1 Introduction
This article focuses on Čech–Stone remainders of locally compact topological spaces, spaces of the form
$X^*=\beta X\setminus X,$
where
$\beta X$
is the Čech–Stone compactification of X, and maps between them.
A natural way to construct a continuous map between Čech–Stone remainders is to consider the restriction of the compactification of a continuous map between the underlying spaces (perhaps forgetting about compact sets). To be precise, if X and Y are locally compact noncompact spaces, and
$V_X\subseteq X$
is an open set with compact closureFootnote
1
, a proper continuous map
$X\setminus V_X\to Y$
extends to a map
$\beta (X\setminus V_X)\to \beta Y$
whose restriction to
$X^*$
gives a continuous map
$X^*\to Y^*$
. Does this construction recover all continuous maps between Čech–Stone remainders, at least without appealing to unnatural back-and-forth constructions? This question was first analyzed for the simplest space among all,
$\mathbb N$
with the discrete topology, and it was given a negative answer: in [Reference Dow2], Dow constructed (in
${\mathsf {ZFC}}$
) a nontrivial copy of
${\mathbb N}^*$
inside
${\mathbb N}^*$
which is nowhere dense, and therefore cannot come from a map
${\mathbb N}\to {\mathbb N}$
as above (further considerations on nontrivial copies of
${\mathbb N}^*$
were recently made in [Reference Dow3, Reference Dow4]). Conjecturally, this is all there is in
${\mathsf {ZFC}}$
, and under some reasonably weak forcing axioms, all maps between Čech–Stone remainders arise as a blend of two maps as above. This intuition was formalised by Farah, who in [Reference Farah6, Section 4] introduced the weak Extension Principle. This principle was stated originally for pairs of zero-dimensional spaces; the following adapts it to the class of locally compact noncompact second countable spaces.
Definition 1.1. Let X and Y be locally compact noncompact second countable topological spaces. We say that X and Y satisfy the weak Extension Principle, and write
${\mathsf {wEP}}(X,Y)$
, if the following happens for all pairs of naturals
$d ,\ell \geq 1$
.
For every continuous map
$F\colon (X^*)^d\to (Y^*)^\ell $
, there exists a partition into clopen sets
$(X^*)^d =U_0\cup U_1$
, an open set with compact closure
$V_X\subseteq X$
such that
$F[U_0]$
is nowhere dense in
$(Y^*)^\ell $
, and a continuous function
$G\colon (\beta (X\setminus V_X))^d\to (\beta Y)^\ell $
, which restricts to F on
$U_1$
.
By
${\mathsf {wEP}},$
we denote the statement “
${\mathsf {wEP}}(X,Y)$
holds whenever X and Y are locally compact, noncompact second countable spaces”.
The weak Extension Principle cannot be a consequence of
${\mathsf {ZFC}}$
, as witnessed by the existence of nontrivial autohomeomorphisms of
$\mathbb N^*$
under the Continuum Hypothesis
${\mathsf {CH}}$
[Reference Rudin20]. More in depth, by a classical result of Parovičenko [Reference Parovičenko19],
${\mathsf {CH}}$
implies that if X is a zero-dimensional locally compact noncompact second countable space, then
$X^*$
is homeomorphic to
${\mathbb N}^*$
. All of these nontrivial homeomorphisms witness the failure of
${\mathsf {wEP}}$
. Further, again under
${\mathsf {CH}}$
, all Čech–Stone remainders (of second countable spaces) are continuous images of
${\mathbb N}^*$
, and all connected ones are continuous images of
$[0,1)^*$
. A simple counting argument shows that most of these maps cannot arise from continuous maps from
${\mathbb N}$
(or
$[0,1)$
) to the locally compact space of interest. In general,
${\mathsf {CH}}$
and back-and-forth constructions make it essentially impossible to classify maps between Čech–Stone remainders.
Is it then possible that
${\mathsf{wEP}}$
is consistent? Instances of
${\mathsf {wEP}}$
were analysed first for
$X=Y={\mathbb N}$
, and more generally to study maps between finite powers of
$\mathbb N^*$
in different models of set theory (see [Reference Dow and Hart5, Reference Farah8, Reference Just15, Reference Just16, Reference van Douwen22] and [Reference van Mill23], or [Reference Farah and Pearl9] for more related questions). Farah, while formally introducing the
${\mathsf {wEP}}$
, showed under some fairly mild forcing axioms (which are Todorcevi
$\grave{c}$
’s
${\mathsf {OCA}}$
and Martin’s Axioms
${\mathsf {MA}_{\aleph _1}}$
),
${\mathsf {wEP}}(X,Y)$
holds for pairs of countable locally compact noncompact spaces. Further results were obtained in [Reference Farah and McKenney13] and [Reference Vignati24], focusing on just homeomorphisms instead of general continuous maps. For a lengthy and detailed treatment on how forcing axioms impact the structure of homomorphisms between quotient structures, see [Reference Farah, Ghasemi, Vaccaro and Vignati12].
The main result of this article is that the weak Extension Principle holds unconditionally on the spaces of interest, again under the same forcing axioms assumptions.
Theorem 1.2. Assume
${\mathsf {OCA}}$
and
${\mathsf {MA}_{\aleph _1}}$
. Then
${\mathsf {wEP}}$
holds.
As a consequence, we show that the thesis of [Reference Farah7, Theorem 5.1] holds (under
${\mathsf {OCA}}$
and
${\mathsf {MA}_{\aleph _1}}$
) for all pairs of locally compact noncompact second countable spaces: if
$\kappa $
and
$\lambda $
are cardinals with
$\kappa <\lambda $
then there is no continuous surjection
$(X^*)^\kappa \to (Y^*)^\lambda $
. This result does not hold under
${\mathsf {CH}}$
, as under this assumption
${\mathbb N}^*$
continuously surjects onto every compact topological space of weight
$\leq \mathfrak c$
, and specifically onto every finite power of every Čech–Stone remainder of a locally compact second countable topological space, while
$[0,1)^*$
continuously surjects onto every finite power of every such connected Čech–Stone remainder.
The proof of our main result proceeds as follows: first, we use a result of Farah [Reference Farah7] on maps between powers of Čech–Stone remainders (more generally,
$\beta {\mathbb N}$
-spaces) to simplify the situation and show that it is enough to show
${\mathsf {wEP}}$
for maps, where
$d=\ell =1$
(Proposition 2.2). Secondly, we rely on Gel’fand’s duality, and study
$^*$
-homomorphisms between coronas of abelian
$\mathrm {C^*}$
-algebras. We build on a strong lifting theorem proved in [Reference Vignati24] (see Proposition 2.3) which gives that, under forcing axioms,
$^*$
-homomorphisms between abelian corona
$\mathrm {C^*}$
-algebras have well-behaved liftings on certain blocked subalgebras. The main content of our proof is to glue and analyse such liftings. This approach is fairly different from the one of [Reference Farah6, Section 4], where Boolean algebras of clopen sets were used. Once the dimension increases, Stone duality is replaced by Gel’fand’s one, as clopen sets do not recover completely the topology in this case.
For all background material on
$\mathrm {C^*}$
-algebras theory, we refer the reader to [Reference Blackadar1], or to [Reference Farah11] for the connections between
$\mathrm {C^*}$
-algebras and set theory. For notions in topology, we refer to the standard book of Munkres [Reference Munkres18] and to [Reference Gillman and Jerison14] more specifically for Čech–Stone compactifications and remainders.
2 Proof of the main result
This main section is fully dedicated to the proof of Theorem 1.2. As mentioned, our first step is to allow us to only consider maps between Čech–Stone remainders. This is due to a powerful result of Farah. Some context: If
$d\geq 1$
, a function between topological spaces
$F \colon X^d \to Y$
is said to depend on at most one coordinate on
$U \subseteq X^d$
if there exists
$j \leq d$
and
${H \colon X \to Y}$
such that
$F \restriction U = H \restriction \pi _j (U)$
,
$\pi _j\colon X^d\to X$
being the projection on the j-th coordinate. A rectangle in
$X^d$
is a set of the form
$\prod _{i\leq d}A_i$
, where
$A_i\subseteq X$
. The following is a reformulation of [Reference Farah7, Theorem 3]Footnote
2
.
Theorem 2.1. Let K be a compact space, Y be locally compact, noncompact, and second countable,
$d\geq 1$
, and let
$F\colon K^d\to Y^*$
be a continuous map. Then
$K^d$
can be covered by finitely many clopen rectangles on each of which F depends on at most one coordinate.
Proposition 2.2. Suppose the
${\mathsf {wEP}}$
holds for all continuous maps between Čech–Stone remainders. Then the
${\mathsf {wEP}}$
holds.
Proof. The statement of the proposition asserts that if we can prove the weakening of the
${\mathsf {wEP}}$
where we only consider maps between Čech–Stone remainders, i.e.,
$d=\ell =1$
, then the
${\mathsf {wEP}}$
holds.
At first, we show that we can assume
$\ell =1$
. For this, assume the
${\mathsf {wEP}}$
holds for all maps
$(X^*)^d\to Y^*$
(where X and Y satisfy the appropriate hypotheses, and
$d\geq 1$
is arbitrary). Let
$F\colon (X^*)^d\to (Y^*)^\ell $
be a continuous map, and, for
$i\leq \ell $
, compose F with the projection on the i-th coordinate maps. This gives continuous maps
$F_i=\pi _i\circ F\colon (X^*)^d\to Y^*$
. Applying the weak Extension principle to these maps, we get, for
$i\leq \ell $
, a clopen decomposition
$(X^*)^d = U_0^i \cup U_1^i$
and an open set with compact closure
$V_X^i$
such that
$F_i[U_0^i]$
is nowhere dense and there is a continuous
$G_i\colon (\beta X\setminus V_X^i)^d\to \beta Y$
which restricts to
$F_i$
on
$U_1^i$
. The sets
$U_0=\bigcup _i U_0^i$
,
$U_1=\bigcap _i U_1^i$
,
$V_X=\bigcup _i V_X^i$
and the map
$G\colon (\beta X\setminus V_X^i)^d\to (\beta Y)^\ell $
sending x to
$(G_1(x),\ldots ,G_\ell (x))$
witness that F satisfies the
${\mathsf {wEP}}$
.
Let us now suppose we have a continuous map of the form
$F \colon (X^*)^d \to Y^*$
, and apply Theorem 2.1. Then there exists a finite cover of
$(X^*)^d$
by clopen rectangles
$R_1,\dots ,R_n$
. Let
$R_i = A_1^i \times \cdots \times A_d^i$
. Since each
$R_i$
is clopen in the compact space
$(X^*)^d$
, we can assume each
$A^i_j\subseteq X^*$
is clopen, hence we can find open sets
$Z_{i,j}\subseteq X$
such that
$A_j^i=Z_{i,j}^*$
. For every i, let
$j(i)$
be such that
$F\restriction R_i$
depends on the
$j(i)$
-th coordinate. Hence we can find a continuous
$H_i\colon A_{j(i)}^i=Z_{i,j(i)}^*\to Y^*$
. Applying the
${\mathsf {wEP}}$
to each
$H_i$
, for
$i\leq n$
, we get open sets with compact closure
$V_i\subseteq Z_{i,j(i)}$
, partitions into clopen sets
$Z_{i,j(i)}^*=U_0^i\cup U_1^i$
and continuous maps
$G_i\colon \beta (Z_{i,j(i)}\setminus V_i)\to \beta Y$
lifting
$H_i$
on
$U_1^i$
. For
$i\leq n$
, define
$$\begin{align*}\tilde G_i\colon \beta Z_{i,1}\times\cdots\times \beta(Z_{i,j(i)}\setminus V_i)\times\cdots\times \beta Z_{i,d}\to \beta Y \end{align*}$$
by mapping a tuple to the
$G_i$
image of its
$j(i)$
-th entry. Fix
$V_X=\bigcup V_i$
. Gluing all of these maps together gives a continuous lift on
$$\begin{align*}U_1=\bigcup_{i\leq n}(Z_{i,1}^*\times\cdots\times \underbrace{U_1^i}_{j(i)}\times\cdots\times Z_{i,d}^*) \end{align*}$$
while the F-image of the complement of
$U_1$
is a finite union of the nowhere dense sets
$F[Z_{i,1}^*\times \cdots \times U_0^i\times \cdots \times Z_{i,d}^*]$
. This concludes the proof.
From now on, we only deal with continuous maps between Čech–Stone remainders (i.e., we assume that
$d=\ell =1$
in the definition of the
${\mathsf {wEP}}$
). We approach these via Gel’fand’s duality, by looking at the
$\mathrm {C^*}$
-algebras
$C(X^*)$
and
$C(Y^*)$
and at
$^*$
-homomorphisms between them. In this article, we only consider abelian
$\mathrm {C^*}$
-algebras, i.e., algebras of continuous functions on locally compact topological spaces where the topology is induced by the
$\sup $
-norm and operations are defined pointwise. For an introduction and basic terminology, see [Reference Blackadar1, Chapter II.2].
Algebras of the form
$C(X^*)$
are typical examples of corona
$\mathrm {C^*}$
-algebras ([Reference Blackadar1, Chapter II.7] or [Reference Farah11, Section 13]). The structure of
$^*$
-homomorphisms between corona algebras and their behaviour under forcing axioms was extensively studied in [Reference McKenney and Vignati17, Reference Vignati24], after the use of forcing axioms to analyse massive quotients of
$\mathrm {C^*}$
-algebras was popularised by the seminal [Reference Farah10].
If X and Y are locally compact noncompact second countable topological spaces then Gelf’and’s duality gives that
$C(\beta X)\cong C_b(X)$
, the latter being the algebra of bounded continuous functions on X, and that
$C(X^*)\cong C_b(X)/C_0(X)$
, where
$C_0(X)$
is the algebra of continuous functions on X vanishing at infinity. We denote by
$\pi _X$
the canonical quotient map
$\pi _X\colon C_b(X)\to C(X^*)$
.
Since our spaces are second countable and locally compact, they are
$\sigma $
-compact (alternatively, the
$\mathrm {C^*}$
-algebras
$C_0(X)$
and
$C_0(Y)$
are
$\sigma $
-unital). The following notation is set up to match the hypotheses of [Reference Vignati24, Theorem 4.3]. We let
-
•
$Y_n$
be an increasing sequence of open subsets of Y such that
$Y=\bigcup Y_n$
, each
$Y_n$
has compact closure, and
$\overline Y_n\subseteq Y_{n+1}$
for all
$n\in {\mathbb N}$
. We assume each
$Y_{n+1}\setminus \overline Y_n$
is not empty, and that
$Y_0=\emptyset $
. -
•
$U_{n}^e=Y_{10n+7}\setminus \overline Y_{10n}$
and
$U_{n}^o=Y_{10n+12}\setminus \overline Y_{10n+5}$
.
The letters e and o are for even and odd partitions. Recall [Reference Blackadar1, Chapter II.3.4] that if A is a
$\mathrm {C^*}$
-algebra, a
$\mathrm {C^*}$
-subalgebra
$B\subseteq A$
is hereditary if for all positive
$a\leq b$
,
$b\in B$
implies that
$a\in B$
, where
$a\in A$
is positive if
$0\leq a$
(in the abelian setting, a takes only values in
$[0,\infty )$
).
We have the following properties:
-
(1) for every
$n\neq m$
and
$i\in \{e,o\}$
,
$\overline U_n^i\cap \overline U_m^i=\emptyset $
; -
(2) for each
$i\in \{e,o\}$
,
$\prod _nC_0(U_n^i)$
is a hereditary
$\mathrm {C^*}$
-subalgebra of
$C(\beta Y)$
such that
$(\prod _nC_0(U_n^i))\cap C_0(Y)=\bigoplus _nC_0(U_n^i)$
. -
(3)
$C_b(Y)=\prod _nC_0(U_n^e)+\prod _nC_0(U_n^o)$
, and consequently
$$\begin{align*}C(Y^*)=\prod_nC_0(U_n^e)/\bigoplus_nC_0(U_n^e)+\prod_nC_0(U_n^o)/\bigoplus_nC_0(U_n^o). \end{align*}$$
Note that the sum above is not a direct sum, as
$U_n^e\cap U_n^o\neq \emptyset $
for all n.
If
$i\in \{e,o\}$
and
$f\in \prod _nC_0(U_n^i)$
, since the sets
$U_n^i$
, for
$n\in {\mathbb N}$
, are pairwise disjoint, we can view f as a sequence
$(f_n)$
, where
$f_n\in C_0(U_n^i)$
. If
$S\subseteq {\mathbb N}$
, we write
$f_S$
for the element of
$\prod _nC_0(U_n^i)$
defined by
$$\begin{align*}(f_S)_n=\begin{cases} f_n&\text{ if }n\in S\\ 0 & \text{ else}. \end{cases} \end{align*}$$
For fixed
$S\subseteq {\mathbb N}$
and
$i\in \{e,o\}$
, the map
$f\mapsto f_S$
gives a
$^*$
-homomorphism
$\prod _nC_0(U_n^i)\to \prod _{n\in S}C_0(U_n^i)$
, dual to the inclusion
$\bigcup _{n\in S}U_n^i\subseteq \bigcup _n U_n^i$
.
Recall, a
$^*$
-homomorphism between nonunital
$\mathrm {C^*}$
-algebras
$\psi \colon A\to B$
is nondegenerate [Reference Blackadar1, Chapter II.7.3.8] if the ideal generated by the image of
$\psi $
is dense (in norm topology). This is equivalent to ask that approximate unit for A is sent to an approximate unit of B. In the abelian setting, Gel’fand’s duality asserts that nondegenerate
$^*$
-homomorphisms are dual to proper continuous maps. (Recall, a map between topological spaces is proper if inverse images of compact subsets are compact.)
The following is derived from [Reference Vignati24]. For the reader’s convenience, we shortly sketch its proof, referring heavily to the notation of [Reference Vignati24].
Proposition 2.3. Assume
${\mathsf {OCA}}$
and
${\mathsf {MA}_{\aleph _1}}$
. Suppose that
$\Phi \colon C(Y^*)\to C(X^*)$
is a
$^*$
-homomorphism. Then there are a natural
$\bar n$
, open subsets of X with compact closure
$V_n^e$
and
$V_n^o$
, for
$n\geq \bar n$
, and nondegenerate
$^*$
-homomorphisms
$$\begin{align*}\psi_n^i\colon C_0(U_n^i)\to C_0(V_n^i)\text{ for }i\in \{e,o\} \end{align*}$$
such that:
-
(1) For
$i\in \{e,o\}$
and
$n\neq m$
,
$\overline V_n^i\cap \overline V_m^i=\emptyset $
, and, if
$n>\bar n$
,
$$\begin{align*}\overline {V_{n}^e}\subseteq V_{n-1}^o\cup V_{n}^e\cup V_n^o, \,\, \overline {V_{n}^o}\subseteq V_{n}^e\cup V_{n}^o\cup V_{n+1}^e. \end{align*}$$
-
(2) Every compact subset of X intersects only finitely many of the sets
$\{V_n^i\mid n\geq \bar n, i\in \{e,o\}\}$
. -
(3) For
$i\in \{e,o\}$
, is a
$$\begin{align*}\Psi^i=\sum_{n\geq \bar n}\psi_n^i\colon \prod_{n\geq \bar n} C_0(U_n^i)\to\prod_{n\geq \bar n} C_0(V_n^i) \end{align*}$$
$^*$
-homomorphism such that for every
$f=(f_n)\in \prod _{n\geq \bar n}C_0(U_n^i)$
there is a nonmeager ideal
$\mathcal I_f\subseteq \mathcal P({\mathbb N})$
such that for all
$$\begin{align*}\pi_X(\Psi^i(f_S))=\Phi(\pi_Y(f_S)) \end{align*}$$
$S\in \mathcal I_f$
, and
-
(4) If
$\gamma _n^i\colon V_n^i\to U_n^i$
is the proper continuous map dual to
$\psi _n^i$
, then
$\gamma _n^e$
and
$\gamma _m^o$
(equivalently, the
$^*$
-homomorphisms
$\psi _n^e$
and
$\psi _m^o$
) agree on their common domain.
Proof. Write X as an increasing union of nonempty open sets
$X=\bigcup X_n,$
where for each
$n,$
we have that
$\overline {X_n}\subseteq X_{n+1}$
,
$X_{n+1}\setminus \overline {X_n}$
is nonempty, and each
$\overline {X_n}$
is compact. We furthermore ask that if
$K\subseteq X$
is compact then
$K\subseteq X_n$
for some n. This can be done since X is
$\sigma $
-compact. Writing B for
$C_0(X)$
, we let
$e_n^B\in C_0(X)$
be a positive contractionFootnote
3
such that
$e_n^B[\overline {X_n}]=1$
and
$e_n^B[X\setminus X_{n+1}]=0$
, so that
$e_{n+1}^Be_n^B=e_n^B$
for all n.
We are ready to construct the sets
$V_n^i$
. For convenience, we only give the details on how to construct the sets
$V_n^e$
; the ‘odd’ sets can be constructed in the same way after appropriately re-indexing.
Let
$e_n^A$
be an approximate identity of positive contractions for
$A=C_0(Y)$
such that
$e_n^A[\overline Y_n]=1$
and
$e_n^A(y)\neq 0$
if and only if
$y\in Y_{n+1}$
, and let
${J_n=[10n-1, 10n+7]}$
, so that the algebra
$A_{J_n}$
given in [Reference Vignati24, Notation 3.3] is precisely
$C_0(U_n^e)$
. Applying the results of Sections 3 and 4 of [Reference Vignati24], and specifically Lemma 4.2 therein, we get mutually orthogonalFootnote
4
functions
$\alpha _n\colon C_0(U_n^e)\to C_0(X)$
such that the function
$$\begin{align*}\Gamma^e=\sum\alpha_n\colon \prod_n C_0(U_n^e)\to C_b(X) \end{align*}$$
has the property that for every
$f=(f_n)\in \prod _n C_0(U_n^e)$
we have a nonmeager ideal
$\mathcal I_f$
such that
$$\begin{align*}\pi_X(\Gamma^i(f_S))=\Phi(\pi_Y(f_S)) \end{align*}$$
for all
$S\in \mathcal I_f$
. The mutually orthogonal
$\alpha _n$
s come with two sequences of naturals
$j_n<k_n$
such that the range of
$\alpha _n$
is contained in
$(e_{k_n}^B-e_{j_n}^B)C_0(X)(e_{k_n}^B-e_{j_n}^B)$
and
$\lim _n j_n= \infty $
. This is the key property in showing condition (2), as it gives that if
$K\subseteq X$
is compact then the set
$$\begin{align*}\{n\mid\exists f\in C_0(U_n^e) \exists x\in K \, (\alpha_n(f)(x)\neq 0)\} \end{align*}$$
is finite.
We will now slightly modify the maps
$\alpha _n$
, which themselves are not
$^*$
-homomorphisms. By Ulam stability for approximate maps between abelian
$\mathrm {C^*}$
-algebras (a result proved by Sěrml in [Reference Sěmrl21], see also [Reference Vignati24, Section 5] or, for details, [Reference Farah, Ghasemi, Vaccaro and Vignati12, Section 5.4 and Theorem 5.18]) shows that we can find
$\bar n$
and mutually orthogonal
$^*$
-homomorphisms
$\psi _n^e\colon C_0(U_n^e)\to (e_{k_n}^B-e_{j_n}^B)C_0(X)(e_{k_n}^B-e_{j_n}^B)$
, for
$n\geq \bar n$
such that
$\Psi ^i=\sum \psi _n^e$
still lifts elements of
$\prod _nC_0(U_n^e)$
on nonmeager ideals. The sets
$V_n^e$
are now constructed from the maps
$\psi _n^e$
as in [Reference Vignati24, Section 4.1]Footnote
5
, by setting
$$\begin{align*}V_n^e=\bigcup_{f\in C_o(U_n^e)}\mathrm{supp}(\psi^e_n(f)). \end{align*}$$
Since
$\lim _n j_n=\infty $
and
$V_n^e\subseteq X_{k_n}\setminus X_{j_n}$
, if a K is compact then K can intersect at most finitely many of the
$V_n^e$
s. This shows condition (2). The other properties are verified exactly as in Section 4.1 in [Reference Vignati24]. This concludes the construction of the sets
$V_n^e$
, and the proof of the proposition.
If f and g are elements of
$C_b(X)$
, we write
$f=_{C_0(X)}g$
for
$f-g\in C_0(X)$
.
Proof of Theorem 1.2
By Proposition 2.2, it is sufficient to prove
${\mathsf {wEP}}$
for maps between Čech–Stone remainders (i.e., when
$d=\ell =1$
). Fix then a continuous function
$F\colon X^*\to Y^*$
, and let
$\Phi \colon C(Y^*)\to C(X^*)$
be the dual
$^*$
-homomorphism. Let
$\tilde \Phi \colon C_b(Y)\to C_b(X)$
be any set theoretic lift of
$\Phi $
. All of our notation will be as in Proposition 2.3 and the discussion preceding it.
Let
$$\begin{align*}V_1=\bigcup_{n\geq \bar n}V_n^e\cup V_n^o\text{ and } W=\bigcup_{n\geq \bar n}U_n^e\cup U_n^o. \end{align*}$$
Claim 2.4.
$V_1$
is, modulo compact, clopen.
Proof. We show that
$\overline V_1\setminus V_1$
is contained in the compact set
$\overline V_{\bar n}^e$
. Combining conditions (1) and (2) of Proposition 2.3 gives that
$$\begin{align*}\overline {V_1}=\overline{\bigcup_{n\geq\bar n} V_n^e\cup V_n^o} = \bigcup_{n\geq\bar n} \overline{V_n^e}\cup\bigcup_{n\geq\bar n}\overline{V_n^o}\subseteq \overline{V_{\bar n}^e}\cup \bigcup_{n\geq\bar n} V_n^e\cup V_n^o, \end{align*}$$
where the equality
$\overline {\bigcup _{n\geq \bar n} V_n^e\cup V_n^o} = \bigcup _{n\geq \bar n} \overline {V_n^e}\cup \bigcup _{n\geq \bar n}\overline {V_n^o}$
is given by the fact that compact subsets of X can only intersect finitely many of the sets
$V_n^e\cup V_n^o$
. This shows that
$\overline V_1\setminus V_1$
is contained in the compact set
$\overline {V_{\bar n}^e}$
.
(As a side note, a deep understanding of [Reference Vignati24, Section 4.1] shows that if the spaces of interest can be written as a union of pairwise disjoint compact open sets,
$V_1$
can be chosen to be clopen. We shall not need this fact.)
As the continuous proper maps
$\gamma _n^i$
agree on their common domains, the function
$$\begin{align*}\Gamma=\bigcup_{n\geq \bar n}\gamma_n^e\cup\gamma_n^o\colon V_1\to W \end{align*}$$
is a well-defined continuous function. Furthermore,
$\Gamma $
is proper: if
$K\subseteq W$
is compact, then there exists
$\bar n'$
such that
$K\subseteq \bigcup _{\bar n\leq n\leq \bar n'}U_n^e\cup U_n^o$
, hence
$\Gamma ^{-1}[K]$
is a closed subset of
$\bigcup _{\bar n\leq n\leq \bar n'}V_n^e\cup V_n^o$
. As the latter has compact closure,
$\Gamma ^{-1}[K]$
is compact. Since
$\Gamma $
is proper, it induces a
$^*$
-homomorphism
$$\begin{align*}\Psi\colon C_b(W)\to C_b(V_1) \end{align*}$$
mapping
$C_0(W)$
to
$C_0(V_1)$
and whose dual is
$\beta \Gamma \colon \beta V_1\to \beta W\subseteq \beta Y$
.
We are ready to define the necessary sets and maps to prove the
${\mathsf {wEP}}$
: first, let
$V_X\subseteq X$
be an open set with compact closure such that
$V_1$
is clopen in
$X\setminus V_X$
, and set
$V_0=(X\setminus V_X)\setminus V_1$
. Note that
$\beta (X\setminus V_X)=\beta V_1\sqcup \beta V_0$
and therefore
$U_0:=V_0^*$
and
$U_1:=V_1^*$
is a clopen partition of
$X^*$
, by compactness of
$\overline {V_X}$
. Pick a point in
$y\in \beta Y$
, let
$G'\colon \beta V_0\to \{y\}$
be the constant map, and let
$G=\beta \Gamma \cup G'$
.
We are now going to show that these objects satisfy
${\mathsf {wEP}}(X,Y)$
, that is, that
$G\restriction U_1=F$
and that
$F[U_0]$
is nowhere dense.
Claim 2.5.
$G\restriction U_1=F$
.
Proof. First, write
$\Phi $
as
$\Phi =\Phi _0\oplus \Phi _1$
, where
$$\begin{align*}\Phi_i(f)=\chi_{U_i}\Phi(f), \,\, i\in \{0,1\}, \end{align*}$$
$\chi _{U_i}\in C(X^*)$
being the characteristic function of
$U_i$
. We have the following diagram:
Both
$\Phi _0$
and
$\Phi _1$
are
$^*$
-homomorphisms, as
$C(X^*)$
is abelian. To show that
$G\restriction U_1=F$
, it is enough to show that the
$^*$
-homomorphism
$\Psi $
lifts
$\Phi _1$
.
Since every function
$f\in C_b(Y)$
can be written (modulo
$C_0(Y)$
) as a sum
${f=f_e+f_o}$
where, for
$i\in \{e,o\}$
,
$f_i\in \prod _{n\geq \bar n}C_0(U_n^i)$
, it is enough to show
$\Psi $
lifts
$\Phi _1$
on
$\prod _{n\geq \bar n}C_0(U_n^i)$
, again for
$i\in \{e,o\}$
. For convenience, let
$i=e$
, and fix mutually orthogonal positive contractions
$g_n\in C_b(X)$
such that for all
$n\geq \bar n$
$$\begin{align*}g_n[V_n^e]=1\text{ and }\mathrm{supp} (g_n) \subseteq V^o_{n-1} \cup V^e_n \cup V^o_n. \end{align*}$$
Let
$g_n=0$
if
$n<\bar n$
. Such functions exist as distinct
$V_n^e$
s have disjoint closures. For
$S\subseteq {\mathbb N}$
, let
$g_S=\sum _{n\in S}g_n$
. Note that, even if
$V_1$
is not clopen, it is clopen modulo compact, and therefore the function
$\chi _{V_1}$
is equal, modulo
$C_0(X)$
, to a continuous function.
Subclaim 2.6. For every
$f\in \prod _{n\geq \bar n}C_0(U_n^e)$
and
$S\subseteq {\mathbb N,}$
we have that
$$\begin{align*}g_S\tilde\Phi(f)=_{C_0(X)}\chi_{V_1}\tilde\Phi(f_S). \end{align*}$$
Proof. Fix
$f\in \prod _{n\geq \bar n}C_0(U_n^e)$
and
$S\subseteq {\mathbb N}$
. We can suppose S is infinite, otherwise both
$g_S$
and
$\tilde \Phi (f_S)$
belong to
$C_0(X)$
. We will show that for every
$\varepsilon>0$
there is a compact set K such that
$$ \begin{align} \text{if }(x\in V_1\setminus K\text{ and } |\tilde\Phi(f_S)(x)|>\varepsilon)\text{ then }x\in \bigcup_{n\in S}V_n^e. \end{align} $$
This gives that
$g_S\tilde \Phi (f_S)=_{C_0(X)}\chi _{V_1}\tilde \Phi (f_S)$
. As S is arbitrary, the same reasoning applied to
${\mathbb N}\setminus S$
, gives that
$g_S\tilde \Phi (f_{{\mathbb N}\setminus S})\in C_0(X)$
. Altogether, we get that
$$\begin{align*}\chi_{V_1}\tilde\Phi(f_S)=_{C_0(X)}g_S\tilde\Phi(f_S)=_{C_0(X)}g_S\tilde\Phi(f_S)+g_S\tilde\Phi(f_{\mathbb N\setminus S})=_{C_0(X)}g_S\tilde\Phi(f), \end{align*}$$
which is the claim.
Suppose (2.1) fails for
$\varepsilon>0$
. Then there is a sequence of points
$x_k\in V_1\setminus (\bigcup _{n\in S}V_n^e)$
such that
$|\tilde \Phi (f_S)(x_k)|>\varepsilon $
and
$x_k\to \infty $
as
$k\to \infty $
(meaning for every compact
$K\subseteq X$
we have that
$K\cap \{x_k\}$
is finite). Passing to a subsequence, we can assume that there is
$i\in \{e,o\}$
such that
$\{x_k\}\subseteq \bigcup _n V_n^i$
. Let
$n_k$
be such that
$x_k\in V_{n_k}^i$
. As
$x_k\to \infty $
as
$k\to \infty $
, we can assume, by further shrinking the sequence
$\{x_k\}$
, that
$\{n_k\}$
is strictly increasing and that
$n_0>\bar n$
.
Since the homomorphisms
$\psi _n^i$
are nondegenerate, then for all
$n>\bar n$
and
$i\in \{e,o\}$
,
$$ \begin{align*} V_n^i&=\bigcup\{\mathrm{supp}(\psi_n^i(h))\mid h\in C_0(U_n^i)\}\\&=\{x\in X\mid\exists h\in C_0(U_n^i), 0\leq h\leq 1, \psi_n^i(h)(x)=1\}\\ &=\{x\in X\mid\exists h\in C_0(U_n^i), 0\leq h\leq 1, \Psi(h)(x)=1\}. \end{align*} $$
Hence, for each
$k,$
we can find positive contractions
$h_{n_k}\in C_0(U_{n_k}^i)$
such that
$$\begin{align*}\psi_{n_k}^i(h_{n_k})(x_k)=\Psi(h_k)(x_k)=1. \end{align*}$$
Since
$x_k\notin \bigcup _{n\in S}V_n^e=\Gamma (\bigcup _{n\in S}U_n^e)$
, we can further assume that
$\mathrm {supp}(h_{n_k})\cap \bigcup _{n\in S}U_n^e=\emptyset $
.
Set
$h_j=0$
if
$j\notin \{n_k\mid k\in \mathbb N\}$
, and let
$$\begin{align*}h=\sum h_k\in \prod_{n\geq \bar n} C_0(U_n^i). \end{align*}$$
Notice that for every
$T\subseteq {\mathbb N}$
we have
$\Psi (h_T)(x_k)=1$
whenever k is such that
${n_k\in T}$
and
$h_Tf_S=0$
, since
$\mathrm {supp}(f_S)\subseteq \bigcup _{n\in S}U_n^e$
which is disjoint from
$\mathrm {supp}(h)\supseteq \mathrm {supp}(h_T)$
.
Let
$\mathcal I_h$
be the nonmeager ideal given to us from Proposition 2.3, and let T be an infinite set in
$\mathcal I_h$
such that
$T\subseteq \{n_k\}$
, so that
$\Psi (h_T)=_{C_0(X)}\tilde \Phi (h_T)$
. Since
$|\tilde \Phi (f_S)(x_k)|>\varepsilon $
then
$$\begin{align*}\limsup_{k\to\infty}|\tilde\Phi(f)\tilde\Phi(h_T)(x_k)|=\limsup_{k\to\infty}|\tilde\Phi(f)\Psi(h_T)(x_k)|\geq \varepsilon. \end{align*}$$
Since
$x_k\to \infty $
as
$k\to \infty $
, this contradicts that
$\tilde \Phi (f)\tilde \Phi (h_T)\in C_0(X)$
, as
$\tilde \Phi $
lifts the
$^*$
-homomorphism
$\Phi $
and
$h_T$
and
$f_S$
are orthogonal.
By the subclaim,
$g_{\mathbb N}\tilde \Phi (f)$
is a lift for
$\Phi _1(f)$
. Let
$$\begin{align*}\mathcal J_f=\{S\subseteq{\mathbb N}\mid g_S(\Psi(f)-\tilde\Phi(f))\in C_0(X)\}. \end{align*}$$
Since
$g_S\Psi (f)=\Psi (f_S)$
for all
$S\subseteq {\mathbb N}$
, and
$\Psi (f_S)=_{C_0(X)}\tilde \Phi (f_S)$
whenever
${S\in \mathcal I_f}$
, then
$\mathcal I_f\subseteq \mathcal J_f$
, and the latter is therefore nonmeager and contains all finite sets. Since the association
$S\mapsto g_S$
is continuous (when
$C_b(X)$
is given the strict topology),
$C_0(X)$
is Borel in
$C_b(X)$
, and
$\Psi (f)$
and
$\tilde \Phi (f)$
are fixed,
$\mathcal J_f$
is a Borel ideal. Since the only Borel nonmeager ideal containing
$\mathrm {Fin}$
is
$\mathcal P({\mathbb N})$
,
$\mathcal J_f=\mathcal P({\mathbb N})$
, hence
$g_{\mathbb N}(\Psi (f)-\tilde \Phi (f))\in C_0(X)$
. Since
$g_{\mathbb N}\Psi (f)=\Psi (f)$
and
$g_{\mathbb N}\tilde \Phi (f)$
lifts
$\Phi _1(f)$
, we have that
$\Psi (f)$
lifts
$\Phi _1(f)$
, which is our thesis. The claim is then proved.
Claim 2.7.
$F\restriction U_0$
has nowhere dense range.
Proof. Since F is a continuous map between compact spaces, it is closed, hence so is
$F[U_0]$
. Suppose for a contradiction that there is a nonempty open
${U \subseteq F[U_0]\subseteq Y^*}$
. Let
$O\subseteq \beta Y$
be open such that
$O\cap Y^*=U$
, and let
$O'=O\cap Y$
. Since the sets
$\{U_n^i\mid i\in \{e,o\}, n\in {\mathbb N}\}$
cover Y, we can find an infinite
$S\subseteq {\mathbb N}$
and
$i\in \{e,o\}$
such that
$O'\cap U_n^i\neq \emptyset $
for all
$n\in S$
. For
$n\in S$
, let
$h_n$
be a positive contraction whose support is in
$U_n^e$
and such that there is
$x\in O'$
with
$h_n(x)=1$
. Let
$h=\sum h_n$
. Note that for every infinite
$T\subseteq S$
we have that
$\left \lVert \Phi (h_T)\chi _{U_0} \right \rVert =1$
. Find
$T\subseteq S$
infinite such that
$\Psi $
lifts
$\Phi $
on
$h_T$
, and notice that
$\chi _{U_0}\pi _X(\Psi (h_T))=0$
. This is a contradiction.
Claims 2.5 and 2.7 together show that G,
$U_0,$
and
$U_1$
satisfy the hypotheses of
${\mathsf {wEP}}(X,Y)$
. This concludes the proof of the theorem.
The following is a consequence of the result of Farah.
Corollary 2.8. Assume
${\mathsf {OCA}}$
and
${\mathsf {MA}_{\aleph _1}}$
. Let
$\kappa <\lambda $
be cardinals, and suppose that X and Y are locally compact noncompact second countable spaces. Then there is no continuous surjection
$(X^*)^\kappa \to (Y^*)^\lambda $
.
Proof. This is a consequence of Theorem 5.1 in [Reference Farah7], which asserts that the thesis holds if we have
${\mathsf {wEP}}$
. The result then follows from Theorem 1.2.
The
${\mathsf {wEP}}$
has consequences on the existence of surjections from Čech–Stone remainders to powers of them: while
${\mathsf {CH}}$
implies that
$\mathbb N^*$
surjects onto every compact space of density
$\aleph _1$
, and therefore onto every power of remainder of every (reasonable small) space, the same cannot happen if
${\mathsf {wEP}}$
holds, where, for example, that
$(\mathbb N^*)^d$
surjects onto
$(\mathbb N^*)^\ell $
if and only if
$d\geq \ell $
(see [Reference Farah6, Section 4.6]). In future work, we intend to explore these phenomena, along with extension principles in the noncommutative setting.
Acknowledgments
The authors would like to thank I. Farah for useful conversations, and the anonymous referee for helpful comments.
Funding
This research originated from D.Y.’s Master’s thesis, written under the guidance of A.V. at Université Paris Cité. In this period, D.Y. was supported by an INSMI (Institut National des Sciences Mathématiques et de leurs Interactions) grant for master’s students.
