interesting hypothesis to represent such
universality. A QCP, which is a continuous phase transition at zero
temperature, has been seen in many systems. However, in metallic ferromagnets,
a QCP at zero field is hardly achieved Brando ; Pfleiderer_MnSi ; Uhlarz ;
Kabeya ; Valentin ; Kotegawa1 ; Aoki ; Kimura ; Shimizu ; Araki ; Kobayashi ;
Wilhelm ; Sullow ; Sidorov ; Moroni ; Niklowitz ; Lausberg ; Kotegawa2 ;
Lengyel ; Valentin_LaCrGe3 ; Kaluarachchi ; Kaluarachchi2 . A notable way to
prevent an FM QCP is the appearance of a tricritical point (TCP), at which the
second-order FM transition changes to a first-order transition Pfleiderer_MnSi
; Uhlarz ; Kabeya ; Valentin ; Kotegawa1 ; Aoki ; Kimura ; Shimizu ; Araki .
The theoretical work has specified its universality; that is, the TCP appears
irrespective of the electronic structure Brando ; Belitz . Another way to
prevent an FM QCP is by switching of the ordered state. In many systems, the
FM transitions are preempted by the transition into a different phase such as
an antiferromagnetic (AFM) phase, before the QCP is reached Brando ; Kobayashi
; Wilhelm ; Sullow ; Sidorov ; Moroni ; Niklowitz ; Lausberg ; Kotegawa2 ;
Lengyel ; Valentin_LaCrGe3 ; Kaluarachchi ; Kaluarachchi2 . The switching of
the ordered state can be caused by a change in the magnetic interaction or a
formation of a modulated magnetic phase induced by quantum fluctuation Conduit
; Efremov ; Karahasanovic ; Abdul .

However, it was found that CeRh6Ge4, which crystalizes in the hexagonal
structure with the space group $P$6̄$m$2, is a possible counterexample
Matsuoka ; Kotegawa_CeRh6Ge4 ; Shen . It has been suggested that the Curie
temperature of 2.5 K reaches a QCP under pressure at zero field. Although
microscopic measurements are highly required to confirm the QCP,
noncentrosymmetry might be a key ingredient to induce an FM QCP Kirkpatrick .
These studies show the importance of surveying FM materials to clarify what
determines the behavior in the boundary of the FM QCP in the actual material.

Meanwhile, the difficulty of such an investigation arises from the fact that
metallic ferromagnets near the quantum phase transition are few. To
investigate the intrinsic behavior of the FM quantum phase transition, high-
quality crystals are essential Sang . In this study, we focus on CeZn that
satisfies these conditions. The CeZn crystallizes in a cubic structure with
the space group $Pm\bar{3}m$ at ambient pressure Chao . It shows an AFM
transition at $T_{N}=30$ K Schmitt , whereas an FM transition appears under
pressure, accompanied by a structural phase transition Kadomatsu ; Shigeoka .
It has been reported that CeZn transforms into a rhombohedral structure above
$\sim 2.5$ GPa at 300 K and $\sim 1.0$ GPa at low temperatures Shigeoka ;
Uwatoko , but a recent measurement revealed that it is a tetragonal structure
with space inversion symmetry ($P4/nmm$) XShen . The pressure-temperature
phase diagram up to $\sim 3$ GPa has been reported in earlier studies
Kadomatsu ; Shigeoka , in which the FM phase survives even at the highest
pressure, and a recent study has confirmed that the magnetically ordered state
disappears above $\sim 3$ GPa XShen .

Figure 1: (color online) Pressure-temperature phase diagram of CeZn, which is
obtained in this study. (a): $0<T<300$ K, (b): $0<T<20$ K. The AFM phase at
ambient pressure changes into the FM phase through the cubic to tetragonal
structural phase transition. The magnetically ordered state terminates at
$P_{c}\sim 3$ GPa. The metamagnetic-like transitions appear in the pressure
region, where the transition temperatures are indicated by the closed
triangles. They suggest that the FM state changes into an AFM-like state just
before a QCP is reached.

In this study, we investigated the quantum phase transition of CeZn using
resistivity measurements under pressure up to $\sim 4.5$ GPa and under
magnetic fields. To give an outlook of this study, the obtained phase diagram
is shown in Fig. 1. The first-order structural phase transition at $T_{S}$ and
the modification from the AFM state to the FM state are consistent with those
of previous studies Kadomatsu ; Shigeoka ; Uwatoko . In contrast to the
previous study Kadomatsu , the FM transition temperature $T_{C}$ in the
tetragonal phase decreases gradually at increasing pressure. The quantum phase
transition into the paramagnetic (PM) phase likely occurs at $P_{c}\sim 3$
GPa. These results are consistent with those of the recent report XShen . We
found that the metamagnetic-like transition suddenly emerges above $\sim 2.1$
GPa, indicating that the ordered state is no longer the FM state. The
experimental observation suggests that the FM state transforms into another
AFM-like state just before a QCP is reached in CeZn.

A single crystal of CeZn was made using a similar procedure to that in Ref.
Shigeoka . The residual resistivity ratio at ambient pressure is approximately
60, ensuring a high-quality crystal. High pressure was applied using an
indenter-type pressure cell indenter , and Daphne7474 was used as a pressure-
transmitting medium Murata . The low temperatures down to 0.4 K were achieved
using a 3He cryostat. For electrical resistivity measurement, the current
flowed along the cubic [110] direction, and in most cases the magnetic field
was applied along the cubic [$\bar{1}$10] direction ($j\perp H$). However, the
sample might be in a multidomain in the tetragonal phase. The cubic [110] and
[$\bar{1}$10] directions correspond to any of the tetragonal [100], [112], and
their equivalent directions XShen .

Figure 2: (color online) Temperature dependence of the electrical resistivity
for CeZn. (a), (b) $P<2.5$ GPa; (c), (d) $P>2.5$ GPa; and (e) the lower
temperature part down to 0.4 K. The AFM transition changes into the FM
transition in the tetragonal phase below $T_{S}$. The magnetically ordered
state vanishes at $\sim 3$ GPa. In the PM state, two local maxima denoted as
$T_{\rm max1}$ and $T_{\rm max2}$ appear, and another characteristic
temperature $T^{*}$ is seen above $\sim 3$ GPa.

Figure 2(a-d) shows the temperature dependence of the electrical resistivity
down to 1.5 K for several pressures: (a,b) $P<2.5$ GPa and (c,d) $P>2.5$ GPa.
The low-temperature data down to 0.4 K are shown in Fig. 2(e). The data of
$P<2.5$ GPa in Fig. 2(a) and (b) are almost consistent with those of a
previous study Kadomatsu . From ambient pressure to 1.02 GPa, the temperature
dependence of the resistivity in the PM state shows monotonous variation,
followed by the first-order AFM transition at $T_{N}\sim 30$ K. The Kondo
effect is unremarkable in this phase, and the ordered moment at ambient
pressure has been reported to be 1.91 $\mu_{B}$/Ce Shigeoka . Above 1.39 GPa,
a first-order structural phase transition appears at $T_{S}$, as seen in the
large jump of the resistivity. In the tetragonal phase below $T_{S}$, the
Kondo effect becomes prominent and the FM transition emerges. First, we
mention the pressure evolution of the resistivity in the tetragonal PM phase.
In the resistivity, two local maxima are seen in the range $2.03-2.47$ GPa.
One is $T_{\rm max1}\sim 120$ K, which is insensitive to pressure. This is
likely related to the crystal electric field (CEF) splitting, which was 65 K
between the ground-state quartet and the excited-state doublet in the cubic
phase Pierre . The quartet is lifted into two doublets in the tetragonal
phase, but the magnitude of the separation is unknown. Another local maximum
denoted as $T_{\rm max2}$ appears at low temperatures. In Fig. 2(c), $T_{\rm
max2}$ is approximately $10$ K at 2.71 GPa, and increases remarkably under
further pressure and merges with $T_{\rm max1}$. Another characteristic
temperature denoted as $T^{*}$ appears, which is the shoulder in the
resistivity at $\sim 3$ GPa, as seen in Fig. 2(d). The $T^{*}$ is determined
by a cross point of two extrapolation lines, as shown in the data at 3.79 GPa.
This is considered a signature of the coherent Kondo effect, which makes the
ground state nonmagnetic and drastically increases under further pressure.
These characteristic temperatures are summarized in Fig. 1. As mentioned
above, $T_{\rm max1}$ is speculated to be related to the highest CEF level. In
the range $T>T_{\rm max1}$, the Kondo effect for all CEF states probably
occurs. In this context, $T_{\rm max2}$ may be related to the Kondo effect for
the lower two doublets, that is, the quasiquartet. Because it is lifted in the
tetragonal phase, it is considered that $T^{*}$ appears as the Kondo effect
for the ground-state doublet.

As seen in Fig. 2(b), the FM transition temperature $T_{C}$ is approximately
12 K at 1.39 GPa, which is consistent with that of previous studies Kadomatsu
; Shigeoka , and the neutron scattering measurement under pressure has
confirmed that the FM transition in this pressure range is of the second order
Shigeoka . The $T_{C}$ decreases gradually with increasing pressure, and it is
suppressed below 1.5 K at 2.47 GPa. Here the transition temperature was
determined by the peak in $-d^{2}\rho/dT^{2}$. The low-temperature data are
shown in Fig. 2(e). In this study, the transition temperature decreased
continuously with decreasing pressure and reached the measurement limit at
2.82 GPa, which is consistent with that of a recent report XShen . There is no
clear indication that the transition changes into the first order.

Figure 3: (color online) (a-e) Temperature dependence of the resistivity under
magnetic fields. At 1.25 and 2.05 GPa, the transition is changed into
crossover as expected for an FM transition. Above 2.1 GPa, the transition
temperatures decrease under magnetic fields, suggesting that the ordered phase
is AFM-like. (c) and (d) show that the field-direction dependence is small at
2.13 GPa. Inset of (e) and (f): Corresponding metamagnetic-like transition
appears in the field sweep of resistivity above 2.1 GPa, and they seem to
reach 0 T at approximately 2.82 GPa. (g) The magnetic-field-temperature phase
diagram of CeZn. The points for $j\parallel H$ at 2.13 GPa are shown by opened
triangles.

To check whether or not the magnetically ordered state terminates at $P_{c}$
while maintaining the FM state, we investigated the field response of the
ordered state. Figures 3 (a-e) show the temperature variation of the
resistivity under several magnetic fields at 1.25, 2.05, 2.13, and 2.63 GPa.
The $d^{2}\rho/dT^{2}$ is shown in the insets of Figs. 3(a-d). At 1.25 GPa,
where the FM state has been confirmed by the neutron scattering measurement
Shigeoka , the peak of $-d^{2}\rho/dT^{2}$ moves to higher temperatures under
magnetic fields and the peak structure is broadened drastically. This behavior
is a typical one for FM systems, and the phase transition is suggested to be
changed into a crossover under magnetic fields. The behavior at 2.05 GPa is
similar to that at 1.25 GPa. However, at 2.13 GPa, the resistivity becomes
insensitive to lower magnetic fields. The peak of $-d^{2}\rho/dT^{2}$ is
robust to the magnetic field and moves to lower temperatures, suggesting that
the ordered state is no longer a simple FM state. If the FM state exhibits
strong anisotropy and the magnetic field is applied along the hard axis of the
FM state, it is possible for this behavior to occur. To check this point, we
changed the direction of the pressure cell against the magnetic fields at 2.13
GPa and confirmed the similar field response, as shown in Figs. 3(c) and (d).
This excludes a possibility that the FM state remains and the magnetic easy
axis is changed above $\sim 2.1$ GPa. At 2.63 GPa, the transition temperature
of $\sim 0.82$ K at zero field clearly decreases to $\sim 0.45$ K at 1 T.
Another important feature appears in the field sweep of the resistivity, which
is shown in the inset of Fig. 3(e). Obviously, the peak structures appear at
approximately 1 T, indicative of the metamagnetic transition.

Figure 3(f) shows the field-sweep of the resistivity including the data under
different pressures. At lower pressures of 1.25, 1.52, and 2.05 GPa, there is
no peak structure. This is consistent with the interpretation that the ordered
state in this pressure range is the FM state. The peak structure starts to
appear above 2.13 GPa and becomes obvious at 2.63 GPa. The peak reaches 0 T at
2.82 GPa. Figure 3(g) shows the magnetic field-temperature phase diagram in
the tetragonal phase of CeZn. The field response is drastically changed beyond
$\sim 2.1$ GPa. Below 2.1 GPa, the temperature showing the anomaly increases,
whereas it decreases above 2.1 GPa. At 2.63 GPa, the peaks of
$-d^{2}\rho/dT^{2}$ and $\rho(H)$ are consistent with one another, and the
magnetically ordered phase is closed under the magnetic field of $\sim 1$ T.
Therefore, it is conjectured that the FM state is modified into the AFM-like
state above $\sim 2.1$ GPa. Figure 1(b) shows the pressure-temperature phase
diagram focusing on the magnetically ordered phase. We distinguished the
region, where the metamagnetic-like behavior is observed, as a different
phase. The phase boundary between the FM phase and the AFM-like phase is not
clearly detected in this study. We consider that the phase boundary between
two ordered phases is of first order, because the field responses for two
phases seem not to be connected smoothly under magnetic fields. If this is the
case, the mixed state may be realized near 2.1 GPa, and the volume fraction of
each phase is probably changed under magnetic fields. The slope of the
pressure dependence of the transition temperature changes at approximately 2.3
GPa. This change indicates that the magnetostriction at the phase transition
is different between the two regions, supporting the modification of the
ordered state.

The switching of the ordered state occurs when the Curie temperature reaches
approximately $3$ K. The obtained phase diagram strongly impresses the
difficulty of the realization of the FM QCP. What prevents the FM QCP in CeZn
is an interesting issue. In disordered systems, the spin-glass phase often
appears just before the FM QCP is reached Brando . This should be excluded in
CeZn, because CeZn is a stoichiometric and clean system, as demonstrated by
the low residual resistivity. A possible scenario is that an inhomogeneous
magnetic phase is formed by quantum fluctuations Conduit ; Efremov ;
Karahasanovic . The appearance of the new phase at low temperature implies a
contribution of the quantum fluctuation for the formation of the phase. The
residual resistivity likely increases above 2.1 GPa, as shown in Fig. 2(e),
suggesting that a complex magnetic structure is realized in the AFM-like
phase. Another scenario is the accidental change of the magnetic interaction
(RKKY interaction) through the enhancement of the hybridization between the
conduction electrons and $f$-electrons. It is crucial to know the magnetic
structure in the new phase to reveal the mechanism of switching of the phase.
It is also interesting to compare with the case in LaCrGe3, where the magnetic
correlations are insensitive irrespective of the change in the ground state
Rana .

In summary, we investigated the quantum phase transition in CeZn, in which the
FM state appears in the pressure-induced tetragonal phase. We observed the
suppression of the magnetically-ordered phase under a pressure of $P_{c}\sim
3$ GPa. However, the sudden change in the field response suggests that the FM
state is changed into another AFM-like state before the QCP is reached. The
switching of the ordered state at low temperature suggests an interesting
mechanism behind it. CeZn is an example to understand how the FM QCP is
avoided in the actual material.

## Acknowledgements

This work was supported by JSPS KAKENHI Grant Numbers JP15H05882, JP15H05885,
JP18H04320, JP18H04321, and JP21K03446.

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