aa r X i v : . [ m a t h . L O ] J a n CORSON REFLECTIONS
ILIJAS FARAH AND MENACHEM MAGIDOR
Abstract.
A reflection principle for Corson compacta holds in theforcing extension obtained by Levy-collapsing a supercompact cardinalto ℵ . In this model, a compact Hausdorff space is Corson if and onlyif all of its continuous images of weight ℵ are Corson compact. Weuse the Gelfand–Naimark duality, and our results are stated in terms ofunital abelian C ∗ -algebras. Before starting, we should thank Alan Dow for pointing our attention to[2]. Our use of C ∗ -algebras is closely related to Bandlow’s use of large Hilbertcubes. Similar methods have been used in [3], [1], [10], [4], [8], [9], [16], [11],and it is possible that the C ∗ -algebraic vantage point may yield additionalapplications. A paper of Kunen ([16]) contains a closely related analysis ofCorson compact spaces. Some of the ideas of this note are contained in theirpapers. Since we were not aware of these results, the present paper shouldbe considered as a survey rather than a research article.A compact Hausdorff space X is a Corson compactum (or shortly, Corson)if it is homeomorphic to a subspace of some Tychonoff cube [0 , κ which hasthe property that for every ξ < κ the set { x ∈ X : x ( ξ ) = 0 } is countable.Every metrizable compactum is homeomorphic to a subspace of [0 , ω andtherefore Corson. In [19] it was proved that if there exists a non-reflectingstationary subset of cofinality ω ordinals in ω , then there exists a compactHausdorff space X all of whose continuous images of weight ℵ are Corson(and even uniform Eberlein; see § X is not Corson. Theorem 1.
Suppose κ is a supercompact cardinal. Then the following re-flection statement holds in V Coll( ℵ ,<κ ) : If X is a compact Hausdorff space,then all continuous images of X of weight at most ℵ are Corson compactif and only if X is Corson. The same principle follows from Martin’s Max-imum. Acknowledgments.
In addition to thanking Alan Dow again, we wouldlike to thank G. Plebanek for helpful remarks on an early draft of thispaper.
Date : January 28, 2020.
1. C ∗ -algebras and elementary submodels Background on C ∗ -algebras. We quickly review the required resultson C ∗ -algebras, and unital, abelian C ∗ -algebras in particular. For additionalinformation see e.g., [13], [21] or [5].A C ∗ -algebra is a complex Banach algebra with involution which is iso-morphic to a norm-closed, self-adjoint, algebra of bounded linear operatorson a complex Hilbert space. If C is a C ∗ -algebra and Z ⊆ C thenC ∗ ( Z )denotes the C ∗ -subalgebra of C generated by Z . We write C ∗ ( a, Z ) forC ∗ ( { a } ∪ Z ).1.1.1. Positivity.
An element a of a C ∗ -algebra A is self-adjoint if a = a ∗ and positive if it is self-adjoint and its spectrum is included in [0 , ∞ ). Astandard argument (see [5, II.3.1.3(ii)]) shows that a is positive if and onlyif a = b ∗ b for some b ∈ A . It is common to write a ≥ a is positive’.For a C ∗ -algebra A we writeA sa = { a ∈ A : a = a ∗ } ,A + = { a ∈ A : a ≥ } ,A + , = { a ∈ A : 0 ≤ a ≤ , k a k = 1 } . On A sa one defines partial ordering by letting a ≤ b if and only if b − a ispositive.1.1.2. States.
A continuous linear functional ϕ on a C ∗ -algebra C is positive if ϕ ( a ) ≥ a ∈ C + . A positive functional ϕ is a state if k ϕ k = 1. If C is a unital C ∗ -algebra, with unit denoted 1 C , then a functional ϕ is positiveif and only if ϕ (1) = k ϕ k . The space of all states of a unital C ∗ -algebra C is convex and weak*-compact.A proof of the following a well-known property of states is included forreader’s convenience (see [13, Proposition 1.7.8] for a more general state-ment). Lemma 1.1.
Suppose that ϕ is a state of a C ∗ -algebra A and ≤ a .If ϕ ( a ) = 0 then ϕ ( ab ) = 0 for all b .If ϕ ( a ) = k a k then ϕ ( ab ) = ϕ ( ba ) = k a k ϕ ( b ) for all b .Proof. Fix a ∈ A + , such that ϕ ( a ) = 1. Then 0 ≤ − a ≤ − a ) ≤ − a . Since ϕ is positive, it satisfies ϕ ((1 − a ) ) ≤ ϕ (1 − a ) = 0.As it satisfies the Cauchy–Schwarz inequality, | ϕ ( d ∗ c ) | ≤ ϕ ( c ∗ c ) ϕ ( d ∗ d ) forall c and d , we have ϕ ( b (1 − a )) ≤ ϕ ((1 − a ) ) ϕ ( b ∗ b ) = 0 . By simplifying the left-hand side, this implies ϕ ( b ) = ϕ ( ba ). The otherequalities follow by a similar argument and by rescaling. (cid:3) ORSON REFLECTIONS 3
Characters. A character of a unital C ∗ -algebra is a unital *-homo-morphism ϕ : C → C . If A is a unital C ∗ -algebra, then its unit has anopen neighbourhood consisting of invertible elements ([13, Lemma 1.2.6]).Therefore, every proper maximal two-sided ideal of A is closed. In par-ticular, the kernel of any character of A is norm-closed. All characters ofa unital C ∗ -algebras are therefore automatically continuous. Since every*-homomorphism is positive and of norm at most 1, every character is astate.1.1.4. The Gelfand–Naimark duality.
The category of compact Hausdorffspaces with respect to continuous maps as morphisms is equivalent to thecategory of unital, abelian C ∗ -algebras with respect to ∗ -homomorphisms(i.e., homomorphisms which respect the adjoint operation) as morphisms.Given a compact Hausdorff space X , the C ∗ -algebra associated with it is the ∗ -algebra of all complex continuous functions on X . To a continuous map f : X → Y one associates the ∗ -homomorphism C ( Y ) ∋ a a ◦ f ∈ C ( X ) . The inverse functor is defined as follows. If A is a unital abelian C ∗ -algebra,then the space X of characters of A is compact in the weak*-topology andit separates points of A . The Gelfand transform identifies A with C ( X ).By the Riesz Representation Theorem, the states on C ( X ) are in a bijectivecorrespondence with the regular Radon probability measures on X , via thecorrespondence ϕ µ ϕ where ϕ ( a ) = Z X a ( x ) dµ ϕ ( x ) . Every unital ∗ -homomorphism Φ : C ( X ) → C ( Y ) is of the form a a ◦ f for a continuous function f : Y → X .The space of states of C is denoted S ( C ). The pure states are the extremepoints of S ( C ). A state ϕ on C ( X ) is pure if and only if the associated proba-bility measure µ ϕ is a point-mass measure (i.e., a measure that concentrateson a single point). In this case ϕ agrees with the evaluation functional a a ( x ) for x ∈ X . The space of pure states of C , also known as the spectrum of C , is denoted P ( C ).When C is abelian, then a state is pure if and only if it is a character.We will also need the following well-known lemma (the weight of a topo-logical space is the smallest cardinality of its basis, and the density characterof a metric space is the minimal cardinality of a dense subspace). Lemma 1.2.
The weight of an infinite compact Hausdorff space X is equalto the density character of C ( X ) .Proof. The weight of an infinite compact Hausdorff space X is equal to theminimal cardinal κ such that X is homeomorphic to a subspace of [0 , κ .The density character of C ( X ) is equal to the cardinality of a minimalgenerating set for C ( X ), which is by the Stone–Weierstrass theorem equal ILIJAS FARAH AND MENACHEM MAGIDOR to the minimal cardinality of a subset of C ( X ) that separates points of X .But this is the minimal κ such that X is homeomorphic to a subspaceof [0 , κ . (cid:3) Continuous Functional Calculus.
The spectrum of an element a of aunital C ∗ -algebra C , is sp( a ) = { λ ∈ C : λ − a is not invertible } . Two nontrivial facts deserve mention. The spectrum of any operator is anonempty compact subset of the field of complex numbers. Second, if A ⊆ B are unital C ∗ -algebras with the same unit and a ∈ A , then sp( a ) as computedin A is equal to sp( a ) as computed in B . A normal element a is positive ifand only if its spectrum is included in [0 , ∞ ).If a ∈ C ( X ) then clearly sp( a ) is equal to the range of a . An element a of a C ∗ -algebra is normal if aa ∗ = a ∗ a . The continuous functional calculus asserts that C (sp( a )) ∼ = C ∗ ( a, f f ( a ).1.2. Corson compacta.
We will need the following standard properties ofCorson compacta.(1) Every closed subspace of a Corson compactum is a Corson com-pactum.(2) Every Corson compactum X has the following property. If Z ⊆ X and x is an accumulation point of Z , then there exists a se-quence z n ∈ Z , for n ∈ N , such that lim n z n = x . A space with thisproperty is said to be Fr´echet or Fr´echet–Urysohn .(3) Every continuous image of a Corson compactum is a Corson com-pactum.The third property is [20, Theorem 6.2], and the first two are straightfor-ward.1.3.
Corson compacta and C ∗ -algebras. The following, almost tauto-logical, lemma provides reformulation of our results in terms of C ∗ -algebras. Lemma 1.3. If X is a compact Hausdorff space then X is Corson compactif and only if there are a cardinal κ and a family a α , α < κ , in C = C ( X ) such that the following conditions hold (see § ≤ ). (1) For every α we have ≤ a α ≤ . (2) For every pure state ϕ on C the set { α < κ : ϕ ( a α ) = 0 } is countable. (3) The C ∗ -algebra C is generated by { a α : α < κ } and .Proof. Suppose X is a Corson compactum. Therefore we may identify X with a subspace of a Hilbert cube [0 , κ such that { α < κ : x ( α ) = 0 } iscountable for all x ∈ X . Then for α < κ the projection a α to the α th coor-dinate is a continuous function from X into [0 , Since C is unital, we identify the scalar multiples of its unit 1 C with the complexnumbers. ORSON REFLECTIONS 5 the points of X , and therefore the complex Stone–Weierstrass theorem im-plies C ( X ) ∼ = C ∗ ( { } ∪ { a λ : λ < κ } ). The condition (2) is clearly equivalentto the assertion that for every λ < κ the set { x ∈ X : x ( λ ) = 0 } is countable.Since the pure states of C ( X ) are exactly the evaluation functions at thepoints of X (see § ϕ vanishes at allbut countably many of a λ .Conversely, if some family a α , for α < κ , in C ( X ) satisfies (1)–(3), thenthe function from X = P ( C ( X )) defined by ϕ
7→ h ϕ ( a α ) : α < κ i is a homeomorphism onto a Corson compact subspace of [0 , κ . (cid:3) An proof analogous to that of Lemma 1.3 gives the following.
Lemma 1.4. If X is a compact Hausdorff space then X is Eberlein compactif and only if there are a cardinal κ and a family a α , α < κ , in C = C ( X ) such that the following hold (see § ≤ ). (1) For every α we have ≤ a α ≤ . (2) The set { α < κ : ϕ ( a α ) > ε } is finite for every pure state ϕ on C and every ε > . (3) The C ∗ -algebra C is generated by { a α : α < κ } and . (cid:3) This is a good moment to introduce the following lemma, needed in theproof of Theorem 1.
Lemma 1.5. If X is a Corson compactum of weight at most λ then thecardinality of X is not greater than λ ℵ . In particular, if the ContinuumHypothesis holds and λ ≤ ℵ , then | X | ≤ ℵ .Proof. Let κ be a cardinal such that X is homeomorphic to a subspace of[0 , κ such that supp( x ) = { α : x ( α ) = 0 } is countable for all x ∈ X . We claim that there exists S ⊆ κ with | S | ≤ λ such that the evaluation functions at the points of S separate the pointsof X . To prove this, fix a pair of basic open sets U and V of X suchthat V ⊆ U . By compactness, there is a finite list of Tychonoff basic opensubsets of [0 , κ , W i = W i ( U, V ), for i < m ( U, V ), such that V ⊆ S i W i and S i W i ∩ X ⊆ U . There are λ such pairs U, V and each W i depends on afinite set of coordinates in κ . Let S be the set of all these coordinates. Thenits cardinality is not greater than λ and it separates the points of X . Theprojection to [0 , S is a continuous injection on X . By compactness (andHausdorffness) of X , it is a homeomorphism. Therefore, X is homeomorphicto a subset of [0 , λ as in the definition of Corson compacta. For a countable A ⊆ λ let X A = { x ∈ X : supp( x ) ⊆ A } (with supp( x ) re-evaluated as asubset of λ ). Then | X A | ≤ | [0 , | | A | = 2 ℵ . Since X is Corson, we have X = S A X A and therefore | X | ≤ λ ℵ · ℵ = λ ℵ , as required. The secondclaim follows immediately. (cid:3) ILIJAS FARAH AND MENACHEM MAGIDOR Elementary submodels
Throughout this section C is a unital C ∗ -algebra. It is assumed to beabelian and equal to C ( X ) for some compact Hausdorff space X unless oth-erwise specified. By § X with the space P ( C ) of allpure states of C . A large enough regular cardinal θ is fixed. The set H θ ofall sets of hereditary cardinality strictly less than θ is a model of a fragmentof ZFC sufficiently large for many practical applications (see e.g., [17]). Al-though elementary submodels are have been an important tool in generaltopology for years (see [7]), this is to the best of our knowledge the firsttime that they are applied to study of compact Hausdorff spaces via theGelfand–Naimark duality. Definition 2.1.
Suppose that C is a C ∗ -algebra (unital and not necessarilyabelian) and M is an elementary submodel (not necessarily countable) of alarge enough H θ such that C ∈ M . We write C M = C ∗ ( M ∩ C ) , P ( C ) M = P ( C ) ∩ M. For a ∈ C and Y ⊆ S ( C ) write k a k Y = sup ϕ ∈ Y | ϕ ( a ) | and let k a k M = k a k M ∩ P ( C ) .By a minor abuse of notation, we denote the annihilator of P ( C ) ∩ M in C by M ⊥ ∩ C , so that M ⊥ ∩ C = { a ∈ C : k a k M = 0 } . The following is a consequence of the definitions.
Lemma 2.2. If M and C = C ( X ) are as in Definition 2.1 then we have M ⊥ ∩ C = { a ∈ C : a ( x ) = 0 for all x ∈ X ∩ M } . (cid:3) Lemma 2.3. If M , C = C ( X ) , and Y ⊆ S ( C ( X )) , are as in Definition 2.1then k · k Y (and k · k M in particular) is a seminorm majorized by k · k on C .Proof. For every state ϕ of C we have k ϕ k = 1 and therefore | ϕ ( · ) | is aseminorm majorized by k · k . Therefore k · k Y is the supremum of a familyof seminorms majorized by k · k . (cid:3) An order ideal in a C ∗ -algebra A is a subset A of A + that is a cone (i.e., closed under the multiplication by positive scalars and addition) and hereditary (i.e., if a ∈ A and 0 ≤ b ≤ a , then b ∈ A ). Lemma 2.4.
Suppose that A is a (not necessarily commutative) C ∗ -algebraand M is an elementary submodel of a large enough H θ such that A ∈ M .Then the annihilator of P ( A ) M , ( P ( A ) ∩ M ) ⊥ = { a ∈ A : k a k M = 0 } ORSON REFLECTIONS 7 is a norm-closed subspace of A and its positive cone, (( P ( A ) ∩ M ) ⊥ ) + , is anorm-closed order ideal in A .If A is in addition abelian, then ( P ( A ) ∩ M ) ⊥ is an ideal of A , and thequotient C/ (( P ( A ) ∩ M ) ⊥ ) is isomorphic to C ( M ∩ X ) , where M ∩ X isconsidered with the subspace topology.Proof. For the first part, the annihilator of any subset of the dual space ofa Banach space Z is a norm-closed subspace of Z .To prove the second part, note that the zero set { a ∈ A + : ϕ ( a ) = 0 } ofany state ϕ is a norm-closed cone. Hence ( P ( A ) ∩ M ) ⊥ is an intersection ofa family of cones, and therefore a norm-closed cone itself.A C ∗ -subalgebra of a C ∗ -algebra is a left ideal if and only if its positivepart is an order ideal (this is a result of Effros, see [23, Theorem 1.5.2]).Now suppose A is abelian. Then every left ideal of A is an ideal of A . It remains to prove C/ (( P ( A ) ∩ M ) ⊥ ) ∼ = C ( M ∩ X ). Consider the *-homomorphism π M : C → C ( M ∩ X ) defined by π M ( a ) = a ↾ M ∩ X. This is a surjection of C onto C ( M ∩ X ), and its kernel is equal to { a ∈ C : a ↾ M ∩ X = 0 } = ( P ( A ) ∩ M ) ⊥ . (cid:3) By Lemma 2.4, if C is a unital abelian C ∗ -algebra and M ≺ H θ has C asan element, then we have an exact sequence0 M ⊥ ∩ C C C ( M ∩ X ) 0 π M What is the relation of C M to the algebras in this exact sequence? With ι M : C M → C denoting the inclusion map, we have the following commuta-tive diagram. C ( M ∩ X ) M ⊥ ∩ C C C M π M ι M π M ◦ ι M A model M that satisfies any of the equivalent conditions in Lemma 2.5 issaid to split X . In § Lemma 2.5.
Suppose X ∈ H θ is a compact Hausdorff space, M ≺ H θ , and X ∈ M . Then the following are equivalent (1) C ∗ ( C M , M ⊥ ∩ C ) = C . (2) If x and y are in M ∩ X and satisfy a ( x ) = a ( y ) for all a ∈ M ∩ C ,then x = y . (3) π M ◦ ι M is a surjection of C M onto C ( M ∩ X ) , and the exact se-quence → M ⊥ ∩ C → C ( X ) → C ( M ∩ X ) → splits. ILIJAS FARAH AND MENACHEM MAGIDOR
Proof. (3) → (1): Suppose π M ◦ ι M is a surjection. Let us first show that thisimplies that the exact sequence in (3) splits. Since C M separates points of M ∩ X , π M ◦ ι M is an injection and therefore an isomorphism between C M and C ( M ∩ X ). With θ : C ( M ∩ X ) → C M denoting the inverse of π M ◦ ι M ,we have a split exact sequence0 M ⊥ ∩ C C C ( M ∩ X ) 0 π M ι M ◦ θ In order to prove (1), fix a ∈ C and let a = ( ι M ◦ θ )( a ). Then a ∈ C M , a = a − a belongs to M ⊥ ∩ C , and therefore a = a + a belongs to C ( C M , M ⊥ ∩ C )(1) → (2): Suppose that (2) fails and fix distinct x and y in M ∩ X suchthat a ( x ) = a ( y ) for all a ∈ M ∩ C . Since x = y , there exists b ∈ C suchthat b ( x ) = b ( y ). But every c ∈ ( M ∩ X ) ⊥ satisfies c ( x ) = 0 = c ( y ), andtherefore b / ∈ C ∗ ( M ∩ C, ( M ∩ X ) ⊥ ), showing that (1) fails.(2) → (3) The assumption asserts that the elements of C M separate pointsof M ∩ X . Therefore ( π M ◦ ι M )( C M ) is a norm-closed, self-adjoint, subalge-bra of C ( M ∩ X ) that separates points and contains all constant functions.By the complex Stone–Weierstrass theorem (e.g., [22, Theorem 4.3.4]), it isequal to C ( M ∩ X ). (cid:3) The following proposition ought to be well-known.
Proposition 2.6.
Suppose that X is a compact Hausdorff space such thatevery continuous image of X of weight at most ℵ is Fr´echet. Then X isFr´echet.Proof. Fix Z ⊆ X and x ∈ Z . In order to find a sequence in Z that convergesto x , fix a large enough regular cardinal θ and M ≺ H θ that contains X, Z ,and x and such that M ω ⊆ M and | M | = 2 ℵ . Then the pure state spaceof C M is, being a continuous image of X of weight at most 2 ℵ , Fr´echet. Wecan identify all z ∈ Z ∩ M and x with pure states of C M . Claim 2.7.
In the weak*-topology induced by C M , x is an accumulationpoint of Z ∩ M .Proof. In the weak*-topology induced by C , x is an accumulation pointof Z . This means that for all n ≥ a j ∈ C , for j < n , the n -tuple( a j ( x ) : j < n ) is an accumulation point of { ( a j ( z ) : j < n ) : z ∈ Z } . Since M ≺ H θ , the following holds for all n :For all a j ∈ C ( X ) ∩ M , for j < n , the n -tuple ( a j ( x ) : j < n ) is anaccumulation point of { ( a j ( z ) : j < n ) : z ∈ Z ∩ M } .Since C ( X ) ∩ M is dense in C M , x is an accumulation point of Z ∩ M inthe weak*-topology induced by C M . (cid:3) ORSON REFLECTIONS 9
Let z n , for n ∈ N , be a sequence in Z ∩ M that converges to x in theweak*-topology of C M . This sequence belongs to M , since M ω ⊆ M . Byelementarity, lim n a ( z n ) = a ( x ) for all a ∈ C . This implies lim n z n = x inthe topology of X .Since Z and x were arbitrary, this proves that X is Fr´echet. (cid:3) The following result appears as [12, Exercise 2.4G] and we include a prooffor reader’s convenience.
Lemma 2.8.
A continuous image of a compact, Hausdorff, and Fr´echetspace is Fr´echet.Proof.
Suppose X is Fr´echet and f : X → Y is a surjection. Fix Z ⊆ Y andan accumulation point x of Z . Let Z ′ = f − ( { x } ) and let T = f − ( Z ). Weclaim that Z ′ ∩ T = ∅ . Assume otherwise, and for every z ∈ T fix an openneighbourhood u z disjoint from Z ′ . Then U = S z ∈ T u z is an open coverof T disjoint from Z ′ , and f [ X \ U ] is a compact subset of Y containing Z that x does not belong to; contradiction.Therefore Z ′ ∩ T = ∅ . Since X is Fr´echet, there exists a sequence ( z ′ n )in Z ′ such that x ′ = lim n z ′ n belongs to T . Then f ( z ′ n ) ∈ Z for all n andlim n f ( z ′ n ) = x .Since Z and x were arbitrary, this proves that Y is Fr´echet. (cid:3) Proof of Theorem 1
Some N ≺ H θ of cardinality ℵ is said to be internally approachable ifit is equal to the union of an increasing ω -chain of countable elementarysubmodels each of whose proper initial segments belongs to the next modelin the sequence. Consider the following reflection principle.(R) If θ is an uncountable regular cardinal and S ⊆ P ℵ ( H θ ) is sta-tionary, then there exists an internally approachable N ≺ H θ ofcardinality ℵ such that S ∩ P ℵ ( N ) is stationary in P ℵ ( N ).This principle was introduced and proved to follow from MM in [15, The-orem 13]. Requiring N to be an elementary submodel of H θ is not a lossof generality, since if every stationary subset of P ℵ ( H θ ) reflects to P ℵ ( Z )for some Z ∈ P ℵ ( H θ ), then every stationary subset of P ℵ ( H θ ) reflects toa stationary set of Z ∈ P ℵ ( H θ ).Proposition 3.1 is standard, but we could not find it in the literature. Aproof is included for reader’s convenience. Proposition 3.1. If κ is a supercompact cardinal then Coll( ℵ , < κ ) forcesthe following strengthening of (R).If θ is an uncountable regular cardinal and S ⊆ P ℵ ( H θ ) is sta-tionary, then there exists N ≺ H θ of cardinality ℵ closed under ω -sequences such that S ∩ P ℵ ( N ) is stationary in P ℵ ( N ) .Proof. Every model of cardinality ℵ closed under ω -sequences is clearlyapproachable. We will prove a strengthening in which the model N is required to be closed under ω -sequences. Let V [ G ] be a forcing exten-sion by Coll( ℵ , < κ ). Suppose θ is an uncountable regular cardinal and S ⊆ P ℵ ( H θ ) is stationary. Let j : V → N be an elementary embeddingwith critical point κ such that j ( κ ) > θ and N is closed under 2 <θ -sequences.(We will be using the fact that | H θ | = 2 <θ .)By the standard methods ([6, Proposition 9.1]), j can be extended to anelementary embedding (also denoted j ) j : V [ G ] → N [ G ]for an N -generic filter G ⊆ Coll( ℵ , < j( κ )) such that G ∩ Coll( ℵ , < κ ) = G. Let Z = ( H θ ) V [ G ] . Since θ ≥ κ is regular and Coll( ℵ , < κ ) has κ -cc, H Vθ is a Coll( ℵ , < κ )-name for Z . As N is closed under 2 <θ -sequences andColl( ℵ , < κ ) has κ -cc, Y = j “ Z belongs to N [ G ]. It is an elementarysubmodel of j ( Z ) = H j ( θ ) . Since Coll( ℵ , < j( κ )) is ℵ -closed, P ℵ ( Y ) ⊆ Y .In V [ G ] it holds that S is a stationary subset of P ℵ ( Z ), and the quotientforcing Coll( ℵ , [ κ, j ( κ ))) is ℵ -closed. Therefore S remains a stationarysubset of P ℵ ( Z ) in N [ G ]. But this is equivalent to j “ S being stationaryin P ℵ ( Y ).In N [ G ] we therefore have | Y | = ℵ , Y ω ⊆ Y , and j ( S ) reflects to Y . Byelementarity, in V there exists X ≺ H θ of cardinality ℵ such that S ∩P ℵ ( Y )is stationary in P ℵ ( X ) and X ω ⊆ X .Since θ and S were arbitrary, we have proved that (R) holds in V [ G ]. (cid:3) Lemma 3.2.
Suppose that (R) holds and X is a compact Hausdorff spacesuch that every continuous image of X of weight not greater than ℵ isCorson. If θ is a regular cardinal such that X ∈ H θ then the set D = { M ∈ P ℵ ( H θ ) : X ∈ M, M does not split X } is nonstationary.Proof. Assume otherwise. By (R) fix an internally approachable N ≺ H θ ofcardinality ℵ such that D ∩ P ℵ ( N ) is stationary in P ℵ ( N ). Then N ∩ X is a closed subspace of X of weight not greater than ℵ , and therefore aCorson compactum.By Lemma 1.3 we can fix a α , for α < ℵ , that together with 1 generate C N so that the set Z ( x ) = { α < ℵ : a α ( x ) = 0 } is countable for all x ∈ N ∩ X .At this point we cannot assert that this sequence of generators belongsto N (we cannot even assert that any of the generators belongs to N ). Since N is internally approachable, we can choose a continuous ℵ -sequence M α ,for α < ℵ , of countable elementary submodels of the expanded structure( N, ( a α : α < ℵ )) such that N = S α M α . We say that an ordinal α < ℵ is good if the following two conditions hold. ORSON REFLECTIONS 11 (1) If β < α then a β ∈ C M α (2) If x ∈ M α ∩ X , then Z ( x ) ⊆ α .A standard closing off argument shows that the set of good α includes a clubin ℵ . Since D ∩ N is stationary, there exists a good α such that M α ∈ D . As M α does not split X , some distinct x and y in M α ∩ X satisfy a ( x ) = a ( y )for all a ∈ M α ∩ C . Since M α ∈ N , by elementarity we can choose x and y to be elements of N ∩ X . Since a ( x ) = a ( y ) for all a ∈ C M α , (1) impliesthat a β ( x ) = a β ( y ) for all β ∈ M α ∩ ℵ .Since N ∩ X is Corson, it is Fr´echet, and there are x n and y n , for n < ω ,in M α ∩ X such that, in the weak*-topology of C N , we have lim n x n = x andlim n y n = y . By (2), Z ( x n ) ⊆ M α and Z ( y n ) ⊆ M α for all n . Therefore for γ ∈ ℵ \ M α we have a γ ( x n ) = a γ ( y n ) = 0 for all n , hence a γ ( x ) = a γ ( y ) = 0.We have proved that a γ ( x ) = a γ ( y ) for all γ < ℵ . However, x and y belong to N , C N separates points of N ∩ X , and C N is generated by a γ , for γ < ℵ ; contradiction. (cid:3) Lemma 3.3.
Suppose that θ is a regular cardinal, X ∈ H θ is a compact,Fr´echet, Hausdorff space, and the set K = { M ∈ P ℵ ( H θ ) : X ∈ M and M splits X } includes a club. If f : H <ωθ → H θ is such that every M ∈ P ℵ ( H θ ) closedunder f is in K , then every N ≺ H θ closed under f and such that X ∈ N splits X .Proof. Fix N ≺ H θ such that X ∈ N and N is closed under f . Supposethat x and y are distinct points of N ∩ X . Since X is Fr´echet, there aresequences ( x n ) and ( y n ) in N ∩ X converging to x and y , respectively. Fixa countable M ≺ N closed under f and such that X and all x n and all y n belong to M . Then x and y belong to M ∩ X . Since M splits X , there is a ∈ C M such that a ( x ) = a ( y ). Since M ⊆ N , we have a ∈ N .Because x and y were arbitrary distinct points of N ∩ X , we concludethat N splits X . (cid:3) Lemma 3.4.
Suppose X ∈ H θ is a compact, Fr´echet, Hausdorff spaceand M α , for α < λ , is a continuous chain of elementary submodels withthe following properties for all α < λ : (1) X ∈ M . (2) M α splits X . (3) M α ∩ X is a Corson compact subspace of X . (4) S α<λ M α ⊇ X .Then X is Corson compact.Proof. We write C = C ( X ), and we also write C α in place of C M α . Let ussay that a sequence ( a α ) α<κ in a unital, abelian, C ∗ -algebra D that satisfies(1)–(3) of Lemma 1.3 is a sequence of Corson generators for D . We willchoose a sequence of Corson generators for C in blocks, by recursion on λ . First choose a sequence A of Corson generators for C . For α < λ , let D α +1 = C ∗ (1 , M ⊥ α ∩ C α +1 ) . This is a unital C ∗ -subalgebra of C α +1 . Since the pure state space of C α +1 is Corson and every continuous image of a Corson compact space is Cor-son, the pure state space of D α +1 is Corson. We can therefore choosea sequence A α +1 of Corson generators for D α +1 . For a limit ordinal α let A α = ∅ .We claim that A = S α A α is a sequence of Corson generators for C ( X ).First we prove that C ∗ ( A ∪ { } ) = C . Towards this end, we use inductionon α ≤ κ to prove that C ∗ ( S β ≤ α A β ∪ { } ) = C α .This is true for α = 0. Suppose that the assertion is true for all β < α .Consider the case when α is a successor ordinal, say α = β + 1. Since M β splits X , we have C β +1 = C ∗ ( C β , M ⊥ β ) = C ∗ ( A β +1 ∪ S γ ≤ β A γ )as required. If α is a limit ordinal, then S β<α C β is dense in C α , and A α = S β<α C β generates C α .Therefore C ∗ ( A ) = C , and it remains to prove that the set Z ( x ) = { a ∈ A : a ( x ) = 0 } is countable for every x ∈ X . Assume otherwise and fix x such that Z ( x )is uncountable. Since A α is a sequence of Corson generators for every α , Z ( x ) ∩ A α is countable for all α . Therefore the set { α : Z ( x ) ∩ A α = ∅} isuncountable. Let β be the least limit point of this set of cofinality ℵ . Let ϕ be the pure state of C β corresponding to x , i.e., ϕ ( a ) = a ( x ) for a ∈ C β .Lemma 2.8 implies that P ( C β ) is Fr´echet. Since cof( β ) is uncountable, thereexists γ < β such that ϕ belongs to M γ ∩ X . By construction, this impliesthat A δ is annihilated by x for all α ≤ δ < β ; contradiction. (cid:3) A discussion on why Lemma 3.4 does not apply to the space constructedin [19] is given in Example 4.3.
Proof of Theorem 1.
By Proposition 3.1, (R) holds in V Coll( ℵ ,<κ ) . Thismodel also satisfies the Continuum Hypothesis. By induction on λ we willprove that if X is a compact Hausdorff space of weight λ such that allcontinuous images of X of weight at most ℵ are Corson, then X is Corson.Suppose that the inductive hypothesis is true for all compact Hausdorffspaces of weight less than λ . Then λ ≥ ℵ . Fix a compact Hausdorff space X such that all continuous images of X of weight less than λ are Corson. Sinceevery Corson space is Fr´echet and since 2 ℵ = ℵ < λ , Proposition 2.6implies that X is Fr´echet.Fix a regular θ such that X and C ( X ) belong to H θ . By Lemma 3.2, theset K = { M ∈ P ℵ ( H θ ) : X ∈ M, M splits X } ORSON REFLECTIONS 13 includes a club. By Kueker’s theorem (see e.g., [14, Theorem 3.4]), thereexists f : H <ωθ → H θ such that every M ∈ P ℵ ( H θ ) closed under f belongsto K . By Lemma 3.3, if M ≺ H θ , X ∈ M , and M is closed under f , then M splits X . Hence if in addition | M | < λ , then M ∩ X is a Corson compactsubspace of X .Let M α , for α < λ , be an increasing chain of elementary submodels of H θ closed under f such that X ∈ M and | M α | < λ for all α . By the previousparagraph, these models satisfy the assumptions of Lemma 3.4. Since inaddition X is Fr´echet, Lemma 3.4 implies that X is Corson. (cid:3) Some remarks on splitting models
In the concluding section we collect a few observations on the notion ofsplitting models not required in the proofs of our main results. In all of thefollowing examples, θ is a cardinal large enough regular to have X ∈ H θ . Example 4.1.
There exists a compact Hausdorff space X such that nocountable M ≺ H θ splits X . Take for example X = β N \ N , the ˇCech–Stoneremainder of N . We only need to know that if Z is a countable discretesubset of X , then the closure of Z is homeomorphic to β N and therefore ofcardinality 2 ℵ .Suppose M ≺ H θ is countable (and certainly X = β N \ N ∈ M if θ is largeenough). A counting argument shows that (3) of Lemma 2.5 fails. First, | M ∩ X | = 2 ℵ . Second, C M is separable and therefore |P ( C M ) | ≤ ℵ . Example 4.2.
There is a compact, Hausdorff, Fr´echet (even first count-able), and separable space X such that no countable M ≺ H θ splits X . Let X be [0 , × { , } with the lexicographical ordering and the order topology.It is easy to check that this space is compact, Hausdorff, first countable, andseparable. Suppose M ≺ H θ is countable. Then M ∩ X = X since a count-able dense set of X belongs to (and is therefore a subset of) M . Choose x ∈ [0 , \ M . Then ( x,
0) and ( x,
1) are not separated by the elementsof C M , and therefore M fails (2) of Lemma 2.5.Our last example is more specific and most relevant to Theorem 1. Example 4.3.
There exists a compact Hausdorff space X and an increasingsequence of elementary submodels M n ≺ H θ such that each M n splits X but S n M n does not split X . This space is based on [19], and we include therelevant details for reader’s convenience.Suppose λ is an ordinal, S ⊆ λ and every α ∈ S is a limit ordinal ofcofinality ω . For each α ∈ S fix a strictly increasing sequence p n ( α ), for n ∈ N , of ordinals such that sup n p n ( α ) = α . Let A α = { p n ( α ) : n ∈ N } andlet A ( S ) be the Boolean algebra of subsets of Z = S α ∈ S A α generated by allfinite subsets of Z and { A α : α ∈ S } .Let X ( S ) be the Stone space of A ( S ), i.e., the space of ultrafilters of A . Itis a compact Hausdorff space. Each ordinal ξ < λ corresponds to a principalultrafilter which is an isolated point x ( ξ ) of X . Each α ∈ S corresponds to a unique nonprincipal ultrafilter y ( α ) that concentrates on A α . It satisfies y ( α ) = lim n x ( p n ( α )). Finally, a unique ultrafilter z in X ( S ) does notconcentrate on any countable set.If S is stationary then X is not Corson (this was proved for λ = ω in [19,Lemma 3.3], but the proof of the general case is identical). On the otherhand, if | λ | ≤ ℵ and S ∩ γ is nonstationary for all γ < λ , then A ( S ) isuniform Eberlein compact ([19, Lemma 3.5]).Therefore the existence of a nonreflecting stationary set S ⊆ S impliesthat there exists a space X = X ( S ) all of whose continuous images of weightnot greater than ℵ are uniform Eberlein compacta, but X is not Corson.Since X ( S ) is the Stone space of the Boolean algebra A ( S ), the relevantC ∗ -algebra C ( X ) has a dense subset consisting of continuous functions withfinite range (i.e., ‘step functions’) and C ( X ) therefore does not provide anymore information than A ( S ). We will therefore work with A ( S ) in thefollowing.Now suppose M ≺ H θ is such that λ and S belong to M . Then M ∩ X ( S ) = { x ( ξ ) : ξ ∈ M } ∪ { y ( α ) : α ∈ M } ∪ { z } and (note that α ∈ M implies A α ⊆ M ) M ∩ X ( S ) = { x ( ξ ) : ξ ∈ M } ∪ { y ( α ) : A α ∩ M is infinite } ∪ { z } . Therefore if M ∩ λ is an ordinal and it belongs to S , then y ( M ∩ λ ) belongsto M ∩ X ( S ) but not to M ∩ X ( S ). In particular, every M ≺ H θ such that M ≺ λ is an ordinal of uncountable cofinality splits X ( S ).Now suppose that S ⊆ S is stationary. We can then choose an increasingsequence M n ≺ H θ for n ∈ N such that M n ∩ ω is an ordinal of cofinality ω ,but M = S n M n satisfies M ∩ ω ∈ S . Then each M n splits X ( S ), but M does not. 5. Concluding Remarks
A compact Hausdorff space X is an Eberlein compactum (or shortly, Eber-lein) if it is homeomorphic to a subspace of some Tychonoff cube [0 , κ which has the property that for every ξ < κ and every ε > { x ∈ X : x ( ξ ) > ε } is finite. Every Eberlein compactum is clearly Corson.We do not know whether analog of Theorem 1 holds for Eberlein compacta.It is not difficult to prove that it holds for strong Eberlein compacta, andthis can be easily extracted from [16, Corollary 3.3].Our original proof of (a special case of) Theorem 1 used a very strongreflection principle obtained by adapting the proof of [18, Theorem 1, (a) ⇒ (c)]. Here S denotes the set of all ordinals below ω of cofinality ω . ORSON REFLECTIONS 15
References
1. I. Bandlow,
A note on applications of the L¨owenheim-Skolem theorem in general topol-ogy , Z. Math. Logik Grundlag. Math. (1989), no. 3, 283–288. MR 10009712. , A characterization of Corson-compact spaces , Comment. Math. Univ. Carolin (1991), no. 3, 545–550.3. , A construction in set-theoretic topology by means of elementary substructures ,Z. Math. Logik Grundlag. Math. (1991), no. 5, 467–480. MR 12701894. , On function spaces of Corson-compact spaces , Comment. Math. Univ. Carolin (1994), no. 2, 347–356.5. B. Blackadar, Operator algebras , Encyclopaedia of Mathematical Sciences, vol. 122,Springer-Verlag, Berlin, 2006, Theory of C ∗ -algebras and von Neumann algebras,Operator Algebras and Non-commutative Geometry, III.6. J. Cummings, Iterated forcing and elementary embeddings , Handbook of set theory,Vol. 2 (M. Foreman and A. Kanamori, eds.), Springer, Dordrecht, 2010, pp. 775–883.7. A. Dow,
An introduction to applications of elementary submodels to topology , TopologyProc. (1988), no. 1, 17–72.8. , Set theory in topology , Recent progress in general topology (Prague, 1991);North-Holland, Amsterdam (1992), 167–197.9. ,
More set-theory for topologists , Top. Appl. (1995), no. 3, 243–300.10. T. Eisworth, Elementary submodels and separable monotonically normal compacta ,arXiv preprint math/0608376 (2006).11. ,
Elementary submodels and separable monotonically normal compacta , Pro-ceedings of the 20th Summer Conference on Topology and its Applications, vol. 30,2006, pp. 431–443. MR 235274212. R. Engelking,
General topology , Heldermann, 1989.13. I. Farah,
Combinatorial set theory and C ∗ -algebras , Springer Monographs in Mathe-matics, Springer, 2019.14. M. Foreman, Ideals and generic elementary embeddings , Handbook of set theory, Vol.2 (M. Foreman and A. Kanamori, eds.), Springer, Dordrecht, 2010, pp. 885–1147.15. M. Foreman, M. Magidor, and S. Shelah,
Martin’s maximum, saturated ideals andnonregular ultrafilters, I , Ann. of Math. (2) (1988), 1–47.16. K. Kunen,
Compact spaces, compact cardinals, and elementary submodels , Topologyand its Applications (2003), no. 2, 99–109.17. ,
Set theory , Studies in Logic (London), vol. 34, College Publications, London,2011.18. M. Magidor,
Reflecting stationary sets , J. Symolic Logic (1982), no. 4, 755–771.19. M. Magidor and G. Plebanek, On properties of compacta that do not reflect in smallcontinuous images , Topo. Appli. (2017), 131–139.20. E. Michael and M.E. Rudin,
A note on Eberlein compacts , Pacific J. Math. (1977),no. 2, 487–495.21. G.J. Murphy, C ∗ -algebras and operator theory , Academic Press Inc., Boston, MA,1990.22. G.K. Pedersen, Analysis now , Graduate Texts in Mathematics, vol. 118, Springer-Verlag, New York, 1989.23. , C ∗ -algebras and their automorphism groups , second ed., Pure and AppliedMathematics, Academic Press, 2018. Department of Mathematics and Statistics, York University, 4700 KeeleStreet, North York, Ontario, Canada, M3J 1P3
E-mail address : [email protected] URL : The Hebrew University of Jerusalem, Einstein Institute of Mathematics,Edmond J. Safra Campus, Givat Ram, Jerusalem 91904, Israel
E-mail address ::