1. Introduction and main theorems

We will use the standard terminology and notation of Banach space theory. For unexplained definitions and notation we refer to [^{1}], [^{2}], [^{3}], [^{4}], [^{5}], [^{6}], [^{7}], [^{8}], [^{9}], [^{10}]. As usual K stands for the field R or C. For a compact Hausdorff space *K*, we denote by *C*(*K*) the Banach space of K-valued continuous functions on *K*, provided with the supremum norm.

The classical Banach-Stone theorem states that the Banach space *C*(*K*) determines the topology of *K* [^{3}], [^{4}], [^{5}], [^{11}]. More precisely, if *T*: *C*(*K*) → *C*(*S*) is an onto isometry, then there are a homeomorphism *h*: *S* → *K* and a continuous function *σ*: *S* → K with |*σ*(*s*)| = 1 for all *s* ∈ *S* such that

The conclusion of the Banach-Stone theorem is too far to be valid when we consider into isomorphisms between *C*(*K*) spaces. Thus it seems natural to ask for topological properties which are preserved under into isomorphisms of *C*(*K*) spaces. In this direction, Holsztyński [^{8}] proved:

**Theorem 1.1.**
*Let K and S be compact Hausdorff spaces. If T*: *C*(*K*) → *C*(*S*) *is an into isometry, then there are a closed subset* Δ *of S, a continuous surjection ψ*: Δ → *K and a continuous function σ*: Δ → K *with* |*σ*(*s*)| = 1 *for all s* ∈ Δ *such that*

In [^{2}], it is established the following generalization of Theorem 1.1 for extremely regular spaces. According to [^{6}], a closed subspace A of *C*(*K*) is called extremely regular if for each *k* ∈ *K* and each neighborhood *U* of *k* and each 0 < *ε* < 1, there exists *f* ∈ *A* satisfying ║*f*║ = *f*(*k*) = 1 and |*f*(*w*)| < *ε* for all *w* ∈ *K* \ *U*.

**Theorem 1.2.**
*Let K and S be compact Hausdorff spaces. Let A be an extremely regular subspace of C*(*K*) *and B a closed subspace of C*(*S*). *Suppose that T*: *A* → *B is an into isometry. Then there exist a closed subset* Δ *of S, a continuous function ψ from* Δ *onto K and a continuous function σ*: Δ → K *with* |*σ*(*s*)| = 1 *for all s* ∈ Δ *such that*

The aim of this note is to give an alternative proof of Theorem 1.2. The paper is divided as follows: in the second section we generalize a result which is proved by Plebanek in the setting of *C*(*K*) spaces (see [^{9, Theorem 3.3}]). In third section, we prove Theorem 1.2.

2. Preliminaries

Following [^{7, p. 222}], we identify dual space *C*(*K*)* with the space of regular countably additive bounded measures, and we denote it by *M*(*K*). We always consider *M*(*K*) equipped with the *weak** topology inherited from *C*(*K*)*. The total variation of a measure *μ* ∈ *M*(*K*) on a Borel set *E* is denoted by |*μ*|(*E*), and its norm by ║*μ*║ = |*μ*|(*K*).

Let *K* and *S* be compact Hausdorff spaces. Throughout the paper *A* denotes an extremely regular subspace of *C*
_{0}(*K*). Also *B* will be a closed subspace of *C*(*S*). If *s* ∈ *S* is fixed and *T*: *A* → *B* is an embedding, *ν _{s}
* will denote any norm-preserving extension to

*C*(

*K*) of the functional

*T**

*δ*:

_{s}*A*→ R defined as

*T**

*δ*(

_{s}*f*) =

*T f*(

*s*) for

*f*∈

*A*. Also let us assume that

*T*satisfies

*r*║

*f*║ ≤ ║

*T*

*f*║ ≤ ║

*f*║ for all

*f*∈

*A*, where

*r*> 0. Analogously if

*E*=

*TA*⊂

*B*and

*k*∈

*K*is given, let

*μk*be any norm-preserving extension to

*C*(

*S*) of the functional (

*T*

^{−1})*

*δ*:

_{k}*E*→ R.

Before stating our first result, we need to establish a notation.

Let *k* ∈ *K* be given and *V _{k}
* any fundamental system of open neighborhoods of

*k*. Consider the set

*C*=

_{k}*V*× (0,∞). In

_{k}*C*we define a partial order as follows: (

_{k}*U, t*) ≺ (

*V, s*) iff

*V*⊂

*U*and

*s*<

*t*. Note that (

*C*,≺) is a directed set. It is easy to see that there exists a net (

_{k}*f*

_{(U,t)})

_{(U,t)∈Ck }in

*A*satisfying

We will write {(*U, t*), *f*
_{(U,t)}}_{(U,t)∈Ck
} ↔ {*k*} to indicate that the above conditions are satisfied.

**Lemma 2.1.**
*Let A be an extremely regular subspace of C*(*K*) *and k* ∈ *K given. Suppose that* {(*U, t*), *f*
_{(U,t)}}_{(U,t)∈Ck
} ↔ {*k*}. *If μ* ∈ *M*(*K*), *then*

*Proof*. The statement is obvious if ║*μ*║ = 0, so we assume that ║*μ*║ ≠ 0. Let *ε* > 0 be given. Since |*μ*| is regular, there is *W* ⊂ *K* open with *k* ∈ *W* such that |*μ*|(*W*\{*k*}) < *ε*/2. Let *U*
_{0} ∈ *V _{k}
* be such that

*U*

_{0}⊂

*W*. If (

*U*

_{0},

*ε*/2║

*μ*║) ≺ (

*V, t*), we have

The next two results are proved in [^{9}] for *C*(*K*) spaces. However, we noted that they are also valid for extremely regular subspaces of *C*(*K*). So, for sake of completeness we include a proof here.

**Lemma 2.2.**
*Let*
*k* ∈ *K*
*be fixed. If*
*μ* = *μ _{k}
*,

*then*║

*ν*║ ≥

_{s}*r*

*μ*-

*almost everywhere*.

*Proof*. Let *N* = {*s* ∈ *S*: ║*δ _{s}
*|

_{ E }║ < 1}. We show that

*μ*(

*N*) = 0. For 0 <

*h*< 1, define

*N*= {

_{h}*s*∈

*S*: ║

*δ*|

_{s}_{ E }║ ≤

*h*}; then

*Nh*is closed and

*N*= ∪

_{ h<1}

*N*. It suffices to prove that |

_{h}*μ*|(

*N*) = 0 for all

_{h}*h*∈ (0, 1). If

*ε*> 0 is given, then there is

*f*∈

*A*with ║

*T f*║ ≤ 1 such that ║

*μ*║ −

*ε*< |

*μ*(

*T f*)|. Thus,

Since ║*μ*║ = |*μ*|(*N _{h}
*) + |

*μ*|(

*S*\

*N*), we infer that |

_{h}*μ*|(

*N*) ≤

_{h}*ε*/1 −

*h*. Thus, |

*μ*|(

*N*) = 0, by the arbitrariness of

_{h}*ε*.

Now let *s* ∈ *S* \ *N*; then ║*δ _{s}
*|

_{ E }║ ≥ 1. For a positive number ε there exists

*f*∈

*A*with ║

*T f*║ ≤ 1 such that |

*T f*(

*s*)| > 1 −

*ε*. From the fact ║

*f*║ ≤ 1/

*r*, we infer that r(1 −

*ε*) < ║

*ν*║. So, the result follows when

_{s}*ε*→ 0. ☑

If *h* is a real valued function defined on a topological space *X*, the oscillation of *h* at *x* on a set *A* is

where the infimum is taken over all open neighborhoods *U* of *x*.

**Lemma 2.3.**
*Let k* ∈ *K and ε* > 0 *be fixed. Consider the measure μ* = *μ _{k}. Suppose that there is a compact subset F of S such that*

*Then, there is*
*s* ∈ *F*
*such that* |*ν _{s}
*({

*k*})| ≥

*r*− 2

*ε*.

*Proof*. Let *δ* > 0 be given and let *U* ⊂ *K* be open with *k* ∈ *K*. Since *A* is extremely regular, there exists *f _{U}
* ∈

*A*such that ║

*f*║ =

_{U}*f*(

_{U}*k*) = 1 and |

*f*(

_{U}*w*)| <

*δ*for all

*w*∈

*K*\

*U*. We will show that there is

*s*∈

_{U}*F*satisfying |

*T*

*f*(

_{U}*s*)| >

_{U}*r*−

*ε*. Indeed, if |

*T*

*f*(

_{U}*s*)| <

*r*−

*ε*for all

*s*∈

*F*, then

which is absurd. Now if *s _{U}
* ∈

*F*satisfies |

*T*

*f*(

_{U}*s*)| >

_{U}*r*−

*ε*, then

since ║*ν _{sU}
* ║ = ║

*T**

*δ*║ ≤ 1. So if

_{sU}*δ*→ 0, then

*r*−

*ε*≤ |

*ν*|(

_{sU}*U*). Let

*V*be a fundamental system of open neighborhoods of

_{k}*k*and consider the net (

*s*)

_{U}_{ U∈}

*in*

_{Vk}*F*. Since

*F*is compact, there is a subnet (

*s*)

_{U}_{ U∈W }converging to

*s*∈

*F*. By (2), so we may assume that ║

*ν*║ ≤ ║

_{sU}*ν*║ +

_{s}*ε*for all

*U*∈

*W*.

Now, if *U* ⊂ *K* is open with *k* ∈ *U*, then we have |*ν _{s}
*|(

*U*) ≥

*r*− 2

*ε*. Indeed, by Urysohn Lemma [

^{7, Proposition 4.32}] there exists

*g*:

*K*→ [0, 1] continuous such that

*g*= 1 on an open set

*V*containing

*k*and

*g*= 0 outside

*U*. Thus, if

*W*∈

*W*satisfies

*W*⊂

*V*, then |

*υ*|(

_{sW}*g*) ≥ |

*υ*|(

_{sW}*W*) ≥

*r*−

*ε*. Whence,

Since *ν _{sW}
* →

*ν*in the weak* topology, by [

_{s}^{9, Lemma 2.1}] and the above inequality we have

Therefore, |*ν _{s}
*|(

*U*) ≥ |

*ν*|(

_{s}*g*) ≥

*r*− 2

*ε*. Regularness of

*ν*implies |

_{s}*ν*({

_{s}*k*})| ≥

*r*− 2

*ε*, and the proof is complete. ☑

The proof of the next result follows as in [^{9, Theorem 3.3}] by using Lemmas 2.2 and 2.3.

**Theorem 2.4.**
*Let K and S be compact Hausdorff spaces. Suppose that T : A* → *B is an embedding. For each k* ∈ *K we have*

3. Proof of Theorem 1.2

Since *T* is an isometry we have ║*T* ║ = ║*T*
^{−1}║ = 1. For *k* ∈ *K* we set

By Theorem 2.4 we have Δ_{
k
} ≠ ∅ for each *k* ∈ *K*.

**Claim 3.1.** If *k*
_{1}, *k*
_{2} ∈ *K* and *k*
_{1} ≠ *k*
_{2}, then Δ_{
k₁} ∩ Δ_{
k₂} = ∅.

If not, let *s* ∈ *S* be such that *s* ∈Δ_{
k₁} ∩ Δ_{
k₂} . Then

By taking *a*, *b* ∈ *K* with *aT***δ _{s}
*({

*k*

_{1}}) = 1 and

*bT**

*δ*({

_{s}*k*

_{2}}) = 1, we infer from definition of variation that

which is absurd. This proves the claim.

**Claim 3.2.** Let *k* ∈ *K* be given. If *s* ∈ Δ_{
k
}, then there is *a _{s}
* ∈

*K*with |

*a*| = 1 such that

_{s}*T f*(

*s*) =

*a*(

_{s}f*k*) for all

*f*∈

*A*.

Indeed, if *s* ∈ Δ_{
k
}, then *a _{s}
* =

*T**

*δ*({

_{s}*k*}) ∈

*K*and |

*a*| = 1. On the other hand,

_{s}*T**

*δ*=

_{s}*a*+

_{s}δ_{k}*μ*, where

*μ*∈

*M*(

*K*) satisfies

*μ*({

*k*}) = 0. So, it follows that

So, ║*μ*║ = 0, which means that *μ* = 0. Hence *T* **δ _{s}
* =

*a*, that is,

_{s}δ_{k}*T f*(

*s*) =

*a*(

_{s}f*k*) for all

*f*∈

*A*, as claimed.

Set Δ = ∪_{
k∈K
} Δ_{
k
}, and let *ψ*: Δ → *K* and *σ*: Δ → *K* be defined as *ψ*(*s*) = k and *σ*(*s*) = *a _{s}
*, respectively, iff

*s*∈ Δ

_{ k }, where

*a*is determined as in Claim 3.2. Note that

_{s}*ψ*is well-defined by Claim 3.1. The surjectivity of

*ψ*is consequence from the fact Δ

_{ k }≠ ∅ for each

*k*∈

*K*. Clearly, |

*σ*(

*s*)| = 1 for all

*s*∈

*S*. Also, by Claim 3.2 we have

**Claim 3.3.**
*ψ*: Δ → *K* and *σ*: Δ → *K* are continuous.

Let *s* ∈ Δ be given and (*s _{α}
*) a net in Δ such that

*s*→

_{α}*s*. Suppose that

*ψ*(

*s*) =

_{α}*k*↛

_{α}*ψ*(

*s*) =

*k*. Thus, there is a compact neighborhood

*V*⊂

*K*of

*k*such that for all

*α*, there is

*α*

_{′}≥

*α*with

*kα*

_{′}∉

*V*. Since

*A*is extremely regular, there exists

*f*∈

*A*such that ║

*f*║ =

*f*(

*k*) = 1 and |

*f*(

*w*)| < 1/2 for all

*w*∈

*K*\

*V*. Note that |

*T f*(

*s*)| = |

*f*(

*ψ*(

*s*))| = |

*f*(

*k*)| = 1. By continuity of

*T f*, there is

*α*0 such that |

*T f*(

*s*)| > 1/2 for all

_{α}*α*≥

*α*

_{0}. By taking

*α*

_{′}≥

*α*

_{0}with

*k*∉

_{α′}*V*, we have 1/2 > |

*f*(

*k*

_{α}_{′})| = |

*f*(

*ψ*(

*s*

_{α}_{′}))| = |

*T f*(

*s*

_{α}_{′})| > 1/2, which is impossible.

Now we prove continuity of *σ*. Let *s* ∈ Δ be given and *ψ*(*s*) = *k*. Take *f* ∈ *A* such that ║*f*║ = *f*(*k*) = 1. By Equation (2) we have *σ*(*s*) = *T f*(*s*), and continuity follows immediately.

**Claim 3.4.** Δ is closed.

Let (*s _{α}
*) be a net in Δ and suppose that

*s*→

_{α}*s*for some

*s*∈

*S*. Write

*ψ*(

*s*) =

_{α}*k*for all

_{α}*α*. By compactness of

*K*, we may assume that

*k*→

_{α}*k*for some

*k*∈

*K*. By Claim 3.2 we have |

*T f*(

*s*)| = |

_{α}*f*(

*ψ*(

*s*))| = |

_{α}*f*(

*k*)| for all

_{α}*f*∈

*A*. Thus, |

*T f*(

*s*)| = |

*f*(

*k*)| for all

*f*∈

*A*. Let (

*f*

_{(U,t)})

_{(U,t)∈Ck }be a net in

*A*such that {(

*U, t*),

*f*

_{(U,t)}}

_{(U,t)∈}

*↔ {*

_{Ck}*k*}. Then |

*T f*

_{(U,t)}(

*s*)| = |

*f*

_{(U,t)}(

*k*)| = 1 for all (

*U, t*) ∈

*C*. Once again by Lemma 2.1, we have

_{k}

So, |*T* **δ _{s}
*({

*k*})| = 1, that is,

*s*∈ Δ.