# Orthogonal forms and orthogonality preservers on real function algebras

Jorge J. Garcés<EMAIL_ADDRESS>Departamento de Análisis Matemático, Facultad
de Ciencias, Universidad de Granada, 18071 Granada, Spain.  and  Antonio M.
Peralta<EMAIL_ADDRESS>Departamento de Análisis Matemático, Facultad de
Ciencias, Universidad de Granada, 18071 Granada, Spain.

###### Abstract.

We initiate the study of orthogonal forms on a real C∗-algebra. Motivated by
previous contributions, due to Ylinen, Jajte, Paszkiewicz and Goldstein, we
prove that for every continuous orthogonal form $V$ on a commutative real
C∗-algebra, $A$, there exist functionals $\varphi_{1}$ and $\varphi_{2}$ in
$A^{*}$ satisfying

$V(x,y)=\varphi_{1}(xy)+\varphi_{2}(xy^{*}),$

for every $x,y$ in $A$. We describe the general form of a (not-necessarily
continuous) orthogonality preserving linear map between unital commutative
real C∗-algebras. As a consequence, we show that every orthogonality
preserving linear bijection between unital commutative real C∗-algebras is
continuous.

###### Key words and phrases:

Orthogonal form, real C∗-algebra, orthogonality preservers, disjointness
preserver, separating map

###### 2010 Mathematics Subject Classification:

Primary 46H40; 4J10, Secondary 47B33; 46L40; 46E15; 47B48.

Authors partially supported by the Spanish Ministry of Economy and
Competitiveness, D.G.I. project no. MTM2011-23843, and Junta de Andalucía
grants FQM0199 and FQM3737.

##  1\. Introduction and preliminaries

Elements $a$ and $b$ in a real or complex C∗-algebra, $A$, are said to be
_orthogonal_ (denoted by $a\perp b$) if $ab^{*}=b^{*}a=0$. A bounded bilinear
form $V:A\times A\to\mathbb{K}$ is called _orthogonal_ (resp., _orthogonal on
self-adjoint elements_) whenever $V(a,b^{*})=0$ for every $a\perp b$ in $A$
(resp., in the self-adjoint part of $A$). All the forms considered in this
paper are assumed to be continuous. Motivated by the seminal contributions by
K Ylinen [51] and R. Jajte and A. Paszkiewicz [29], S. Goldstein proved that
every orthogonal form $V$ on a (complex) C∗-algebra, $A,$ is of the form

$V(x,y)=\phi(xy)+\psi(xy)\ \ (x,y\in A),$

where $\phi$ and $\psi$ are two functionals in $A^{*}$ (cf. [21, Theorem
1.10]). A simplified proof of Goldstein’s theorem was published by U. Haagerup
and N.J. Laustsen in [24]. This characterisation has emerged as a very useful
tool in the study of bounded linear operators between C∗-algebras which are
orthogonality or disjointness preserving (see, for example, [11, 12]).

The first aim of this paper is to study orthogonal forms on the wider class of
real C∗-algebras. Little or nothing is known about the structure of an
orthogonal form $V$ on a real C∗-algebra. At first look, one is tempted to
consider the canonical complex bilinear extension of $V$ to a form on the
complexification, $A_{\mathbb{C}}=A\oplus iA$, of $A$ and, when the latter is
orthogonal, to apply Goldstein’s theorem. However, the complex bilinear
extension of $V$ to $A_{\mathbb{C}}\times A_{\mathbb{C}}$, need not be, in
general, orthogonal (see Example 2.7). The study of orthogonal forms on real
C∗-algebras requires a completely independent strategy; surprisingly the
resulting forms will enjoy a different structure to that established by S.
Goldstein in the complex setting.

In section 2 we establish some structure results for orthogonal forms on a
general real C∗-algebra, showing, among other properties, that every
orthogonal form on a real C∗-algebra extends to an orthogonal form on its
multiplier algebra (see Proposition 1.3). It is also proved that, for each
orthogonal and symmetric form $V$ on a real C∗-algebra, $A,$ there exists a
functional $\phi\in A^{*}$ satisfying $V(a,b)=\phi(ab+ba),$ for every $a,b\in
A$ with $a=a^{*}$, $b^{*}=b$ (cf. Proposition 1.5). In the real setting, the
skew-symmetric part of a real C∗-algebra, $A,$ is not determined by the self-
adjoint part of $A,$ so the information about the behavior of $V$ on the rest
of $A$ is very limited.

Section 3 contains one of the main results of the paper: the characterisation
of all orthogonal forms on a commutative real C∗-algebra. Concretely, we prove
that a form $V$ on a commutative real C∗-algebra $A$ is orthogonal if, and
only if, there exist functionals $\varphi_{1}$ and $\varphi_{2}$ in $A^{*}$
satisfying

$V(x,y)=\varphi_{1}(xy)+\varphi_{2}(xy^{*}),$

for every $x,y\in A$ (see Theorem 2.4). Among the consequences, it follows
that the complex bilinear extension of $V$ to the complexification of $A$ is
orthogonal if, and only if, we can take $\varphi_{2}=0$ in the above
representation.

We recall that a mapping $T:A\to B$ between real or complex C∗-algebras is
said to be _orthogonality or disjointness preserving_ (also called
_separating_) whenever $a\perp b$ in $A$ implies $T(a)\perp T(b)$ in $B.$ The
mapping $T$ is _bi-orthogonality preserving_ whenever the equivalence

$a\perp b\Leftrightarrow T(a)\perp T(b)$

holds for all $a,b$ in $A$. As noticed in [13], every bi-orthogonality
preserving linear surjection, $T:A\to B$ between two C∗-algebras is injective.

The study of orthogonality preserving operators between C∗-algebras started
with the work of W. Arendt [1] in the setting of unital abelian C∗-algebras.
Subsequent contributions by K. Jarosz [30] extended the study to the setting
of orthogonality preserving (not necessarily bounded) linear mappings between
abelian C∗-algebras. The first study on orthogonality preserving symmetric
(bounded) linear operators between general (complex) C∗-algebras is originally
due to M. Wolff (cf. [49]). Orthogonality preserving bounded linear maps
between C∗-algebras, JB∗-algebras and JB∗-triples were completely described in
[11] and [12].

The pioneer works of E. Beckenstein, L. Narici, and A.R. Todd in [8] and [9]
(see also [7]) were applied by K. Jarosz to prove that every orthogonality
preserving linear bijection between $C(K)$-spaces is (automatically)
continuous (see [30]). More recently, M. Burgos and the authors of this note
proved in [13] that every bi-orthogonality preserving linear surjection
between two von Neumann algebras (or between two compact C∗-algebras) is
automatically continuous (compare [40], [41] for recent additional
generalisations).

The main goal of section 4 is to describe the orthogonality preserving linear
mappings between unital commutative real C∗-algebras (see Theorem 3.2). As a
consequence, we shall prove that every orthogonality preserving linear
bijection between unital commutative real C∗-algebras is automatically
continuous. We shall exhibit some examples illustrating that the results in
the real setting are completely independent from those established for complex
C∗-algebras. We further give a characterisation of those linear mappings
between real forms of $C(K)$-spaces which are bi-orthogonality preserving.

###  1.1. Preliminary results

Let us now introduce some basic facts and definitions required later. A _real
C ∗-algebra_ is a real Banach *-algebra $A$ which satisfies the standard
C∗-identity, $\|a^{*}a\|=\|a\|^{2}$, and which also has the property that
$1+a^{\ast}a$ is invertible in the unitization of $A$ for every $a\in A$. It
is known that a real Banach *-algebra, $A,$ is a real C∗-algebra if, and only
if, it is isometrically *-isomorphic to a norm-closed real *-subalgebra of
bounded operators on a real Hilbert space (cf. [39, Corollary 5.2.11]).

Clearly, every (complex) C∗-algebra is a real C∗-algebra when scalar
multiplication is restricted to the real field. If $A$ is a real C∗-algebra
whose algebraic complexification is denoted by ${B}=A\oplus iA,$ then there
exists a C∗-norm on ${B}$ extending the norm of $A$. It is further known that
there exists an involutive conjugate-linear ∗-automorphism $\tau$ on ${B}$
such that $A={B}^{\tau}:=\\{x\in B:\tau(x)=x\\}$ (compare [39, Proposition
5.1.3] or [47, Lemma 4.1.13], and [22, Corollary 15.4]). The dual space of a
real or complex C∗-algebra $A$ will be denoted by $A^{*}$. Let
$\widetilde{\tau}:B^{*}\rightarrow B^{*}$ denote the map defined by

$\widetilde{\tau}(\phi)(b)=\overline{\phi(\tau(b))}\qquad(\phi\in B^{*},\,b\in
B).$

Then $\widetilde{\tau}$ is a conjugate-linear isometry of period 2 and the
mapping

$(B^{*})^{\widetilde{\tau}}\to A^{*}$ $\varphi\mapsto\varphi|_{A}$

is a surjective linear isometry. We shall identify
$(B^{*})^{\widetilde{\tau}}$ and $A^{*}$ without making any explicit mention.

When $A$ is a real or complex C∗-algebra, then $A_{sa}$ and $A_{skew}$ will
stand for the set of all self-adjoint and skew-symmetric elements in $A$,
respectively. We shall make use of standard notation in C∗-algebra theory.

Given Banach spaces $X$ and $Y$, $L(X,Y)$ will denote the space of all bounded
linear mappings from $X$ to $Y$. We shall write $L(X)$ for the space $L(X,X)$.
Throughout the paper the word “operator” (respectively, multilinear or
sesquilinear operator) will always mean bounded linear mapping (respectively
bounded multilinear or sesquilinear mapping). The dual space of a Banach space
$X$ is always denoted by $X^{*}$.

Let us recall that a series $\sum_{n}x_{n}$ in a Banach space is called
_weakly unconditionally Cauchy (w.u.C.)_ if there exists $C>0$ such that for
any finite subset $F\subset{\mathbb{N}}$ and $\varepsilon_{n}=\pm 1$ we have
$\displaystyle\left\|\sum_{n\in F}\varepsilon_{n}x_{n}\right\|\leq C$. A
(linear) operator $T:X\longrightarrow Y$ is _unconditionally converging_ if
for every w.u.C. series $\sum_{n}x_{n}$ in $X,$ the series $\sum_{n}T(x_{n})$
is unconditionally convergent in $Y$, that is, every subseries of
$\sum_{n}T(x_{n})$ is norm converging. It is known that $T:X\to Y$ is
unconditionally converging if, and only if, for every w.u.C. series
$\sum_{n}x_{n}$ in $X,$ we have $\|T(x_{n})\|\to 0$ (compare, for example,
[44, page 1257])

Let us also recall that a Banach space $X$ is said to have _Pełczyński’s_
_property_ (V) if, for every Banach space $Y$, every unconditionally
converging operator $T:X\to Y$ is weakly compact.

The proof of the following elementary lemma is left to the reader.

######  Lemma 1.1.

Let $X$ be a complex Banach space, $\tau:X\to X$ a conjugate-linear period-2
isometry. Then the real Banach space $X^{\tau}:=\\{x\in X:\tau(x)=x\\}$
satisfies property (V) whenever $X$ does. $\hfill\Box$

We shall require, for later use, some results on extensions of multilinear
operators. Let $X_{1},\dots,X_{n}$, and $X$ be Banach spaces,
$T:X_{1}\times\cdots\times X_{n}\to X$ a (continuous) $n$-linear operator, and
$\pi:\\{1,\dots,n\\}\to\\{1,\dots,n\\}$ a permutation. It is known that there
exists a unique $n$-linear extension $AB(T)_{\pi}:X_{1}^{**}\times\cdots\times
X_{n}^{**}\to X^{**}$ such that for every $z_{i}\in X_{i}^{**}$ and every net
$(x_{\alpha_{i}}^{i})\in X_{i}$ ($1\leq i\leq n$), converging to $z_{i}$ in
the weak* topology we have

$AB(T)_{\pi}(z_{1},\ldots,z_{n})=\mbox{weak*-}\lim_{\alpha_{\pi(1)}}\cdots\mbox{weak*-}\lim_{\alpha_{\pi(n)}}T(x_{\alpha_{1}}^{1},\ldots,x_{\alpha_{n}}^{n}).$

Moreover, $AB(T)_{\pi}$ is bounded and has the same norm as $T$. The
extensions $AB(T)_{\pi}$ coincide with those considered by Arens in [2, 3] and
by Aron and Berner for polynomials in [4]. The $n$-linear operators
$AB(T)_{\pi}$ are usually called the Arens or _Aron-Berner extensions_ of $T$.

Under some additional hypothesis, the Arens extension of a multilinear
operator also is separately weak∗ continuous. Indeed, if every operator from
$X_{i}$ to $X_{j}^{*}$ is weakly compact ($i\not=j)$ the Arens extensions of
$T$ defined above do not depend on the chosen permutation $\pi$ and they are
all separately weak∗ continuous (see [5], and Theorem 1 in [10]). In
particular, the above requirements always hold when every $X_{i}$ satisfies
Pelczynski’s property $(V)$ (in such case $X_{i}^{*}$ contains no copies of
$c_{0}$, therefore every operator from $X_{i}$ to $X_{j}^{*}$ is
unconditionally converging, and hence weakly compact by property $(V)$, see
[43]). When all the Arens extensions of $T$ coincide, the symbol
$AB(T)=T^{**}$ will denote any of them.

We should note at this point that every C∗-algebra satisfies property $(V)$
(cf. Corollary 6 in [46]). Since every real C∗-algebra is, in particular, a
real form of a (complex) C∗-algebra, it follows from Lemma 1.1 that every real
C∗-algebra satisifes property $(V)$. We therefore have:

######  Lemma 1.2.

Let $A_{1},\ldots,A_{k}$ be real C∗-algebras and let $T$ be a multilinear
continuous operator from $A_{1}\times\ldots\times A_{k}$ to a real Banach
space $X$. Then $T$ admits a unique Arens extension
$T^{**}:A_{1}^{**}\times\ldots\times A_{k}^{**}\to X^{**}$ which is separately
weak∗ continuous.$\hfill\Box$

Given a real or complex C∗-algebra, $A$, the _multiplier algebra_ of $A$,
$M(A)$, is the set of all elements $x\in A^{**}$ such that, for each element
$a\in A$, $xa$ and $ax$ both lie in $A$. We notice that $M(A)$ is a C∗-algebra
and contains the unit element of $A^{**}$. It should be recalled here that
$A=M(A)$ whenever $A$ is unital.

######  Proposition 1.3.

Let $A$ be a real C∗-algebra. Suppose that $V:A\times A\to\mathbb{R}$ is an
orthogonal bounded bilinear form. Then the continuous bilinear form

$\tilde{V}:M(A)\times M(A)\to\mathbb{R},\ \ \tilde{V}(a,b):=V^{**}(a,b)$

is orthogonal.

###### Proof.

Let $a$ and $b$ be two orthogonal elements in $M(A)$. Let $a^{[\frac{1}{3}]}$
(resp., $b^{[\frac{1}{3}]}$) denote the unique element $z$ in $M(A)$
satisfying $zz^{*}z=a$ (resp., $zz^{*}z=b$). We notice that
$a^{[\frac{1}{3}]}$ and $b^{[\frac{1}{3}]}$ are orthogonal, so, for each pair
$x,y$ in $A$, $a^{[\frac{1}{3}]}xa^{[\frac{1}{3}]}$ and
$b^{[\frac{1}{3}]}yb^{[\frac{1}{3}]}$ are orthogonal elements in $A$. Since
$V$ is orthogonal, we have

$V(a^{[\frac{1}{3}]}xa^{[\frac{1}{3}]},(b^{[\frac{1}{3}]})^{*}y(b^{[\frac{1}{3}]})^{*})=0$

for every $x,y\in A$.

Goldstine’s theorem (cf. Theorem V.4.2.5 in [18]) guarantees that the closed
unit ball of $A$ is weak*-dense in the closed unit ball of $A^{**}$. Therefore
we can pick two bounded nets $(x_{\lambda})$ and $(y_{\mu})$ in $A$,
converging in the weak∗ topology of $A^{**}$ to $(a^{[\frac{1}{3}]})^{*}$ and
$b^{[\frac{1}{3}]}$, respectively.

We have already mentioned that $V^{**}:A^{**}\times A^{**}\to\mathbb{R}$ is
separately weak∗ continuous. Since
$0=V(a^{[\frac{1}{3}]}x_{\lambda}a^{[\frac{1}{3}]},(b^{[\frac{1}{3}]})^{*}y_{\mu}(b^{[\frac{1}{3}]})^{*}),$
for every $\lambda$ and $\mu$, taking limits, first in $\lambda$ and
subsequently in $\mu$, we deduce that

$V^{**}(a^{[\frac{1}{3}]}(a^{[\frac{1}{3}]})^{*}a^{[\frac{1}{3}]},(b^{[\frac{1}{3}]})^{*}b^{[\frac{1}{3}]}(b^{[\frac{1}{3}]})^{*}))=\tilde{V}(a,b^{*})=0,$

which shows that $\tilde{V}$ is orthogonal. ∎

Since the multiplier algebra of a real or complex C∗-algebra always has a unit
element, Proposition 1.3 allows us to restrict our study on orthogonal
bilinear forms on a real C∗-algebra $A$ to the case in which $A$ is unital.

A _real von Neumann algebra_ is a real C∗-algebra which is also a dual Banach
space (cf. [28] or [39, §6.1]). Clearly, the self adjoint part of a real von
Neumann algebra is a JW-algebra in the terminology employed in [25], so every
self-adjoint element in a real von Neumann algebra $W$ can be approximated in
norm by a finite real linear combination of mutually orthogonal projections in
$W$ (cf. [25, Proposition 4.2.3]). We shall explore now the validity in the
real setting of some of the results established by S. Goldstein in [21].

######  Lemma 1.4.

Let $A$ be a real von Neumann algebra with unit $1$. Suppose that $V:A\times
A\to\mathbb{R}$ is a bounded bilinear form. The following are equivalent:

  1. $(a)$

$V$ is orthogonal on $A_{sa}$;

  2. $(b)$

$V(p,q)=0$, whenever $p$ and $q$ are two orthogonal projections in $A$;

  3. $(c)$

$V(a,b)=V(ab,1)$ for every $a,b\in A_{sa}$ with $ab=ba$.

If any of the above statements holds and $V$ is symmetric, then defining
$\phi_{1}(x):=V(x,1)$ ($x\in A$), we have $V(a,b)=\phi_{1}(\frac{ab+ba}{2})$,
for every $a,b\in A_{sa}$.

###### Proof.

Applying the existence of spectral resolutions for self-adjoint elements in a
real von Neumann algebra, the argument given by S. Goldstein in [21,
Proposition 1.2] remains valid to prove the equivalence of $(a)$, $(b)$ and
$(c)$.

Suppose now that $V$ is symmetric. Let $a=\sum_{j=1}^{m}\lambda_{j}p_{j}$ be
an algebraic element in $A_{sa}$, where the $\lambda_{j}$’s belong to
$\mathbb{R}$ and $p_{1},\ldots,p_{m}$ are mutually orthogonal projections in
$A$. Since $V$ is orthogonal, for every projection $p\in A$, we have

$V(p,1)=V(p,1-p)+V(p,p)=V(p,p).$

Thus,

$V(a,a)=\sum_{j=1}^{m}\lambda_{j}^{2}V(p_{j},p_{j})=\sum_{j=1}^{m}\lambda_{j}^{2}V(p_{j},1)=V\left(\sum_{j=1}^{m}\lambda_{j}^{2}p_{j},1\right)=V(a^{2},1).$

The (norm) density of algebraic elements in $A_{sa}$ and the continuity of $V$
imply that $V(a,a)=V(a^{2},1),$ for every $a\in A_{sa}$. Finally, applying
that $V$ is symmetric we have

$V(a^{2},1)+V(b^{2},1)+V(ab+ba,1)=V((a+b)^{2},1)$
$=V(a+b,a+b)=V(a,a)+V(b,b)+2V(a,b),$

for every $a,b\in A_{sa}$, and hence $V(a,b)=V(\frac{ab+ba}{2},1),$ for all
$a,b\in A_{sa}$. ∎

The above result holds for every monotone $\sigma$-complete unital real
C∗-algebra $A$ (that is, each upper bounded, monotone increasing sequence of
selfadjoint elements of $A$ has a least upper bound).

Surprisingly, the final conclusion of the above Lemma can be established for
unital real C∗-algebras with independent basic techniques.

######  Proposition 1.5.

Let $A$ be a unital real C∗-algebra with unit $1$. Suppose that $V:A\times
A\to\mathbb{R}$ is an orthogonal, symmetric, bounded, bilinear form. Then
defining $\phi_{1}(x):=V(x,1)$ ($x\in A$), we have
$V(a,b)=\phi_{1}(\frac{ab+ba}{2})$, for every $a,b\in A_{sa}$.

###### Proof.

Let $a$ be a selfadjoint element in $A$. The real C∗-subalgebra, $C$, of $A$
generated by $1$ and $a$ is isometrically isomorphic to the space
$C(K,\mathbb{R})$ of all real-valued continuous functions on a compact
Hausdorff space $K$. The restriction of $V$ to $C\times C$ is orthogonal,
therefore the mapping $x\mapsto V(x,x)$ is a $2$-homogeneous orthogonally
additive polynomial on $C$. The main result in [45] implies the existence of a
functional $\varphi_{a}\in C^{*}$ such that $V(x,x)=\varphi_{a}(x^{2})$, for
every $x\in C$. It is clear that $\varphi_{a}(x)=V(x,1)$ for every $x\in C$.
In particular

$V(a,a)=\varphi_{a}(a^{2})=V(a^{2},1).$

The argument given at the end of the proof of Lemma 1.4 gives the desired
statement. ∎

The above proposition shows that we can control the form of a symmetric
orthogonal form on the self adjoint part of a (unital) real C∗-algebra. The
form on the skew-symmetric part remains out of control for the moment.

##  2\. Orthogonal forms on abelian real C∗-algebras

Throughout this section, $A$ will denote a unital, abelian, real C∗-algebra
whose complexification will be denoted by $B$. It is clear that $B$ is a
unital, abelian C∗-algebra. It is known that there exists a period-2
conjugate-linear ∗-automorphism $\tau:B\to B$ such that $A=B^{\tau}:=\\{x\in
B:\tau(x)=x\\}$ (cf. [47, 4.1.13] and [22, 15.4] or [39, §5.2]).

By the commutative Gelfand theory, there exists a compact Hausdorff space $K$
such that $B$ is C∗-isomorphic to the C∗-algebra $C(K)$ of all complex valued
continuous functions on $K$. The Banach-Stone Theorem implies the existence of
a homeomorphism $\sigma:K\to K$ such that $\sigma^{2}(t)=t$, and

$\tau(a)(t)=\overline{a(\sigma(t))},$

for all $t\in K$, $a\in C(K)$. Real function algebras of the form
$C(K)^{\tau}$ have been studied by its own right and are interesting in some
other settings (cf. [37]).

Henceforth, the symbol $\mathfrak{B}$ will stand for the $\sigma$-algebra of
all Borel subsets of $K$, $S(K)$ will denote the space of
$\mathfrak{B}$-simple scalar functions defined on $K$, while the _Borel
algebra over $K$_, $B(K)$, is defined as the completion of $S(K)$ under the
supremum norm. It is known that $B=C(K)\subset B(K)\subset C(K)^{**}.$ The
mapping $\tau^{**}:C(K)^{**}\to C(K)^{**}$ is a period-2 conjugate-linear
∗-automorphism on $B^{**}=C(K)^{**}.$ It is easy to see that
$\tau^{**}(B(K))=B(K),$ and hence $\tau^{**}|_{B(K)}:B(K)\to B(K)$ defines a
period-2 conjugate-linear ∗-automorphism on $B(K)$. By an abuse of notation,
the symbol $\tau$ will denote $\tau$, $\tau^{**}$ and $\tau^{**}|_{B(K)}$
indistinctly. It is clear that, for each Borel set $B\in\mathfrak{B},$
$\tau(\chi_{{}_{B}})=\chi_{{}_{\sigma(B)}}$.

Let $a$ be an element in $B(K)$. For each $\varepsilon>0,$ there exist complex
numbers $\lambda_{1},\ldots,\lambda_{r}$ and disjoint Borel sets
$B_{1},\ldots,B_{r}$ such that
$\displaystyle{\left\|a-\sum_{k=1}^{r}\lambda_{k}\chi_{{}_{B_{k}}}\right\|<{\varepsilon}.}$
When $a\in A$ is $\tau$-symmetric (i.e. $\tau(a)=a$) then, since
$a=\frac{1}{2}(a+\tau(a)),$ we have

$\left\|a-\frac{1}{2}\sum_{k=1}^{r}\lambda_{k}\chi_{{}_{B_{k}}}+\overline{\lambda_{k}}\chi_{{}_{\sigma(B_{k})}}\right\|\leq\frac{1}{2}\left\|a-\sum_{k=1}^{r}\lambda_{k}\chi_{{}_{B_{k}}}\right\|+\frac{1}{2}\left\|a-\sum_{k=1}^{r}\overline{\lambda_{k}}\chi_{{}_{\sigma(B_{k})}}\right\|$
$\leq\frac{1}{2}\left\|a-\sum_{k=1}^{r}\lambda_{k}\chi_{{}_{B_{k}}}\right\|+\frac{1}{2}\left\|\tau\left(a-\sum_{k=1}^{r}\lambda_{k}\chi_{{}_{B_{k}}}\right)\right\|<\varepsilon.$

Consequently, every element in $B(K)^{\tau}$ can be approximated in norm by
finite linear combinations of the form
$\sum_{k}\alpha_{k}\chi_{{}_{B_{k}}}+\overline{\alpha_{k}}\chi_{{}_{\sigma(B_{k})}},$
where $\alpha_{1},\ldots,\alpha_{n}$ are complex numbers and
$B_{1},\ldots,B_{n}$ are mutually disjoint Borel sets. Having in mind that,
for each Borel set $B\in\mathfrak{B}$ and each $\alpha\in\mathbb{C},$
$\left(\alpha\chi_{{}_{B}}+\overline{\alpha}\chi_{{}_{\sigma(B)}}\right)^{*}=\overline{\alpha}\chi_{{}_{B}}+{\alpha}\chi_{{}_{\sigma(B)}},$
we have

$\left(\alpha\chi_{{}_{B}}+\overline{\alpha}\chi_{{}_{\sigma(B)}}\right)+\left(\alpha\chi_{{}_{B}}+\overline{\alpha}\chi_{{}_{\sigma(B)}}\right)^{*}=2\Re\hbox{e}(\alpha)\left(2\chi_{{}_{\sigma(B)\cap
B}}+\chi_{{}_{\sigma(B)\backslash B}}+\chi_{{}_{B\backslash\sigma(B)}}\right)$
$=2\Re\hbox{e}(\alpha)\left(2\chi_{{}_{\sigma(B)\cap
B}}+\chi_{{}_{(\sigma(B)\backslash B)\cup\sigma(\sigma(B)\backslash
B)}}\right),$

and

$\left(\alpha\chi_{{}_{B}}+\overline{\alpha}\chi_{{}_{\sigma(B)}}\right)-\left(\alpha\chi_{{}_{B}}+\overline{\alpha}\chi_{{}_{\sigma(B)}}\right)^{*}=2i\Im\hbox{m}(\alpha)\left(\chi_{{}_{B\backslash\sigma(B)}}-\chi_{{}_{\sigma(B)\backslash
B}}\right).$

Suppose now that $a\in B(K)^{\tau}$ is *-symmetric (i.e. $a^{*}=a$). It
follows from the above that $a$ can be approximated in norm by linear
combinations of the form
$\displaystyle{\sum_{k=1}^{r}\alpha_{k}\chi_{{}_{E_{k}}},}$ where
$\alpha_{k}\in\mathbb{R}$ and $E_{1},\ldots,E_{r}$ are mutually disjoint Borel
subsets of $K$ with $\sigma(E_{i})=E_{i}$. Let $b$ be an element in
$B(K)^{\tau}$ satisfying $b^{*}=-b$. Similar arguments to those given for
*-symmetric elements, allow us to show that $b$ can be approximated in norm by
finite linear combinations of the form $\displaystyle{\sum_{k=1}^{r}i\
\alpha_{k}(\chi_{{}_{E_{k}}}-\chi_{{}_{\sigma(E_{k})}}),}$ where
$\alpha_{k}\in\mathbb{R}$ and $E_{1},\ldots,E_{r}$ are mutually disjoint Borel
subsets of $K$ with $\sigma(E_{i})\cap E_{i}=\emptyset$.

######  Lemma 2.1.

Let $A$ be a unital, abelian, real C∗-algebra whose complexification is
denoted by $B=C(K)$, for a suitable compact Hausdorff space $K$. Let
$\tau:B\to B$ be a period-2 conjugate-linear ∗-automorphism satisfying
$A=B^{\tau}$ and $\tau(a)(t)=\overline{a(\sigma(t))},$ for all $t\in K$, $a\in
C(K)$, where $\sigma:K\to K$ is a period-2 homeomorphism. Then the set
$N=\\{t\in K:\sigma(t)\neq t\\}$ is an open subset of $K$, $F=\\{t\in
K:\sigma(t)=t\\}$ is a closed subset of $K$ and there exists an open subset
$\mathcal{O}\subset K$ maximal with respect to the property
$\mathcal{O}\cap\sigma(\mathcal{O})=\emptyset.$

###### Proof.

That $F$ is closed follows easily from the continuity of $\sigma,$ and
consequently, $N=K/F$ is open.

Let $\mathcal{F}$ be the family of all open subsets $O\subseteq K$ such that
$O\cap\sigma(O)=\emptyset$ ordered by inclusion. Let
$S=\\{O_{\lambda}\\}_{\lambda}$ be a totally ordered subset of $\mathcal{F}.$
We shall see that $O=\bigcup_{\lambda}O_{\lambda}$ is an open set which also
lies in $\mathcal{F},$ that is, $O\cap\sigma(O)=\emptyset.$

Let us suppose, on the contrary, that there exists $t\in
O\cap\sigma(O)\neq\emptyset.$ Then there exist $\lambda,\beta$ such that $t\in
O_{\lambda}$ and $t\in\sigma(O_{\beta}).$ Since $S$ is totally ordered,
$O_{\lambda}\subseteq O_{\beta}$ or $O_{\beta}\subseteq O_{\lambda}.$ We shall
assume that $O_{\lambda}\subseteq O_{\beta}.$ Then
$\sigma(O_{\lambda})\subseteq\sigma(O_{\beta})$ and $t$ lies in
$O_{\beta}\cap\sigma(O_{\beta})=\emptyset,$ which is a contradiction. Finally,
Zorn’s Lemma gives the existence of a maximal element $\mathcal{O}$ in
$\mathcal{F}.$ ∎

It should be noticed here that, in Lemma 2.1,
$\mathcal{O}\cup\sigma(\mathcal{O})=N$, an equality which follows from the
maximality of $\mathcal{O}$.

Our next lemma analyses the “spectral resolution” of a *-skew-symmetric
element in $B(K)^{\tau}.$

######  Lemma 2.2.

In the notation of Lemma 2.1, let $B(A)=B(K)^{\tau}$, let $a\in
B(K)^{\tau}_{sa},$ and let $b$ be an element in $B(A)_{skew}$. Then the
following statements hold:

  1. $a)$

$b|F=0$;

  2. $b)$

For each $\varepsilon>0$, there exist mutually disjoint Borel sets
$B_{1},\ldots,B_{m}\subset\mathcal{O}$ and real numbers
$\lambda_{1},\ldots,\lambda_{m}$ satisfying
$\displaystyle{\left\|b-\sum_{j=1}^{m}i\
\lambda_{j}(\chi_{{}_{B_{j}}}-\chi_{{}_{\sigma(B_{j})}})\right\|<\varepsilon;}$

  3. $c)$

For each $\varepsilon>0$, there exist mutually disjoint Borel sets
$C_{1},\ldots,C_{m}\subset K$ and real numbers $\mu_{1},\ldots,\mu_{m}$
satisfying $\sigma(C_{j})=C_{j},$ and
$\displaystyle{\left\|a-\sum_{j=1}^{m}\mu_{j}\chi_{{}_{C_{j}}}\right\|<\varepsilon.}$

###### Proof.

$a)$ Since $b^{*}=-b$, we have $Re(b(t))=0,\forall t\in K.$ Now, let $t\in F$,
applying $\sigma(t)=t$ and $\tau(b)=b$ we get
$\overline{b(t)}=\overline{b(\sigma(t))}=b(t),$ and hence
$\Im\hbox{m}(b(t))=0.$

Statements $b)$ and $c)$ follow from the comments prior to Lemma 2.1 and the
maximality of $\mathcal{O}$ in that Lemma. ∎

It is clear that in a commutative real (or complex) C∗-algebra, $A,$ two
elements $a,b$ are orthogonal if and only if they have zero-product, that is,
$ab=0.$ Therefore, $V(a,b^{*})=0=V(a,b)$ whenever $V:A\times A\to{\mathbb{R}}$
is an orthogonal bilinear form on an abelian real C∗-algebra and $a,b$ are two
orthogonal elements in $A$. We shall make use of this property without an
explicit mention.

We shall keep the notation of Lemma 2.1 throughout the section. Henceforth,
for each $C\subseteq\mathcal{O}$ we shall write $u_{{}_{C}}=i\
(\chi_{{}_{C}}-\chi_{{}_{\sigma(C)}}).$ The symbol $u_{{}_{0}}$ will stand for
the element $u_{{}_{\mathcal{O}}}.$ It is easy to check
$1=\chi_{{}_{F}}+u_{{}_{0}}u_{{}_{0}}^{*},$ where $1$ is the unit element in
$B(K)^{\tau}.$ By Lemma 2.2 $a)$, for each $b\in B(K)^{\tau}_{skew}$ we have
$b\perp\chi_{{}_{F}},$ and so $b=bu_{{}_{0}}u_{{}_{0}}^{*}.$

######  Proposition 2.3.

Let $K$ be a compact Hausdorff space, $\tau$ a period-2 conjugate-linear
isometric ∗-homomorphism on $C(K)$, $A=C(K)^{\tau}$, and $V:A\times
A\to\mathbb{R}$ be an orthogonal bounded bilinear form whose Arens extension
is denoted by $V^{**}:A^{**}\times A^{**}\to\mathbb{R}$. Let $\sigma:K\to K$
be a period-2 homeomorphism satisfying $\tau(a)(t)=\overline{a(\sigma(t))},$
for all $t\in K$, $a\in C(K)$. Then the following assertions hold for all
Borel subsets $D,B,C$ of $K$ with $\sigma(B)\cap B=\sigma(C)\cap C=\emptyset$
and $\sigma(D)=D$:

  1. $a)$

$V(\chi_{{}_{D}},u_{{}_{B}})=V(u_{{}_{B}},\chi_{D})=0,$ whenever $D\cap
B=\emptyset$;

  2. $b)$

$V(u_{{}_{B}},u_{{}_{C}})=0,$ whenever $B\cap C=\emptyset;$

  3. $c)$

$V((u_{{}_{0}}u_{{}_{0}}^{*}-u_{{}_{C}}u^{*}_{{}_{C}})u_{{}_{B}},u_{{}_{C}})=V(u_{{}_{C}},(u_{{}_{0}}u_{{}_{0}}^{*}-u_{{}_{C}}u^{*}_{{}_{C}})u_{{}_{B}})=0.$

###### Proof.

By an abuse of notation, we write $V$ for $V$ and $V^{**}$.

Let $K_{1},K_{2}$ be compact subsets of $K$ such that $K_{1},K_{2}$ and
$\sigma(K_{2})$ are mutually disjoint. By regularity and Urysohn’s Lemma there
exist nets $(f_{\lambda})_{{}_{\lambda}}$, $(g_{\gamma})_{{}_{\gamma}}$ in
$C(K)^{+}$ such that $\chi_{{}_{K_{1}}}\leq
f_{\lambda}\leq\chi_{{}_{K\backslash(K_{2}\cup\sigma(K_{2}))}},$
$\chi_{{}_{K_{2}}}\leq
g_{\gamma}\leq\chi_{{}_{K\backslash(K_{1}\cup\sigma(K_{1})\cup\sigma(K_{2}))}},$
$(f_{\lambda})_{{}_{\lambda}}$ (respectively, $(g_{\gamma})_{{}_{\gamma}}$)
converges to $\chi_{{}_{K_{1}}}$ (resp., to $\chi_{{}_{K_{2}}}$) in the weak∗
topology of $C(K)^{**}$.

The nets $\widetilde{f}_{\lambda}=\frac{1}{2}(f_{\lambda}+\tau(f_{\lambda}))$
and $\widetilde{g}_{\gamma}=i(g_{\gamma}-\tau(g_{\gamma}))$ lie in
$C(K)^{\tau}$ and converge in the weak∗ topology of $C(K)^{**}$ to
$\frac{1}{2}(\chi_{{}_{K_{1}}}+\chi_{{}_{\sigma(K_{1})}})$ and
$u_{{}_{K_{2}}},$ respectively. It is also clear that $f_{\lambda}\perp
g_{\gamma},$ $\tau(f_{\lambda})\perp g_{\gamma},$ and hence
$\widetilde{f}_{\lambda}\perp\widetilde{g}_{\gamma},$ for every
$\lambda,\gamma.$

By the separate weak∗ continuity of $V^{**}\equiv V$ we have

(1)
$V\left(\frac{1}{2}(\chi_{{}_{K_{1}}}+\chi_{{}_{\sigma(K_{1})}}),u_{{}_{K_{2}}}\right)=w^{*}-\lim_{\lambda}\left(w^{*}-\lim_{\gamma}V\left(\widetilde{f_{\lambda}},\widetilde{g}_{\gamma}\right)\right)=0,$

and

$V\left(u_{{}_{K_{2}}},\frac{1}{2}(\chi_{{}_{K_{1}}}+\chi_{{}_{\sigma(K_{1})}})\right)=0.$

We can similarly prove that

(2) $V\left(u_{{}_{K_{1}}},u_{{}_{K_{2}}}\right)=0,$

whenever $K_{1}$ and $K_{2}$ are two compact subsets of $K$ such that
$K_{1},K_{2},\sigma(K_{1})$ and $\sigma(K_{2})$ are pairwise disjoint.

$a)$ Let now $D,B$ be two disjoint Borel subsets of $K$ such that
$\sigma(D)=D$ and $B\subseteq\mathcal{O}.$ By inner regularity there exist
nets of the form $(\chi_{{}_{K_{\lambda}^{{}^{D}}}})_{\lambda}$ and
$(\chi_{{}_{K_{\gamma}^{{}^{B}}}})_{\gamma}$ such that
$(\chi_{{}_{K_{\lambda}^{{}^{D}}}})_{\lambda}$ and
$(\chi_{{}_{K_{\gamma}^{{}^{B}}}})_{\gamma}$ converge in the weak∗ topology of
$C(K)^{**}$ to $\chi_{{}_{D}}$ and $\chi_{{}_{B}}$, respectively, where each
$K_{\lambda}^{{}^{D}}\subseteq D$ and each $K_{\gamma}^{{}^{B}}\subseteq B$ is
compact subset of $K$. By the assumptions made on $D$ and $B$ we have that
$K_{\lambda}^{{}^{D}}\cap
K_{\gamma}^{{}^{B}}=K_{\lambda}^{{}^{D}}\cap\sigma(K_{\gamma}^{{}^{B}})=\emptyset$
and $K_{\gamma}^{{}^{B}}\subseteq\mathcal{O}$ for all $\lambda$ and $\gamma.$
By (1) and the separate weak∗ continuity of $V$ we have

(3)
$V(\chi_{{}_{D}},u_{{}_{B}})=w^{*}-\lim_{\lambda}\left(w^{*}-\lim_{\gamma}V\left(\frac{\chi_{{}_{K_{\lambda}^{{}^{D}}}}+\chi_{{}_{\sigma(K_{\lambda}^{{}^{D}})}}}{2},u_{{}_{K_{\gamma}^{{}^{B}}}}\right)\right)=0,$

and

(4) $V(u_{{}_{B}},\chi_{{}_{{}_{D}}})=0.$

A similar argument, but replacing (1) with (2), applies to prove $b)$.

To prove the last statement, we observe that

$(u_{{}_{0}}u_{{}_{0}}^{*}-u_{c}u_{c}^{*})u_{{}_{B}}=(\chi_{{}_{\mathcal{O}}}+\chi_{{}_{\sigma(\mathcal{O}})}-\chi_{{}_{C}}-\chi_{{}_{\sigma(C)}})u_{{}_{B}}=(\chi_{{}_{\mathcal{O}\setminus
C}}+\chi_{{}_{\sigma(\mathcal{O}\setminus
C)}})u_{{}_{B}}=u_{{}_{(\mathcal{O}\setminus C)\cap B}},$

and hence the statement $c)$ follows from $b)$. ∎

We can now establish the description of all orthogonal forms on a commutative
real C∗-algebra.

######  Theorem 2.4.

Let $V:A\times A\to\mathbb{R}$ be a continuous orthogonal form on a
commutative real C∗-algebra, then there exist $\varphi_{1},$ and $\varphi_{2}$
in $A^{*}$ satisfying

$V(x,y)=\varphi_{1}(xy)+\varphi_{2}(xy^{*}),$

for every $x,y\in A.$

###### Proof.

We may assume, without loss of generality, that $A$ is unital (compare
Proposition 1.3). Let $B$ denote the complexification of $A$. In this case $B$
identifies with $C(K)$ for a suitable compact Hausdorff space $K$ and
$A=C(K)^{\tau}$, where $\tau$ is a conjugate-linear period-2 *-homomorphism on
$C(K)$. We shall follow the notation employed in the rest of this section.

The form $V:A\times A\to\mathbb{R}$ extends to a continuous form
$V^{**}:A^{**}\times A^{**}\to\mathbb{R}$ which is separately weak∗ continuous
(cf. Lemma 1.2). The restriction $V^{**}|_{B(K)^{\tau}\times
B(K)^{\tau}}:B(K)^{\tau}\times B(K)^{\tau}\to\mathbb{R}$ also is a continuous
extension of $V$. We shall prove the statement for $V^{**}|_{B(K)^{\tau}\times
B(K)^{\tau}}.$ Henceforth, the symbol $V$ will stand for $V$, $V^{**}$ and
$V^{**}|_{B(K)^{\tau}\times B(K)^{\tau}}$ indistinctly.

Let us first take two self-adjoint elements $a_{1},a_{2}$ in $B(K)^{\tau}$. By
Proposition 1.5,

(5) $V(a_{1},a_{2})=V(a_{1}a_{2},1).$

To deal with the skew-symmetric part, let $D,B,C$ be Borel subsets of $K$
with, $D=\sigma(D)$ and $B,C\subseteq\mathcal{O}.$ From Proposition 2.3 $a)$,
we have

(6)
$V(\chi_{{}_{D}},u_{{}_{B}})=V(\chi_{{}_{D}},u_{{}_{B}}(1-\chi_{{}_{D}}+\chi_{{}_{D}}))=V(\chi_{{}_{D}},u_{{}_{B\cap(K\backslash
D)}})+V(\chi_{{}_{D}},u_{{}_{B}}\chi_{{}_{D}})$
$=V(\chi_{{}_{D}}-1+1,u_{{}_{B}}\chi_{{}_{D}})=V(-\chi_{{}_{(K\backslash
D)}}+1,u_{{}_{(B\cap D)}})=V(1,u_{{}_{B}}\chi_{{}_{D}}).$

Similarly,

(7) $V(u_{{}_{B}},\chi_{{}_{D}})=V(u_{{}_{B}}\chi_{{}_{D}},1).$

Now, Proposition 2.3 $b)$ and $c)$, repeatedly applied give:

$V(u_{{}_{B}},u_{{}_{C}})=V(u_{{}_{B}}(\chi_{{}_{F}}+u_{{}_{0}}u_{{}_{0}}^{*}),u_{{}_{C}})=V(u_{{}_{B}}u_{{}_{0}}u_{{}_{0}}^{*},u_{{}_{C}})$
$=V(u_{{}_{B}}(u_{{}_{0}}u_{{}_{0}}^{*}+u_{{}_{C}}u^{*}_{{}_{C}}-u_{{}_{C}}u^{*}_{{}_{C}}),u_{{}_{C}})=V(u_{{}_{B}}u_{{}_{C}}u^{*}_{{}_{C}},u_{{}_{C}})$
$=V(u_{{}_{B}}u_{{}_{C}}u^{*}_{{}_{C}},u_{{}_{C}}-u_{{}_{0}}+u_{{}_{0}})=V(u_{{}_{(B\cap
C)}},-u_{{}_{(\mathcal{O}\backslash C)}}+u_{{}_{0}})=V(u_{{}_{(B\cap
C)}},u_{{}_{0}})$
$=V(u_{{}_{B}}u_{{}_{C}}(u^{*}_{{}_{C}}-u_{{}_{0}}^{*}+u_{{}_{0}}^{*}),u_{{}_{0}})=V(u_{{}_{B}}u_{{}_{C}}u_{{}_{0}}^{*},u_{{}_{0}}).$

Thus, we have

(8)
$V(u_{{}_{B}},u_{{}_{C}})=V(u_{{}_{B}}u_{{}_{C}}u_{{}_{0}}^{*},u_{{}_{0}}),$

and similarly

(9)
$V(u_{{}_{B}},u_{{}_{C}})=V(u_{{}_{0}},u_{{}_{B}}u_{{}_{C}}u_{{}_{0}}^{*}).$

Let $\displaystyle{a_{l}=\sum_{j=1}^{m_{l}}\mu_{l,j}\chi_{{}_{D^{l}_{j}}},}$
$\displaystyle{b_{l}=\sum_{k=1}^{p_{l}}\lambda_{l,k}u_{{}_{B^{l}_{k}}}}$
($l\in\\{1,2\\}$) be two simple elements in $B(K)^{\tau}_{sa}$ and
$B(K)^{\tau}_{skew}$, respectively, where
$\lambda_{l,k},\mu_{l,j}\in\mathbb{R},$ for each $l\in\\{1,2\\},$
$\\{D^{l}_{1},\ldots,D^{l}_{m_{l}}\\}$ and
$\\{B^{l}_{1},\ldots,B^{l}_{p_{l}}\\}$ are families of mutually disjoint Borel
subsets of $K$ with $\sigma(D^{l}_{j})=D^{l}_{j}$ and
$B^{l}_{i}\subseteq\mathcal{O}.$ By $(\ref{eq theorem -1})$, $(\ref{eq sym
against anti})$, $(\ref{eq anti against sym})$, and $(\ref{eq anti anti 1}),$
we have

$V(a_{1}+b_{1},a_{2}+b_{2})=V(a_{1}a_{2},1)+\sum_{j=1}^{m_{1}}\sum_{k=1}^{p_{2}}\mu_{1,j}\lambda_{2,k}V\left(\chi_{{}_{D^{1}_{j}}},u_{{}_{B^{2}_{k}}}\right)$
$+\sum_{k=1}^{p_{1}}\sum_{j=1}^{m_{2}}\mu_{2,j}\lambda_{1,k}V\left(u_{{}_{B^{1}_{k}}},\chi_{{}_{D^{2}_{j}}}\right)+\sum_{k=1}^{p_{1}}\sum_{k=1}^{p_{2}}\lambda_{2,k}\lambda_{1,k}V\left(u_{{}_{B^{1}_{k}}},u_{{}_{B^{2}_{k}}}\right)$
$=V(a_{1}a_{2},1)+\sum_{j=1}^{m_{1}}\sum_{k=1}^{p_{2}}\mu_{1,j}\lambda_{2,k}V\left(1,\chi_{{}_{D^{1}_{j}}}u_{{}_{B^{2}_{k}}}\right)$
$+\sum_{k=1}^{p_{1}}\sum_{j=1}^{m_{2}}\mu_{2,j}\lambda_{1,k}V\left(u_{{}_{B^{1}_{k}}}\chi_{{}_{D^{2}_{j}}},1\right)+\sum_{k=1}^{p_{1}}\sum_{k=1}^{p_{2}}\lambda_{2,k}\lambda_{1,k}V\left(u_{{}_{B^{1}_{k}}}u_{{}_{B^{2}_{k}}}u_{{}_{0}}^{*},u_{{}_{0}}\right)$
$=V(a_{1}a_{2},1)+V\left(1,a_{1}b_{2}\right)+V\left(b_{1}a_{2},1\right)+V\left(b_{1}b_{2}u_{{}_{0}}^{*},u_{{}_{0}}\right)$
$=\psi_{1}(a_{1}a_{2})+\psi_{2}\left(a_{1}b_{2}\right)+\psi_{1}\left(b_{1}a_{2}\right)+\psi_{4}\left(b_{1}b_{2}\right),$

where $\psi_{1},\psi_{2},$ and $\psi_{4}$ are the functionals in $A^{*}$
defined by $\psi_{1}(x)=V(x,1),$ $\psi_{2}(x)=V(1,x),$ and
$\psi_{4}(x)=V(xu_{{}_{0}}^{*},u_{{}_{0}}),$ respectively. Since, by
Proposition 2.2, simple elements of the above form are norm-dense in
$B(K)^{\tau}_{sa}$ and $B(K)^{\tau}_{skew}$, respectively, and $V$ is
continuous, we deduce that

$V(a_{1}+b_{1},a_{2}+b_{2})=\psi_{1}(a_{1}a_{2})+\psi_{2}\left(a_{1}b_{2}\right)+\psi_{1}\left(b_{1}a_{2}\right)+\psi_{4}\left(b_{1}b_{2}\right),$

for every $a_{1},a_{2}\in B(K)^{\tau}_{sa}$, $b_{1},b_{2}\in
B(K)^{\tau}_{skew}.$

Now, taking $\phi_{1}=\frac{1}{4}(2\psi_{1}+\psi_{2}+\psi_{4}),$
$\phi_{2}=\frac{1}{4}(2\psi_{1}-\psi_{2}-\psi_{4}),$
$\phi_{3}=\frac{1}{4}(\psi_{2}-\psi_{4}),$ and
$\phi_{4}=\frac{1}{4}(\psi_{4}-\psi_{2}),$ we get

$V(a_{1}+b_{1},a_{2}+b_{2})=\phi_{1}((a_{1}+b_{1})(a_{2}+b_{2}))+\phi_{2}\left((a_{1}+b_{1})(a_{2}+b_{2})^{*}\right)$
$+\phi_{3}\left((a_{1}+b_{1})^{*}(a_{2}+b_{2})\right)+\phi_{4}\left((a_{1}+b_{1})^{*}(a_{2}+b_{2})^{*}\right),$

for every $a_{1},a_{2}\in B(K)^{\tau}_{sa}$, $b_{1},b_{2}\in
B(K)^{\tau}_{skew}.$

Finally, defining $\varphi_{1}(x)=\phi_{1}(x)+\phi_{4}(x^{*})$ and
$\varphi_{2}(x)=\phi_{2}(x)+\phi_{3}(x^{*})$ $(x\in A)$, we get the desired
statement. ∎

######  Remark 2.5.

The functionals $\varphi_{1}$ and $\varphi_{2}$ appearing in Theorem 2.4 need
not be unique. For example, let $(\varphi_{1},\varphi_{2})$ and
$(\phi_{1},\phi_{2})$ be two couples of elements in the dual of a commutative
real C∗-algebra $A$. It is not hard to check that

$\varphi_{1}(xy)+\varphi_{2}(xy^{*})=\phi_{1}(xy)+\phi_{2}(xy^{*}),$

for every $x,y\in A$ if, and only if,
$\varphi_{1}+\varphi_{2}=\phi_{1}+\phi_{2}$,
$(\varphi_{1}-\varphi_{2})(z)=(\phi_{1}-\phi_{2})(z)$ and
$(\varphi_{1}-\varphi_{2})(zw)=(\phi_{1}-\phi_{2})(zw),$ for every $z,w\in
A_{skew}.$ These conditions are not enough to guarantee that
$\phi_{i}=\varphi_{i}$. Take, for example,
$A=\mathbb{R}\oplus^{\infty}\mathbb{C}_{\mathbb{R}},$
$\phi_{1}(a,b)=a+\Re\hbox{e}(b)+\Im\hbox{m}(b),$ $\phi_{2}(a,b)=0,$
$\varphi_{1}(a,b)=\frac{a}{2}+\Re\hbox{e}(b)+\Im\hbox{m}(b),$ and
$\varphi_{2}(a,b)=\frac{a}{2}.$

######  Corollary 2.6.

Let $V:A\times A\to\mathbb{R}$ be a continuous orthogonal form on a
commutative real C∗-algebra, then its (unique) Arens extension
$V^{**}:A^{**}\times A^{**}\to\mathbb{R}$ is an orthogonal form.$\hfill\Box$

Clearly, the statement of the above Theorem 2.4 doesn’t hold for bilinear
forms on a commutative (complex) C∗-algebra. The real version established in
this paper is completely independent to the result proved by K. Ylinen for
commutative complex C∗-algebras in [51] and [21]. It seems natural to ask
whether the real result follows from the complex one by a mere argument of
complexification. Our next example shows that the (canonical) extension of an
orthogonal form on a commutative real C∗-algebra need not be an orthogonal
form on the complexification.

######  Example 2.7.

Let $K=\\{t_{1},t_{2}\\}.$ We define $\sigma:K\rightarrow K$ by
$\sigma(t_{1})=t_{2}.$ Let $A=C(K)^{\tau}$ be the real C∗-algebra whose
complexification is $C(K)$ and let $V:A\times A\to\mathbb{R},$ be the
orthogonal form defined by
$V(x,y)=\phi_{{}_{t_{1}}}(xy^{*})=\Re\hbox{e}(x(t_{1})\overline{y(t_{1})})=\Re\hbox{e}(x(t_{1})y(t_{2})),$
where $\phi_{t_{1}}=\Re\hbox{e}(\delta_{{}_{t_{1}}}).$ In this case, the
canonical complex bilinear extension $\widetilde{V}:C(K)\times
C(K)\to\mathbb{C}$ is given by
$\widetilde{V}(x,y)=\phi_{{}_{t_{1}}}(x\tau(y)^{*})=x(t_{1})y(t_{2})$ $(x,y\in
C(K)).$ It is clear that $\chi_{{}_{t_{1}}}\perp\chi_{{}_{t_{2}}}$ in $C(K),$
however $\widetilde{V}(\chi_{{}_{t_{1}}},\chi_{{}_{t_{2}}})=1\neq 0,$ which
implies that $\widetilde{V}$ is not orthogonal.

The (complex) bilinear extension of an orthogonal form $V$ on a real
C∗-algebra to its complexification is orthogonal precisely when $V$ satisfies
the generic form of an orthogonal form on a (complex) C∗-algebra given by the
main result in [21].

######  Corollary 2.8.

Let $V:A\times A\to\mathbb{R}$ be a continuous orthogonal form on a
commutative real C∗-algebra, let $B$ denote the complexification of $A$ and
let $\widetilde{V}:B\times B\to\mathbb{R}$ be the (complex) bilinear extension
of $V$. Then the form $\widetilde{V}$ is orthogonal if, and only if, $V$
writes in the form $V(x,y)=\varphi_{1}(xy)$ $(x,y\in A),$ where $\varphi_{1}$
is a functional in $A^{*}$.

###### Proof.

Let $\tau$ be the period-2 ∗-automorphism on $B$ satisfying that $B^{\tau}=B$
and let $\widetilde{\tau}:B^{*}\to B^{*}$ be the involution defined by
$\widetilde{\tau}(\phi)(b)=\overline{\phi(\tau(b))}$.

Suppose $\widetilde{V}$ is orthogonal. By the main result in [21] (see also
[51]), there exists $\phi\in B^{*}$ satisfying $\widetilde{V}(x,y)=\phi(xy),$
for every $x,y\in B$. Since $\widetilde{V}$ is an extension of $V$, we get
$V(a,b)=\Re\hbox{e}\phi(ab)=\phi(ab),$ for every $a,b\in A$. In particular,
$\phi(a)\in\mathbb{R}$, for every $a\in A$ and hence
$\widetilde{\tau}(\phi)=\phi$ lies in $(B^{*})^{\widetilde{\tau}}\equiv
A^{*}$.

Let us assume that $V$ writes in the form $V(x,y)=\varphi_{1}(xy)$ $(x,y\in
A)$, where $\varphi_{1}$ is a functional in $A^{*}$. The functional
$\varphi_{1}$ can be regarded as an element in $B^{*}$ satisfying
$\widetilde{\tau}(\varphi_{1})=\varphi_{1}.$ It is easy to check that
$\widetilde{V}(x,y)=\varphi_{1}(xy),$ for every $x,y\in B$. ∎

##  3\. Orthogonality preservers between commutative real C∗-algebras

Throughout this section, $A_{1}=C(K_{1})^{\tau_{1}}$ and
$A_{2}=C(K_{2})^{\tau_{2}}$ will denote two unital commutative real
C∗-algebras, $K_{1}$ and $K_{2}$ will be two compact Hausdorff spaces and
$\tau_{i}$ will denote a conjugate-linear period-2 ∗-automorphism on
$C(K_{i})$ given by $\tau_{i}(f)(t)=\overline{f(\sigma_{i}(t))}$ ($t\in
K_{i}$, $f\in C(K_{i})$), where $\sigma_{i}:K_{i}\to K_{i}$ is a period-2
homeomorphism. We shall write $B_{1}=C(K_{1})$ and $B_{2}=C(K_{2})$ for the
corresponding complexifications of $A_{1}$ and $A_{2}$, respectively.

By the Banach-Stone theorem, every surjective isometry $T:C(K_{1})\to
C(K_{2})$ is a composition operator, that is, there exist a unitary element
$u$ in $C(K_{2})$ and a homeomorphism $\sigma:K_{2}\to K_{1}$ such that
$T(f)(t)=(uC_{\sigma})(f)(t):=u(t)\ f(\sigma(t))$ ($t\in K_{2}$, $f\in
C(K_{1})$). This result led to the study of the so-called Banach-Stone
theorems in different classes of Banach spaces containing $C(K)$-spaces, in
which their algebraic and geometric properties are mutually determined. That
is the case of general C∗-algebras (R. Kadison [31] and Paterson and Sinclair
[42]), JB- and JB∗-algebras (Wright and M. Youngson [50] and Isidro and A.
Rodríguez [27]), JB∗-triples (Kaup [33] and Dang, Friedman and Russo [15]),
real C∗-algebras (Grzesiak [23], Kulkarni and Arundhathi [36], Kulkarni and
Limaye [37] and Chu, Dang, Russo and Ventura [14]) and real JB∗-triples
(Isidro, Kaup and Rodrguez [26], Kaup [34] and Fernndez-Polo, Martnez and
Peralta [19]). In what concerns us, we highlight that any surjective linear
isometry $T:C(K_{1})^{\tau_{1}}\to C(K_{2})^{\tau_{2}}$ is a composition
operator given by a homeomorphism $\phi:K_{2}\to K_{1}$ which satisfies
$\sigma_{1}\circ\phi=\phi\circ\sigma_{2}$ (cf. [23] or [36] or [37, Corollary
5.2.4]).

The class of orthogonality preserving (continuous) operators between
$C(K)$-spaces is strictly bigger than the class of surjective isometries.
Actually, a bounded linear operator $T:C(K_{1})\to C(K_{2})$ is orthogonality
preserving (equivalently, disjointness preserving) if, and only if, there
exist $u$ in $C(K_{2})$ and a mapping $\varphi:K_{2}\to K_{1}$ which is
continuous on $\\{t\in K_{2}:u(t)\neq 0\\}$ such that
$T(f)(t)=(uC_{\varphi})(f)(t)=u(t)\ f(\varphi(t))$ (compare [1, Example
2.2.1]).

Developing ideas given by E. Beckenstein, L. Narici, and A.R. Todd in [8] and
[9] (see also [7]), K. Jarosz showed, in [30], that the above hypothesis of
$T$ being continuous can be, in some sense, relaxed. More concretely, for
every orthogonality preserving linear mapping $T:C(K_{1})\to C(K_{2})$, there
exists a disjoint decomposition $K_{2}=S_{1}\cup S_{2}\cup S_{3}$ (with
$S_{2}$ open, $S_{3}$ closed), and a continuous mapping $\varphi$ from
$S_{1}\cup S_{2}$ into $K_{1}$ such that $T(f)(s)=\chi(s)f(\varphi(s))$ for
all $s\in S_{1}$ (where $\chi$ is a continuous, bounded, non-vanishing,
scalar-valued function on $S_{1}$), $T(f)(s)=0$ for all $s\in S_{3}$,
$\varphi(S_{2})$ is finite and, for each $s\in S_{2}$, the mapping $f\mapsto
T(f)(s)$ is not continuous. As a consequence, every orthogonality preserving
linear bijection between $C(K)$-spaces is (automatically) continuous. More
recently, M. Burgos and the authors of this note prove, in [13], that every
bi-orthogonality preserving linear surjection between two von Neumann algebras
(or between two compact C∗-algebras) is automatically continuous (compare
[40], [41] for recent additional generalisations).

The main goal of this section is to describe the orthogonality preserving
linear mappings between $C(K)^{\tau}$-spaces. Among the consequences, we
establish that every orthogonality preserving linear bijection between unital
commutative real C∗-algebras is automatically continuous. We shall provide an
example of an orthogonality preserving linear bijection between
$C(K)^{\tau}$-spaces which is not bi-orthogonality preserving and give a
characterisation of bi-orthogonality preserving linear maps.

We shall borrow and adapt some of the ideas developed in those previously
mentioned papers (cf. [8, 9] and [30]). In order to have a good balance
between completeness and conciseness, we just give some sketch of the
refinements needed in our setting. In any case, the results presented here are
independent innovations and extensions of those proved by Beckenstein, Narici,
and Todd and Jarosz for $C(K)$-spaces.

Let $T:C(K_{1})^{\tau_{1}}\to C(K_{2})^{\tau_{2}}$ be an orthogonality
preserving linear mapping. Keeping in mind the notation in the previous
section, we write $L_{i}:=\mathcal{O}_{i}\cup F_{i},$ where $\mathcal{O}_{i}$
and $F_{i}$ are the subsets of $K_{i}$ given by Lemma 2.1. The map sending
each $f$ in $C(Ki)^{\tau_{i}}$ to its restriction to $L_{i}$ is a
C∗-isomorphism (and hence a surjective linear isometry) from
$C(Ki)^{\tau_{i}}$ onto the real C∗-algebra $C_{r}(L_{i})$ of all continuous
functions $f:L_{i}\to\mathbb{C}$ taking real values on $F_{i}$. Thus, studying
orthogonality preserving linear maps between $C(K)^{\tau}$ spaces is
equivalent to study orthogonality preserving linear mappings between the
corresponding $C_{r}(L)$-spaces.

Henceforth, we consider an orthogonality preserving (not necessarily
continuous) linear map $T:C_{r}(L_{1})\to C_{r}(L_{2})$, where $L_{1}$ and
$L_{2}$ are two compact Hausdorff spaces and each $F_{i}$ is a closed subset
of $L_{i}$. Let us consider the sets

$Z_{1}=\\{s\in L_{2}:{\delta}_{s}T\hbox{ is a non-zero bounded real-linear
mapping}\\},$ $Z_{3}=\\{s\in L_{2}:{\delta}_{s}T=0\\},\hbox{ and
}Z_{2}=L_{2}\backslash(Z_{1}\cup Z_{3}).$

It is easy to see that $Z_{3}$ is closed. Following a very usual technique
(see, for example, [8, 9, 30, 16] and [17]), we can define a continuous
support map $\varphi:Z_{1}\cup Z_{2}\to L_{1}$. More concretely, for each
$s\in Z_{1}\cup Z_{2}$, we write $\hbox{supp}(\delta_{s}T)$ for the set of all
$t\in L_{1}$ such that for each open set $U\subseteq L_{1}$ with $t\in U$
there exists $f\in C_{r}(L_{1})$ with $\hbox{coz}(f)\subseteq U$ and
$\delta_{s}(T(f))\neq 0$. Actually, following a standard argument, it can be
shown that, for each $s\in Z_{1}\cup Z_{2},$ $\hbox{supp}(\delta_{s}T)$ is
non-empty and reduces exactly to one point $\varphi(s)\in L_{1}$, and the
assignment $s\mapsto\varphi(s)$ defines a continuous map from $Z_{1}\cup
Z_{2}$ to $L_{1}$. Furthermore, the value of $T(f)$ at every $s\in Z_{1}$
depends strictly on the value $f(\varphi(s))$. More precisely, for each $s\in
Z_{1}$ with $\varphi(s)\notin F_{1}$, the value $T(g)(s)$ is the same for
every function $g\in C_{r}(L_{1})$ with $g\equiv i$ on a neighborhood of
$\varphi(s).$ Thus, defining $T(i)(s):=0$ for every $s\in Z_{3}\cup Z_{2}$ and
for every $s\in Z_{1}$ with $\varphi(s)\in F_{1}$, and $T(i)(s):=T(g)(s)$ for
every $s\in Z_{1}\cup Z_{2}$ with $\varphi(s)\notin F_{1}$, where $g$ is any
element in $C_{r}(L_{1})$ with $g\equiv i$ on a neighborhood of $\varphi(s),$
we get a (well-defined) mapping $T(i):L_{2}\to\mathbb{C}.$ It should be
noticed that “$T(i)$” is just a symbol to denoted the above mapping and not an
element in the image of $T$. In this setting, the identity

$T(f)(s)=T(1)(s)\ \Re\hbox{e}f(\varphi(s))+T(i)(s)\ \Im\hbox{m}f(\varphi(s)),$

holds for every $s\in Z_{1}.$ Clearly, $T(1)(s),T(i)(s)\in\mathbb{R}$, for
every $s\in F_{2}$ and $\left|T(1)(s)\right|+\left|T(i)(s)\right|\neq 0$, for
every $s\in Z_{1}$.

The following property also follows from the definition of $\varphi$ by
standard arguments: Under the above conditions, let $s$ be an element in
$Z_{1}\cup Z_{2},$ then

(10) $\delta_{s}T(f)=0\hbox{ for every }f\in C_{r}(L_{1})\hbox{ with
}\varphi(s)\notin\overline{\hbox{coz}(f)}.$

######  Lemma 3.1.

The mapping $T(i)$ is bounded on the set $\varphi^{-1}(\mathcal{O}_{1})$.
Furthermore, the inequality

$\left|T(f)(s)\right|\leq\|T(1)\|+\sup_{\widetilde{s}\in\varphi^{-1}(\mathcal{O}_{1})}|T(i)(\widetilde{s})|$

holds for all $s\in Z_{1}$ and all $f\in C_{r}(L_{1})$ with
$|\Re\hbox{e}(f)|,|\Im\hbox{m}(f)|\leq 1.$

###### Proof.

Arguing by contradiction, we suppose that, for each natural $n$, there exists
$s_{n}\in\varphi^{-1}(\mathcal{O}_{1})$ such that
$\left|T(i)(s_{n})\right|>n^{3}$. The elements $s_{n}^{\prime}s$ can be chosen
so that $\varphi(s_{n})\neq\varphi(s_{m})$ for $n\neq m$, and consequently we
can find a sequence of pairwise disjoint open subsets $(U_{n})$ of
$\mathcal{O}_{1}$ with $\varphi(s_{n})\in U_{n}$. It is easily seen that we
can define a function $\displaystyle g=\sum_{n=1}^{\infty}i\ g_{n}\in
C_{r}(L_{1})$ with coz$(g_{n})\subset U_{n}$, $0\leq
g_{n}\leq\frac{1}{n^{2}}$, and $g_{n}\equiv\frac{1}{n^{2}}$ on a neighborhood
of $s_{n}$, for all $n$. By the form of $g$, and since $T$ is orthogonality
preserving, we have $|T(g)(s_{n})|=n^{2}|T(i)(s_{n})|>n$ for all $n$, which is
absurd. ∎

We can easily show now that $Z_{2}$ is an open subset of $L_{2}$. With this
aim, we consider an element $s_{0}$ in $Z_{2}$. We can pick a function $f\in
C_{r}(L_{1})$ such that $\|f\|\leq 1$ and

$|T(f)(s_{0})|>1+\|T(1)\|+\sup_{\widetilde{s}\in\varphi^{-1}(\mathcal{O}_{1})}|T(i)(\widetilde{s})|.$

Since
$\displaystyle\left|T(f)(s)\right|\leq\|T(1)\|+\sup_{\widetilde{s}\in\varphi^{-1}(\mathcal{O}_{1})}|T(i)(\widetilde{s})|<|T(f)(s_{0})|-1$,
for every $s\in Z_{1}\cup Z_{3},$ we conclude that there exists an open
neighborhood of $s_{0}$ contained in $Z_{2}$.

The next theorem resumes the above discussion.

######  Theorem 3.2.

In the notation above, let $T:C_{r}(L_{1})\to C_{r}(L_{2})$ be an
orthogonality preserving linear mapping. Then $L_{2}$ decomposes as the union
of three mutually disjoint subsets $Z_{1},Z_{2},$ and $Z_{3}$, where $Z_{2}$
is open and $Z_{3}$ is closed, there exist a continuous support map
$\varphi:Z_{1}\cup Z_{2}\to L_{1}$, and a bounded mapping
$T(i):L_{2}\to\mathbb{C}$ which is continuous on
$\varphi^{-1}(\mathcal{O}_{1})$ satisfying:

$T(i)(s)\in\mathbb{R}\ (\forall s\in F_{2}),\ T(i)(s)=0,\ (\forall s\in
Z_{3}\cup Z_{2}\hbox{ and }\forall s\in Z_{1}\hbox{ with }\varphi(s)\in
F_{1}),$ (11) $\left|T(1)(s)\right|+\left|T(i)(s)\right|\neq 0,\ (\forall s\in
Z_{1}),$ (12) $T(f)(s)=T(1)(s)\ \Re\hbox{e}f(\varphi(s))+T(i)(s)\
\Im\hbox{m}f(\varphi(s)),\hbox{ {\rm(}$\forall s\in Z_{1},f\in
C_{r}(L_{1}){\rm)},$}$ $T(f)(s)=0,\hbox{ {\rm(}$\forall s\in Z_{3},f\in
C_{r}(L_{1})${\rm),}}$

and for each $s\in L_{2}$, the mapping $C_{r}(L_{1})\to\mathbb{C}$, $f\mapsto
T(f(s)),$ is unbounded if, and only if, $s\in Z_{2}$. Furthermore, the set
$\varphi(Z_{2})$ is finite.

###### Proof.

Everything has been substantiated except perhaps the statement concerning the
set $\varphi(Z_{2})$. Arguing by contradiction, we assume the existence of a
sequence $(s_{n})$ in $Z_{2}$ such that $\varphi(s_{n})\neq\varphi(s_{m})$ for
every $n\neq m$. Find a sequence $(U_{n})$ of mutually disjoint open subsets
of $L_{1}$ satisfying $\varphi(s_{n})\in U_{n}$ and a sequence
$(f_{n})\subseteq C_{r}(L_{1})$ such that $\|f_{n}\|\leq\frac{1}{n}$,
$\hbox{coz}(f_{n})\subseteq U_{n}$ and $|\delta_{s_{n}}T(f_{n})|>n$, for every
$n\in\mathbb{N}$. The element $\displaystyle f=\sum_{n=1}^{\infty}f_{n}$ lies
in $C_{r}(L_{1}),$ and for each natural $n_{0}$,
$\displaystyle{f_{n_{0}}\perp\sum_{n=1,n\neq n_{0}}^{\infty}f_{n}}.$ Thus,
$|\delta_{s_{n_{0}}}T(f)|\geq|\delta_{s_{n_{0}}}T(f_{n_{0}})|>n_{0},$ which is
impossible. ∎

######  Remark 3.3.

The mapping $T(i):L_{2}\to\mathbb{C}$ has been defined to satisfy $T(i)(s)=0,$
for all $s\in Z_{3}\cup Z_{2}$ and for all $s\in Z_{1}$ with $\varphi(s)\in
F_{1}.$ It should be noticed here that the value $T(i)(s)$ is uniquely
determined only when $s\in Z_{1}$ and $\varphi(s)\notin F_{1}$. There are some
other choices for the values of $T(i)(s)$ at $s\in Z_{3}\cup Z_{2}$ and at
$s\in Z_{1}$ with $\varphi(s)\in F_{1}$ under which conditions $(\ref{eq 1 thm
32})$ and $(\ref{eq 2 thm 32})$ are satisfied.

######  Remark 3.4.

We shall now explore some of the consequences derived from Theorem 3.2. Let
$T:C_{r}(L_{1})\to C_{r}(L_{2})$ be an orthogonality preserving linear
mapping.

  1. $(a)$

The set $Z_{3}$ is empty whenever $T$ is surjective.

  2. $(b)$

$Z_{3}=\emptyset$ implies that $Z_{1}=L_{2}\backslash Z_{2}$ is a compact
subset of $L_{2}.$

  3. $(c)$

$\varphi(Z_{2})$ is a finite set of non-isolated points in $L_{1}$. Indeed, if
$\varphi(s_{0})=t_{0}$ is isolated for some $s_{0}\in Z_{2}$, then we can find
an open set $U\subseteq L_{1}$ such that $U\cap K_{1}=\\{t_{0}\\}$. Therefore,
for each $f\in C_{r}(L_{1})$ with $f(t_{0})=0$ we have $\delta_{s_{0}}T(f)=0$.
Pick an arbitrary $h\in C_{r}(L_{1})$. Clearly, $\chi_{{}_{t_{0}}}\in
C_{r}(L_{1})$, while $i\chi_{{}_{t_{0}}}$ lies in $C_{r}(L_{1})$ if, and only
if, $t_{0}\notin F_{1}$. Therefore,

$h_{0}=\Re\hbox{e}(h(t_{0}))\chi_{{}_{t_{0}}}+\Im\hbox{m}(h(t_{0}))\
i\chi_{{}_{t_{0}}}$

lies in $C_{r}(L_{1})$ and $(h-h_{0})(t_{0})=0$.

Assume first that $t_{0}\notin F_{1}$. Denoting
$\lambda_{0}=\delta_{s_{0}}T(\chi_{{}_{t_{0}}})$ and
$\mu_{{}_{0}}=\delta_{s_{0}}T(i\chi_{{}_{t_{0}}}),$ we have

$\delta_{s_{0}}T(h)=\delta_{s_{0}}T(h_{0})=\lambda_{0}\Re\hbox{e}(h(t_{0}))+\mu_{{}_{0}}\Im\hbox{m}(h(t_{0}))$
$=\frac{\lambda_{0}-i\mu_{{}_{0}}}{2}\delta_{t_{0}}(h)+\frac{\lambda_{0}+i\mu_{{}_{0}}}{2}\overline{\delta_{t_{0}}}(h).$

This shows that
$\delta_{s_{0}}T=\frac{\lambda_{0}-i\mu_{{}_{0}}}{2}\delta_{t_{0}}+\frac{\lambda_{0}+i\mu_{{}_{0}}}{2}\overline{\delta_{t_{0}}}$
is a continuous mapping from $C_{r}(L_{1})$ to $\mathbb{C}$, which is
impossible.

When $t_{0}\in F_{1}$ we have $\delta_{s_{0}}T=\lambda_{0}\delta_{t_{0}}$ is a
continuous mapping from $C_{r}(L_{1})$ to $\mathbb{R}$, which is also
impossible.

  4. $(d)$

$T$ surjective implies
$\varphi(Z_{1}\cap\mathcal{O}_{2})\subseteq\mathcal{O}_{1}$. Suppose, on the
contrary that there exists $s_{0}\in Z_{1}\cap\mathcal{O}_{2}$ with
$\varphi(s_{0})\in F_{1}$. By $(\ref{eq 2 thm 32})$,

$T(f)(s_{0})=T(1)(s_{0})\Re\hbox{e}f(\varphi(s_{0})),$

for every $f\in C_{r}(L_{1})$. It follows from the surjectivity of $T$,
together with the condition $s_{0}\in\mathcal{O}_{2},$ that for every complex
number $\omega$ there exists a real $\lambda$ satisfying
$\omega=T(1)(s_{0})\lambda,$ which is impossible.

  5. $(e)$

Suppose $T$ is surjective and fix $s_{0}\in Z_{1}\cap\mathcal{O}_{2}$. The
mapping $\delta_{s_{0}}T$ is a bounded real-linear mapping from $C_{r}(L_{1})$
onto $\mathbb{C}$. On the other hand, by $(\ref{eq 2 thm 32})$,

$\delta_{s_{0}}T(f)=T(1)(s_{0})\Re\hbox{e}f(\varphi(s_{0}))+T(i)(s_{0})\Im\hbox{m}f(\varphi(s_{0})),\
(\forall f\in C_{r}(L_{1})).$

Thus, $T$ being surjective implies that the space
$\mathbb{C}_{\mathbb{R}}=\mathbb{R}\times\mathbb{R}$ is linearly spanned by
the elements $T(1)(s_{0})$ and $T(i)(s_{0})$. Therefore, for each $s_{0}\in
Z_{1}\cap\mathcal{O}_{2},$ the set $\\{T(1)(s_{0}),T(i)(s_{0})\\}$ is a basis
of $\mathbb{C}_{\mathbb{R}}=\mathbb{R}\times\mathbb{R}.$ Consequently, when
$T$ is surjective and $s_{0}\in Z_{1}\cap\mathcal{O}_{2},$ the condition
$T(f)(s_{0})=0$ implies $f(\varphi(s_{0}))=0.$ For any other $s_{1}\in
Z_{1}\cap\mathcal{O}_{2}$ with $\varphi(s_{0})=\varphi(s_{1})$, we have:

$T(f)(s_{0})=0\Rightarrow f(\varphi(s_{0}))=0\Rightarrow T(f)(s_{1})=0.$

The fact that $C_{r}(L_{2})$ separates points implies that $s_{1}=s_{0}$.
Thus, $\varphi$ is injective on $Z_{1}\cap\mathcal{O}_{2}$.

We can now state the main result of this section which affirms that every
orthogonality preserving linear bijection between unital commutative real
C∗-algebras is (automatically) continuous.

######  Theorem 3.5.

Every orthogonality preserving linear bijection between unital commutative
(real) C∗-algebras is (automatically) continuous.

###### Proof.

Since $T$ is surjective, $Z_{3}=\emptyset$, and hence $Z_{1}=L_{2}\backslash
Z_{2}$ is a compact subset of $L_{2}.$ It is also clear that $\varphi(L_{2})$
is compact. We claim that $\varphi(L_{2})=L_{1}.$ Otherwise, there would exist
a non-zero function $f\in C_{r}(L_{1})$ with
$\overline{\hbox{coz}(f)}\subseteq L_{1}\backslash\varphi(L_{2})$. Thus, by
$(\ref{eq varphi notin cozero})$, $T(f)=0,$ contradicting the injectivity of
$T$. By Remark 3.4$(c)$,
$\varphi(Z_{1})=\overline{\varphi(Z_{1})}=\varphi(L_{2})=\varphi(Z_{1})\cup\varphi(Z_{2})=L_{1}.$

We next see that $Z_{2}=\emptyset.$ Otherwise we can take $g\in C_{r}(L_{2})$
with $\emptyset\neq coz(g)\subset Z_{2}.$ Let $h=T^{-1}(g).$ Obviously
$Th(s)=0$ whenever $s\in Z_{1}.$ We claim that $h(t)=0,$ for every
$t\in\varphi(Z_{1})\setminus\varphi(Z_{2}).$ Let us fix
$t\in\varphi(Z_{1})\setminus\varphi(Z_{2}).$ Since $\varphi(Z_{2})$ is a
finite set there are disjoint open sets $U_{1},U_{2}$ such that $t\in
U_{1},\varphi(Z_{2})\subset U_{2}.$ Let $f\in C(L_{1},{\mathbb{R}})$ be such
that $f(t)\neq 0$ and $\overline{coz(f)}\subset U_{1}.$ We see that $T(fh)=0.$
Indeed, let $s\in L_{2}=Z_{1}\cup Z_{2}.$ If $s$ lies in $Z_{1},$ then the
maps $fh$ and $f(\varphi(s))h$ lie in $C_{r}(L_{1})$ and coincide at
$\varphi(s).$ Since $T$ is linear over ${\mathbb{R}}$ and $f$ takes real
values, we deduce, by $(\ref{eq 2 thm 32})$, that
$T(fh)(s)=f(\varphi(s))Th(s)=0.$ If $s\in Z_{2}$ then, since
$\varphi(s)\notin\overline{coz(fh)}$, then $\delta_{s}T(fh)=T(fh)(s)=0$.

We have shown that $T(fh)=0$. Thus, since $T$ is injective, $fh=0$ and
therefore $h(t)=0.$ We have therefore proved that
$coz(h)\subset\varphi(Z_{2})$ which is a finite set. This means that $h$ must
be a finite linear combination of characteristic function on points of
$\varphi(Z_{2})$ and these points must be isolated which is impossible, since
by $c)$ in Remark 3.4 no point in $\varphi(Z_{2})$ can be isolated. We have
proved that $Z_{2}=\emptyset.$ Now the fact that $T$ is continuous follows
easily. ∎

The above theorem is the first step toward extending, to the real setting,
those results proved in [30], [6], [13], [40], [38] and [48] for (complex)
C∗-algebras.

Orthogonality preserving linear bijections enjoy an interesting additional
property.

######  Proposition 3.6.

In the notation of this section, let $T:C_{r}(L_{1})\to C_{r}(L_{2})$ be an
orthogonality preserving linear bijection. Then $T^{-1}$ preserves invertible
elements, that is, $T^{-1}(g)$ is invertible whenever $g$ is an invertible
element in $C_{r}(L_{2})$.

###### Proof.

Take an invertible element $g\in C_{r}(L_{2})$. Let $f$ be the unique element
in $C_{r}(L_{1})$ satisfying $T(f)=g$. Theorem 3.2 implies that

$0\neq g(s)=T(f)(s)=T(1)(s)\ \Re\hbox{e}f({\varphi(s)})+T(i)(s)\
\Im\hbox{m}f({\varphi(s)}),$

for every $s\in Z_{1}$. This assures that $f({\varphi(s)})\neq 0,$ for every
$s\in Z_{1},$ and since $\varphi(Z_{1})=L_{1},$ $f=T^{-1}(g)$ must be
invertible in $C_{r}(L_{1})$. ∎

In the setting of complex Banach algebras, it follows from the Gleason-Kahane-
Żelazko theorem that a linear transformation $\phi$ from a unital,
commutative, complex Banach algebra $A$ into $\mathbb{C}$ satisfying
$\phi(1)=1$ and $\phi(a)\neq 0$ for every invertible element $a$ in $A$ is
multiplicative, that is, $\phi(ab)=\phi(a)\phi(b)$ (see [20, 32]). Although,
the Gleason-Kahane-Żelazko theorem fails for real Banach algebras, S.H.
Kulkarni found in [35] the following reformulation: a linear map $\phi$ from a
real unital Banach algebra ${A}$ into the complex numbers is multiplicative if
$\varphi(1)=1$ and $\phi(a)^{2}+\phi(b)^{2}\neq 0$ for every $a,b\in{A}$ with
$ab=ba$ and $a^{2}+b^{2}$ invertible. It is not clear that statement $(b)$ in
the above proposition can be improved to get the hypothesis of Kulkarni’s
theorem. The structure of orthogonality preserving linear mappings between
$C_{r}(L)$-spaces described in Theorem 3.2 invites us to affirm that they are
not necessarily multiplicative.

###  3.1. Bi-orthogonality preservers

As a consequence of the description of orthogonality preserving linear maps
given in [30], it can be shown that an orthogonality preserving linear
bijection between (complex) $C(K)$-spaces is bi-orthogonality preserving. It
is natural to ask wether every orthogonality preserving linear bijection
between commutative (unital) real C∗-algebras is bi-orthogonality preserving.

This is known to be true in two cases: first, between spaces
$C_{\mathbb{R}}(K)$ of real (and also complex) valued functions on a compact
Hausdorff space $K$, as it is well-known; second, between spaces of the type
$C_{\mathbb{R}}(K;\mathbb{R}^{n})$ (compare [16, Section 3]). Spaces like
those we are dealing with in this paper need not satisfy this property, that
is, there exists an orthogonality preserving linear bijection
$T:C_{r}(L_{1})\to C_{r}(L_{2})$ which is not bi-orthogonality preserving (and
even $L_{1}$ and $L_{2}$ are not homeomorphic either).

######  Example 3.7.

Let $L_{1}=\\{t_{1},t_{2},t_{3}\\}$ $L_{2}=\\{s_{1},s_{2},s_{3},s_{4}\\}$ with
$\mathcal{O}_{1}=\\{t_{1},t_{3}\\}$, $\mathcal{O}_{2}=\\{s_{1}\\}$,
$F_{1}=\\{t_{2}\\}$ and $F_{2}=\\{s_{2},s_{3},s_{4}\\}$. Define
$\varphi:L_{2}\to L_{1}$ by $\varphi(s_{i})=t_{i},$ for $i=1,2,$ and
$\varphi(s_{i})=t_{3},$ for $i=3,4$. It is easy to check that
$T(f)(s_{i})=f(\varphi(s_{i}))$ if $i=1,2$, and
$T(f)(s_{3})=\Re\hbox{e}f(t_{3}),$ $T(f)(s_{4})=\Im\hbox{m}f(t_{3})$ is an
orthogonality preserving linear bijection, but $T^{-1}$ is not orthogonality
preserving.

In the above example, $\varphi^{-1}(\mathcal{O}_{1})\cap F_{2}$ is non-empty.
Our next result shows that a topological condition on $F_{2}$ assures that an
orthogonality preserving linear bijection between unital commutative real
C∗-algebras is bi-orthogonality preserving.

######  Proposition 3.8.

In the notation of this section, let $T:C_{r}(L_{1})\to C_{r}(L_{2})$ be an
orthogonality preserving linear bijection (not assumed to be bounded). The
following statements hold:

  1. $(a)$

If $T$ is bi-orthogonality preserving then $\varphi:L_{2}\to L_{1}$ is a
(surjective) homeomorphism, $\varphi(F_{2})=F_{1},$ and
$\varphi(\mathcal{O}_{2})=\mathcal{O}_{1}$. In particular,
$\varphi^{-1}(\mathcal{O}_{1})\cap F_{2}=\emptyset$.

  2. $(b)$

If $F_{2}$ has empty interior then $T$ is biorthogonality preserving.

###### Proof.

$(a)$ If $T$ is bi-orthogonality preserving, it can be easily seen that
$\varphi:L_{2}\to L_{1}$ is a homeomorphism, and for each $s\in L_{2}$,
supp$(\delta_{\varphi(s)}T^{-1})=\\{s\\}$. By Remark 3.4$(d)$, applied to $T$
and $T^{-1}$, we have $\varphi(F_{2})=F_{1}$ and
$\varphi(\mathcal{O}_{2})=\mathcal{O}_{1}$. Then
$\varphi^{-1}(\mathcal{O}_{1})\cap F_{2}=\emptyset.$ So, $a)$ is clear.

$(b).$ Let us assume that $F_{2}$ has empty interior. Arguing by contradiction
we suppose that $T^{-1}$ is not orthogonality preserving. Then there exist
$f_{1},f_{2}\in C_{r}(L_{1})$ with $f_{1}f_{2}\neq 0$, but $T(f_{1})\perp
T(f_{2})$. Thus $U:=\hbox{coz}(f_{1})\cap\hbox{coz}(f_{2})$ is a non-empty
open subset of $L_{1}.$ Keeping again the notation of Theorem 3.2 for $T$, we
recall that, by Theorem 3.5 and Remark 3.4, $Z_{3}=\emptyset,$
$Z_{2}=\emptyset$, $\varphi(L_{2})=L_{1}$,
$\varphi(\mathcal{O}_{2})\subset\mathcal{O}_{1},$ $\varphi|_{\mathcal{O}_{2}}$
is injective, and for each $s\in\mathcal{O}_{2}$, and $\\{T(1)(s),T(i)(s)\\}$
is a basis of $\mathbb{C}_{\mathbb{R}}=\mathbb{R}\times\mathbb{R}.$

By the form of $T$, there are no points of $\varphi(\mathcal{O}_{2})$ in
$U=\hbox{coz}(f_{1})\cap\hbox{coz}(f_{2})$ (because for each
$s\in\mathcal{O}_{2},$ $T(f)(s)\neq 0$ when $f(\varphi(s))\neq 0$). Now, let
$k$ be a non-zero element in $C(L_{1},\mathbb{R})$, with
coz$(k)\subseteq\hbox{coz}(f_{1})\cap\hbox{coz}(f_{2})$. By Theorem 3.2
$(\ref{eq 2 thm 32})$, it is clear that
$\varphi(\hbox{coz}(T(k)))\subseteq\hbox{coz}(k)$, and hence, since
$\varphi(\mathcal{O}_{2})\subseteq\mathcal{O}_{1}$, $\hbox{coz}(T(k))$ is a
non-empty subset of $F_{2}$, against our hypotheses. ∎

As we have already seen, an orthogonality preserving linear bijection between
$C_{r}(L)$-spaces needs not to be biorthogonality preserving. Example 3.7 also
shows that, unlike in the complex case, the existence of an orthogonality
preserving linear bijection between $C_{r}(L)$-spaces does not guarantee that
the corresponding compacts spaces are homeomorphic. We next provide a
characterisation of those (linear) mappings which are bi-orthogonality
preserving. As a consequence, we shall see that if there exists a bi-
orthogonality preserving linear map $T:C_{r}(L_{1})\to C_{r}(L_{2})$ then
$L_{1}$ and $L_{2}$ are homeomorphic.

######  Theorem 3.9.

Let $T:C_{r}(L_{1})\to C_{r}(L_{2})$ be a mapping. The following statements
are equivalent:

  1. $(a)$

$T$ is a bi-orthogonality preserving linear surjection;

  2. $(b)$

There exists a (surjective) homeomorphism $\varphi:L_{2}\to L_{1}$ with
$\varphi(\mathcal{O}_{2})=\mathcal{O}_{1},$ a function
$a_{1}=\gamma_{1}+i\gamma_{2}$ in $C_{r}(L_{2})$ with $a_{1}(s)\neq 0$ for all
$s\in L_{2},$ and a function $a_{2}=\eta_{1}+i\eta_{2}:L_{2}\to{\mathbb{C}}$
continuous on $\mathcal{O}_{2}$ with the property that

$0<\inf_{s\in\mathcal{O}_{2}}\left|\det\left(\begin{array}[]{cc}\gamma_{1}(s)&\eta_{1}(s)\\\
\gamma_{2}(s)&\eta_{2}(s)\\\
\end{array}\right)\right|\leq\sup_{s\in\mathcal{O}_{2}}\left|\det\left(\begin{array}[]{cc}\gamma_{1}(s)&\eta_{1}(s)\\\
\gamma_{2}(s)&\eta_{2}(s)\\\ \end{array}\right)\right|<+\infty,$

such that

$T(f)(s)=a_{1}(s)\ \Re\hbox{e}f({\varphi(s)})+a_{2}(s)\
\Im\hbox{m}f({\varphi(s)})$

for all $s\in L_{2}$ and $f\in C_{r}(L_{1}).$

###### Proof.

$(a)\Rightarrow(b)$. Since every bi-orthogonality preserving linear mapping is
injective, we can assume that $T:C_{r}(L_{1})\to C_{r}(L_{2})$ is a bi-
orthogonality preserving linear bijection. We keep the notation given in
Theorem 3.2. We have already shown that $Z_{3}=\emptyset,$ $Z_{2}=\emptyset$,
$\varphi:L_{2}\to L_{1}$ is a surjective homeomorphism,
$\varphi(\mathcal{O}_{2})=\mathcal{O}_{1},$ and for each
$s\in\mathcal{O}_{2}$, $\\{T(1)(s),T(i)(s)\\}$ is a basis of
$\mathbb{C}_{\mathbb{R}}=\mathbb{R}\times\mathbb{R}$ (compare Theorem 3.5,
Remark 3.4 and Proposition 3.8). Taking $a_{1}=T(1)=\gamma_{1}+i\gamma_{2}$
and $a_{2}=T(i)=\eta_{1}+i\eta_{2}$ we only have to show that

$0<\inf_{s\in\mathcal{O}_{2}}\left|\det\left(\begin{array}[]{cc}\gamma_{1}(s)&\eta_{1}(s)\\\
\gamma_{2}(s)&\eta_{2}(s)\\\
\end{array}\right)\right|\leq\sup_{s\in\mathcal{O}_{2}}\left|\det\left(\begin{array}[]{cc}\gamma_{1}(s)&\eta_{1}(s)\\\
\gamma_{2}(s)&\eta_{2}(s)\\\ \end{array}\right)\right|<+\infty.$

Let us denote $M_{s}=\left(\begin{array}[]{cc}\gamma_{1}(s)&\eta_{1}(s)\\\
i\gamma_{2}(s)&i\eta_{2}(s)\\\ \end{array}\right).$ Clearly $\det(M_{s})\neq
0$, for every $s\in\mathcal{O}_{2}$ and
$T(f)(s)=M_{s}\cdot\left(\begin{array}[]{c}\Re\hbox{e}f(\varphi(s))\\\
\Im\hbox{m}f(\varphi(s))\\\ \end{array}\right),$ for every $f\in
C_{r}(L_{1}),s\in L_{2}$. By the boundedness of $T(1):L_{2}\to\mathbb{C}$ and
$T(i)|_{\mathcal{O}_{2}}:\mathcal{O}_{2}\to\mathbb{C}$ (see Lemma 3.1) there
exists $M>0$ such that $\left|det(M_{s})\right|\leq M$ for all
$s\in\mathcal{O}_{2}.$

Applying the above arguments to the mapping $T^{-1}$ we find a surjective
homeomorphism $\psi=\varphi^{-1}:L_{1}\to L_{2}$, a mapping
$T^{-1}(i):L_{1}\to L_{2}$ and $m>0,$ such that
$\psi(\mathcal{O}_{1})=\mathcal{O}_{2}$, for each $t\in\mathcal{O}_{1}$,
$\\{T^{-1}(1)(t),T^{-1}(i)(t)\\}$ is a basis of
$\mathbb{C}_{\mathbb{R}}=\mathbb{R}\times\mathbb{R},$
$T^{-1}(g)(t)=N_{t}\cdot\left(\begin{array}[]{c}\Re\hbox{e}g(\psi(t))\\\
\Im\hbox{m}g(\psi(t))\\\ \end{array}\right)$ $(g\in C_{r}(L_{2}),t\in L_{1})$,
$\left|\det(N_{t})\right|\leq m,$ for all $t\in\mathcal{O}_{1},$ where
$N_{t}=\left(\begin{array}[]{cc}\Re\hbox{e}T^{-1}(1)(t)&\Re\hbox{e}T^{-1}(i)(t)\\\
i\Im\hbox{m}T^{-1}(1)(t)&i\Im\hbox{m}T^{-1}(i)(t)\\\ \end{array}\right).$ It
can be easily seen that, for each $s\in\mathcal{O}_{2},$
$N_{\varphi(s)}=M_{s}^{-1},$ which shows that
$\left|det(M_{s})\right|\geq\frac{1}{m}$, for all $s\in\mathcal{O}_{2}.$

$(b)\Rightarrow(a)$. Let $T:C_{r}(L_{1}):\to C_{r}(L_{2})$ be a mapping
satisfying the hypothesis in $(b).$ Clearly, $T$ is linear, and since
$\varphi(F_{2})=F_{1},$ $Tf(s)\in{\mathbb{R}}$ for all $s\in F_{2}$ and $f\in
C_{r}(L_{1})$ (that is, $T(f)\in C_{r}(L_{2})$). We can easily check that,
under these hypothesis, $T$ is injective and preserves orthogonality.

We shall now prove that $T$ is surjective. Indeed, for each
$s\in\mathcal{O}_{2}$

$T(f)(s)=\left(\begin{array}[]{c}\Re\hbox{e}g(s)\\\ \Im\hbox{m}g(s)\\\
\end{array}\right)=\left(\begin{array}[]{cc}\gamma_{1}(s)&\eta_{1}(s)\\\
i\gamma_{2}(s)&i\eta_{2}(s)\\\
\end{array}\right)\cdot\left(\begin{array}[]{c}\Re\hbox{e}f({\varphi(s)})\\\
\Im\hbox{m}f({\varphi(s)})\\\ \end{array}\right)$
$=M_{s}\cdot\left(\begin{array}[]{c}\Re\hbox{e}f({\varphi(s)})\\\
\Im\hbox{m}f({\varphi(s)})\\\ \end{array}\right),$

thus,

$\left(\begin{array}[]{c}\Re\hbox{e}f({\varphi(s)})\\\
\Im\hbox{m}f({\varphi(s)})\\\
\end{array}\right)=M_{s}^{-1}\cdot\left(\begin{array}[]{c}\Re\hbox{e}g(s)\\\
\Im\hbox{m}g(s)\\\ \end{array}\right).$

We define $b_{1}(t):L_{1}\to{\mathbb{C}}$ and
$b_{2}:\mathcal{O}_{1}\to{\mathbb{C}}$ by
$b_{1}(t)=\widetilde{\gamma}_{1}(t)+i\widetilde{\gamma}_{2}(t)$ and
$b_{2}=\widetilde{\eta}_{1}(t)+i\widetilde{\eta}_{2}(t)$
($t\in\mathcal{O}_{1}$), where
$M_{\varphi^{-1}(t)}^{-1}=\left(\begin{array}[]{cc}\widetilde{\gamma}_{1}(t)&\widetilde{\eta}_{1}(t)\\\
i\widetilde{\gamma}_{2}(t)&i\widetilde{\eta}_{2}(t)\\\ \end{array}\right),$
and $b_{1}(t)=\frac{1}{\gamma_{1}(\varphi^{-1}(t))},$ for every $t\in F_{1}$.
Then $S:C_{r}(L_{2})\to C_{r}(L_{1}),$ defined by $S(g)(t)=b_{1}(t)\
\Re\hbox{e}g(\varphi^{-1}(t))+b_{2}(t)\ \Im\hbox{m}g(\varphi^{-1}(t)),$ is
linear, preserves orthogonality and it is easy to check that $S=T^{-1}.$ It
follows that $T$ is bi-orthogonality preserving. ∎

Let $T$ be a bi-orthogonality preserving linear mapping with associated
homeomorphism $\varphi:L_{2}\to L_{1}$. Clearly, the operator
$S:C_{r}(L_{1})\to C_{r}(L_{2})$, $S(f)(s):=f(\varphi(s))$ is a ∗-isomorphism.
Having in mind that a linear mapping $T:A\to B$ between real C∗-algebras is a
∗-isomorphism if, and only if, the complex linear extension
$\widetilde{T}:A\oplus iA\to B\oplus iB$, $\widetilde{T}(a+ib)=T(a)+iT(b)$ is
a ∗-isomorphism, we get the following corollary.

######  Corollary 3.10.

The following statements are equivalent:

  1. $(a)$

There exists a bi-orthogonality preserving linear bijection $T:C_{r}(L_{1})\to
C_{r}(L_{2})$;

  2. $(b)$

There exists a C∗-isomorphism $S:C_{r}(L_{1})\to C_{r}(L_{2})$;

  3. $(c)$

There exists a C∗-isomorphism $\widetilde{S}:C(L_{1})\to C(L_{2})$;

  4. $(d)$

$L_{1}$ and $L_{2}$ are homeomorphic.$\hfill\Box$

Acknowledgements: The authors gratefully thank to the Referee for the
constructive comments and detailed recommendations which definitely helped to
improve the readability and quality of the paper.

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