# On the relation between Preference Reversal and Strategy-Proofness

K. P. S. Bhaskara Rao
Department of Computer Information Systems
Indiana University Northwest, Gary, IN 46408
E-mail<EMAIL_ADDRESS>
Achille Basile
Dipartimento di Scienze Economiche e Statistiche
Università Federico II, 80126 Napoli, Italy
E-mail<EMAIL_ADDRESS>
Surekha Rao
School of Business and Economics
Indiana University Northwest, Gary, IN 46408
E-mail<EMAIL_ADDRESS>Corresponding author.

###### Abstract

We analyze the relation between strategy-proofness and preference reversal in
the case that agents may declare indifference. Interestingly, Berga and Moreno
(2020), have recently derived preference reversal from group strategy-
proofness of social choice functions on strict preferences domains if the
range has no more than three elements. We extend this result and at the same
time simplify it. Our analysis points out the role of individual strategy-
proofness in deriving the preference reversal property, giving back to the
latter its original individual nature (cfr. Eliaz, 2004).

Moreover, we show that the difficulties Berga and Moreno highlighted relaxing
the assumption on the cardinality of the range, disappear under a proper
assumption on the domain. We introduce the concept of complete sets of
preferences and show that individual strategy-proofness is sufficient to
obtain the preference reversal property when the agents’ feasible set of
orderings is complete. This covers interesting cases like single peaked
preferences, rich domains admitting regular social choice functions, and
universal domains. The fact that we use individual rather than group strategy-
proofness, allows to get immediately some of the known, and some new,
equivalences between individual and group strategy-proofness. Finally, we show
that group strategy-proofness is only really needed to obtain preference
reversal if there are infinitely many voters.

JEL Code: D71

Mathematics Subject Classification: 91B14

Keywords: Social choice function, Preference reversal, Strategy-proofness.

##  1 Introduction

In a framework where individuals from a society $V=\\{1,2,\dots,n\\}$ express
strict preferences on a set $A=\\{a,b,c,\dots\\}$ of alternatives, Berga and
Moreno [3] recently conducted a study on the relation between group strategy-
proofness and a property of a social choice function, namely preference
reversal, introduced by Eliaz [4].

Group strategy-proofness is a well known strengthening of the usual (i.e.
individual) strategy-proofness: it prevents a social choice rule from being
manipulated by groups of individuals (see Barberà et al. [2] for a sharp
distinction between group and individual strategy-proofness). Preference
reversal has been introduced, for strict preference orders, as a property of a
social choice function that looks rather natural and innocuous, one is tempted
to say. A social choice function satisfies the preference reversal property
if: given that $\,a\,$ is the social choice corresponding to a profile $P$ of
preferences and $\,b\,$ is chosen instead if a different profile $Q$ prevails,
then at least one individual that was preferring $a$ to $b$ in the profile $P$
must have moved to prefer $b$ to $a$ under $Q$.

The interesting finding of Berga and Moreno is that group strategy-proofness
is implied by the preference reversal property [3, Theorem 3] and that the two
properties are equivalent in case the set $A$ has at most three alternatives
[3, Theorem 4]. The equivalence does not hold true in general, as they show in
[3, Example 5]

Our purpose in this short note is twofold. First, in the case that agents may
declare indifference, a model more general than the one considered by Berga
and Moreno [3], we formulate a definition of the property of preference
reversal and, through a simple and short induction argument, we show the
direct role played by the individual strategy-proofness in obtaining the
conclusion of [3, Theorem 4]. To be precise we show that: if the range of a
social choice function has cardinality at most three, then individual
strategy-proofness is sufficient to obtain the preference reversal property.
The Berga and Moreno [3, Theorem 4] is a consequence of our result. The
Barberà et al. [2, Corollary 1], according to which individual and group
strategy-proofness are equivalent when the range of a social choice function
has cardinality at most three, also follows from result. In our framework, we
also introduce the notion of almost preference reversal property, and observe
that it is equivalent to group strategy-proofness.

Secondly, we show that the difficulties highlighted by [3, Example 5] when the
range of the social choice function has cardinality more than 3, disappear
under an appropriate assumption on the domain. We introduce the concept of
complete sets of preferences and show that individual strategy-proofness is
sufficient to obtain the preference reversal property when the agents’
feasible set of preferences is complete. This covers the cases of universal
domain, singe peaked preferences, and the case of rich domains admitting
regular social choice functions ([6] and [2]). Finally, we describe a simple
trick and show that our results can be extended to infinite sets of
individuals when there are finitely many alternatives in $A,$ and group
strategy-proofness is assumed. This marks a distinction between group and
individual strategy-proofness as a sufficient condition for the preference
reversal property.

The paper proceeds by presenting the model in Section 2, the results and their
discussion in Section 3. Conclusions are presented in Section 4.

##  2 The Model

Let $A$ and $V$ be arbitrary sets that represent, respectively, the set of the
available alternatives $\,\,a,b,c,\dots\,\,$ and the set of agents $\,v\,$.
Let ${\cal W}$ be the set of weak orderings, or weak preferences, or simply
preferences, for short, over the set $A$. Its subset consisting of strict
orderings over $A$ will be denoted by ${\cal S}$. With reference to an element
$W\in{\cal W}$, the notation $x\quad{\underset{\sim}{\succ}}_{W}y$ stands for
$(x,y)\in W$, the notation $x\,\underset{W}{\succ}\,y$ stands for
$\big{[}(x,y)\in W$ and $(y,x)\notin W\big{]}$, the notation
$x\,\underset{W}{\sim}\,y$ stands for $\big{[}(x,y)\in W$ and $(y,x)\in
W\big{]}$.

A preference profile is $P=(P_{v})_{v\in V}$ where $P_{v}\in{\cal W}$ for
every voter $v\in V$.

Let ${\cal P}$ be a subset of the class ${\cal W}^{V}$ of all preference
profiles. A social choice function (scf, for short) is a mapping
$\phi:\mbox{\boldmath${\cal P}$}\to A.$ The model assumes that for every agent
$v$ there is a feasible set ${\cal W}_{v}\subseteq{\cal W}$ of preference
relations to be used in contributing to the formation of a preference profile.
In other words we assume that ${\cal P}$ is a cartesian product

$\mbox{\boldmath${\cal P}$}=\\{P\in{\cal W}^{V}:P_{v}\in{\cal W}_{v},\,\forall
v\in V\\}=\underset{v\in V}{\times}\,{\cal W}_{v},$

the sets ${\cal W}_{v}$ of preferences being primitives of the model.

The literature refers to the special cases

\- ${\cal W}_{v}={\cal W}$ for all $v\in V$, and

\- ${\cal W}_{v}={\cal S}$ for all $v\in V$

as to the Universal Domain Hypothesis (one may wish to add the distinction
weak or strict, with obvious meaning). The, indeed more general, case we
consider here is referred to as the Restricted Domain Hypothesis.

For a profile $P=(P_{v})_{v\in V}$ we shall also use the notation
$P=[P_{I},P_{I^{c}}]$ if $I$ is a subset of $V$ and $P_{I}$, $P_{I^{c}}$ are
the obvious restrictions $P_{I}=(P_{v})_{v\in I}$, $P_{I^{c}}=(P_{v})_{v\notin
V}$ of $P$111We denote by the superscript c the complement of a set..

We also shorten $\phi([P_{I},P_{I^{c}}])$ as $\phi(P_{I},P_{I^{c}})$, and, if
the set $I$ consists of one player $\\{v\\}$, we write $P_{v}$ and $P_{-v}$
instead of $P_{\\{v\\}}$ and $P_{\\{v\\}^{c}}$, as it is usual.

We now introduce the notion of Preference Reversal property in the case of
weak orders.

###### Definition 2.1

A scf $\phi$ satisfies the Preference Reversal property if:

$\phi(P)\neq\phi(Q)\implies\exists v\in V$ such that
$\left\\{\begin{array}[]{ll}\phi(P)&\,{{\underset{\sim}{\succ}}}_{P_{v}}\phi(Q)\\\
\phi(Q)&\,{{\underset{\sim}{\succ}}}_{Q_{v}}\phi(P)\\\
P_{v}&\neq\,Q_{v}\end{array}\right.$

When individuals are not allowed to declare indifference, i.e. ${\cal
W}_{v}\subseteq{\cal S}$ for all $v\in V$, then Definition 2.1 coincides with
[3, Definition 2].

The notion of almost preference reversal property is introduced next with the
aim of characterizing group strategy-proofness, as the Proposition 3.1 shows.

###### Definition 2.2

A scf $\phi$ satisfies the Almost Preference Reversal property if:

$\phi(P)\neq\phi(Q)\implies\exists v\in V$ such that
$\left\\{\begin{array}[]{ll}\phi(P)&{{\underset{\sim}{\succ}}}_{P_{v}}\phi(Q)\\\
P_{v}&\neq\,Q_{v}\end{array}\right.$

Note that Almost Preference Reversal Property requires only that there exist
two voters $v$ and $w$ such that
$\phi(P)\,\,{{\underset{\sim}{\succ}}}_{P_{v}}\phi(Q)$ and
$\phi(Q)\,\,{{\underset{\sim}{\succ}}}_{Q_{w}}\phi(P)$, and that $v$ need not
be same as $w.$ In the Preference Reversal property we are demanding that
there is a unique $v$ that satisfies both the properties.

Let $\phi$ be scf, $D$ a coalition of voters, and $P\in\mbox{\boldmath${\cal
P}$}$ a profile of preferences. We recall that if there is a partial profile
$Q_{D}=(Q_{v})_{v\in D}$, with $Q_{v}\in{\cal W}_{v}$, such that every
individual $v$ of the coalition D prefers $\phi(Q_{D},P_{D^{c}})$ to $\phi(P)$
according to $P_{v}$, i.e.
$\phi(Q_{D},P_{D^{c}})\underset{P_{v}}{\succ}\,\phi(P),\,\forall v\in D$, then
one says that the coalition $D$ can manipulate $P$ under $\phi$ by presenting
the profile $Q_{D}$.

###### Definition 2.3

A scf $\phi$ is said to be group strategy-proof (GSP, for short) if no
coalition of voters can manipulate any profile under $\phi$.

If in the definition of GSP we replace arbitrary coalitions with singleton
sets, we have the notion of individual strategy-proofness (ISP, for short).

##  3 Results

We start by showing that almost preference reversal property characterizes
group strategy-proofness.

Evidently, the definition of group strategy-proofness can be reformulated by
requiring the validity of the following implication for a scf $\phi$:

$\big{[}D\mbox{ is a coalition, }P,Q\in\mbox{\boldmath${\cal P}$}\mbox{ are
profiles identical on }D^{c}\big{]}\Rightarrow\big{[}\exists v\in
D:\,\phi(P){{\underset{\sim}{\succ}}}_{P_{v}}\phi(Q)\big{]}.$

and this allows us to state the following proposition.

###### Proposition 3.1

A scf is group strategy-proof if and only if it satisfies the almost
preference reversal property.

proof: That group strategy-proofness gives the almost preference reversal
property, as a matter of fact, has been already observed above.

For the converse, let us negate GSP for $\phi$: for a coalition $D$, two
profiles $P$ and $Q=[Q_{D},P_{D^{c}}]$ we have
$\phi(Q_{D},P_{D^{c}})\underset{P_{v}}{\succ}\phi(P),\,\forall v\in D$. If
$\phi$ satisfies the almost preference reversal property, there is a voter
$v$, necessarily in $D$ since the condition $P_{v}\neq Q_{v}$ is also assumed,
with $\phi(P){{\underset{\sim}{\succ}}}_{P_{v}}\phi(Q)$, a contradiction.
$\Box$

In [3, Theorem 3] it was shown that group strategy-proofness follows from
preference reversal property. We have provided above the missing link in the
sense that our definition of almost preference reversal property does in fact
characterize group strategy-proofness. Proposition 3.1, when restricted to
strict profiles, supplements [3, Theorem 3] with a converse result.

Evidently, the property of preference reversal gives the individual strategy-
proofness of a scf. The following theorem provides the first of our results
about reversing the implication. It generalizes Berga and Moreno [3, Theorem
4] and, at the same time, provides a very short induction argument to show the
validity of the result.

###### Theorem 3.2

Let $\phi$ be a scf. If $V$ is finite and the range of $\phi$ has cardinality
at most 3, then:

$\phi$ is ISP $\Rightarrow\phi$ satisfies the Preference Reversal property.

proof: We prove the theorem by induction on the cardinality of the society
$V$. Clearly the theorem is true when $|V|=1.$ So let us suppose that the
theorem is true when $|V|=n$ and prove that it is also true when $|V|=n+1$.

Assume $a=\phi(P)\neq\phi(Q)=b$ with $P,Q\in{\cal W}_{1}\times\dots\times{\cal
W}_{n}\times{\cal W}_{n+1}$. Take a voter $v$ for which $P_{v}\neq Q_{v}.$

Since the range of our scf is at most of cardinality three,
$\phi(P_{v},Q_{-v})$ can be either $a,$ or $b$ or, possibly, a third value
$c.$

According to the values of $\phi(P_{v},Q_{-v})$ we discuss the following
cases:

Case 1: $\phi(P_{v},Q_{-v})=a.$ We also know that $\phi(Q_{v},Q_{-v})=b.$ An
application of ISP property of $\phi$ gives us that
$a\,\,{{\underset{\sim}{\succ}}}_{P_{v}}b$ and
$b\,\,{{\underset{\sim}{\succ}}}_{Q_{v}}a$.

Case 2: $\phi(P_{v},Q_{-v})=b.$ We also know that $\phi(P_{v},P_{-v})=a.$ We
consider the function $\phi(P_{v},\ldots).$ This is ISP. Hence by the
induction hypothesis there is voter $w\in V\setminus\\{v\\}$ that witnesses
preference reversal.

Case 3: $\phi(P_{v},Q_{-v})=c.$ We will divide this into three sub-cases
according to the values of $\phi(Q_{v},P_{-v}).$ If we have
$\phi(Q_{v},P_{-v})=a,$ we consider the function $\phi(Q_{v},\ldots)$ and use
induction as in Case 2 above.

The subcase ( $\phi(P_{v},Q_{-v})=c$ and ) $\phi(Q_{v},P_{-v})=b,$ is similar
to Case 1 above since $\phi(P_{v},P_{-v})=a.$

Finally we have to analyze the subcase: $\phi(P_{v},Q_{-v})=c$ and
$\phi(Q_{v},P_{-v})=c.$

Since $\phi(P_{v},Q_{-v})=c$ and $\phi(Q_{v},Q_{-v})=b,$ by ISP we have
$c\,\,{{\underset{\sim}{\succ}}}_{P_{v}}b$ and
$b\,\,{{\underset{\sim}{\succ}}}_{Q_{v}}c$.

Since $\phi(Q_{v},P_{-v})=c$ and $\phi(P_{v},P_{-v})=a,$ by ISP we have
$c\,\,{{\underset{\sim}{\succ}}}_{Q_{v}}a$ and
$a\,\,{{\underset{\sim}{\succ}}}_{P_{v}}c$. By transitivity, we see that
$a\,\,{{\underset{\sim}{\succ}}}_{P_{v}}b$ and
$b\,\,{{\underset{\sim}{\succ}}}_{Q_{v}}a$. Thus we have preference reversal.
$\Box$

Comparing with [3, Theorem 4], we see that our result

\- clearly points out the individual nature of the Preference Reversal
property,

\- it is given for the more general setting of weak orderings,

\- and at the same time presents a very simple proof.

If we consider only strict orderings as in [3], formally [3, Theorem 4] and
Theorem 3.2 can be seen as equivalent. The latter, indeed, can be obtained
from the former because when the range of $\phi$ has cardinality at most
three, it is known from [2, Corollary 1] that
$\phi\,\,ISP\Rightarrow\phi\,\,GSP$. Yet, as discussed in Remark 3.15, our
direct approach gives some insights about the implication
$\phi\,\,ISP\Rightarrow\phi\,\,GSP$, and in particular [2, Corollary 1] is a
corollary of Theorem 3.2.

###### Remark 3.3

Theorem 3.2 cannot be extended without the restriction on the range of the
scf. This is clarified by the example [3, Example 5] of Berga and Moreno.
However, if we assume some properties for the domain of the scf, we can give
up on the condition on the range of $\phi.$

The simplest possibility is to assume the Universal Domain hypothesis.
Certainly, the Preference Reversal property is a trivial consequence of the
Gibbard-Satterthwaite Theorem if one remains confined to use strict
preferences. However dealing with weak orderings one must be a bit more
accurate in the verification that an ISP scf satisfies the Preference Reversal
property.222The following would be a possible argument. If we suppose that:
$\phi:{\cal W}^{n}\to A$ is ISP, its range has more than three elements, and
$a=\phi(P)\neq\phi(Q)=b$, let $v$ be a dictator. By definition  ([5,
Definition 6.4]) : $a$ is one of the top elements for $P_{v}$ and $b$ is one
of the top elements for $Q_{v}$. The conclusion that the Preference Reversal
is satisfied would be achieved if it were true that one has $P_{v}\neq Q_{v}$.
But this cannot be guaranteed in general (unless the domain of $\phi$ is
${\cal S}^{n}$ instead of ${\cal W}^{n}$). So, let us suppose that this is not
the case and $P_{v}=Q_{v}$. In such a case we could repeat the same induction
argument used in the proof of Theorem 3.2 to obtain the Preference Reversal
property.

Remarkably, finer results can be obtained in a surprisingly simple and direct
way, that is without appealing to the Gibbard-Satterthwaite Theorem. Again
induction will be used to achieve this.

Let us introduce some useful definitions, and provide illustrative examples.

###### Definition 3.4

Given a weak ordering $P$ and an alternative $a,$ we say that a weak ordering
$W$ resolves the $P$-indifference at $a$ if

$a\,\,{{\underset{\sim}{\succ}}}_{P}x,\,x\neq a\Rightarrow
a\underset{W}{\succ}x.$

###### Definition 3.5

Given two alternatives $a\neq b,\in A$ and the weak orderings $P,Q,W$, we say
that $W$ is an $(a,b)$-resolvent of $(P,Q)$ if $W$ at the same time resolves
the $P$-indifference at $a$ and the $Q$-indifference at $b$.

Clearly an $(a,b)$-resolvent of a pair of profiles $(P,Q)$ can only exist if
the conditions $a\,\,{{\underset{\sim}{\succ}}}_{P}b$ and
$b\,\,{{\underset{\sim}{\succ}}}_{Q}a$ are not simultaneously true.

###### Example 3.6

Let us denote by $W_{(a,b)}$ a weak ordering such that $a\underset{W}{\succ}b$
and all the other alternatives are, according to $W$, strictly below $b$, i.e.
$W_{(a,b)}$ denotes an arbitrary element of the set ${\cal
O}_{(a,b)}=\\{W\in{\cal
W}:a\underset{W}{\succ}b\underset{W}{\succ}A\setminus\\{a,b\\}\\}.$

Let us suppose that the conditions $a\,\,{{\underset{\sim}{\succ}}}_{P}b$ and
$b\,\,{{\underset{\sim}{\succ}}}_{Q}a$ are not simultaneously true. Then we
immediately see that:

\- if $a\underset{Q}{\succ}b$, then $W_{(a,b)}$ is a resolvent of $(P,Q)$, for
all $P$.

\- if $b\underset{P}{\succ}a$, then $W_{(b,a)}$ is a resolvent of $(P,Q)$, for
all $Q$. $\Box$

###### Example 3.7

Let us suppose that the conditions $a\,\,{{\underset{\sim}{\succ}}}_{P}b$ and
$b\,\,{{\underset{\sim}{\succ}}}_{Q}a$ are not simultaneously true, for given
pairs $(a,b)$ of distinct alternatives, and $(P,Q)$ of preferences. If
$a\underset{Q}{\succ}b$, let us denote by $W^{\prime}(a,b,P,Q)$ a weak
ordering such that, omitting $(a,b,P,Q)$ for simplicity below,

  * •

the alternatives in the set $L(Q,b):=\\{x\in A:x\neq
b,b\,\,{{\underset{\sim}{\succ}}}_{Q}x\\}$ are $W^{\prime}$-indifferent

  * •

the alternatives in the set $L(P,a)\setminus L(Q,b)$ are
$W^{\prime}$-indifferent

  * •

$a\underset{W^{\prime}}{\succ}L(P,a)\setminus
L(Q,b)\underset{W^{\prime}}{\succ}b\underset{W^{\prime}}{\succ}L(Q,b)$

Then we immediately see that $W^{\prime}(a,b,P,Q)$ is an $(a,b)$-resolvent of
$(P,Q)$.

If $b\underset{P}{\succ}a$, then $W^{\prime}(b,a,Q,P)$ is an $(a,b)$-resolvent
of $(P,Q)$. $\Box$

###### Example 3.8

Examples can be also presented with reference to single peaked preferences.
Let us suppose that the set $A$ of alternatives is identified with the ordered
set of the first $k$ natural numbers $1,2,3,\dots,k$. Let $P,Q$ be single
peaked preferences and $a$ and $b$ be two alternatives. We will see that there
is a single peaked $W$ that is an $(a,b)$-resolvent of $(P,Q)$. Under the
assumption that the conditions $a\,\,{{\underset{\sim}{\succ}}}_{P}b$ and
$b\,\,{{\underset{\sim}{\succ}}}_{Q}a$ are not simultaneously true, a single
peaked resolvent can be defined as follows.

In case $b\underset{P}{\succ}a$ a desired resolvent $W$ has $b$ as unique
$W$-top element, and further, in case $b<a$,

  * •

$a\underset{W}{\succ}b-1$

  * •

$b\underset{W}{\succ}b+1\underset{W}{\succ}b+2\underset{W}{\succ}\dots\underset{W}{\succ}a\underset{W}{\succ}a+1\underset{W}{\succ}\dots\underset{W}{\succ}k$

  * •

$b\underset{W}{\succ}b-1\underset{W}{\succ}b-2\underset{W}{\succ}\dots\underset{W}{\succ}1$

Remaining in the case $b\underset{P}{\succ}a$, but with $b>a$, $W$ has $b$ as
unique $W$-top element, and further,

  * •

$a\underset{W}{\succ}b+1$

  * •

$b\underset{W}{\succ}b+1\underset{W}{\succ}b+2\underset{W}{\succ}\dots\underset{W}{\succ}k$

  * •

$b\underset{W}{\succ}b-1\underset{W}{\succ}b-2\underset{W}{\succ}\dots\underset{W}{\succ}a\underset{W}{\succ}a-1\underset{W}{\succ}a-2\underset{W}{\succ}\dots\underset{W}{\succ}1$.

In case we have $a\underset{Q}{\succ}b$, the example we can provide is as
above, just replacing the role of $a$ and $b$.$\Box$

###### Definition 3.9

A set ${\cal O}\subseteq{\cal W}$ of weak orderings is said to be complete if
it contains an $(a,b)$-resolvent $W$ of $(P,Q)$ whenever $P,Q\in{\cal
O},\,a\neq b\in A,\,$ and the conditions
$a\,\,{{\underset{\sim}{\succ}}}_{P}b$ and
$b\,\,{{\underset{\sim}{\succ}}}_{Q}a$ are not simultaneously true.

On the basis of Examples 3.6, 3.7 and 3.8, and by direct comparison with the
definitions of

\- rich domains ([6, Definition 6], [2, Definition 9])

\- regular scfs ([6, Definition 3], [2, Definition 10])

we can state what follows.

###### Proposition 3.10

The following are complete sets of preferences:

  * •

the set ${\cal W}$, of all preferences (and so is the set ${\cal S}$);

  * •

the set of all single-peaked preferences;

  * •

every set ${\cal O}$ of preferences such that for every $a\neq b$ a preference
$W_{(a,b)}\in\cal O$.

  * •

every set ${\cal O}$ of preferences such that for every pair $(a,b)$ of
distinct alternatives and every pair $(P,Q)$ of preferences, then a preference
$W^{\prime}(a,b,P,Q)\in\cal O$.

  * •

every rich set ${\cal O}$ of preferences such that ${\cal O}^{V}$ is the
domain of an onto, regular scf.

When agents may select their preferences from a complete subset of ${\cal W}$,
the restriction of no more than three alternatives in the range can be relaxed
in order to obtain the preference reversal property from the strategy-
proofness. This is the content of the next theorem.

###### Theorem 3.11

Suppose $V$ is finite. Let $\phi:{\cal W}_{1}\times\dots\times{\cal W}_{n}\to
A$ be an ISP scf. If for every agent $v$ the feasible set ${\cal W}_{v}$ of
preferences is complete, then $\phi$ satisfies the Preference Reversal
property.

proof: We prove the theorem by induction on the cardinality of the society
$V$. Clearly the theorem is true when $|V|=1.$ So let us suppose that the
theorem is true when $|V|=n$ and prove that it is also true when $|V|=n+1$.

Assume $a=\phi(P)\neq\phi(Q)=b$ with $P,Q\in{\cal W}_{1}\times\dots\times{\cal
W}_{n}\times{\cal W}_{n+1}$.

If for the first voter we have $P_{1}=Q_{1}$, since the function
$\phi(P_{1},\dots)$ is ISP, by the induction hypothesis we shall get
immediately a voter $v\geq 2$ that witnesses preference reversal. So we can
suppose that $P_{1}\neq Q_{1}$. If the conditions
$a\,\,{{\underset{\sim}{\succ}}}_{P_{1}}b$ and
$b\,\,{{\underset{\sim}{\succ}}}_{Q_{1}}a$ are simultaneously true, then voter
1 itself witnesses preference reversal. If this is not the case, let
$W\in{\cal W}_{1}$ be an $(a,b)$-resolvent of $(P_{1},Q_{1})$.

Since $\phi$ is ISP, then also $\phi(W,P_{-1})=a$. Let us show this.

Suppose $\phi(W,P_{-1})=x$.

For the value $x$ it is not possible that $x\underset{P_{1}}{\succ}a$, because
we also have $\phi(W,P_{-1})=x$, $\phi(P_{1},P_{-1})=a$. Therefore we
necessarily have $a\,\,{{\underset{\sim}{\succ}}}_{P_{1}}x$, and if $x\neq a$,
the resolvent properties say that $a\underset{W}{\succ}x$. This together with
$\phi(W,P_{-1})=x$, $\phi(P_{1},P_{-1})=a$, violates ISP, hence the proof that
$\phi(W,P_{-1})=a$ is concluded.

Also we can say that $\phi(W,Q_{-1})=b$. Indeed assume $\phi(W,Q_{-1})=x$.

For the value $x$ it is not possible that $x\underset{Q_{1}}{\succ}b$, because
we also have $\phi(W,Q_{-1})=x$, $\phi(Q_{1},Q_{-1})=b$.

Therefore we necessarily have $b\,\,{{\underset{\sim}{\succ}}}_{Q_{1}}x$, and
if $x\neq b$, the resolvent properties say that $b\underset{W}{\succ}x$. This
together with $\phi(W,Q_{-1})=x$, $\phi(Q_{1},Q_{-1})=b$, violates ISP. So it
remains only the possibility that $\phi(W,Q_{-1})=b$. Consequently, applying
the induction hypothesis to $\phi(W,\dots)$ again we shall get immediately a
voter $v\geq 2$ that witnesses preference reversal. $\Box$

After Proposition 3.1 and Theorems 3.2 and 3.11 we can summarize as follows
our results up to now.

###### Theorem 3.12

For a social choice function $\phi:\underset{v\in V}{\times}\,{\cal W}_{v}\to
A$, the following is true:

  * (i)

$\phi$ satisfies the Preference Reversal property $\Rightarrow\phi$ satisfies
the Almost Preference Reversal property $\Leftrightarrow\phi$ is GSP
$\Rightarrow\phi$ is ISP.

  * (ii)

If $V$ is finite and $\phi$ either takes at most three values or if the
feasible set of orderings of every voter is complete, then we have further
that:

$\phi$ is ISP $\Rightarrow$ $\phi$ satisfies the Preference Reversal property,
that is in this case Preference Reversal, Almost Preference Reversal, Group
Strategy-Proofness, Individual Strategy-proofness are equivalent properties of
a scf.

Since in our model agents may declare indifference, we cannot add, as a
further equivalence, that $\phi$ is a dictatorial function ([5, Definition
6.4]) as the following example shows.

###### Example 3.13

Consider a society with voters $\\{1,2\\}$ and alternatives $A=\\{a,b,c\\}$.
Define on the universal domain a scf $\phi$ as follows.

$\phi(P)$ is the $P_{1}$-top alternative, if such an alternative is unique.
Otherwise among the $P_{1}$-top alternatives we select as $\phi(P)$, the
alphabetical first element of the ${P_{2}}^{-1}$\- top elements.

This scf is dictatorial by definition. But it is not ISP 333Indeed consider
$P=[P_{1},P_{2}],Q=[P_{1},Q_{2}]$ where
$a\underset{P_{1}}{\sim}b\underset{P_{1}}{\succ}c$,
$b\underset{P_{2}}{\succ}\,a$, $\,Q_{2}=P_{2}^{-1}$.. Also, $\phi$ does not
satisfy the Preference Reversal property. So the obviousness that a
dictatorial scf satisfies the Preference Reversal property for the case of
strict orderings, is not true when dealing with weak orderings.

The finiteness of $V$ plays a crucial role in the arguments given before.
However, as we are going to show, the results can be extended to infinite
societies if the individual strategy-proofness is strengthened by assuming
group strategy-proofness. We will reduce, in the way shown by the proof of the
following Theorem, the infinite case to the finite one.

###### Theorem 3.14

Suppose $V$ is infinite and the set of alternatives is finite. Let $\phi$ be a
GSP scf. Then, $\phi$ satisfies the Preference Reversal property if either one
of the following hypothesis is assumed:

(1) the range of $\phi$ is of cardinality at most three,

(2) every voter has the same complete set ${\cal O}$ of feasible orderings.

proof: Since we want to show that the Preference Reversal property is true,
let us suppose $a=\phi(\bar{P})\neq\phi(\bar{Q})=b$.

Let us consider the function $v\mapsto(\bar{P}_{v},\bar{Q}_{v})$. Since its
range is finite because of our assumption on $A$, let us list the elements of
this range in the following way: $(W_{1},W^{\prime}_{1}),$
$(W_{2},W^{\prime}_{2}),\dots,$ $(W_{\alpha},W^{\prime}_{\alpha})$ and the
corresponding inverse images as $V_{1},V_{2},...,V_{\alpha}$.

For the society whose elements are $\\{1,2,\dots,\alpha\\}$, let us consider
the scf

$\Phi:\\{W_{1},W^{\prime}_{1}\\}\times\\{W_{2},W^{\prime}_{2}\\}\times\dots\times\\{W_{\alpha},W^{\prime}_{\alpha}\\}\to
A,\mbox{ in case (1) }$ $\Phi:{\cal O}^{\alpha}\to A,\mbox{ in case (2)}$

given as follows. From the profile $\pi=(\pi_{1},\dots,\pi_{\alpha})$, define
the profile $P=(P_{v})_{v\in V}$ by $P_{v}=\pi_{i},\mbox{ for }v\in V_{i}$ and
then, since $P\in\mbox{\boldmath${\cal P}$}$ by construction,
$\Phi(\pi)=\phi(P).$ Notice that $\Phi$ is GSP, and

$a=\phi(\bar{P})=\Phi(W_{1},W_{2},\dots,W_{\alpha})\mbox{ and
}b=\phi(\bar{Q})=\Phi(W^{\prime}_{1},W^{\prime}_{2},\dots,W^{\prime}_{\alpha}).$

Duly applying Theorem 3.2 or Theorem 3.11, we get the Preference Reversal
property of $\Phi$. Hence there is an $i\leq\alpha$ with $W_{i}\neq
W^{\prime}_{i}$, $a\,\,{{\underset{\sim}{\succ}}}_{W_{i}}b$, and
$b\,\,{{\underset{\sim}{\succ}}}_{W^{\prime}_{i}}a$. Since for every $v\in
V_{i}$ we have $(\bar{P}_{v},\bar{Q}_{v})=(W_{i},W^{\prime}_{i})$, we are
done. $\Box$

###### Remark 3.15

As a final remark we note that our analysis not only extends [3, Theorems 3
and 4] but also provides some insights concerning the equivalence of
individual and group strategy-proofness. Precisely:

  * -

The fact that in general an ISP three-valued scf is GSP, is a consequence of
the Preference Reversal property derived from the individual strategy-
proofness. Compare with [2, Corollary 1], where the authors go through [2,
Proposition 1 and Theorem 1] to prove the result.

  * -

In the same way we show the equivalence under the Universal Domain hypothesis.
Specifically: neither using the Gibbard-Satterthwaite Theorem444What in the
weak order case, anyway requires an extra argument, as we have seen in Remark
3.3. nor using the notion of a domain satisfying the indirect sequential
inclusion property ([2, Section 4.4] combined with [2, Theorem 2]).

  * -

We can go deeper regarding this point. Indeed, for proving the sufficiency of
ISP for the Preference Reversal property, a weaker than Universal Domain
hypothesis can be used. As proved by Theorem 3.11, it is enough to assume that
the feasible set of orderings of every voter is complete, in the sense fixed
by the Definition 3.9. In this case again individual implies group strategy-
proofness.

##  4 Conclusions

For a social choice function, we have analyzed the relation between non-
manipulability and preference reversal. Our analysis extends the recent
findings of Berga and Moreno [3] and points out the role of individual
strategy-proofness in deriving the preference reversal property. At the same
time we have clarified that the strictness of the preferences does not play a
role in the findings, whereas it is the finiteness or not of the set of agents
that determines the use of individual versus group strategy-proofness. The
fact that we were able to use individual rather than group strategy-proofness
has also allowed to get immediately some of the known, and some new,
equivalences between individual and group strategy-proofness. Further we have
characterized group strategy-proof by introducing the notion of almost
preference reversal property.

## References

  * [1]
  * [2] Barberà, S., Berga, D. and Moreno, B., Individual versus group strategy-proofness: When do they coincide?, J. of Economic Theory, 145(2010), 1648–1674. https://doi.org/10.1016/j.jet.2010.03.006
  * [3] Berga, D. and Moreno, B., Preference reversal and group strategy-proofness, Economics Letters, 196(2020). https://doi.org/10.1016/j.econlet.2020.109493.
  * [4] Eliaz, K., Social Aggregators, Soc. Choice Welf., 22(2004), 317–330. https://doi.org/10.1007/s00355-003-0291-1
  * [5] Jehle, G. A. and Reny, P. J., Advanced Microeconomic Theory, Pearson, 2011.
  * [6] Le Breton, M. and Zaporozhets, V., On the equivalence of coalitional and individual strategy-proofness properties Soc. Choice Welf., 33(2009), 287–309. https://doi.org/10.1007/s00355-008-0358-0


# Orthogonally additive, orthogonality preserving, holomorphic mappings
between C∗-algebras

Jorge J. Garcés<EMAIL_ADDRESS>Departamento de Análisis Matemático, Facultad
de Ciencias, Universidad de Granada, 18071 Granada, Spain.  ,  Antonio M.
Peralta<EMAIL_ADDRESS>Departamento de Análisis Matemático, Facultad de
Ciencias, Universidad de Granada, 18071 Granada, Spain.  ,  Daniele Puglisi
<EMAIL_ADDRESS>Department of Mathematics and Computer Sciences,
University of Catania, Catania, 95125, Italy  and  María Isabel Ramírez
Departamento de Algebra y Análisis Matemático, Universidad de Almería, 04120
Almería, Spain<EMAIL_ADDRESS>
###### Abstract.

We study holomorphic maps between C∗-algebras $A$ and $B$. When
$f:B_{A}(0,\varrho)\longrightarrow B$ is a holomorphic mapping whose Taylor
series at zero is uniformly converging in some open unit ball
$U=B_{A}(0,\delta)$ and we assume that $f$ is orthogonality preserving on
$A_{sa}\cap U$, orthogonally additive on $U$ and $f(U)$ contains an invertible
element in $B$, then there exist a sequence $(h_{n})$ in $B^{**}$ and Jordan
∗-homomorphisms $\Theta,\widetilde{\Theta}:M(A)\to B^{**}$ such that

$f(x)=\sum_{n=1}^{\infty}h_{n}\widetilde{\Theta}(a^{n})=\sum_{n=1}^{\infty}{\Theta}(a^{n})h_{n},$

uniformly in $a\in U$. When $B$ is abelian the hypothesis of $B$ being unital
and $f(U)\cap\hbox{inv}(B)\neq\emptyset$ can be relaxed to get the same
statement.

Authors partially partially supported by the Spanish Ministry of Economy and
Competitiveness, D.G.I. project no. MTM2011-23843, and Junta de Andalucía
grant FQM3737.

2010 MSC: Primary 46G20, 46L05; Secondary 46L51, 46E15, 46E50.
Keywords and phrases: C∗-algebra, von Neumann algebra, orthogonally additive
holomorphic functions, orthogonality preservers, orthomorphism, non-
commutative $L_{1}$-spaces.

##  1\. Introduction

The description of orthogonally additive $n$-homogeneous polynomial on
$C(K)$-spaces and on general C∗-algebras, developed by Y. Benyamini, S.
Lassalle, J.L.G. Llavona [1] and D. Pérez, and I. Villanueva [14] and C.
Palazuelos, A.M. Peralta and I. Villanueva [12], respectively (see also [5]
and [4, §3]), led Functional Analysts to study and explore orthogonally
additive holomorphic functions on $C(K)$-spaces (see [6, 10]) and subsequently
on general C∗-algebras (cf. [13]).

We recall that a mapping $f$ from a C∗-algebra $A$ into a Banach space $B$ is
said to be _orthogonally additive_ on a subset $U\subseteq A$ if for every
$a,b$ in $U$ with $a\perp b$, and $a+b\in U$ we have $f(a+b)=f(a)+f(b)$, where
elements $a,$ $b$ in $A$ are said to be _orthogonal_ (denoted by $a\perp b$)
whenever $ab^{*}=b^{*}a=0$. We shall say that $f$ is _additive on elements
having zero-product_ if for every $a,b$ in $A$ with $ab=0$ we have
$f(a+b)=f(a)+f(b)$. Having this terminology in mind, the description of all
$n$-homogeneous polynomials on a general C∗-algebra, $A,$ which are
orthogonally additive on the self adjoint part, $A_{sa}$, of $A$ reads as
follows (see section §2 for concrete definitions not explained here).

######  Theorem 1.

[12] Let $A$ be a C∗-algebra, $B$ a Banach space, $n\in\mathbb{N},$ and let
$P:A\to B$ be an $n$-homogeneous polynomial. The following statements are
equivalent:

  1. $(a)$

There exists a bounded linear operator $T:A\to X$ satisfying

$P(a)=T(a^{n}),$

for every $a\in A,$ and $\|P\|\leq\|T\|\leq 2\|P\|$.

  2. $(b)$

$P$ is additive on elements having zero-products.

  3. $(c)$

$P$ is orthogonally additive on $A_{sa}$.$\hfill\Box$

The task of replacing $n$-homogeneous polynomials by polynomials or by
holomorphic functions involves a higher difficulty. For example, as noticed by
D. Carando, S. Lassalle and I. Zalduendo [6, Example 2.2.], when $K$ denotes
the closed unit disc in $\mathbb{C}$, there is no entire function
$\Phi:{\mathbb{C}}\to{\mathbb{C}}$ such that the mapping $h:C(K)\to C(K)$,
$h(f)=\Phi\circ f$ factors all degree-2 orthogonally additive scalar
polynomials over $C(K)$. Furthermore, similar arguments show that, defining
$P:C([0,1])\to\mathbb{C}$, $P(f)=f(0)+f(1)^{2}$, we cannot find a triplet
$(\Phi,\alpha_{1},\alpha_{2})$, where $\Phi:C[0,1]\to\mathbb{C}$ is a
∗-homomorphism and $\alpha_{1},\alpha_{2}\in\mathbb{C}$, satisfying that
$P(f)=\alpha_{1}\Phi(f)+\alpha_{2}\Phi(f^{2})$ for every $f\in C([0,1])$.

To avoid the difficulties commented above, Carando, Lassalle and Zalduendo
introduce a factorization through an $L_{1}(\mu)$ space. More concretely, for
each compact Hausdorff space $K$, a holomorphic mapping of bounded type
$f:C(K)\to\mathbb{C}$ is orthogonally additive if and only if there exist a
Borel regular measure $\mu$ on $K$, a sequence $(g_{k})_{k}\subseteq
L_{1}(\mu)$ and a holomorphic function of bounded type $h:C(K)\to L_{1}(\mu)$
such that $\displaystyle h(a)=\sum_{k=0}^{\infty}g_{k}~{}a^{k},$ and

$f(a)=\int_{K}h(a)~{}d\mu,$

for every $a\in C(K)$ (cf. [6, Theorem 3.3]).

When $C(K)$ is replaced with a general C∗-algebra $A$, a holomorphic function
of bounded type $f:A\to\mathbb{C}$ is orthogonally additive on $A_{sa}$ if and
only if there exist a positive functional $\varphi$ in $A^{*}$, a sequence
$(\psi_{n})$ in $L_{1}(A^{**},\varphi)$ and a power series holomorphic
function $h$ in $\mathcal{H}_{b}(A,A^{*})$ such that

$h(a)=\sum_{k=1}^{\infty}\psi_{k}\cdot a^{k}\hbox{ and }f(a)=\langle
1_{{}_{A^{**}}},h(a)\rangle=\int h(a)\ d\varphi,$

for every $a$ in $A$, where $1_{{}_{A^{**}}}$ denotes the unit element in
$A^{**}$ and $L_{1}(A^{**},\varphi)$ is a non-commutative $L_{1}$-space (cf.
[13]).

A very recent contribution due to Q. Bu, M.-H. Hsu, and N.-Ch. Wong [2], shows
that, for holomorphic mappings between $C(K)$, we can avoid the factorization
through an $L_{1}(\mu)$-space by imposing additional hypothesis. Before
stating the detailed result, we shall set down some definitions.

Let $A$ and $B$ be C∗-algebras. When $f:U\subseteq A\to B$ is a map and the
condition

(1) $a\perp b\Rightarrow f(a)\perp f(b)$

(respectively,

(2) $ab=0\Rightarrow f(a)f(b)=0\ )$

holds for every $a,b\in U$, we shall say that $f$ _preserves orthogonality_ or
is _orthogonality preserving_ (respectively, $f$ _preserves zero products_) on
$U$. In the case $A=U$ we shall simply say that $f$ is _orthogonality
preserving_ (respectively, $f$ _preserves zero products_). Orthogonality
preserving bounded linear maps between C∗-algebras were completely described
in [3, Theorem 17] (see [4] for completeness).

The following Banach-Stone type theorem for zero product preserving or
orthogonality preserving holomorphic functions between $C_{0}(L)$ spaces is
established by Bu, Hsu and Wong in [2, Theorem 3.4].

######  Theorem 2.

[2] Let $L_{1}$ and $L_{2}$ be locally compact Hausdorff spaces and let
$H:B_{C_{0}(L_{1})}(0,r)\to C_{0}(L_{2})$ be a bounded orthogonally additive
holomorphic function. If $H$ is zero product preserving or orthogonality
preserving, then there exist a sequence $(\mathcal{O}_{n})$ of open subsets of
$L_{2}$, a sequence $(h_{n})$ of bounded functions from
$L_{2}\cup\\{\infty\\}$ into $\mathbb{C}$ and a mapping $\varphi:L_{2}\to
L_{1}$ such that for each natural $n$ the function $h_{n}$ is continuous and
nonvanishing on $\mathcal{O}_{n}$ and

$f(a)(t)=\sum_{n=1}^{\infty}h_{n}(t)\left(a(\varphi(t))\right)^{n},(t\in
L_{2}),$

uniformly in $a\in B_{C_{0}(L_{1})}(0,r)$.$\hfill\Box$

The study developed by Bu, Hsu and Wong restricts to commutative C∗-algebras
or to orthogonality preserving and orthogonally additive, $n$-homogeneous
polynomials between general C∗-algebras. The aim of this paper is to extend
their study to holomorphic maps between general C∗-algebras. In Section 4, we
determine the form of every orthogonality preserving, orthogonally additive
holomorphic function from a general C∗-algebra into a commutative C∗-algebra
(see Theorem 16).

In the wider setting of holomorphic mappings between general C∗-algebras, we
prove the following: Let $A$ and $B$ be C∗-algebras with $B$ unital and let
$f:B_{A}(0,\varrho)\longrightarrow B$ be a holomorphic mapping whose Taylor
series at zero is uniformly converging in some open unit ball
$U=B_{A}(0,\delta)$. Suppose $f$ is orthogonality preserving on $A_{sa}\cap
U$, orthogonally additive on $U$ and $f(U)$ contains an invertible element.
Then there exist a sequence $(h_{n})$ in $B^{**}$ and Jordan ∗-homomorphisms
$\Theta,\widetilde{\Theta}:M(A)\to B^{**}$ such that

$f(x)=\sum_{n=1}^{\infty}h_{n}\widetilde{\Theta}(a^{n})=\sum_{n=1}^{\infty}{\Theta}(a^{n})h_{n},$

uniformly in $a\in U$ (see Theorem 18).

The main tool to establish our main results is a newfangled investigation on
orthogonality preserving pairs of operators between C∗-algebras developed in
Section 3. Among the novelties presented in Section 3, we find an innovating
alternative characterization of orthogonality preserving operators between
C∗-algebras which complements the original one established in [3] (see
Proposition 14). Orthogonality preserving pairs of operators are also valid to
determine orthogonality preserving operators and orthomorphisms or local
operators on C∗-algebras in the sense employed by A.C. Zaanen [19] and B.E.
Johnson [11], respectively.

##  2\. Orthogonally additive, orthogonality preserving, holomorphic mappings
on C∗-algebras

Let $X$ and $Y$ be Banach spaces. Given a natural $n$, a (continuous)
$n$-homogeneous polynomial $P$ from $X$ to $Y$ is a mapping
$P:X\longrightarrow Y$ for which there is a (continuous) multilinear symmetric
operator $A:X\times\ldots\times X\to Y$ such that $P(x)=A(x,\ldots,x),\
\text{for every}\ x\in X.$ All the polynomials considered in this paper are
assumed to be continuous. By a $0$-homogeneous polynomial we mean a constant
function. The symbol $\mathcal{P}(^{n}X,Y)$ will denote the Banach space of
all continuous $n$-homogeneous polynomials from $X$ to $Y$, with norm given by
$\displaystyle\|P\|=\sup_{\|x\|\leq 1}\|P(x)\|.$

Throughout the paper, the word operator will always stand for a bounded linear
mapping.

We recall that, given a domain $U$ in a complex Banach space $X$ (i.e. an
open, connected subset), a function $f$ from $U$ to another complex Banach
space $Y$ is said to be _holomorphic_ if the Frchet derivative of $f$ at
$z_{0}$ exists for every point $z_{0}$ in $U$. It is known that $f$ is
holomorphic in $U$ if and only if for each $z_{0}\in X$ there exists a
sequence $\left(P_{k}(z_{0})\right)_{k}$ of polynomials from $X$ into $Y$,
where each $P_{k}(z_{0})$ is $k$-homogeneous, and a neighborhood $V_{z_{0}}$
of $z_{0}$ such that the series

$\sum_{k=0}^{\infty}P_{k}(z_{0})(y-z_{0})$

converges uniformly to $f(y)$ for every $y\in V_{z_{0}}$. Homogeneous
polynomials on a C∗-algebra $A$ constitute the most basic examples of
holomorphic functions on $A$. A holomorphic function $f:X\longrightarrow Y$ is
said to be of bounded type if it is bounded on all bounded subsets of $X$, in
this case its Taylor series at zero, $f=\sum_{k=0}^{\infty}P_{k},$ has
infinite radius of uniform convergence, i.e.
$\limsup_{k\rightarrow\infty}\|P_{k}\|^{\frac{1}{k}}=0$ (compare [7, §6.2],
see also [8]).

Suppose $f:B_{X}(0,\delta)\to Y$ is a holomorphic function and let
$\displaystyle f=\sum_{k=0}^{\infty}P_{k}$ be its Taylor series at zero which
is assumed to be uniformly convergent in $U=B_{X}(0,\delta)$. Given
$\varphi\in Y^{*}$, it follows from Cauchy’s integral formula that, for each
$a\in U$, we have:

$\varphi P_{n}(a)=\frac{1}{2\pi i}\int_{\gamma}\frac{\varphi f(\lambda
a)}{\lambda^{n+1}}d\lambda,$

where $\gamma$ is the circle forming the boundary of a disc in the complex
plane $D_{\mathbb{C}}(0,r_{1}),$ taken counter-clockwise, such that
$a+D_{\mathbb{C}}(0,r_{1})a\subseteq U$. We refer to [7] for the basic facts
and definitions used in this paper.

In this section we shall study orthogonally additive, orthogonality
preserving, holomorphic mappings between C∗-algebras. We begin with an
observation which can be directly derived from Cauchy’s integral formula. The
statement in the next lemma was originally stated by D. Carando, S. Lassalle
and I. Zalduendo in [6, Lemma 1.1] (see also [13, Lemma 3]).

######  Lemma 3.

Let $f:B_{A}(0,\varrho)\longrightarrow B$ be a holomorphic mapping, where $A$
is a C∗-algebra and $B$ is a complex Banach space, and let $\displaystyle
f=\sum_{k=0}^{\infty}P_{k}$ be its Taylor series at zero, which is uniformly
converging in $U=B_{A}(0,\delta)$. Then the mapping $f$ is orthogonally
additive on $U$ (respectively, orthogonally additive on $A_{sa}\cap U$ or
additive on elements having zero-product in $U$) if, and only if, all the
$P_{k}$’s satisfy the same property. In such a case, $P_{0}=0$.$\hfill\Box$

We recall that a functional $\varphi$ in the dual of a C∗-algebra $A$ is
_symmetric_ when $\varphi(a)\in\mathbb{R}$, for every $a\in A_{sa}$.
Reciprocally, if $\varphi(b)\in\mathbb{R}$ for every symmetric functional
$\varphi\in A^{*}$, the element $b$ lies in $A_{sa}$. Having this in mind, our
next lemma also is a direct consequence of the Cauchy’s integral formula. A
mapping $f:A\to B$ between C∗-algebras is called _symmetric_ whenever
$f(A_{sa})\subseteq B_{sa}$, or equivalently, $f(a)=f(a)^{*}$, whenever $a\in
A_{sa}$.

######  Lemma 4.

Let $f:B_{A}(0,\varrho)\longrightarrow B$ be a holomorphic mapping, where $A$
and $B$ are C∗-algebras, and let $\displaystyle f=\sum_{k=0}^{\infty}P_{k}$ be
its Taylor series at zero, which is uniformly converging in
$U=B_{A}(0,\delta)$. Then the mapping $f$ is symmetric on $U$ (i.e.
$f(A_{sa}\cap U)\subseteq B_{sa}$) if, and only if, $P_{k}$ is symmetric (i.e.
$P_{k}(A_{sa})\subseteq B_{sa}$) for every
$k\in\mathbb{N}\cup\\{0\\}$.$\hfill\Box$

######  Definition 5.

Let $S,T:A\to B$ be a couple of mappings between two C∗-algebras. We shall say
that the pair $(S,T)$ is orthogonality preserving on a subset $U\subseteq A$
if $S(a)\perp T(b)$ whenever $a\perp b$ in $U$. When $ab=0$ in $U$ implies
$S(a)T(b)=0$ in $B$, we shall say that $(S,T)$ preserves zero products on $U$.

We observe that a mapping $T:A\to B$ is orthogonality preserving in the usual
sense if and only if the pair $(T,T)$ is orthogonality preserving. We also
notice that $(S,T)$ is orthogonality preserving (on $A_{sa}$) if and only if
$(T,S)$ is orthogonality preserving (on $A_{sa}$).

Our next result assures that the $n$-homogeneous polynomials appearing in the
Taylor series of an orthogonality preserving holomorphic mapping between
C∗-algebras are pairwise orthogonality preserving.

######  Proposition 6.

Let $f:B_{A}(0,\varrho)\longrightarrow B$ be a holomorphic mapping, where $A$
and $B$ are C∗-algebras, and let $\displaystyle f=\sum_{k=0}^{\infty}P_{k}$ be
its Taylor series at zero, which is uniformly converging in
$U=B_{A}(0,\delta)$. The following statements hold:

  1. $(a)$

The mapping $f$ is orthogonally preserving on $U$ (respectively, orthogonally
preserving on $A_{sa}\cap U$) if, and only if, $P_{0}=0$ and the pair
$(P_{n},P_{m})$ is orthogonality preserving (respectively, orthogonally
preserving on $A_{sa}$) for every $n,m\in\mathbb{N}$.

  2. $(b)$

The mapping $f$ preserves zero products on $U$ if, and only if, $P_{0}=0$ and
for every $n,m\in\mathbb{N},$ the pair $(P_{n},P_{m})$ preserves zero
products.

###### Proof.

$(a)$ The “if” implication is clear. To prove the ”only if” implication, let
us fix $a,b\in U$ with $a\perp b$. Let us find two positive scalars $r,C$ such
that $a,b\in B(0,r)$, and $\|f(x)\|\leq C$ for every $x\in
B(0,r)\subset\overline{B}(0,r)\subseteq U$. From the Cauchy estimates we have
$\|P_{m}\|\leq\frac{C}{r^{m}},$ for every $m\in\mathbb{N}\cup\\{0\\}.$ By
hypothesis $f(ta)\perp f(tb)$, for every $r>t>0$, and hence

$P_{0}(ta)P_{0}(tb)^{*}+P_{0}(ta)\left(\sum_{k=1}^{\infty}P_{k}(tb)\right)^{*}+\left(\sum_{k=1}^{\infty}P_{k}(ta)\right)\left(\sum_{k=0}^{\infty}P_{k}(tb)\right)^{*}=0,$

and by homogeneity

$P_{0}(a)P_{0}(b)^{*}=-P_{0}(a)\left(\sum_{k=1}^{\infty}t^{k}P_{k}(b)\right)^{*}+\left(\sum_{k=1}^{\infty}t^{k}P_{k}(a)\right)\left(\sum_{k=0}^{\infty}t^{k}P_{k}(b)\right)^{*}.$

Letting $t\to 0$, we have $P_{0}(a)P_{0}(b)^{*}=0$. In particular, $P_{0}=0$.

We shall prove by induction on $n$ that the pair $(P_{j},P_{k})$ is
orthogonality preserving on $U$ for every $1\leq j,k\leq n$. Since
$f(ta)f(tb)^{*}=0$, we also deduce that

$P_{1}(ta)P_{1}(tb)^{*}+P_{1}(ta)\left(\sum_{k=2}^{\infty}P_{k}(tb)\right)^{*}+\left(\sum_{k=2}^{\infty}P_{k}(ta)\right)\left(\sum_{k=1}^{\infty}P_{k}(tb)\right)^{*}=0,$

for every $\frac{\min\\{\|a\|,\|b\|\\}}{r}>t>0,$ which implies that

$t^{2}P_{1}(a)P_{1}(b)^{*}=-tP_{1}(a)\left(\sum_{k=2}^{\infty}t^{k}P_{k}(b)\right)^{*}-\left(\sum_{k=2}^{\infty}t^{k}P_{k}(a)\right)\left(\sum_{k=1}^{\infty}t^{k}P_{k}(b)\right)^{*},$

for every $\frac{\min\\{\|a\|,\|b\|\\}}{r}>t>0$, and hence

$\left\|P_{1}(a)P_{1}(b)^{*}\right\|\leq
tC\|P_{1}(a)\|\sum_{k=2}^{\infty}\frac{\|b\|^{k}}{r^{k}}t^{k-2}$
$+tC^{2}\left(\sum_{k=2}^{\infty}\frac{\|a\|^{k}}{r^{k}}t^{k-2}\right)\left(\sum_{k=1}^{\infty}\frac{\|b\|^{k}}{r^{k}}t^{k-1}\right).$

Taking limit in $t\to 0$, we get $P_{1}(a)P_{1}(b)^{*}=0$. Let us assume that
$(P_{j},P_{k})$ is orthogonality preserving on $U$ for every $1\leq j,k\leq
n$. Following the argument above we deduce that

$P_{1}(a)P_{n+1}(b)^{*}+P_{n+1}(a)P_{1}(b)^{*}=-tP_{1}(a)\left(\sum_{j=n+2}^{\infty}t^{j-n-2}P_{j}(b)\right)^{*}$
$-t\sum_{k=2}^{n}t^{k-2}P_{k}(a)\left(\sum_{j=n+1}^{\infty}t^{j-n-1}P_{j}(b)\right)^{*}-tP_{n+1}(a)\left(\sum_{j=2}^{\infty}t^{j-2}P_{j}(b)\right)^{*}$
$-t\left(\sum_{k=n+2}^{\infty}t^{k-n-2}P_{k}(a)\right)\left(\sum_{j=1}^{\infty}t^{j-1}P_{j}(b)\right)^{*},$

for every $\frac{\min\\{\|a\|,\|b\|\\}}{r}>|t|>0$. Taking limit in $t\to 0$,
we have

$P_{1}(a)P_{n+1}(b)^{*}+P_{n+1}(a)P_{1}(b)^{*}=0.$

Replacing $a$ with $sa$ ($s>0$) we get

$sP_{1}(a)P_{n+1}(b)^{*}+s^{n+1}P_{n+1}(a)P_{1}(b)^{*}=0$

for every $s>0$, which implies that

$P_{1}(a)P_{n+1}(b)^{*}=0.$

In a similar manner we prove that $P_{k}(a)P_{n+1}(b)^{*}=0$, for every $1\leq
k\leq n+1$. The equalities $P_{k}(b)^{*}P_{j}(a)=0$ ($1\leq j,k\leq n+1$)
follow similarly.

We have shown that for each $n,m\in\mathbb{N}$, $P_{n}(a)\perp P_{m}(b)$
whenever $a,b\in U$ with $a\perp b$. Finally, taking $a,b\in A$ with $a\perp
b$, we can find a positive $\rho$ such that $\rho a,\rho b\in U$ and $\rho
a\perp\rho b$, which implies that $P_{n}(\rho a)\perp P_{m}(\rho b)$ for every
$n,m\in\mathbb{N}$, witnessing that $(P_{n},P_{m})$ is orthogonality
preserving for every $n,m\in\mathbb{N}$.

The proof of $(b)$ follows in a similar manner. ∎

We can obtain now a corollary which is a first step toward the description of
orthogonality preserving, orthogonally additive, holomorphic mappings between
C∗-algebras.

######  Corollary 7.

Let $f:B_{A}(0,\varrho)\longrightarrow B$ be a holomorphic mapping, where $A$
and $B$ are C∗-algebras, and let $\displaystyle f=\sum_{k=0}^{\infty}P_{k}$ be
its Taylor series at zero, which is uniformly converging in
$U=B_{A}(0,\delta)$. Suppose $f$ is orthogonality preserving on $A_{sa}\cap U$
and orthogonally additive (respectively, orthogonally additive and zero
products preserving). Then there exists a sequence $(T_{n})$ of operators from
$A$ into $B$ satisfying that the pair $(T_{n},T_{m})$ is orthogonality
preserving on $A_{sa}$ (respectively, zero products preserving on $A_{sa}$)
for every $n,m\in\mathbb{N}$ and

(3) $f(x)=\sum_{n=1}^{\infty}T_{n}(x^{n}),$

uniformly in $x\in U$. In particular every $T_{n}$ is orthogonality preserving
(respectively, zero products preserving) on $A_{sa}$. Furthermore, $f$ is
symmetric if and only if every $T_{n}$ is symmetric.

###### Proof.

Combining Lemma 3 and Proposition 6, we deduce that $P_{0}=0$, $P_{n}$ is
orthogonally additive and $(P_{n},P_{m})$ is orthogonality preserving on
$A_{sa}$ for every $n,m$ in $\mathbb{N}.$ By Theorem 1, for each natural $n$
there exists an operator $T_{n}:A\to B$ such that $\|P_{n}\|\leq\|T_{n}\|\leq
2\|P_{n}\|$ and

$P_{n}(a)=T_{n}(a^{n}),$

for every $a\in A$.

Consider now two positive elements $a,b\in A$ with $a\perp b$ and fix
$n,m\in\mathbb{N}$. In this case there exist positive elements $c,d$ in $A$
with $c^{n}=a$ and $d^{m}=b$ and $c\perp d$. Since the pair $(P_{n},P_{m})$ is
orthogonality preserving on $A_{sa}$, we have
$T_{n}(a)=T_{n}(c^{n})=P_{n}(c)\perp P_{m}(d)=T_{m}(d^{m})=T_{m}(b).$ Now,
noticing that given $a,b$ in $A_{sa}$ with $a\perp b$, we can write
$a=a^{+}-b^{-}$ and $b=b^{+}-b^{-}$, where $a^{\sigma},b^{\tau}$ are positive,
$a^{+}\perp a^{-},$ $b^{+}\perp b^{-}$ and $a^{\sigma}\perp b^{\tau},$ for
every $\sigma,\tau\in\\{+,-\\},$ we deduce that $T_{n}(a)\perp T_{m}(b)$. This
shows that the pair $(T_{n},T_{m})$ is orthogonality preserving on $A_{sa}$.

When $f$ orthogonally additive and zero products preserving the pair
$(T_{n},T_{m})$ is zero products preserving on $A_{sa}$ for every
$n,m\in\mathbb{N}$. The final statement is clear from Lemma 4. ∎

It should be remarked here that if a mapping $f:B_{A}(0,\delta)\longrightarrow
B$ is given by an expression of the form in (3) which uniformly converging in
$U=B_{A}(0,\delta)$ where $(T_{n})$ is a sequence of operators from $A$ into
$B$ such that the pair $(T_{n},T_{m})$ is orthogonality preserving on $A_{sa}$
(respectively, zero products preserving on $A_{sa}$) for every
$n,m\in\mathbb{N}$, then $f$ is orthogonally additive and orthogonality
preserving on $A_{sa}\cap U$ (respectively, orthogonally additive and zero
products preserving).

##  3\. Orthogonality preserving pairs of operators

Let $A$ and $B$ be two C∗-algebras. In this section we shall study those pairs
of operators $S,T:A\to B$ satisfying that $S,T$ and the pair $(S,T)$ preserve
orthogonality on $A_{sa}$. Our description generalizes some of the results
obtained by M. Wolff in [17] because a (symmetric) mapping $T:A\to B$ is
orthogonality preserving on $A_{sa}$ if and only if the pair $(T,T)$ enjoys
the same property. In particular, for every ∗-homomorphism $\Phi:A\to B$, the
pair $(\Phi,\Phi)$ preservers orthogonality. The same statement is true
whenever $\Phi$ is a ∗-anti-homomorphism, or a Jordan ∗-homomorphism, or a
triple homomorphism for the triple product
$\left\\{a,b,c\right\\}=\frac{1}{2}(ab^{*}c+cb^{*}a)$.

We observe that $S,T$ being symmetric implies that $(S,T)$ is orthogonality
preserving on $A_{sa}$ if and only if $(S,T)$ is zero products preserving on
$A_{sa}$. We shall offer here a newfangled and simplified proof which is also
valid for pairs of operators.

Let $a$ be an element in a von Neumann algebra $M$. We recall that the _left_
and _right_ _support projections_ of $a$ (denoted by $l(a)$ and $d(a)$) are
defined as follows: $l(a)$ (respectively, $d(a)$) is the smallest projection
$p\in M$ (respectively, $q\in M$) with the property that $pa=a$ (respectively,
$aq=a$). It is known that when $a$ is hermitian $d(a)=l(a)$ is called the
_support_ or _range projection_ of $a$ and is denoted by $s(a)$. It is also
known that, for each $a=a^{*}$, the sequence $(a^{\frac{1}{3^{n}}})$ converges
in the strong∗-topology of $M$ to $s(a)$ (cf. [15, §1.10 and 1.11]).

An element $e$ in a C∗-algebra $A$ is said to be a _partial isometry_ whenever
$ee^{*}e=e$ (equivalently, $ee^{*}$ or $e^{*}e$ is a projection in $A$). For
each partial isometry $e$, the projections $ee^{*}$ and $e^{*}e$ are called
the left and right support projections associated to $e$, respectively. Every
partial isometry $e$ in $A$ defines a Jordan product and an involution on
$A_{e}(e):=ee^{*}Ae^{*}e$ given by
$a\bullet_{{}_{e}}b=\frac{1}{2}(ae^{*}b+be^{*}a)$ and
$a^{\sharp_{{}_{e}}}=ea^{*}e$ ($a,b\in A_{2}(e)$). It is known that
$(A_{2}(e),\bullet_{{}_{e}},{\sharp_{{}_{e}}})$ is a unital JB∗-algebra with
respect to its natural norm and $e$ is the unit element for the Jordan product
$\bullet_{{}_{e}}$.

Every element $a$ in a C∗-algebra $A$ admits a _polar decomposition_ in
$A^{**}$, that is, $a$ decomposes uniquely as follows: $a=u|a|$, where
$|a|=(a^{*}a)^{\frac{1}{2}}$ and $u$ is a partial isometry in $A^{**}$ such
that $u^{*}u=s(|a|)$ and $uu^{*}=s(|a^{*}|)$ (compare [15, Theorem 1.12.1]).
Observe that $uu^{*}a=au^{*}u=u$. The unique partial isometry $u$ appearing in
the polar decomposition of $a$ is called the range partial isometry of $a$ and
is denoted by $r(a)$. Let us observe that taking $c=r(a)|a|^{\frac{1}{3}}$, we
have $cc^{*}c=a$. It is also easy to check that for each $b\in A$ with
$b=r(a)r(a)^{*}b$ (respectively, $b=br(a)^{*}r(a)$) the condition $a^{*}b=0$
(respectively, $ba^{*}=0$) implies $b=0$. Furthermore, $a\perp b$ in $A$ if
and only if $r(a)\perp r(b)$ in $A^{**}$.

We begin with a basic argument in the study of orthogonality preserving
operators between C∗-algebras whose proof is inserted here for completeness
reasons. Let us recall that for every C∗-algebra $A$, the _multiplier algebra_
of $A$, $M(A)$, is the set of all elements $x\in A^{**}$ such that for each
$Ax,xA\subseteq A$. We notice that $M(A)$ is a C∗-algebra and contains the
unit element of $A^{**}$.

######  Lemma 8.

Let $A$ and $B$ be C∗-algebras and let $S,T:A\to B$ be a pair of operators.

  1. $(a)$

The pair $(S,T)$ preserves orthogonality (on $A_{sa}$) if and only if the pair
$(S^{**}|_{M(A)},T^{**}|_{M(A)})$ preserves orthogonality (on $M(A)_{sa}$);

  2. $(b)$

The pair $(S,T)$ preserves zero products (on $A_{sa}$) if and only if the pair
$(S^{**}|_{M(A)},T^{**}|_{M(A)})$ preserves zero products (on $M(A)_{sa}$).

###### Proof.

$(a)$ The “if” implication is clear. Let $a,b$ be two elements in $M(A)$ with
$a\perp b$. We can find two elements $c$ and $d$ in $M(A)$ satisfying
$cc^{*}c=a$, $dd^{*}d=b$ and $c\perp d$. Since $cxc\perp dyd$, for every $x,y$
in $A$, we have $T(cxc)\perp T(dyd)$ for every $x,y\in A$. By Goldstine’s
theorem we find two bounded nets $(x_{\lambda})$ and $(y_{\mu})$ in $A$,
converging in the weak∗ topology of $A^{**}$ to $c^{*}$ and $d^{*}$,
respectively. Since
$T(cx_{\lambda}c)T(dy_{\mu}d)^{*}=T(dy_{\mu}d)^{*}T(cx_{\lambda}c)=0$, for
every $\lambda,\mu$, $T^{**}$ is weak∗-continuous, the product of $A^{**}$ is
separately weak∗-continuous and the involution of $A^{**}$ also is
weak∗-continuous, we get
$T^{**}(cc^{*}c)T^{**}(dd^{*}d)=T^{**}(a)T^{**}(b)^{*}=0=T^{**}(b)^{*}T^{**}(a),$
and hence $T^{**}(a)\perp T^{**}(b)$, as desired.

The proof of $(b)$ follows by a similar argument. ∎

######  Proposition 9.

Let $S,T:A\to B$ be operators between C∗-algebras such that $(S,T)$ is
orthogonality preserving on $A_{sa}$. Let us denote $h:=S^{**}(1)$ and
$k:=T^{**}(1)$. Then the identities

$S(a)T(a^{*})^{*}=S(a^{2})k^{*}=hT((a^{2})^{*})^{*},$
$T(a^{*})^{*}S(a)=k^{*}S(a^{2})=hT((a^{2})^{*})^{*}h,$
$S(a)k^{*}=hT(a^{*})^{*},\hbox{ and, }k^{*}S(a)=T(a^{*})^{*}h$

hold for every $a\in A$.

###### Proof.

By Lemma 8, we may assume, without loss of generality, that $A$ is unital.
$(a)$ For each $\varphi\in B^{*}$, the continuous bilinear form
$V_{\varphi}:A\times A\to\mathbb{C}$,
$V_{\varphi}(a,b)=\varphi(S(a)T(b^{*})^{*})$ is orthogonal, that is,
$V_{\varphi}(a,b)=0$, whenever $ab=0$ in $A_{sa}$. By Goldstein’s theorem [9,
Theorem 1.10] there exist functionals $\omega_{1},\omega_{2}\in A^{*}$
satisfying that

$V_{\varphi}(a,b)=\omega_{1}(ab)+\omega_{2}(ba),$

for all $a,b\in A$. Taking $b=1$ and $a=b$ we have

$\varphi(S(a)k^{*})=V_{\varphi}(a,1)=V_{\varphi}(1,a)=\varphi(hT(a)^{*})$

and

$\varphi(S(a)T(a)^{*})=\varphi(S(a^{2})k^{*})=\varphi(hT(a^{2})^{*}),$

for every $a\in A_{sa}$, respectively. Since $\varphi$ was arbitrarily chosen,
we get, by linearity, $S(a)k^{*}=hT(a^{*})^{*}$ and
$S(a)T(a^{*})^{*}=S(a^{2})k^{*}=hT((a^{2})^{*})^{*}$, for every $a\in A$. The
other identities follow in a similar way, but replacing
$V_{\varphi}(a,b)=\varphi(S(a)T(b^{*})^{*})$ with
$V_{\varphi}(a,b)=\varphi(T(b^{*})^{*}S(a))$. ∎

######  Lemma 10.

Let $J_{1},J_{2}:A\to B$ be Jordan ∗-homomorphism between C∗-algebras. The
following statements are equivalent:

  1. $(a)$

The pair $(J_{1},J_{2})$ is orthogonality preserving on $A_{sa}$;

  2. $(b)$

The identity

$J_{1}(a)J_{2}(a)=J_{1}(a^{2})J_{2}^{**}(1)=J_{1}^{**}(1)J_{2}(a^{2}),$

holds for every $a\in A_{sa}$;

  3. $(c)$

The identity

$J_{1}^{**}(1)J_{2}(a)=J_{1}(a)J_{2}^{**}(1),$

holds for every $a\in A_{sa}$.

Furthermore, when $J_{1}^{**}$ is unital,
$J_{2}(a)=J_{1}(a)J_{2}^{**}(1)=J_{2}^{**}(1)J_{1}(a),$ for every $a$ in $A.$

###### Proof.

The implications $(a)\Rightarrow(b)\Rightarrow(c)$ have been established in
Proposition 9. To see $(c)\Rightarrow(a)$, we observe that
$J_{i}(x)=J_{i}^{**}(1)J_{i}(x)J_{i}^{**}(1)=J_{i}(x)J_{i}^{**}(1)=J_{i}^{**}(1)J_{i}(x)$,
for every $x\in A$. Therefore, given $a,b\in A_{sa}$ with $a\perp b$, we have
$J_{1}(a)J_{2}(b)=J_{1}(a)J_{1}^{**}(1)J_{2}(b)=J_{1}(a)J_{1}(b)J_{2}^{**}(1)=0$.
∎

In [17, Proposition 2.5], M. Wolff establishes a uniqueness result for
∗-homomorphisms between C∗-algebras showing that for each pair $(U,V)$ of
unital ∗-homomorphisms from a unital C∗-algebra $A$ into a unital C∗-algebra
$B$, the condition $(U,V)$ orthogonality preserving on $A_{sa}$ implies $U=V$.
This uniqueness result is a direct consequence of our previous lemma.

Orthogonality preserving pairs of operators can be also used to rediscover the
notion of orthomorphism in the sense introduced by Zaanen in [19]. We recall
that an operator $T$ on a C∗-algebra $A$ is said to be an _orthomorphism_ or a
_band preserving_ operator when the implication $a\perp b\Rightarrow T(a)\perp
b$ holds for every $a,b\in A$. We notice that when $A$ is regarded as an
$A$-bimodule, an operator $T:A\to A$ is an orthomorphism if and only if it is
a _local operator_ in the sense used by B.E. Johnson in [11, §3]. Clearly, an
operator $T:A\to A$ is an orthomorphism if and only if $(T,Id_{A})$ is
orthogonality preserving. The following non-commutative extension of [19,
THEOREM 5] follows from Proposition 9.

######  Corollary 11.

Let $T$ be an operator on a C∗-algebra $A$. Then $T$ is an orthomorphism if
and only if $T(a)=T^{**}(1)a=aT^{**}(1)$, for every $a$ in $A$, that is, $T$
is a multiple of the identity on $A$ by an element in its center.$\hfill\Box$

We recall that two elements $a,$ $b$ in a JB∗-algebra $A$ are said to
_operator commute_ in $A$ if the multiplication operators $M_{a}$ and $M_{b}$
commute, where $M_{a}$ is defined by $M_{a}(x):=a\circ x$. That is, $a$ and
$b$ operator commute if and only if $(a\circ x)\circ b=a\circ(x\circ b)$ for
all $x$ in $A$. An useful result in Jordan theory assures that self-adjoint
elements $a$ and $b$ in $A$ generate a JB∗-subalgebra that can be realized as
a JC∗-subalgebra of some $B(H)$ (compare [18]), and, under this
identification, $a$ and $b$ commute as elements in $L(H)$ whenever they
operator commute in $A$, equivalently $a^{2}\circ b=2(a\circ b)\circ
a-a^{2}\circ b$ (cf. Proposition 1 in [16]).

The next lemma contains a property which is probably known in C∗-algebra, we
include an sketch of the proof because we were unable to find an explicit
reference.

######  Lemma 12.

Let $e$ be a partial isometry in a C∗-algebra $A$ and let $a,b$ be two
elements in $A_{2}(e)=ee^{*}Ae^{*}e$. Then $a$, $b$ operator commute in the
JB∗-algebra $(A_{2}(e),\bullet_{{}_{e}},{\sharp_{{}_{e}}})$ if and only if
$ae^{*}$ and $be^{*}$ operator commute in the JB∗-algebra
$(A_{2}(ee^{*}),\bullet_{{}_{ee^{*}}},{\sharp_{{}_{ee^{*}}}})$, where
$x\bullet_{{}_{ee^{*}}}y=x\circ y=\frac{1}{2}(xy+yx)$, for every $x,y\in
A_{2}(ee^{*})$. Furthermore, when $a$ and $b$ are hermitian elements in
$(A_{2}(e),\bullet_{{}_{e}},{\sharp_{{}_{e}}})$, $a$, $b$ operator commute if
and only if $ae^{*}$ and $be^{*}$ commute in the usual sense (i.e.
$ae^{*}be^{*}=be^{*}ae^{*}$).

###### Proof.

We observe that the mapping
$R_{e^{*}}:(A_{2}(e),\bullet_{{}_{e}})\to(A_{2}(ee^{*}),\bullet_{{}_{ee^{*}}})$,
$x\mapsto xe^{*}$ is a Jordan ∗-isomorphism between the above JB∗-algebras.
So, the first equivalence is clear. The second one has been commented before.
∎

Our next corollary relies on the following description of orthogonality
preserving operators between C∗-algebras obtained in [3] (see also [4]).

######  Theorem 13.

[3, Theorem 17], [4, Theorem 4.1 and Corollary 4.2] Let $T$ be an operator
from a C∗-algebra $A$ into another C∗-algebra $B$ the following are
equivalent:

  1. $a)$

$T$ is orthogonality preserving (on $A_{sa}$).

  2. $b)$

There exits a unital Jordan ∗-homomorphism $J:M(A)\to B_{2}^{**}(r(h))$ such
that $J(x)$ and $h=T^{**}(1)$ operator commute and

$T(x)=h\bullet_{{}_{r(h)}}J(x),\hbox{ for every $x\in A$},$

where $M(A)$ is the multiplier algebra of $A$, $r(h)$ is the range partial
isometry of $h$ in $B^{**}$, $B_{2}^{**}(r(h))=r(h)r(h)^{*}B^{**}r(h)^{*}r(h)$
and $\bullet_{{}_{r(h)}}$ is the natural product making $B_{2}^{**}(r(h))$ a
JB∗-algebra.

Furthermore, when $T$ is symmetric, $h$ is hermitian and hence $r(h)$
decomposes as orthogonal sum of two projections in $B^{**}$.$\hfill\Box$

Our next result gives a new perspective for the study of orthogonality
preserving (pairs of) operators between C∗-algebras.

######  Proposition 14.

Let $A$ and $B$ be C∗-algebras. Let $S,T:A\to B$ be operators and let
$h=S^{**}(1)$ and $k=T^{**}(1)$. Then the following statements hold:

  1. $(a)$

The operator $S$ is orthogonality preserving if and only if there exit two
Jordan ∗-homomorphisms $\Phi,\widetilde{\Phi}:M(A)\to B^{**}$ satisfying
$\Phi(1)=r(h)r(h)^{*}$, $\widetilde{\Phi}(1)=r(h)^{*}r(h),$ and
$S(a)=\Phi(a)h=h\widetilde{\Phi}(a),$ for every $a\in A$.

  2. $(b)$

$S,T$ and $(S,T)$ are orthogonality preserving on $A_{sa}$ if and only if the
following statements hold:

    1. $(b1)$

There exit Jordan ∗-homomorphisms
$\Phi_{1},\widetilde{\Phi}_{1},\Phi_{2},\widetilde{\Phi}_{2}:M(A)\to B^{**}$
satisfying $\Phi_{1}(1)=r(h)r(h)^{*}$, $\widetilde{\Phi}_{1}(1)=r(h)^{*}r(h),$
$\Phi_{2}(1)=r(k)r(k)^{*}$, $\widetilde{\Phi}_{2}(1)=r(k)^{*}r(k),$
$S(a)=\Phi_{1}(a)h=h\widetilde{\Phi}_{1}(a),$ and
$T(a)=\Phi_{2}(a)k=k\widetilde{\Phi}_{2}(a),$ for every $a\in A$;

    2. $(b2)$

The pairs $({\Phi}_{1},{\Phi}_{2})$ and
$(\widetilde{\Phi}_{1},\widetilde{\Phi}_{2})$ are orthogonality preserving on
$A_{sa}$.

###### Proof.

The “if” implications are clear in both statements. We shall only prove the
“only if” implication.

$(a)$. By Theorem 13, there exits a unital Jordan ∗-homomorphism
$J_{1}:M(A)\to B_{2}^{**}(r(h))$ such that $J_{1}(x)$ and $h$ operator commute
in the JB∗-algebra $(B_{2}^{**}(r(h)),\bullet_{{}_{r(h)}})$ and

$S(x)=h\bullet_{{}_{r(a)}}J_{1}(a)\hbox{ for every $a\in A$}.$

Fix $a\in A_{sa}$. Since $h$ and $J_{1}(a)$ are hermitian elements in
$(B_{2}^{**}(r(h)),\bullet_{{}_{r(h)}})$ which operator commute, Lemma 12
assures that $hr(h)^{*}$ and $J_{1}(a)r(h)^{*}$ commute in the usual sense of
$B^{**}$, that is,

$hr(h)^{*}J_{1}(a)r(h)^{*}=J_{1}(a)r(h)^{*}hr(h)^{*},$

or equivalently,

$hr(h)^{*}J_{1}(a)=J_{1}(a)r(h)^{*}h.$

Consequently, we have

$S(a)=h\bullet_{{}_{r(h)}}J_{1}(a)=hr(h)^{*}J_{1}(a)=J_{1}(a)r(h)^{*}h,$

for every $a\in A$. The desired statement follows by considering
$\Phi_{1}(a)=J_{1}(a)r(h)^{*}$ and $\widetilde{\Phi}_{1}(a)=r(h)^{*}J_{1}(a).$

$(b)$ The statement in $(b1)$ follows from $(a)$. We shall prove $(b2)$.

By hypothesis, given $a,b$ in $A_{sa}$ with $a\perp b$, we have

$0=S(a)T(b)^{*}=\left(h\widetilde{\Phi}_{1}(a)\right)\left(k\widetilde{\Phi}_{2}(b)\right)^{*}$
$=h\widetilde{\Phi}_{1}(a)\widetilde{\Phi}_{2}(b)^{*}k^{*}$

Having in mind that $\widetilde{\Phi}_{1}(A)\subseteq r(h)^{*}r(h)B^{**}$ and
$\widetilde{\Phi}_{2}(A)\subseteq B^{**}r(k)^{*}r(k)$, we deduce that
$\widetilde{\Phi}_{1}(a)\widetilde{\Phi}_{2}(b)^{*}=0$ (compare the comments
before Lemma 8), as we desired. In a similar fashion we prove
$\widetilde{\Phi}_{2}(b)^{*}\widetilde{\Phi}_{1}(a)=0$,
${\Phi}_{2}(b)^{*}{\Phi}_{1}(a)=0={\Phi}_{1}(a){\Phi}_{2}(b)^{*}.$ ∎

##  4\. Holomorphic mappings valued in a commutative C∗-algebra

The particular setting in which a holomorphic function is valued in a
commutative C∗-algebra provides enough advantages to establish a full
description of the orthogonally additive, orthogonality preserving,
holomorphic mappings which are valued in a commutatively C∗-algebra.

######  Proposition 15.

Let $S,T:A\to B$ be operators between C∗-algebras with $B$ commutative.
Suppose that $S$, $T$ and $(S,T)$ are orthogonality preserving, and let us
denote $h=S^{**}(1)$ and $k=T^{**}(1)$. Then there exits a Jordan
∗-homomorphism $\Phi:M(A)\to B^{**}$ satisfying $\Phi(1)=r(|h|+|k|)$,
$S(a)=\Phi(a)h,$ and $T(a)=\Phi(a)k,$ for every $a\in A$.

###### Proof.

Let $\Phi_{1},\widetilde{\Phi}_{1},\Phi_{2},\widetilde{\Phi}_{2}:M(A)\to
B^{**}$ be the Jordan ∗-homomorphisms satisfying $(b1)$ and $(b2)$ in
Proposition 14. By hypothesis, $B$ is commutative, and hence
$\Phi_{i}=\widetilde{\Phi}_{i}$ for every $i=1,2$ (compare the proof of
Proposition 14). Since the pair $({\Phi}_{1},{\Phi}_{2})$ is orthogonality
preserving on $A_{sa}$, Lemma 10 assures that

${\Phi}_{1}^{**}(1){\Phi}_{2}(a)={\Phi}_{1}(a){\Phi}_{2}^{**}(1),$

for every $a\in A_{sa}$. In order to simplify notation, let us denote
$p={\Phi}_{1}^{**}(1)$ and $q={\Phi}_{2}^{**}(1)$.

We define an operator ${\Phi}:M(A)\to B^{**}$, defined by

$\Phi(a)=pq\Phi_{1}(a)+p(1-q)\Phi_{1}(a)+q(1-p)\Phi_{2}(a).$

Since $p{\Phi}_{2}(a)={\Phi}_{1}(a)q$, it can be easily checked that $\Phi$ is
a Jordan ∗-homomorphism such that $S(a)=\Phi(a)h,$ and $T(a)=\Phi(a)k,$ for
every $a\in A$. ∎

######  Theorem 16.

Let $f:B_{A}(0,\varrho)\longrightarrow B$ be a holomorphic mapping, where $A$
and $B$ are C∗-algebras with $B$ commutative, and let $\displaystyle
f=\sum_{k=0}^{\infty}P_{k}$ be its Taylor series at zero, which is uniformly
converging in $U=B_{A}(0,\delta)$. Suppose $f$ is orthogonality preserving on
$A_{sa}\cap U$ and orthogonally additive (equivalently, orthogonally additive
and zero products preserving). Then there exist a sequence $(h_{n})$ in
$B^{**}$ and a Jordan ∗-homomorphism $\Phi:M(A)\to B^{**}$ such that

$f(x)=\sum_{n=1}^{\infty}h_{n}\Phi(a^{n})=\sum_{n=1}^{\infty}h_{n}\Phi(a^{n}),$

uniformly in $a\in U$.

###### Proof.

By Corollary 7, there exists a sequence $(T_{n})$ of operators from $A$ into
$B$ satisfying that the pair $(T_{n},T_{m})$ is orthogonality preserving on
$A_{sa}$ (equivalently, zero products preserving on $A_{sa}$) for every
$n,m\in\mathbb{N}$ and

$f(x)=\sum_{n=1}^{\infty}T_{n}(x^{n}),$

uniformly in $x\in U$. Denote $h_{n}=T_{n}^{**}(1)$.

We shall prove now the existence of the Jordan ∗-homomorphism $\Phi$. We
prove, by induction, that for each natural $n$, there exists a Jordan
∗-homomorphism $\Psi_{n}:M(A)\to B^{**}$ such that
$r(\Psi_{n}(1))=r(|h_{1}|+\ldots+|h_{n}|)$ and $T_{k}(a)=h_{k}\Psi_{n}(a)$ for
every $k\leq n$, $a\in A$. The statement for $n=1$ follows from Corollary 7
and Proposition 14. Let us assume that our statement is true for $n$. Since
for every $k,m$ in $\mathbb{N}$, $T_{k}$, $T_{m}$ and the pair $(T_{k},T_{m})$
are orthogonality preserving, we can easily check that $T_{n+1}$,
$T_{1}+\ldots+T_{n}$ and
$(T_{n+1},T_{1}+\ldots+T_{n})=(T_{n+1},(h_{1}+\ldots+h_{n})\Psi_{n})$ are
orthogonality preserving. By Proposition 15, there exists a Jordan
∗-homomorphism $\Psi_{n+1}:M(A)\to B^{**}$ satisfying
$r(\Psi_{n+1}(1))=r(|h_{1}|+\ldots+|h_{n}|+|h_{n+1}|)$,
$T_{n+1}(a)=h_{n+1}\Psi_{n+1}(a^{n+1})$ and
$(T_{1}+\ldots+T_{n})(a)=(h_{1}+\ldots+h_{n})\Psi_{n+1}(a)$ for every $k\leq
n$, $a\in A$. Since for each, $1\leq k\leq n$,

$h_{k}\Psi_{n+1}(a)=h_{k}r(|h_{1}|+\ldots+|h_{n}|+|h_{n+1}|)\Psi_{n+1}(a)$
$=h_{k}(|h_{1}|+\ldots+|h_{n}|)\Psi_{n+1}(a)$
$=h_{k}(|h_{1}|+\ldots+|h_{n}|)\Psi_{n}(a)=h_{k}\Psi_{n}=T_{k}(a),$

for every $a\in A$, as desired.

Let us consider a free ultrafilter $\mathcal{U}$ on ${\mathbb{N}}$. By the
Banach-Alaoglu theorem, any bounded set in $B^{**}$ is relatively
weak∗-compact, and thus the assignment
$a\mapsto\Phi(a):=w^{*}-\lim_{\mathcal{U}}\Psi_{n}(a)$ defines a Jordan
∗-homomorphism from $M(A)$ into $B^{**}$. If we fix a natural $k$, we know
that $T_{k}(a)=h_{k}\Psi_{n}(a)$, for every $n\geq k$ and $a\in A$. Then it
can be easily checked that $T_{k}(a)=h_{k}\Phi(a),$ for every $a\in A$, which
concludes the proof. ∎

The Banach-Stone type theorem for orthogonally additive, orthogonality
preserving, holomorphic mappings between commutative C∗-algebras, established
in Theorem 2 (see [2, Theorem 3.4]) is a direct consequence of our previous
result.

##  5\. Banach-Stone type theorems for holomorphic mappings between general
C∗-algebras

In this section we deal with holomorphic functions between general
C∗-algebras. In this more general setting we shall require additional
hypothesis to establish a result in the line of the above Theorem 16.

Given a unital C∗-algebra $A$, the symbol inv$(A)$ will denote the set of
invertible elements in $A$. The next lemma is a technical tool which is needed
later. The proof is left to the reader and follows easily from the fact that
inv$(A)$ is an open subset of $A$.

######  Lemma 17.

Let $f:B_{A}(0,\varrho)\longrightarrow B$ be a holomorphic mapping, where $A$
and $B$ are C∗-algebras with $B$ unital, and let $\displaystyle
f=\sum_{k=0}^{\infty}P_{k}$ be its Taylor series at zero, which is uniformly
converging in $U=B_{A}(0,\delta)$. Let us assume that there exists $a_{0}\in
U$ with $f(a_{0})\in\hbox{inv}(B)$. Then there exists $m_{0}\in\mathbb{N}$
such that
$\displaystyle\sum_{k=0}^{m_{0}}P_{k}(a_{0})\in\hbox{inv}(B)$.$\hfill\Box$

We can now state a description of those orthogonally additive, orthogonality
preserving, holomorphic mappings between C∗-algebras whose image contains an
invertible element.

######  Theorem 18.

Let $f:B_{A}(0,\varrho)\longrightarrow B$ be a holomorphic mapping, where $A$
and $B$ are C∗-algebras with $B$ unital, and let $\displaystyle
f=\sum_{k=0}^{\infty}P_{k}$ be its Taylor series at zero, which is uniformly
converging in $U=B_{A}(0,\delta)$. Suppose $f$ is orthogonality preserving on
$A_{sa}\cap U$, orthogonally additive on $U$ and
$f(U)\cap\hbox{inv}(B)\neq\emptyset$. Then there exist a sequence $(h_{n})$ in
$B^{**}$ and Jordan ∗-homomorphisms $\Theta,\widetilde{\Theta}:M(A)\to B^{**}$
such that

$f(a)=\sum_{n=1}^{\infty}h_{n}\widetilde{\Theta}(a^{n})=\sum_{n=1}^{\infty}{\Theta}(a^{n})h_{n},$

uniformly in $a\in U$.

###### Proof.

By Corollary 7 there exists a sequence $(T_{n})$ of operators from $A$ into
$B$ satisfying that the pair $(T_{n},T_{m})$ is orthogonality preserving on
$A_{sa}$ for every $n,m\in\mathbb{N}$ and

$f(x)=\sum_{n=1}^{\infty}T_{n}(x^{n}),$

uniformly in $x\in U$.

Now, Proposition 14 $(a)$, applied to $T_{n}$ ($n\in\mathbb{N}$), implies the
existence of sequences $(\Phi_{n})$ and $(\widetilde{\Phi}_{n})$ of Jordan
∗-homomorphisms from $M(A)$ into $B^{**}$ satisfying
$\Phi_{n}(1)=r(h_{n})r(h_{n})^{*}$,
$\widetilde{\Phi}_{n}(1)=r(h_{n})^{*}r(h_{n}),$ where $h_{n}=T_{n}^{**}(1)$,
and

$T_{n}(a)=\Phi_{n}(a)h_{n}=h_{n}\widetilde{\Phi}_{n}(a),$

for every $a\in A$, $n\in\mathbb{N}$. Moreover, from Proposition 14 $(b)$, the
pairs $({\Phi}_{n},{\Phi}_{m})$ and
$(\widetilde{\Phi}_{n},\widetilde{\Phi}_{m})$ are orthogonality preserving on
$A_{sa}$, for every $n,m\in\mathbb{N}$.

Since $f(U)\cap\hbox{inv}(B)\neq\emptyset$, it follows from Lemma 17 that
there exists a natural $m_{0}$ and $a_{0}\in A$ such that

$\displaystyle\sum_{k=1}^{m_{0}}P_{k}(a_{0})=\sum_{k=1}^{m_{0}}\Phi_{k}(a_{0}^{k})h_{k}=\sum_{k=1}^{m_{0}}h_{k}\widetilde{\Phi}_{k}(a_{0}^{k})\in\hbox{inv}(B).$

We claim that $r(h_{1})^{*}r(h_{1})+\ldots+r(h_{m_{0}})^{*}r(h_{m_{0}})$ is
invertible in $B^{+}$ (and in $B^{**}$). Otherwise, we could find a projection
$q\in B^{**}$ satisfying
$(r(h_{1})^{*}r(h_{1})+\ldots+r(h_{m_{0}})^{*}r(h_{m_{0}}))q=0$. This would
imply that

$\left(\sum_{k=1}^{m_{0}}P_{k}(a_{0})\right)q=\left(\sum_{k=1}^{m_{0}}\Phi_{k}(a_{0}^{k})h_{k}\right)q=0,$

contradicting that
$\displaystyle\sum_{k=1}^{m_{0}}P_{k}(a_{0})=\sum_{k=1}^{m_{0}}\Phi_{k}(a_{0}^{k})h_{k}$
is invertible in $B$.

Consider now the mapping $\Psi=\sum_{k=1}^{m_{0}}\widetilde{\Phi}_{k}$. It is
clear that, for each natural $n$, $\Psi$, $\widetilde{\Phi}_{n}$ and the pair
$(\Psi,\widetilde{\Phi}_{n})$ are orthogonality preserving. Applying
Proposition 14 $(b)$, we deduce the existence of Jordan ∗-homomorphisms
$\Theta,\widetilde{\Theta},\Theta_{n},\widetilde{\Theta}_{n}:M(A)\to B^{**}$
such that $(\Theta,\Theta_{n})$ and
$(\widetilde{\Theta},\widetilde{\Theta}_{n})$ are orthogonality preserving,
$\Theta(1)=r(k)r(k)^{*}$, $\widetilde{\Theta}(1)=r(k)^{*}r(k)$,
$\Theta_{n}(1)=r(h_{n})r(h_{n})^{*}$,
$\widetilde{\Theta}_{n}(1)=r(h_{n})^{*}r(h_{n})$,

$\Psi(a)=\Theta(a)k=k\widetilde{\Theta}(a)$

and

$\widetilde{\Phi}_{n}(a)=\Theta_{n}(a)r(h_{n})^{*}r(h_{n})=r(h_{n})^{*}r(h_{n})\widetilde{\Theta}_{n}(a),$

for every $a\in A$, where
$k=\Psi(1)=r(h_{1})^{*}r(h_{1})+\ldots+r(h_{m_{0}})^{*}r(h_{m_{0}})$. The
invertibility of $k$, proved in the previous paragraph, shows that
$\Theta(1)=1$. Thus, since $(\widetilde{\Theta},\widetilde{\Theta}_{n})$ is
orthogonality preserving, the last statement in Lemma 10 proves that

$\widetilde{\Theta}_{n}(a)=\widetilde{\Theta}_{n}(1)\widetilde{\Theta}(a)=\widetilde{\Theta}(a)\widetilde{\Theta}_{n}(1),$

for every $a\in A$, $n\in\mathbb{N}$. The above identities guarantee that

$\widetilde{\Phi}_{n}(a)=\Theta(a)r(h_{n})^{*}r(h_{n})=r(h_{n})^{*}r(h_{n})\widetilde{\Theta}(a),$

for every $a\in A$, $n\in\mathbb{N}$.

A similar argument to the one given above, but replacing
$\widetilde{\Phi}_{k}$ with ${\Phi}_{k}$, shows the existence of a Jordan
∗-homomorphism $\Theta:M(A)\to B^{**}$ such that

${\Phi}_{n}(a)=\Theta(a)r(h_{n})r(h_{n})^{*}=r(h_{n})r(h_{n})^{*}{\Theta}(a),$

for every $a\in A$, $n\in\mathbb{N}$, which concludes the proof. ∎

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