Cent. Eur. J. Math. • 11(6) • 2013 • 1020-1033
DOI: 10.2478/s11533-013-0224-x
Central European Journal of Mathematics
Maps between Banach function algebras
satisfying certain norm conditions
Research Article
Maliheh Hosseini1∗ , Fereshteh Sady2†
1 Department of Mathematics, K.N. Toosi University of Technology, 16315-1618, Tehran, Iran
2 Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, 14115-134, Tehran, Iran
Received 28 December 2011; accepted 21 June 2012
Abstract: Let A and B be Banach function algebras on compact Hausdorff spaces X and Y , respectively, and let A and B
be their uniform closures. Let I, I 0 be arbitrary non-empty sets, α ∈ C \ {0}, ρ : I → A, τ : I 0 → A and S : I → B,
T : I 0 → B be maps such that ρ(I), τ(I 0 ) and S(I), T (I 0 ) are closed under multiplications and contain exp A and exp B,
respectively. We show that if kS(p)T (p0 ) − αkY = kρ(p)τ(p0 ) − αkX for all p ∈ I and p0 ∈ I 0 , then there exist a real
algebra isomorphism S : A → B, a clopen subset K of MB and a homeomorphism φ : MB → MA between the
maximal ideal spaces of B and A such that for all f ∈ A,
b =
Sf
(
bf ◦ φ
bf ◦ φ
on
K,
on
MB \ K,
where b· denotes the Gelfand transformation. Moreover, S can be extended to a real algebra isomorphism from A
onto B inducing a homeomorphism between MB and MA . We also show that under an additional assumption
related to the peripheral range, S is complex linear, that is A and B are algebraically isomorphic. We also consider
the case where α = 0 and X and Y are locally compact.
MSC:
46J10, 47B48, 46J20
Keywords: Banach function algebras • Uniform algebras • Norm-preserving • Peripheral range • Choquet boundary
© Versita Sp. z o.o.
∗
†
1020
E-mail: [email protected]
E-mail: [email protected]
M. Hosseini, F. Sady
1.
Introduction
Investigating the conditions under which a spectrum-preserving map T : A → B between Banach algebras A and B is
linear and multiplicative has attracted a considerable attention for many years. The case where T is assumed to be
linear dates back to the classical Gleason–Kahane–Żelazko theorem. A theorem by Kowalski and Słodkowski provides
a similar result without assuming the linearity assumption. Indeed, by this theorem, for semisimple Banach algebras A
and B every surjective map T : A → B satisfying T (0) = 0 and σ (T (a) − T (b)) ⊆ σ (a − b), a, b ∈ A, where σ ( · ) denotes
the spectrum of algebra elements, is linear and multiplicative [12].
Multiplicatively spectrum-preserving maps were first considered by Molnár in [17]. He proved that if X is a first countable
compact Hausdorff space and C (X ) is the Banach algebra of all continuous complex-valued functions on X , then any
surjective map T : C (X ) → C (X ) preserving multiplicatively the spectrum of functions, i.e., σ (T (f)T (g)) = σ (fg) for all
f, g ∈ C (X ), is a weighted composition operator, or in other words a multiplication of an algebra isomorphism by a
continuous function. Generalizations of Molnár’s result were given in [5, 18, 19] for arbitrary uniform algebras rather
than C (X ) and in [6, 8] for commutative semisimple Banach algebras. Considering a similar multiplicativity condition for
some parts of the spectrum (called peripheral spectrum) instead of the whole spectrum, the above result was extended
in [9, 11, 14, 16].
Clearly, for uniform algebras A and B on compact Hausdorff spaces X and Y every multiplicatively spectrum-preserving
map T : A → B is multiplicatively norm-preserving, that is kT f T gkY = kfgkX , where k · kX and k · kY are the sup-norms
on X and Y . We note that under this weaker condition such map need not be a weighted composition operator or even
a real linear map. However, as it was shown in [4, 14, 23], such maps induce homeomorphisms between the Choquet
boundaries of the uniform algebras under consideration. A mapping T : A → B between Banach algebras A and B is
called a non-symmetric multiplicatively norm-preserving map if kT x T y − αk = kxy − αk holds for all x, y ∈ A, where α
is a non-zero complex number. The general form of non-symmetric multiplicatively norm-preserving maps between (unital)
uniform algebras was characterized in [7, 14, 15], and a similar result for certain subalgebras of continuous functions on
compact Hausdorff spaces which are Banach algebras under some norms was given by the authors in [9]. Generalizations
of these results given in [4, Theorem 1] characterize the general form of arbitrary pair of maps T , S : A0 → B0 between
certain multiplicative semigroups A0 and B0 of (unital) uniform algebras A and B on X and Y , respectively, satisfying
kT (f)S(g) − αkY = kfg − αkX , f, g ∈ A0 , for some non-zero complex number α. More generally, in [21, Theorem 3.1]
Shindo gives the general form of certain maps T , S : A1 → B, where A1 is a subset of a uniform algebra A on a compact
Hausdorff space X and B is a uniform algebra on a compact Hausdorff space Y , such that
kS(f)T (g) − αkY = kρ(f)τ(g) − αkX
for all f, g ∈ A1 , where ρ, τ : A1 → A are certain maps whose ranges are multiplicative semigroups of A and α ∈ C \ {0}.
For a survey on the recent results concerning spectral preserver maps between commutative Banach algebras we refer
the reader to [3].
In this paper, after introducing definitions and preliminaries in Section 2, we will extend the results of [21] in Section 3
and characterize the general form of maps T and S satisfying the above norm condition for the case where A and B are
subalgebras of C (X ) and C (Y ) (for some compact Hausdorff spaces X and Y ) that are complete under some norms (not
necessarily supremum norms) and A1 is replaced by arbitrary non-empty sets I and I 0 , Theorems 3.1 and 3.2. We should
note that, since T and S are not assumed to be linear, we cannot extend them simply to A and B and use directly the
known results for the uniform algebra case. In particular, we see that, under an additional assumption related to the
peripheral ranges of algebra elements, such a map induces an algebra isomorphism between A and B. Then in Section 4
we consider a generalization of multiplicative norm-preserving maps between certain subalgebras A and B of C0 (X )
and C0 (Y ), for locally compact Hausdorff spaces X and Y , endowed with some Banach algebra norms. We show that if
I and I 0 are non-empty sets and ρ : I → A, τ : I 0 → A and S : I → B, T : I 0 → B are certain maps satisfying
kS(p)kY = kρ(p)kX ,
kT (p0 )kY = kτ(p0 )kX ,
kS(p)T (p0 )kY = kρ(p)τ(p0 )kX ,
p ∈ I,
p0 ∈ I 0 ,
then there exists a homeomorphism between some quotients of certain boundaries of A and B, which is, in the uniform
algebra case, a homeomorphism between the Choquet boundaries, Theorem 4.2. This extends [4, Proposition 2] under
a weaker condition.
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Maps between Banach function algebras satisfying certain norm conditions
2.
Preliminaries
Let X be a locally compact Hausdorff space and C0 (X ) be the algebra of all continuous complex-valued functions on X
vanishing at infinity. We denote the supremum norm of f ∈ C0 (X ) by kfkX . A subalgebra A of C0 (X ) is called a function
algebra on X if A separates strongly the points of X in the sense that for distinct points x, y ∈ X there exists a
function f ∈ A with f(x) =
6 f(y) and for each x ∈ X there exists f ∈ A with f(x) 6= 0. A Banach function algebra on X is
a function algebra on X which is a Banach algebra under a norm. A uniformly closed function algebra on X is called a
uniform algebra on X . In the case where X is compact we assume that all function algebras on X contain the constants
and in this case we use the notation A−1 for the group of invertible elements in A. Furthermore, for a subset A0 of A−1
we set I(A0 ) = {f −1 : f ∈ A0 }. Clearly, if A0 is a multiplicative subsemigroup (respectively, subgroup) of A−1 then I(A0 ) is
also a multiplicative subsemigroup (respectively, subgroup) of A−1 .
Let A be a function algebra on a locally compact Hausdorff space X . A subset E of X is called a boundary for A if every
f ∈ A attains its maximum modulus at some point of E. The unique minimal closed boundary for A, denoted by ∂A,
is called the Šilov boundary of A. The Choquet boundary c(A) of A is the set of all x ∈ X for which the evaluation
functional δx at x is an extreme point of the unit ball of the dual space of (A, k · kX ). It is well known that c(A) is dense
in ∂A, see [22] and [1].
For an element f in a Banach function algebra A on a locally compact Hausdorff space X , the peripheral range and the
peripheral spectrum of f are defined, respectively, by Rπ (f) = {λ ∈ f(X ) : |λ| = kfkX } and σπ (f) = {λ ∈ σ (f) : |λ| = r(f)},
where σ ( · ) and r( · ) denote the spectrum and the spectral radius of the algebra elements. We should note that the
peripheral spectrum and the peripheral range of elements in a uniform algebra on a compact Hausdorff space X are the
same by [16, Lemma 1].
Let A be a Banach function algebra on a locally compact Hausdorff space X . A function f ∈ A is called a peaking
function of A if Rπ (f) = {1} and a subset K of X is called a peak set for A if K = {x ∈ X : f(x) = 1} for some peaking
function f. Given a subalgebra A of C0 (X ), by a strong boundary point for A we mean a point x ∈ X such that for every
neighborhood V of x there exists a function f ∈ A with kfkX = f(x) = 1 and |f| < 1 on X \ V . In the case where A is
a Banach function algebra on X , it is easy to see that the function f in this definition can be chosen to be a peaking
function (and in exp A, in compact case). It is well known that if A is a uniform algebra on a locally compact Hausdorff
space X , then c(A) is, indeed, the set of all strong boundary points for A, see [22, Theorem 7.30] for the compact case
and [19, Theorem 2.1] for the general case.
Let X be compact and A be a uniform algebra on X . Following [4, Definition 2], a subset M of A is said to be of type (P)
if it satisfies the following conditions:
(i) for each ε > 0, x ∈ c(A) and open neighborhood U of x, there is a peaking function h ∈ M such that h(x) = 1 and
|h| < ε on X \ U;
(ii) for each x ∈ c(A) and f ∈ A with f(x) 6= 0, there exists a peaking function h ∈ M with h(x) = 1 and Rπ (fh) = {f(x)}.
We note that if X is compact, then by [15, Corollary 1.1] the subset exp A of a uniform algebra A on X (and consequently A−1 ) is a set of type (P).
It is worth mentioning that since there exists a Banach function algebra on a compact metric space with a peak set that
contains no strong boundary point, see, for example, [2], it follows that for a Banach function algebra A the points in c(A)
are not necessarily strong boundary points, i.e. in this case the condition (i) above does not hold necessarily for A itself.
We use the following results, proved by Shindo, concerning the general form of certain preserving maps between some
multiplicative semigroups of invertible elements of uniform algebras.
Theorem 2.1 ([21, Proposition 2.2]).
Let A and B be uniform algebras on compact Hausdorff spaces X and Y , respectively, and let A0 and B0 be multiplicative
subsemigroups of A−1 and B−1 containing the constants. Assume that A0 , I(A0 ), B0 , I(B0 ) are of type (P), S : A0 → B0 is
a surjection satisfying S(1) = 1 and
S(f)
= f − 1
−
1
S(g)
g
Y
1022
X
M. Hosseini, F. Sady
for all f, g ∈ A0 . Then there exist a homeomorphism φ : c(B) → c(A) and a clopen subset K of Y such that for every
f ∈ A0 and y ∈ c(B),
(
f(φ(y)),
y ∈ K,
S(f)(y) =
f(φ(y)),
y ∈ c(B) \ K .
We should note that the subset K given in the the above theorem is defined by K = {y ∈ Y : S(i)(y) = i}.
Theorem 2.2.
Let A and B be uniform algebras on compact Hausdorff spaces X and Y , respectively. Let A1 and A2 be subsets of A
and ρ : A1 → A, τ : A2 → A, S : A1 → B and T : A2 → B be mappings such that the images of ρ, τ and respectively the
images of S, T are multiplicative subsemigroups of A and B, containing exp A and exp B, respectively. If S(e1 )−1 ∈ S(A1 )
and S(e1 ) ∈ T (A2 ) for some e1 ∈ A1 with ρ(e1 ) = 1 and
kS(f)T (g) − αkY = kρ(f)τ(g) − αkX ,
f ∈ A1 ,
g ∈ A2 ,
where α is a non-zero complex number, then there exist a real-algebra isomorphism S : A → B and a homeomorphism
φ : MB → MA such that
(
bf ◦ φ
on K ,
d
S(f) =
bf ◦ φ
on MB \ K ,
d
for every f ∈ A and S(ρ(f)) = S(e1 )−1 S(f) for every f ∈ A1 , where K = {y ∈ MB : S(i)(y)
= i}.
The proof is exactly the same as that of [21, Theorem 3.2], where A1 = A2 .
3.
Norm conditions for unital case
In this section we assume that X and Y are compact Hausdorff spaces and A, B are Banach function algebras on X
and Y , respectively. As we mentioned before, descriptions of non-symmetric multiplicatively norm-preserving maps
between (unital) uniform algebras and, in general, (unital) Banach function algebras were given in [7, 14, 15] and [9],
respectively. A generalization of this result (in the uniform algebra case) given in [21, Theorem 3.2] characterizes, under
certain conditions, the form of maps S, T : A1 → B, from a subset A1 of a unital uniform algebra A to a unital uniform
algebra B satisfying the norm condition
kS(f)T (g) − αkY = kρ(f)τ(g) − αkX ,
f, g ∈ A1 ,
where α is a non-zero complex number and ρ, τ : A1 → A are certain maps.
The aim of this section is to generalize the results of [21] for Banach function algebras on compact Hausdorff spaces.
We should note that the main condition considered in [21] states that certain subsets of the invertible groups are of
type (P). Since for a Banach function algebra A, the set A itself (and consequently A−1 ) is not necessarily of type (P),
it seems that this condition is not good enough for the Banach function algebra case. Indeed, assuming that S, T and
ρ, τ satisfy the above norm-condition in the Banach function algebra case, the key idea in the proof of Theorem 3.1
e according to S, T and ρ, τ, between particular subsets of the uniform closures of A
below is to define a certain map S
and B. Then, using [21, Proposition 2.2] the desired description will be obtained.
Before stating the results we need to fix some notations. Let A be the closure of A in (C (X ), k · kX ) and let I, I 0 be
arbitrary non-empty sets. For maps ρ : I → A and τ : I 0 → A we set
ρ(I)ρτ = {f ∈ ρ(I) : f −1 ∈ τ(I 0 )}.
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Maps between Banach function algebras satisfying certain norm conditions
ρ
Hence for an element p ∈ I, we have ρ(p) ∈ ρ(I)τ if and only if there is an element p0 ∈ I 0 with ρ(p)τ(p0 ) = 1. The idea
of considering such subsets comes from [21].
ρ
Let A0 denote the set all invertible elements f ∈ A for which there exists a sequence {fn } in ρ(I)τ such that
ρ
limn→∞ kfn − fkX = 0, i.e., A0 = (A)−1 ∩ ρ(I)τ , where the closures are taken in (C (X ), k · kX ). Hence for f ∈ A0 and
such a sequence {fn } there exist sequences {pn } and {p0n } in I and I 0 , respectively, with fn = ρ(pn ), ρ(pn )τ(p0n ) = 1,
ρ
ρ
n ∈ N, and limn→∞ kρ(pn ) − fkX = 0. Obviously, A0 contains the subset ρ(I)τ of A−1 and hence ρ(I)τ ⊆ A0 ⊆ (A)−1 . It is
ρ
easy to see that if both ρ(I) and τ(I 0 ) are multiplicative subsemigroups (respectively subgroups) of A, then ρ(I)τ and A0
are subsemigroups (respectively subgroups) of A−1 and (A)−1 .
In the sequel we assume that ρ : I → A, τ : I 0 → A and S : I → B, T : I 0 → B are maps whose ranges ρ(I), τ(I 0 ) and
S(I), T (I 0 ) are multiplicative subsemigroups of A and B, respectively. Similarly, the multiplicative semigroup S(I)ST is
defined by S(I)ST = {F ∈ S(I) : F −1 ∈ T (I 0 )} and we use the notation B0 for the subset (B)−1 ∩ S(I)ST of B.
Theorem 3.1.
Let ρ(I), τ(I 0 ) contain exp A and S(I), T (I 0 ) contain exp B. If S(e1 )−1 ∈ S(I)ST for some e1 ∈ I with ρ(e1 ) = 1 and
kS(p)T (p0 ) − αkY = kρ(p)τ(p0 ) − αkX ,
p ∈ I,
p0 ∈ I 0 ,
where α is a non-zero complex number, then there exist a homeomorphism φ from c(B) onto c(A) and a clopen subset K
ρ
of Y such that for each p ∈ ρ−1 (ρ(I)τ ) and y ∈ c(B),
S(p)(y)
=
S(e1 )(y)
(
ρ(p)(φ(y)),
ρ(p)(φ(y)),
y ∈ K,
y ∈ c(B) \ K .
ρ
ρ
We first show that ρ−1 (ρ(I)τ ) = S −1 (S(I)ST ). Suppose p ∈ ρ−1 (ρ(I)τ ). Then there exists an element p0 ∈ I
such that ρ(p)τ(p0 ) = 1. Since τ(I) is assumed to be a multiplicative semigroup containing the constants, we can choose
p0α ∈ I 0 so that τ(p0α ) = ατ(p0 ). Hence, by the assumption,
Proof.
kS(p)T (p0α ) − αkY = kρ(p)τ(p0α ) − αkX = kαρ(p)τ(p0 ) − αkX = 0,
that is, S(p)T (p0α ) = α. Now taking p00 ∈ I 0 with T (p00 ) = α −1 T (p0α ) we get S(p)T (p00 ) = 1 and consequently,
ρ
ρ
p ∈ S −1 (S(I)ST ). Therefore ρ−1 (ρ(I)τ ) ⊆ S −1 (S(I)ST ). A similar argument implies the opposite inclusion, that is, ρ−1 (ρ(I)τ ) =
S −1 (S(I)ST ).
ρ
Let now p ∈ I and q ∈ ρ−1 (ρ(I)τ ). Then by the above argument there exists q0α ∈ I 0 with ρ(q)τ(q0α ) = α and
S(q)T (q0α ) = α. Thus
S(p)
= 1 S(p) α − α =
−
1
S(q)
|α|
S(q)
Y
Y
1 α
=
ρ(p)
− α
=
|α| ρ(q)
X
Hence
S(p)
S(q) − 1 =
Y
1
1
kS(p)T (q0α ) − αkY =
kρ(p)τ(q0α ) − αkX
|α|
|α|
ρ(p)
.
−
1
ρ(q)
ρ(p)
ρ(q) − 1 ,
X
X
p ∈ I,
q ∈ ρ−1 (ρ(I)ρτ ).
(1)
e : A0 → B0 satisfying the following norm condition:
Using the above equality, we define a surjective map S
e
S(f)
=
−
1
e
S(g)
Y
f
− 1 ,
g
X
ρ
for all f, g ∈ A0 . For this, let f be an arbitrary element of A0 and let {fn } be a sequence in ρ(I)τ converging uniformly
on X to f. Clearly, {fn−1 }, as a sequence in the uniform algebra A, converges uniformly on X to f −1 . For each n ∈ N,
1024
M. Hosseini, F. Sady
ρ
take pn ∈ I with fn = ρ(pn ). Then, since for each n ∈ N, pn ∈ ρ−1 (ρ(I)τ ), it follows from (1) that kS(pn )/S(pm ) − 1kY =
kρ(pn )/ρ(pm ) − 1kX = kfn /fm − 1kX → 0 as n, m → ∞. On the other hand, since ρ(e1 ) = 1, it follows from (1) that
kS(pn )/S(e1 ) − 1kY = kρ(pn ) − 1kX and, consequently,
S(pn )
− 1 + S(e1 )
kS(pn )kY = S(e1 )
≤ kS(e1 )kY kρ(pn ) − 1kX + kS(e1 )kY
S(e1 )
Y
≤ kS(e1 )kY kρ(pn )kX + 2kS(e1 )kY = kS(e1 )kY kfn kX + 2kS(e1 )kY .
This shows that {kS(pn )kY } is a bounded sequence in B. Let M be a positive number such that for all n ∈ N,
kS(pn )kY ≤ M. Then kS(pn ) − S(pm )kY ≤ MkS(pn )/S(pm ) − 1kY for all n, m ∈ N, and therefore {S(pn )} is a Cauchy
sequence in the uniform closure B of B. Hence there exists a function F ∈ B with kS(pn ) − F kY → 0 as n → ∞. It is
ρ
easy to see that the function F , obtained in this way, is independent of the choice of the sequence {fn } in ρ(I)τ converging
ρ
0
−1
−1
uniformly to f. We claim that F is an element of B . Note that since for each n ∈ N, pn ∈ ρ (ρ(I)τ ) = S (S(I)ST ) and
kS(pn ) − F kY → 0, it follows that F is in the uniform closure of S(I)ST . Therefore it suffices to show that F is invertible
/ (B)−1 . Then, since {S(pn )} converges uniformly to F , and for each n ∈ N,
in B. Assume on the contrary that F ∈
−1
−1
−1
S(pn ) ∈ B ⊆ (B) , we have k(S(pn )) kY → ∞. On the other hand, since S(I) contains the constants, there exists an
element e2 ∈ I with S(e2 ) = 1. Hence, using (1) once again, we have
ρ(e2 )
ρ(pn ) − 1 =
X
S(e2 )
−1
−1
S(pn ) − 1 = kS(pn ) − 1kY ≥ kS(pn ) kY − 1 → ∞,
Y
that is, kρ(e2 )/ρ(pn )kX → ∞ while ρ(e2 )/ρ(pn ) = ρ(e2 )fn−1 → ρ(e2 )f −1 in A, as n → ∞. This contradiction shows that
e : A0 → B0 be defined by S(f)
e = limn→∞ S(pn ) where for f ∈ A0 , {pn } is a sequence in I such that
F ∈ B0 . Now let S
ρ
ρ(pn ) ∈ ρ(I)τ and kρ(pn ) − fkX → 0. We note that by the above argument this limit exists and is an element of B0 and,
ρ
e is well defined.
moreover, this limit is independent of the choice of the sequence in ρ(I)τ converging to f. That is, S
It is now easy to see that for each f, g ∈ A0 ,
e
S(f)
=
−
1
e
S(g)
Y
f
− 1 .
g
X
(2)
e : A0 → B0 is surjective. Let g ∈ B0 and let {gn } be a sequence in S(I)S such that kgn − gkY → 0.
Now we show that S
T
ρ
ρ
Then for each n ∈ N, gn = S(pn ) for some pn ∈ I. Hence pn ∈ S −1 (S(I)ST ) = ρ−1 (ρ(I)τ ) and consequently ρ(pn ) ∈ ρ(I)τ .
ρ
Setting fn = ρ(pn ) we see that fn ∈ ρ(I)τ and for all n, m ∈ N,
fn
− 1 =
fm
X
ρ(pn )
=
−
1
ρ(pm )
X
S(pn )
=
−
1
S(pm )
Y
gn
,
−
1
gm
Y
by (1). As before, this implies that {fn } is a Cauchy sequence in A and a similar argument as above shows that
e = g, i.e., S
e is surjective.
kfn − fkX → 0 for some f ∈ A0 . It is easy to see that S(f)
0
−1
e1 : A0 → B0 be defined by S
e1 (f) = S(f)/S(e
e
Now let S
∈ S(I)ST ⊆
1 ), f ∈ A . We note that, since by the assumption S(e1 )
0
0
0
0
e
e
e
B and B is a multiplicative semigroup, S1 (f) ∈ B for all f ∈ A . Clearly S1 is surjective and since S(1) = S(e1 ),
e1 (1) = 1. Clearly, by (2), kS
e1 (f)/S
e1 (g) − 1kY = kf/g − 1kX for all f, g ∈ A0 . We also note that ρ(I)ρτ
it follows that S
ρ
ρ
contains exp A, since ρ(I) and τ(I 0 ) contain exp A. Hence exp A ⊆ exp A ⊆ ρ(I)τ and therefore exp A ⊆ (A)−1 ∩ ρ(I)τ = A0 .
This implies that A0 is a set of type (P) for the uniform algebra A. Similarly, B0 is of type (P) for B. Hence, since A
and B are uniform algebras and A0 , B0 are multiplicative subsemigroups of (A)−1 and (B)−1 containing the constants, it
follows from Theorem 2.1 that there exist a homeomorphism φ : c(B) → c(A) and a clopen subset K of Y such that
e1 (f)(y) =
S
(
f(φ(y)),
f(φ(y)),
y ∈ K,
y ∈ c(B) \ K ,
e1 (i)(y) = i}. Since for each p ∈ ρ−1 (ρ(I)ρτ ) we have ρ(p) ∈ ρ(I)ρτ ⊆ A0
for every f ∈ A0 , y ∈ c(B), and K = {y ∈ Y : S
e
and S(ρ(p)) = S(p), we obtain the desired representation for S.
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Maps between Banach function algebras satisfying certain norm conditions
We should note that in Theorem 3.1, the hypothesis that S(e1 )−1 ∈ S(I)ST for some e1 ∈ I with ρ(e1 ) = 1 has been used
e1 . So under the hypotheses of this theorem, we can define similarly a surjective
for the first time in the definition of S
00
00
00
−1
e
map T : A → B where A = (A) ∩ τ(I 0 )τρ and B00 = (B)−1 ∩ T (I 0 )TS , satisfying
e
T
(f) − 1 =
e
T (g)
Y
f
− 1 ,
g
X
f, g ∈ A00 .
e T
e (αg−1 ) = α for g ∈ A0 , and moreover,
It is easy to see that A00 = I(A0 ), B00 = I(B0 ) and S(g)
e T
e (g) − αkY = kfg − αkX
kS(f)
holds for all f ∈ A0 and g ∈ I(A0 ). Using this fact, we show that there exists a real algebra isomorphism S between A
and B which induces a homeomorphism between MB and MA .
Theorem 3.2.
Let ρ, τ and S, T satisfy the hypotheses of Theorem 3.1. Then there exist a real algebra isomorphism S : A → B, a clopen
subset K of MB and a homeomorphism φ : MB → MA such that
d =
S(f)
(
bf ◦ φ
bf ◦ φ
on
K,
on
MB \ K,
d
for every f ∈ A and S(ρ(p)) = S(p)/S(e1 ) for every p ∈ I, where K = {y ∈ MB : S(i)(y)
= i}. Furthermore, the restriction
map S = SA is a real algebra isomorphism from A onto B and φ can be extended to a homeomorphism from MB onto MA .
e : A0 → B 0
e : A0 → A and e
Let ρ
τ : I(A0 ) → A be the inclusion maps. As we noted before, the surjective maps S
0
0
0
0
e
e
e
and T : I(A ) → I(B ) satisfy kS(f) T (g) − αkY = ke
ρ(f)e
τ (g) − αkX for all f ∈ A and g ∈ I(A ). Since A0 and B0
e
are multiplicative subsemigroups of uniform algebras A and B containing exp A and exp B and S(1)
= S(e1 ) (which
e
−1
0 S
−1
S
e
e
implies easily that S(1) ∈ S(A )Te , since S(e1 ) ∈ S(I)T ), it follows from Theorem 2.2 that there exist a real algebra
isomorphism S : A → B and a homeomorphism φ : MB → MA such that
Proof.
b =
Sf
(
bf ◦ φ
bf ◦ φ
on
K,
on
MB \ K,
d
0
e
for all f ∈ A and S(f) = S(e
ρ(f)) = S(f)/S(e
1 ) for each f ∈ A , where K = {y ∈ MB : S(i)(y) = i}. Hence for every
ρ
e
p ∈ ρ−1 (ρ(I)τ ), S(ρ(p)) = S(ρ(p))/S(e
1 ) = S(p)/S(e1 ). A similar discussion as in [21, Theorem 3.2] shows that for each
p ∈ I, kS(ρ(p))k − 1kY = kS(e1 )−1 S(p)k − 1kY for every k ∈ exp B (and, in particular, for every k ∈ exp B). Therefore,
by [20, Lemma 3.1], we get S(ρ(p)) = S(p)/S(e1 ) for all p ∈ I.
We now show that the restriction map S = SA maps A onto B. Let f ∈ A. Then f can be written as f = f0 + γ
for some f0 ∈ exp A ⊆ ρ(I) and nonzero constant γ. Since S(ρ(p)) = S(p)/S(e1 ) ∈ B for all p ∈ I, it follows that
−1
S(f0 ), S(γ) ∈ B, and so S(f) ∈ B. Similarly S (g) ∈ A for all g ∈ B. Therefore, the restriction map S = SA is a real
algebra isomorphism from A onto B.
Now for each y ∈ MB let φ(y) : A → C be defined by
d e(y) + S(f)(y)(1
d
φ(y)(f) = S(f)(y)b
−b
e)(y),
f ∈ A,
where as before e = (S(i) + i)/(2i) = (S(i) + i)/(2i). We note that here e is indeed an idempotent element in B and b· is
the Gelfand transformation of elements in B. It is easy to see that φ(y) ∈ MA and the same proof as in [21, Theorem 3.2]
can be applied to show that the map φ : MB → MA , y 7→ φ(y) is a homeomorphism. Obviously, φ(yB ) = φ(y) for every
y ∈ MB .
1026
M. Hosseini, F. Sady
Corollary 3.3.
Under the hypotheses of Theorem 3.1 there exist a real algebra isomorphism S : A → B, a clopen subset K of MB and
a homeomorphism φ from MB onto MA such that for all f ∈ A,
b =
Sf
(
bf ◦ φ
bf ◦ φ
on
K,
on
MB \ K,
d
where K = {y ∈ MB : S(i)(y)
= i}. Moreover, S(ρ(p)) = S(p)/S(e1 ) for every p ∈ I, K ∩ c(B) = K and φK = φ, where
K and φ are as in Theorem 3.1.
d
Let S and φ be as in the above theorem and set K = {y ∈ MB : S(i)(y)
= i}. Then clearly K is a clopen
b
d
d
subset of MB and the definition of φ implies that for each f ∈ A, f(φ(y)) = S(f)(y) for y ∈ K and bf(φ(y)) = S(f)(y)
for
y ∈ MB \ K. It is also easy to see that K ∩ c(B) = K and φK = φ.
Proof.
The same argument as in [21, Corollary 3.1] can be applied to prove the following corollary. Alternatively, we may
benefit from [21, Corollary 3.1] to provide a shorter proof.
Corollary 3.4.
Under the hypotheses of Theorem 3.1, if furthermore, T (e2 )−1 ∈ T (I 0 )TS for some e2 ∈ I 0 with τ(e2 ) = 1 then for each
p ∈ I and p0 ∈ I 0 ,
S(ρ(p)) = S(e1 )−1 S(p)
and
S(τ(p0 )) = T (e2 )−1 T (p0 ),
where S is the real algebra isomorphism given in Theorem 3.2. Moreover,

1
S(e\
1 )T (e2 ) =
α
α
on
K,
on
MB \ K,
where K is as in the above corollary.
e : A0 → B0 and T
e : I(A0 ) → I(B0 ) satisfy kS(f)
e T
e (g) − αkY =
As before, since the surjective maps S
0
0
0
0
e : A → A and e
ke
ρ(f)e
τ (g) − αkX for all f ∈ A and g ∈ I(A ) where ρ
τ : I(A ) → A are inclusion maps, it follows
from [21, Corollary 3.1] (for the uniform algebras A and B), that
Proof.
e −1 S(f)
e
S(f) = S(e
ρ(f)) = S(1)
and
e (1)−1 T
e (g)
S(g) = S(e
τ (g)) = T
holds for all f ∈ A0 and g ∈ I(A0 ), where S : A → B is the real algebra isomorphism given in Theorem 3.2. We should note
that [21, Corollary 3.1] can be applied (with the same proof) for the case the domains of mappings under consideration are
not necessarily the same. By Theorem 3.2, S(ρ(p)) = S(ρ(p)) = S(e1 )−1 S(p) for each p ∈ I. For the other equality, since
e (1) T
e (τ(p0 )) = T (e2 )−1 T
e (τ(p0 )) = T (e2 )−1 T (p0 )
for each p0 ∈ τ −1 (τ(I))τρ ), τ(p0 ) ∈ τ(I)τρ ⊆ I(A0 ), it follows that S(τ(p0 )) = T
0
−1
0 τ
−1
0
0
for each p ∈ τ (τ(I ))ρ ). It is now easy to see that kT (e2 ) T (p )k − 1kY = kS(τ(p ))k − 1kY for all p0 ∈ I 0 and
k ∈ exp B. As before, this implies that S(τ(p0 )) = T (e2 )−1 T (p0 ) for all p0 ∈ I 0 .
The following result, which is an easy consequence of the above corollary (for details see [21, Example 3.1]), is a
generalization of [20, Theorem 1.1], [9, Theorem 3.2] and [10].
Corollary 3.5.
(i) Let m, n ∈ N and let S, T : A0 → B0 be surjections between multiplicative subsemigroups A0 and B0 of A−1 and
B−1 containing exp A and exp B, respectively, such that
1027
Maps between Banach function algebras satisfying certain norm conditions
kS(f)m T (g)n − αkY = kf m gn − αkX ,
f, g ∈ A.
Then there exist a real algebra isomorphism S : A → B, a clopen subset K of MB and a homeomorphism φ : MB → MA
such that
(
bf ◦ φ
on K,
d
S(f) =
bf ◦ φ
on MB \ K,
and S(f)m = (S(f)/S(1))m, S(f)n = (T (f)/T (1))n for all f ∈ A0 . In particular, if S = T then S(f)d = (S(f)/S(1))d
where d is the greatest common divisor of m and n.
(ii) If S, T : A → B are surjective maps satisfying kS(f)T (g) − αkY = kfg − αkX for all f, g ∈ A, then
S(f)
T (f)
=
=
S(1)
T (1)
(
bf ◦ φ
bf ◦ φ
on
K,
on
MB \ K,
where K and φ are as in (i).
Imposing an additional assumption, in the next theorem we show that the real algebra isomorphism S given in Theorem 3.2
is complex linear, that is A and B are algebraically isomorphic.
Theorem 3.6.
Let ρ(I), τ(I 0 ) contain exp A and S(I), T (I 0 ) contain exp B. If S(e1 )−1 ∈ S(I)ST for some e1 ∈ I with ρ(e1 ) = 1 and
Rπ (S(p)T (p0 ) − α) ∩ Rπ (ρ(p)τ(p0 ) − α) =
6 ∅
for all p ∈ I and p0 ∈ I 0 . Then there exist an algebra isomorphism S : A → B and a homeomorphism φ : MB → MA such
b = bf ◦ φ on MB and S(ρ(p)) = S(p)/S(e1 ) for every p ∈ I.
that for all f ∈ A, Sf
Proof.
Clearly, the assumptions imply
kS(p)T (p0 ) − αkY = kρ(p)τ(p0 ) − αkX ,
p ∈ I,
p0 ∈ I 0 .
e1 : A0 → B0 and a homeomorphism
Therefore, as it was shown in the proof of Theorem 3.1, there exist a surjective map S
ρ
−1
φ : c(B) → c(A) such that for each p ∈ ρ (ρ(I)τ ) and y ∈ c(B),
e1 (ρ(p))(y) = S(p) (y) =
S
S(e1 )
(
ρ(p)(φ(y)),
ρ(p)(φ(y)),
y ∈ K,
y ∈ c(B) \ K ,
f1 (i)(y) = i}. Furthermore, kS
e1 (f)/S
e1 (g) − 1kY = kf/g − 1kX for all f, g ∈ A0 .
where K = {y ∈ Y : S
ρ
Now let p, q ∈ ρ−1 (ρ(I)τ ) and choose a function q0α ∈ I 0 with ρ(q)τ(q0α ) = α and S(q)T (q0α ) = α, see the proof of
Theorem 3.1. Then
S(p)
1
α
1
− 1 = Rπ S(p)
− α = Rπ S(p)T (q0α ) − α ,
Rπ
S(q)
α
S(q)
α
ρ(p)
1
α
1
Rπ
− 1 = Rπ ρ(p)
− α = Rπ (ρ(p)τ(q0α ) − α)
ρ(q)
α
ρ(q)
α
ρ
and it follows from the hypotheses that Rπ (S(p)/S(q) − 1) ∩ Rπ (ρ(p)/ρ(q) − 1) 6= ∅, for all p, q ∈ ρ−1 (ρ(I)τ ). In particular,
letting q = e1 , we have
e1 (ρ(p)) − 1) ∩ Rπ (ρ(p) − 1) =
Rπ ( S
6 ∅.
Now a minor modification of the proof of [9, Theorem 3.9] can be applied to show that K ∩ c(B) = c(B). In particular, it
e1 (i) = i on c(B) and consequently, S
e1 (i) = i. Therefore K = Y and K = MB , where K is the clopen subset
follows that S
b = bf ◦ φ on MB
of MB given in Corollary 3.3. Hence for the real algebra isomorphism S given in Theorem 3.2, we have Sf
for all f ∈ A which clearly concludes that S is complex-linear. Moreover, by Theorem 3.2, S(ρ(p)) = S(p)/S(e1 ) for every
p ∈ I. Therefore, S is the desired algebra isomorphism.
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M. Hosseini, F. Sady
4.
Generalized multilplicatively norm-preserving maps
Let A be a Banach function algebra on a locally compact Hausdorff space X . For a function f ∈ A we set Mf = {x ∈
S
S
X : |f(x)| = kfkX } and for a point x ∈ f∈A Mf we set Fx (A) = {f ∈ A : |f(x)| = kfkX = 1}. For x ∈ f∈A Mf
T
T
we also set Ix = f∈Fx (A) Mf = x∈Mf Mf . Then it is easy to see that the relation ∼ defined by x ∼ z if and only
e = x ∈ S Mf :
if Fx (A) = Fz (A) (or equivalently, Ix = Iz ) defines an equivalence relation on the subset X
f∈A
Ix is a minimal element between such intersections with [x] = Ix . In particular, if each point in c(A) is a strong boundary
e = c(A) and [x] = {x} for all x ∈ c(A).
point for A (this holds, for example, when A is a uniform algebra on X ), then X
It should be noted that for a locally compact Hausdorff space X and A ⊆ C0 (X ) such intersections were considered
in [13]. Indeed, in [13] a non-empty subset E of X is called an m-set for A if there exists a subset F of A such that
T
E = f∈F Mf . In particular, the subsets Ix of X considered above are all m-sets. By [13, Lemma 3.2, Corollary 3.3]
every m-set E for a subset A of C0 (X ) contains a minimal m-set of A and the sets containing at least one point of each
minimal m-set (for any choice) are boundaries for A. Moreover, by [13] in the case where A is a subalgebra of C0 (X ),
T
an m-set E for A is minimal if and only if it is a p-set, that is E = α Mfα for a family of peaking functions {fα } of A.
Thus, in this case, for each x ∈ E, fα ∈ Fx (A) and consequently Ix ⊆ E, and it follows from the minimality of E that
e and E = Ix . This shows, in particular, that when A is a Banach function algebra on a
E = Ix . In particular, x ∈ X
e is a boundary for A.
locally compact Hausdorff space X , X
In this section we study generalized multilplicatively norm-preserving maps in the sense of Theorem 4.2. As we noted
before, the norm-multilplicativity condition is too weak for linearity and such maps need not be weighted composition
operators or even real-linear. In the uniform algebra case, Hatori and et al. proved in [4, Proposition 2] that under
certain conditions, a generalized multiplicatively norm-preserving map induces a homeomorphism between the Choquet
boundaries. More precisely, they proved that if I is a non-empty set, A, B are uniform algebras on compact Hausdorff
spaces X , Y , and ρ, τ : I → A and S, T : I → B are maps whose ranges are absolutely multiplicative sets of type (P’)
satisfying
kS(p)kY = kρ(p)kX ,
kT (p)kY = kτ(p)kX ,
kS(p)T (q)kY = kρ(p)τ(q)kX ,
p, q ∈ I,
then there exists a homeomorphism φ : c(B) → c(A) such that
|S(p)(y)| = |ρ(p)(φ(y))|,
|T (p)(y)| = |τ(p)(φ(y))|
for all p ∈ I and y ∈ c(B). Here a subset N of a uniform algebra A on X is called absolutely multiplicative if for each
f, g ∈ N there exists a function h ∈ N such that |h| = |fg| on c(A) and a subset M of A is said to be of type (P’) if
(i) for each ε > 0, x ∈ c(A) and a neighborhood U of x, there exists u ∈ M ∩ Fx (A) such that |u| < ε on X \ U;
(ii) for each x ∈ c(A) and f ∈ A with f(x) =
6 0 there exists u ∈ M ∩ Fx (A) such that kfukX = |f(x)|.
We note that since in the uniform algebra case every point in c(A) is a strong boundary point, condition (i) above holds
for M = A. Moreover, by [5, Lemma 2.3], condition (ii) holds for A itself, as well. Therefore every uniform algebra A
is a set of type (P’) for itself. Unfortunately, this is no longer true for the Banach function algebra case, since as we
noted before, (i) does not hold, in general, in that case. However, if A is a Banach function algebra on a locally compact
S
space X , then by [1, Lemma 3] for each x ∈ f∈A Mf and an open neighborhood U of Ix , there exists g ∈ A (which, by
replacing by geg /e can be assumed to be a peaking function) such that kgkX = 1 = g(x) and |g(z)| < 1 for all z ∈ X \ U,
and by [23, Lemma 2.2] for each strong boundary point x0 of A and f ∈ A, |f(x0 )| = inf {kfhkX : h ∈ Fx0 (A)}. Motivated
by this, we give the following definition of sets called approximately of type (P’) and extend the above mentioned result
e we may use the same notation [x]
of [4] to the Banach function algebra case. We note that in the following, for each x ∈ X
e
for both an element of X /∼ and the compact subset Ix of X . Hence, for f ∈ A, kfk[x] denotes the supremum of |f| on this
compact set.
Definition 4.1.
Let A be a Banach function algebra on a locally compact Hausdorff space X . We say that a subset M of A is approximately
of type (P’) if
1029
Maps between Banach function algebras satisfying certain norm conditions
e and an open neighborhood U of [x0 ] in X , there exists a function u ∈ M ∩ Fx (A) such that
(i’) for each ε > 0, x0 ∈ X
0
|u| < ε on X \ U;
e and f ∈ A, kfk[x ] = inf {kfhkX : h ∈ M ∩ Fx (A)}.
(ii’) for each x0 ∈ X
0
0
In a Banach function algebra A, as we noted before, condition (i’) holds for A itself. Using the same proof as in
e and f ∈ A, and consider the open
[23, Lemma 2.2] we can show that (ii’) also holds for M = A. For this, let x0 ∈ X
subset U = {x ∈ X : |f(x)| < kfk[x0 ] + ε} of [x0 ]. Then, as we noted before, there exists a peaking function h ∈ Fx0 (A) such
that Mh ⊆ U. Replacing h with a suitable power of h we see that kfhkX ≤ kfk[x0 ] + ε. On the other hand, since [x0 ] is a
compact subset of X , we can find x1 ∈ [x0 ] such that kfk[x0 ] = |f(x1 )|. Since for each h ∈ Fx0 (A), 1 = |h(x0 )| = |h(x1 )| we
get kfhkX ≥ |f(x1 )| = kfk[x0 ] . Therefore, kfk[x0 ] = inf {kfhkX : h ∈ Fx0 (A)}. That is, A is approximately of type (P’) for
itself. The above argument shows, indeed, that the set of all peaking functions of A is approximately of type (P’) for A.
More generally, the same proof can be applied to show that every absolutely multiplicative set satisfying condition (i’)
is approximately of type (P’).
Using condition (i’), it is easy to see that if M is approximately of type (P’) for a Banach function algebra A, then for
e and x2 ∈ X with M ∩ Fx (A) ⊆ M ∩ Fx (A) we have Ix = Ix , and consequently Fx = Fx . In particular, x2 ∈ X
e
x1 ∈ X
1
2
1
2
1
2
and so x1 ∼ x2 , i.e., [x1 ] = [x2 ].
In the next theorem, by τq we mean the quotient topology on a given quotient space and by τo we mean the weakest
topology under which the corresponding quotient map is open.
Theorem 4.2.
Let I, I 0 be non-empty sets and A, B be Banach function algebras on locally compact Hausdorff spaces X , Y . Let ρ : I → A,
τ : I 0 → A and S : I → B, T : I 0 → B be maps whose ranges are absolutely multiplicative and approximately of type (P’).
If
kS(p)kY = kρ(p)kX ,
kT (p0 )kY = kτ(p0 )kX ,
kS(p)T (p0 )kY = kρ(p)τ(p0 )kX ,
p ∈ I, p0 ∈ I 0 ,
e /∼, τq ) such that φ−1 : (X
e /∼, τo ) → (Ye /∼, τq ) is continuous
then there exists a bijective continuous map φ : (Ye /∼, τo ) → (X
and
kS(p)k[y] = kρ(p)kφ([y]) ,
kT (p0 )k[y] = kτ(p0 )kφ([y])
for all p ∈ I, p0 ∈ I 0 and y ∈ Ye .
Proof.
We prove the theorem through the following steps:
e such that [x] ⊆ T −1
e
For each y ∈ Ye , there exists x ∈ X
p∈S (Fy (B)) Mρ(p) ∩ X .
T
We first show that the intersection p∈S−1 (Fy (B)) Mρ(p) is not empty. By compactness of the maximizing sets we need
only to show that the family {Mρ(p) : p ∈ S −1 (Fy (B))} has finite intersection property. Let p1 , p2 , . . . , pn be elements
in S −1
(Fy (B)). Then by assumption, |S(pi )(y)| = 1 = kS(pi )kY = kρ(pi )kX for each i = 1, . . . , n.
implies
Q
Q This easily
that n1 S(pi )Y = 1. Since S(I) is absolutely multiplicative, there exists q ∈ I with |S(q)| = n1 S(pi ) on c(B). In
e is a boundary for A, it follows that there exists a point x0 ∈ X
e such that
particular, kρ(q)kX = kS(q)kY = 1. Since X
Tn
|ρ(q)(x0 )| = 1. We shall show that x0 ∈ 1 Mρ(pi ) , i.e., ρ(pi ) ∈ Fx0 (A). Since Fx0 (A) = Fx1 (A) for all x1 ∈ [x0 ], it follows
that |ρ(pi )(x0 )| = 1 if and only if kρ(pi )k[x0 ] = 1. Hence we need only to show that for each i = 1, . . . , n, kρ(pi )k[x0 ] = 1.
Assume on the contrary that kρ(pi )k[x0 ] < 1 for some 1 ≤ i ≤ n. Then there exists a neighborhood V of [x0 ] in X such
that |ρ(pi )| < 1 on V . Since τ(I 0 ) is approximately of type (P’), there is, by condition (i’), a function τ(p0 ) ∈ τ(I 0 ) ∩ Fx0 (A)
such that |τ(p0 )| < 1 on X \ V . Hence kρ(pi )τ(p0 )kX < 1 while 1 = kρ(q)τ(p0 )kX = kS(q)T (p0 )kY ≤ kS(pi )T (p0 )kY =
kρ(pi )τ(p0 )kX . This contradiction concludes that |ρ(pi )(x0 )| = 1 for i = 1, . . . , n, as desired.
T
Now since the intersection p∈S−1 (Fy (B)) Mρ(p) is an m-set, it follows from [13, Lemma 3.2, Corollary 3.3] that it contains
e . This implies that the subset
a minimal m-set E for A. As we noted before, for such E we have E = Ix for some x ∈ X
Step 1.
[x] = Ix of X is contained in this intersection. This completes the proof of this step.
1030
M. Hosseini, F. Sady
e such that S −1 (Fy (B)) = ρ−1 (Fx (A)).
For each y ∈ Ye there exists x ∈ X
T
e . We show that for such x we have
Let y ∈ Ye . Then by Step 1 there exists a point x ∈ p∈S−1 (Fy (B)) Mρ(p) ∩ X
−1
−1
−1
S (Fy (B)) ⊆ ρ (Fx (A)). Indeed, let p ∈ S (Fy (B)). Then x ∈ Mρ(p) , that is |ρ(p)(x)| = 1 = kρ(p)kX . Hence
ρ(p) ∈ Fx (A) and, consequently, S −1 (Fy (B)) ⊆ ρ−1 (Fx (A)). A similar argument shows that for this x there exists y0 ∈ Ye
such that ρ−1 (Fx (A)) ⊆ S −1 (Fy0 (B)). Therefore, S −1 (Fy (B)) ⊆ S −1 (Fy0 (B) and so
Step 2.
S(I) ∩ Fy (B) = S S −1 (Fy (B)) ⊆ S S −1 (Fy0 (B)) = S(I) ∩ Fy0 (B).
Since, by assumption, S(I) is approximately of type (P’), it follows that y ∼ y0 , that is [y] = [y0 ]. In particular, Fy (B) =
Fy0 (B) which implies that S −1 (Fy (B)) ⊆ ρ−1 (Fx (A)) ⊆ S −1 (Fy (B)). Hence S −1 (Fy (B)) = ρ−1 (Fx (A)) as we claimed.
T
We note that if y and x are as in the above step and if x 0 ∈ X is an arbitrary point in the intersection p∈S−1 (Fy (B)) Mρ(p) ,
then S −1 (Fy (B)) ⊆ ρ−1 (Fx 0 (A)), and consequently ρ−1 (Fx (A)) ⊆ ρ−1 (Fx 0 (A)). Therefore, ρ(I) ∩ Fx (A) ⊆ ρ(I) ∩ Fx 0 (A). Thus,
as we noted before, this implies that x 0 ∈ [x] since ρ(I) is approximately of type (P’). This shows, indeed, that for
e such that T −1
each y ∈ Ye there exists a point x ∈ X
p∈S (Fy (B)) Mρ(p) = [x]. Now, by the previous step, we can define
e /∼ so that for each y ∈ Ye , φ([y]) = [x] where x ∈ X
e satisfies S −1 (Fy (B)) = ρ−1 (Fx (A)). Clearly φ is well
φ : Ye /∼ → X
defined.
Step 3.
e /∼ is a bijection.
The map φ : Ye /∼ → X
We first show that φ is injective. Indeed, let y, y0 ∈ Ye and φ([y]) = φ([y0 ]). Then S −1 (Fy (B)) = S −1 (Fy0 (B)) and so
S(I) ∩ Fy (B) = S(I) ∩ Fy0 (B), which implies that [y] = [y0 ] since S(I) is approximately of type (P’).
e be an arbitrary point. Then, by an argument similar to the one given
We now show that φ is surjective. Let x ∈ X
e
in Step 2, there exists a point y ∈ Y such that ρ−1 (Fx (A)) ⊆ S −1 (Fy (B)) = ρ−1 (Fx 0 (A)) for all x 0 ∈ φ([y]). Hence,
ρ(I) ∩ Fx (A) ⊆ ρ(I) ∩ Fx 0 (A) which concludes that x ∼ x 0 . Therefore, φ([y]) = [x], i.e., φ is surjective.
Step 4.
For each y ∈ Ye and p ∈ I we have kS(p)k[y] = kρ(p)kφ([y]) .
By a minor modification of [4, Lemma 8], we first show that if y ∈ Ye and x ∈ φ([y]), then T −1 (Fy (B)) = τ −1 (Fx (A)). Let
p0 ∈ T −1 (Fy (B)), then |T p0 (y)| = 1 = kT p0 kY = kτp0 kX . If k ∈ I is such that ρ(k) ∈ ρ(I) ∩ Fx (A), then k ∈ ρ−1 (Fx (A)) =
S −1 (Fy (B)) and so |Sk(y)| = 1 = kSkkY . Hence, 1 = kSkT p0 kY = kρ(k)τ(p0 )kX . Therefore, since ρ(I) is approximately
of type (P’),
kτ(p0 )k[x] = inf kρ(k)τ(p0 )kX : ρ(k) ∈ ρ(I) ∩ Fx (A) = 1,
which implies easily that τ(p0 ) ∈ Fx (A). Consequently, T −1 (Fy (B)) ⊆ τ −1 (Fx (A)). Since the conditions are symmetric,
the converse inclusion will be obtained in a similar manner, that is T −1 (Fy (B)) = τ −1 (Fx (A)).
Let now y ∈ Ye and p ∈ I. Since kS(p)k[y] = inf {kS(p)ukY : u ∈ T (I 0 ) ∩ Fy (B)} it follows that for a given ε > 0 we can
find u ∈ T (I 0 ) ∩ Fy (B) such that kS(p)ukY < kS(p)k[y] + ε. Let p0 ∈ I 0 be such that u = T (p0 ). By the above argument,
p0 ∈ T −1 (Fy (B)) = τ −1 (Fx (A)), where x is an arbitrary element of φ([y]). Therefore, τ(p0 ) ∈ Fx (A), that is |τ(p0 )(x)| = 1.
Hence
|ρ(p)(x)| = |ρ(p)(x)τ(p0 )(x)| ≤ kρ(p)τ(p0 )kX = kS(p)T (p0 )kY = kS(p)ukY .
Thus |ρ(p)(x)| ≤ kS(p)k[y] + ε. Since this inequality holds for all x ∈ φ([y]) and ε > 0, we get kρ(p)kφ([y]) ≤ kS(p)k[y] .
A similar argument for φ−1 instead of φ shows that kS(p)k[y] ≤ kρ(p)kφ([y]) . Therefore kS(p)k[y] = kρ(p)kφ([y]) .
The proof of kT (p0 )k[y] = kτ(p0 )kφ([y]) , p0 ∈ I 0 , y ∈ Ye , is similar to Step 4.
e /∼, τq ) is continuous. Let y0 ∈ Ye and φ([y0 ]) = [x0 ] where x0 ∈ X
e and let V
e
We now show that φ : (Ye /∼, τo ) → (X
e
e
e
e
e
be an open neighborhood of φ([y0 ]) in (X /∼, τq ). Let also π1 : X → X /∼ and π2 : Y → Y /∼ be quotient maps. Then
e) = X
e ∩ V for some open neighborhood V of x0 in X . Since V is, indeed, a neighborhood of [x0 ] as a subset
π1−1 (V
of X and ρ(I) is approximately of type (P’), we can find a function f ∈ ρ(I) ∩ Fx0 (A) such that |f| < 1/2 on X \ V .
e = {[y] ∈ Ye /∼ : kρ(p)kφ([y]) > 1/2}, then by Step 4,
Let p ∈ I be such that ρ(p) = f. Then kρ(p)k[x0 ] = 1. Set U
e
e
U = {[y] ∈ Y /∼ : kS(p)k[y] > 1/2}. Since kS(p)k[y0 ] = kρ(p)kφ([y0 ]) = 1, there exists y1 ∈ [y0 ] with |S(p)(y1 )| = 1 and so
S(p) ∈ Fy1 (B) = Fy0 (B), that is |S(p)(y0 )| = 1. Hence the set U = {y ∈ Y : |S(p)(y)| > 1/2} is an open neighborhood
e This shows that U
e is a neighborhood of [y0 ] in (Ye /∼, τo ) and, since φ(U)
e ⊆V
e , it follows
of y0 in Y with π2 (U ∩ Ye ) ⊆ U.
−1
e
e
e
e
that φ : (Y /∼, τo ) → (X /∼, τq ) is continuous. Similarly, φ : (X /∼, τo ) → (Y /∼, τq ) is continuous as well.
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Maps between Banach function algebras satisfying certain norm conditions
Remark.
(i) Clearly, in the cases where A and B are uniform algebras or Banach function algebras on locally compact Hausdorff
spaces which are completely regular, that is every point in their Choquet boundaries is a strong boundary point,
e = c(A) and Ye = c(B). Since in these cases the quotient maps are identity maps, the above theorem
we have X
implies that there exists a homeomorphism φ : c(B) → c(A) such that
|S(p)(y)| = |ρ(p)(φ(y))|,
|T (p0 )(y)| = |τ(p0 )(φ(y))|
for all p ∈ I, p0 ∈ I 0 and y ∈ c(B).
(ii) Since absolutely multiplicative sets satisfying condition (i’) in Definition 4.1 are approximately of type (P’), the
above theorem shows that [21, Proposition 2.1] and [4, Theorem 1] are valid under the weaker condition that the
ranges of mappings under consideration satisfy just condition (i) in definition of sets of type (P).
For some examples of completely regular Banach function algebras we can refer to the Figà–Talamanca–Herz algebra Ap (G), 1 < p < ∞, of a locally compact group G and the algebras Lip(X , α) of all Lipschitz functions of order α
on X , where 0 < α ≤ 1 and X is a compact metric space, endowed with the Lipschitz norm, and, in general, the algebra
A = C0 (X ) ∩ Lip(X , α) where 0 < α ≤ 1 and (X , d) is a locally compact metric space with the following Lipschitz norm:
kfk = kfkX + sup
x6=y
|f(x) − f(y)|
,
dα (x, y)
f ∈ A;
see [8]. In particular, in this case φ will be a bi-Lipschitz homeomorphism between underlying metric spaces.
Now we state the following corollary which is a similar result to [23, Theorem 2.9].
Corollary 4.3.
Let A and B be dense subalgebras of some uniform algebras A and B on locally compact Hausdorff spaces X and Y ,
such that c(A) and c(B) consist of strong boundary points for A and B, respectively. If T : A → B is a surjective
multilplicatively norm-preserving map, then there exists a homeomorphism φ : c(B) → c(A) such that for each f ∈ A
and y ∈ c(B), |T f(y)| = |f(φ(y))|.
Let I = I 0 = A, ρ = τ = id : A → A, S = T : A → B. Then, since the maps ρ, τ and S, T are onto A and B,
respectively, the hypotheses on the Choquet boundaries of A and B easily imply that the ranges of these maps are
approximately of type (P’). So the result follows from Theorem 4.2.
Proof.
Acknowledgements
The authors would like to thank the referees for their invaluable comments.
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