Discontinuous Homomorphisms of $C(X)$ with $2^{\aleph_0}>\aleph_2$

0.28 MB·

by  Bob A. Dumas

Discontinuous Homomorphisms of C(X) with
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1 2ℵ0 > ℵ
0 2
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a Bob A. Dumas
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Department of Philosophy
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University of Washington

] Seattle, Washington 98195
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January 27, 2017
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Abstract
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1 Assume that M is a c.t.m. of ZFC +CH containing a simplified
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(ω ,2)-morass, P ∈ M is the poset adding ℵ generic reals and G
2 1 3
6 is P-generic over M. In M we construct a function between sets
6 of terms in the forcing language, that interpreted in M[G] is an R-
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linear order-preserving monomorphism from the finite elements of an
0
. ultrapower of the reals, over a non-principal ultrafilter on ω, into the
1
0 Esterle algebra of formal power series. Therefore it is consistent that
7 2ℵ0 = ℵ and,forany infinitecompactHausdorffspaceX, thereexists
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a discontinuous homomorphism of C(X), the algebra of continuous
:
v real-valued functions on X. For n ∈ N, If M contains a simplified
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X (ω ,n)-morass, then in the Cohen extension of M adding ℵ generic
1 n
r reals there exists a discontinuous homomorphism of C(X), for any
a
infinite compact Hausdorff space X.
1 Introduction
This paper addresses a problem in the theory of Banach Algebras concerning
the existence of discontinuous homomorphisms of C(X), the algebra of con-
tinuous real-valued functions with domain X, where X is an infinite compact
1
Hausdorff space. In [11], B. Johnson proved that there is a discontinuous ho-
momorphism of C(X) provided that there is a nontrivial submultiplicative
norm on the finite elements of an ultrapower of R over ω. In [6], J. Esterle
constructs an algebra of formal power series, E, and shows in [7] that the
infinitesimal elements of E admit a nontrivial submultiplicative norm. By
results of Esterle in [8], it is known that E is an η -ordering of cardinality
1
2ℵ0. Furthermore, E is a totally ordered field by a result of Hahn in 1907 [9],
and is real-closed by a result of Maclane [12].
It is a theorem of P. Erd¨os, L. Gillman and M. Henriksen in [5] that any
pairofη -orderedreal-closedfieldsofcardinalityℵ areisomorphicasordered
1 1
fields. In fact, it is shown using a back-and-forth argument that any order-
preserving field isomorphism between countable subsets of η -ordered real-
1
closedfieldsmaybeextendedtoanorder-isomorphism. Itisastandardresult
of model theory that for any non-principal ultrafilter U on ω, Rω/U is an ℵ -
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saturated real-closed field (and hence an η -ordering). By a result of Johnson
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[11], between any pair of η -ordered real-closed fields of cardinality ℵ there
1 1
is an R-linear order-preserving field isomorphism (hereafter R-isomorphism).
This implies, in a model of the continuum hypothesis (CH), that there is
an R-linear order-preserving monomorphism (hereafter R-monomorphism)
from the finite elements of Rω/U to E, and hence in models of ZFC+CH
there exists a discontinuous homomorphism of C(X). The proof that in a
model of ZFC+CH there exists a discontinuous homomorphism of C(X) is
due independently to Dales [1] and Esterle [7].
Shortly thereafter R. Solovay found a model of ZFC+¬CH in which all
homomorphisms of C(X) are continuous. Later, in his Ph.D. thesis, W.H.
Woodin constructed a model of ZFC+Martin’s Axiom in which all homo-
morphisms of C(X) are continuous. This naturally gave rise to the question
of whether there is a model of set theory in which CH fails and there is a
discontinuous homomorphism of C(X). Woodinsubsequently showed that in
the Cohen extension of a model of ZFC+CH by generic reals indexed by ω ,
2
2
there is a discontinuous homomorphism of C(X) [16]. Woodin shows that in
this model the gaps in E that must be witnessed in a classical back-and-forth
construction are always countable. He observes that this construction may
not be extended to a Cohen extension by more than ℵ generic reals. He
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suggests the plausibility of using morasses to construct an R-monomorphism
from the finite elements of an ultrapower of the reals to the Esterle algebra
in generic extensions with more than ℵ generic reals. Woodin’s argument
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does not extend to higher powers of the continuum and leaves open the
question of whether there exists a discontinuous homomorphism of C(X) in
models of set theory in which 2ℵ0 > ℵ . In this paper we show that the
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existence of a simplified (ω ,2)-morass in a model of ZFC+CH is sufficient
1
for the existence of a discontinuous homomorphism of C(X) in a model in
which 2ℵ0 = ℵ . Furthermore, the existence of a simplified (ω ,n)-morass
3 1
in a model of ZFC + CH implies the existence of a discontinuous homo-
morphism of C(X) in the Cohen extension of the model adding ℵ generic
n
reals.
We show that in the Cohen extension adding ℵ generic reals to a model
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of ZFC+CH containing a simplified (ω ,1)-morass, there is a level, morass-
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commutative term function that interpreted in the Cohen extension is an
R-monomorphism of the finite elements of an ultrapower of R over ω into
theEsterle Algebra. This is achieved witha transfinite construction oflength
ω , utilizing the morass functions from the gap-one morass to complete the
1
construction of size ℵ by commutativity with morass maps. In the primary
2
result of this paper, we construct a term function with a transfinite argument
of length ω and utilize morass-commutativity with the embeddings of a gap-
1
2 morassto complete the construction of anR-monomorphism fromthe finite
elements of a standard ultrapower of R over ω to the Esterle Algebra in the
Cohen extension adding ℵ generic reals. The construction along the gap-1
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morass is necessary for the construction using an (ω ,2)-morass, by virtue of
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the inductive character of higher gap morasses. We state the generalization
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to gap-n morasses without detailed proof, since the argument is essentially
identical to the gap-2 case.
The technical obstacles to such a construction may be reduced to condi-
tions we call morass-extendability. This paper is strongly dependent on the
results of [3] and [4], in which term functions are constructed that are forced
to be order-preserving functions. The arguments here are very similar, with
the additional requirement that the functions are also field homomorphisms.
We borrow freely from the definitions, terminology and conventions of that
paper. Some of these will be briefly reviewed in the next section.
2 Preliminaries
Inour initial construction we use a simplified (ω ,1)-morass. Inparticular we
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wish to construct a function on terms in the forcing language for adding ℵ
2
genericrealsthatisforcedinallgenericextensionstobeanR-monomorphism
from the finite elements of Rω/U, where U is a standard non-principal ul-
trafilter (see Definition 6.14 [3]) in the generic extension, into the Esterle
Algebra, E. In some sense we follow the classical route to such constructions,
using a back-and-forth argument of length ω . We will require commutativ-
1
ity with morass maps to construct a function on a domain of cardinality ℵ
2
making only ℵ many explicit commitments. However with each commit-
1
ment of the construction, there are uncountably many future commitments
implied by commutativity with morass maps.
In [14] D. Velleman defines a simplified (ω ,1)-morass.
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Definition 2.1 (Simplified (ω ,1)-morass) A simplified (ω,1)-morass is a
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structure
M = h(θ | α ≤ ω ),(F | α < β ≤ ω )i
α 1 αβ 1
that satisfies the following conditions:
(P0) (a) θ = 1, θ = ω , (∀α < ω ) 0 < θ < ω .
0 ω1 2 1 α 1
(b) F is a set of order-preserving functions f : θ → θ .
αβ α β
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(P1) | F |≤ ω for all α < β < ω .
αβ 1
(P2) If α < β < γ, then F = {f ◦g | f ∈ F , g ∈ F }.
αγ βγ αβ
(P3) If α < ω , then F = {id ↾ θ ,f } where f satisfies:
1 α(α+1) α α α
(∃δ < θ ) f ↾ δ = id ↾ δ and f (δ ) ≥ θ .
α α α α α α α α
(P4) If α ≤ ω is a limit ordinal, β ,β < α, f ∈ F and f ∈ F , then
1 1 2 1 β1α 2 β2α
there is γ < α, γ > β ,β , and there is f′ ∈ F , f′ ∈ F , g ∈ F such
1 2 1 β1γ 2 β2γ γα
that f = g ◦f′ and f = g ◦f′.
1 1 2 2
(P5) For all α > 0, θ = {f[θ ] | β < α, f ∈ F }.
α S β βα
Simplified gap-1 morasses (as well as higher gap simplified morasses) are
known to exist in L.
We will construct, by an inductive argument of length ω , a function
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between sets of terms in the forcing language adding ℵ generic reals. We
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interpret the morass functions on ordinals as functions between terms in in
the forcing language and require that the set of terms under construction
satisfy certain commutativity constraints with the morass functions. It is
implicit that any commitment to an ordered pair of terms in the construction
is de facto a commitment to uncountably many commitments to ordered
pairs in mutually generic extensions. In [3] we worked explicitly with terms
in the forcing language. We wish to simplify the details of the construction
by working with objects in a forcing extension. We use the notions of term
complexity and strict level, from [3], and apply it to objects in a forcing
extension. A discerning term, τ, in the forcing language adding generic reals
indexed by β has strict level α ≤ β provided that there is a term in the
forcing language adding generic reals indexed by α that is forced equal to τ
in all generic extensions, and there is no γ < α for which this is true. Not
all terms have strict levels.
Suppose α < β and P(α) (resp. P(β)) is the poset adding generic reals
indexed by α (resp. β). We consider P(β) as the product forcing P(α) ×
P∗(α). If G(α) is P(α)-generic over M and G∗(α) is P∗(α)-generic over
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M[G(α)], then we say an object, t ∈ M[G(α)][G∗(α)] has strict level α if
t ∈ M[G(α)], and t is not in the corresponding generic extension adding
generic reals indexed by an ordinal less than α. So an object has strict level
α just in case there is a term for t in the forcing language with strict level α,
and if τ is a term of strict level α, then in any generic extension the value of
τ has strict level α. Consequently in our construction we pass freely between
objects of strict level α in a generic extension and term of strict level α in
the forcing language.
Many of the constraints required for commutativity with morass maps
are expressed in terms of the strict level of objects in a forcing extension (or
correspondingly, terms in the forcing language). For instance, in [3] we define
a term function to be level if the strict level of any term in the domain equals
thestrict term ofitsimageunder thefunction. Such mapswill commute with
morass maps in the manner required by our construction.
3 Constructing an R-monomorphism on a real-
closed field
We wish to construct a set of terms in the forcing language (for adding ℵ
2
genericreals)thatisforcedinallgenericextensionstobeanR-monomorphism
on the finite elements of an ultrapower of R. It is a result of B. Johnson
[11] that η -ordered real-closed fields with cardinality ℵ are R-isomorphic
1 1
in models of ZFC+CH. This result strengthens the classical result that ℵ -
1
saturated real closed fields of cardinality ℵ are isomorphic. It is conceivable
1
that in a naive back-and-forth construction of an order-isomorphism between
η -ordered real-closed fields that a choice is made for an image (or pre-image)
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of the function under construction that precludes satisfaction of R-linearity.
Definition 3.1 (Full real-closed field) A real-closed field D is full iff for
every finite element, r +δ, where r ∈ R and δ is infinitesimal, r ∈ D.
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We will need to extend two results due to B. Johnson [11] to meet the re-
quirement of R-linearity in the context of constructing term functions using
a morass.
Lemma 3.2 (B.Johnson) Assume that D and I are full real-closed sub-
fields of η -ordered real-closed fields D∗ and I∗ (resp.), φ : D → I is an
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R-monomorphism, and r ∈ R. Then there is an extension of φ, φ∗, that
is an R-monomorphism of the real closure of the field generated by D and
r, F(D,r), onto the real closure of the field generated by I and r, F(I,r).
Furthermore F(D,r) (and consequently, F(I,r)) is full.
Lemma 3.3 (B.Johnson) Let D, D∗, I, I∗ and φ be as in Lemma 3.2,
x ∈ D∗ and assume that the real closure of the field generated by D and d,
F(D,d), is full. Let y ∈ I∗ be such that
(∀d ∈ D)(d < x ⇐⇒ φ(d) < y).
Then there is an R-monomorphism extending φ, φ∗ : F(D,r) → I∗, such
that φ∗(x) = y.
Definition 3.4 (Archimedean valuation) If x and y are non-zero elements
of a real-closed field, they have the same Archimedean valuation, x ∼ y,
provided that there are m,n ∈ N such that
| x |< n | y |
and
| y |< m | x | .
If | x |<| y | and x ≁ y, then x has Archimedean valuation greater than y,
x (cid:23) y.
Archimedean valuation is an equivalence relation on the non-zero elements of
areal-closedfield(RCF).Thenon-zerorealnumbershavethesamevaluation.
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Elements with the same valuation as a real number are said to have real
valuation. In a nonstandard real-closed field, elements with valuation greater
than a real valuation are infinitesimal. The finite elements of a real closed
field are the infinitesimal elements and those with real valuation.
4 The Esterle algebra
We define the Esterle algebra [7] and review some basic properties.
Definition 4.1 (S ) S is the lexicographic linear-ordering with domain
ω1 ω1
{s : ω → 2 | s has countable support and the support of s has a largest
1
element }.
Definition 4.2 (G ) G is the ordered group with domain {g : S →
ω1 ω1 ω1
R | g has countable well-ordered support}, lexicographic ordering, and group
operation pointwise addition.
We define an ordered algebra of formal power series, E. The universe of E is
the set of formal power series, α xaλ, where:
Pλ<γ λ
1. γ < ω .
1
2. (∀λ < γ)α ∈ R.
λ
3. {a | λ < γ} is a countable well-ordered subset of G and λ < λ <
λ ω1 1 2
γ ⇒ a < a .
λ1 λ2
The ordered algebra, E, is isomorphic to the set of functions, with count-
able well-ordered support, from G to R. The lexicographic order linearly-
ω1
orders E. Addition is pointwise and multiplication is defined as follows:
Suppose a = α xaλ and b = β xbκ are members of E. Let
Pλ<γ1 λ Pκ<λ2 κ
C = {c | (∃λ < γ )(∃κ < γ ) c = a +b }.
1 2 λ κ
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Then
a·b = X(( X αλ ·βκ)xc).
c∈C aλ+bκ=c
Definition 4.3 (Esterle algebra, E) The Esterle algebra, E, is {f : G →
ω1
R | f has countable well-ordered support}. E is lexicographically ordered, with
pointwise addition, and multiplication defined above.
R may be embedded in E by α 7−→ αxe, where e is the group identity in
G . Exponents in G larger than e (called positive exponents) correspond
ω1 ω1
to infinitesimal Archimedean valuations, and those smaller than e (called
negative exponents) correspond to infinite valuations. The finite elements of
E are those with leading exponent ≥ e.
Theorem 4.4 (J. Esterle [7]) E is an η -orderded real-closed field.
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A norm, kk, on an algebra A is submultiplicative if for any a,b ∈ A,
ka·bk ≤ kak·kbk.
Theorem 4.5 (G. Dales [1], J. Esterle [8]) The set of finite elements of E
bears a submultiplicative norm.
It is a standard result of model theory that if U is a non-principal ultrafilter
on ω, the ultrapower Rω/U is an ℵ -saturated real-closed field. Any two ℵ -
1 1
saturated, or η -ordered, real-closed fields with cardinality of the continuum
1
are isomorphic in models of ZFC+CH. Hence CH implies that there is a
discontinuous homomorphism of C(X).
Theorem 4.6 (B.Johnson [11]) (CH) If U is a non-principal ultrafilter,
there is an R-monomorphism from the finite elements of Rω/U into E.
We turn our attention to terms in a forcing language MP that are forced
to be members of the Esterle algebra.
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Notation 4.7 (P(A)) If A is a set of ordinals, we let P(A) be the poset
adding generic reals indexed by the ordinals of A. That is,
P(A) := Fn(A×ω,2),
the finite partial functions from A×ω to 2.
In [3] and [4], we found sufficient conditions for morass constructions. The
aggregate of these conditions were characterized as morass-definability and
gap-2 morass-definability. The central theorem of the papers were that
morass-definable η -orderings are order-isomorphic in the Cohen extension
1
adding ℵ generic reals of a model of ZFC + CH containing a simplified
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(ω ,1)-morass; and gap-2 morass definable η -orderings are order-isomorphic
1 1
in the Cohen extension adding ℵ generic reals of a model of ZFC + CH
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containing a simplified (ω ,2)-morass.
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We will prove that morass-definable η -ordered real-closed fields are iso-
1
morphic by anR-isomorphism intheCohen extension adding ℵ generic reals
2
of a model of ZFC +CH containing a simplified (ω ,1)-morass. We repeat
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the definitions of morass-commutativity and morass-definability.
Definition 4.8 (Morass-Commutative) Suppose h(θ | α ≤ ω ),(F | α <
α 1 αβ
β ≤ ω )i is a simplified (ω ,1)-morass, λ ≤ ω and X ⊆ MP(ω1). We say
1 1 1
that X is morass-commutative beneath λ provided that for any ζ < ξ ≤ λ and
f ∈ F , x ∈ X∩MP(θζ) iff g(x) ∈ X. We say that X is morass-commutative
ζξ
if X is morass-commutative beneath ω
1
Definition 4.9 (Morass-definable) Let hX,<i ∈ M[G] be a linear ordering.
X is morass definable if there is a set of terms T ⊆ MP
1. T is a morass-commutative set of discerning terms of strict level, and
val (T) = hX,<i.
G
2. T is level dense and upward level dense.
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