aa r X i v : . [ m a t h . L O ] J un CAN YOU TAKE AKEMANN–WEAVER’S ♦ AWAY?
DANIEL CALDERÓN AND ILIJAS FARAH
A Carlos Di Prisco en su cumpleaños número 70.
Abstract.
A counterexample to Naimark’s problem can be constructedwithout using Jensen’s diamond principle. We also construct, using ourweakening of diamond, a separably represented, simple C ∗ -algebra withexactly m inequivalent irreducible representations for all m ≥ . Ourprincipal technical contribution is the introduction of a forcing notionthat generically adds an automorphism of a given C ∗ -algebra with aprescribed action on its space of pure states. Introduction
A major early result in the theory of operator algebras (and, at the time,probably the deepest result in the theory, see [2, §IV.1.5]) was Glimm’s 1960dichotomy theorem. It states (among other things) that a separable andsimple C ∗ -algebra A either has a unique irreducible representation up to theunitary equivalence, or it has ℵ inequivalent irreducible representations.The former condition is equivalent to A being isomorphic to the algebra ofcompact operators on a separable Hilbert space, while the latter is equivalentto A not being of type I (see [2, Theorem IV.1.5.1] for the full statement).Parts of Glimm’s theorem were extended to non-separable C ∗ -algebrasby Sakai (see [2, IV.1.5.8] for a discussion). In the 1970s further progresson extending Glimm’s theorem to all simple C ∗ -algebras slowed down to ahalt. The most obvious question, asked by Naimark already in the 1950s,was whether a C ∗ -algebra with a unique irreducible representation up tothe unitary equivalence is necessarily isomorphic to the algebra of compactoperators. A counterexample to Naimark’s problem is a C ∗ -algebra that isnot isomorphic to the algebra of compact operators on some Hilbert space,yet still has only one irreducible representation up to unitary equivalence.In a seminal paper (see [1]), Akemann and Weaver constructed a coun-terexample to Naimark’s problem using Jensen’s ♦ axiom. By related proofs, Date : June 15, 2020.2010
Mathematics Subject Classification.
Primary: 03E35, 46L30.
Key words and phrases.
Representations of C ∗ -algebras, forcing, Jensen’s diamond,Naimark’s problem.Partially supported by NSERC.Corresponding author: Ilijas Farah. ORCID: 0000-0001-7703-6931. also conditioned on Jensen’s ♦ on ℵ , several counterexamples with addi-tional properties (a prescribed tracial simplex [27], not isomorphic to its op-posite algebra [11]) were obtained. N.C. Phillips observed that the Akemann–Weaver construction provides a nuclear C ∗ -algebra. The range of applica-tions of this construction was extended to other problems: In [11] it wasshown that Glimm’s dichotomy can fail: Assuming ♦ , there exists a simpleC ∗ -algebra with exactly m inequivalent irreducible representations for all m ≤ ℵ (the latter was announced in [8, §8.2]) and a hyperfinite II factornot isomorphic to its opposite was constructed in [12].The following is a special case of our Theorem 6.3. Theorem A.
It is relatively consistent with
ZFC that there exists a coun-terexample to Naimark’s problem while ♦ fails. More specifically, we isolate a combinatorial principle ♦ Cohen that, to-gether with the Continuum Hypothesis ( CH ), implies the existence of a coun-terexample to Naimark’s problem and show that ♦ Cohen + CH is consistentwith the failure of ♦ (see §9).A simple C ∗ -algebra with a unique irreducible representation up to theunitary equivalence that is represented on a Hilbert space of density characterstrictly smaller than ℵ is necessarily isomorphic to the algebra of compactoperators on some Hilbert space (see e.g., [9, Corollary 5.5.6]). We prove the‘next best thing.’ Theorem B.
For any m ≥ , ♦ Cohen + CH implies that there exists a sepa-rably represented, simple, unital C ∗ -algebra with exactly m irreducible repre-sentations up to the spatial equivalence. This is a special case of Theorem 8.1 proved below.Our principal technical contribution is the introduction of a forcing no-tion that generically adds an automorphism of a given C ∗ -algebra with aprescribed action on its space of pure states (see §3, §4, and §7). Acknowledgments.
Some of the results of this paper come from the first au-thor’s masters thesis written under the second author’s supervision. We areindebted to Ryszard Nest and Chris Schafhauser for enlightening discussionsof Remark 7.4 and its extensions. We also wish to thank Andrea Vaccaro,and to Assaf Rinot for precious comments on precious stones that consider-ably improved the presentation of the transfinite constructions in this paper.2. Preliminaries and notation
The reader is assumed to be familiar with the basics of forcing and thebasics of C ∗ -algebras. Our notation and terminology follow [2] for operatoralgebras, [17] and [21] for set theory (in particular, forcing), and [9] for both(except forcing). It is understood that [9] is used as a reference only for the Diamonds.
AN YOU TAKE AKEMANN–WEAVER’S ♦ AWAY? 3 reader’s (as well as the author’s) convenience; none of the results referred toare claimed to be due to the author.2.1. C ∗ -algebras and their representations. C ∗ -algebras are complexBanach algebras with involution ∗ that satisfy the C ∗ -equality, k aa ∗ k = k a k .By a result of Gelfand and Naimark, these are exactly the algebras isomor-phic to a norm-closed subalgebra of the algebra B ( H ) of bounded linearoperators on a complex Hilbert space H . A homomorphism between C ∗ -algebras that preserves the involution is called a ∗ -homomorphism , and a ∗ -homomorphism into B ( H ) is a representation . A representation π : A → B ( H ) is irreducible if H has no nontrivial closed subspaces invariant underthe image of A . A representation is faithful if it is injective. Every faithfulrepresentation is necessarily isometric. More generally, all ∗ -homomorphismsare contractive (take note that in the theory of operator algebras ‘contrac-tive’ is synonymous with ‘1-Lipshitz.’) When A is unital, U ( A ) is the set ofunitary elements of A , i.e., the set of u ∈ A such that uu ∗ = u ∗ u = 1 A .2.2. States.
A bounded linear functional on a C ∗ -algebra A is a state if it ispositive (see [9, §1.7] for definitions) and of norm 1. The space of all states on A is denoted S ( A ) . Via the Gelfand–Naimark–Segal (GNS) construction (see[9, §1.10]), every state ϕ on A is associated with a representation π ϕ : A → B ( H ϕ ) such that a unique unit vector ξ ϕ in H ϕ satisfies ϕ ( a ) = ( π ϕ ( a ) ξ ϕ | ξ ϕ ) for all a ∈ A . The triplet ( π ϕ , H ϕ , ξ ϕ ) is the GNS triplet associated with ϕ .Conversely, every representation ( π, H ) of A with a cyclic vector (i.e., ξ ∈ H of norm 1 and such that the π [ A ] -orbit of ξ is dense in H ) is of the form π ϕ for the unique state ϕ on S ( A ) . When A is unital, the states space of A is aweak ∗ -compact and convex set. The extreme points of S ( A ) are called purestates and a state is pure if and only if the corresponding GNS representationis irreducible , if and only if π ϕ [ A ] is dense in B ( H ϕ ) in the weak operatortopology (see [9, §3.6]). The space of all pure states on A is denoted P ( A ) .2.3. Automorphisms and crossed products.
An automorphism Φ of aC ∗ -algebra A is inner if it is of the form Ad u ( a ) := uau ∗ for a unitary u in the unitization of A . It is approximately inner if there exists a net ofunitaries ( u p : p ∈ G ) in the unitization of A such that Φ( a ) = lim G Ad u p ( a ) for each a ∈ A . The automorphism group of A is denoted Aut( A ) .An automorphism Φ of A determines a continuous action of Z on A givenby n.a := Φ n ( a ) . To such a non-commutative dynamical system one canassociate a reduced crossed product , A ⋊ Φ Z . This C ∗ -algebra is generatedby an isomorphic copy of A (routinely denoted A ) and a unitary u thatimplements Φ on A , in the sense that Ad u ( a ) = Φ( a ) for all a ∈ A . Formore details see e.g., [3, §4.1] or [9, §2.4.2].2.4. Equivalences of states and representations.
We say that two rep-resentations ( π , H ) and ( π , H ) of a C ∗ -algebra are spatially equivalent ,and we write π ∼ π , if there exists an *-isomorphism Φ : B ( H ) → B ( H ) such that Φ ◦ π = π . Two pure states ϕ and ψ of A are called conjugate if DANIEL CALDERÓN AND ILIJAS FARAH there exists an automorphism Φ of A such that ϕ ◦ Φ = ψ . If Φ can be chosento be inner, we say that ϕ and ψ are unitarily equivalent and write ϕ ∼ ψ .Using a GNS argument plus the Kadison transitivity theorem, it can beshown (see [9, Lemma 3.8.1]) that ϕ ∼ ψ if and only if π ϕ ∼ π ψ , if and onlyif there is a unitary u in the unitization of A such that k ϕ ◦ Ad u − ψ k < .This implies that Naimark’s problem as stated in the previous section isequivalent to asking whether every C ∗ -algebra with a unique pure state upto unitary equivalence is isomorphic to some algebra of compact operators.2.5. The space P m ( A ) . Following [9, §5.6], for m ∈ N , the space of m -tuples of pairwise inequivalent pure states of A is denoted P m ( A ) . A typicalelement of P m ( A ) is of the form ¯ ϕ = ( ϕ i : i < m ) . If ¯ ϕ ∈ P m ( A ) and Φ ∈ Aut ( A ) then ¯ ϕ ◦ Φ := ( ϕ i ◦ Φ : i < m ) . We write G ⋐ A if G is a finitesubset of A . If G ⋐ A and δ > , we define B ( ¯ ϕ, G, δ ) := (cid:26) ¯ ψ ∈ P m ( A ) : max b ∈ G (cid:18) max i ZFC - P . Our ambient theory is ZFC , the Zermelo–Fraenkel set theory with the Axiom of Choice (see e.g., [9, §A.1]). Because ofmetamathematical obstructions of no direct relevance to the present paper,while working in ZFC one cannot prove the existence of a model of ZFC .Fortunately, for any uncountable regular cardinal κ the set H κ of all setswhose hereditary closure has cardinality smaller than κ (see e.g., [9, §A.7])is a model of ZFC - P , the theory obtained by removing the Power Set axiomfrom ZFC . This fragment of ZFC suffices for our purposes.Borel subsets of a Polish space with a fixed countable basis can be codedby elements of N N (see e.g., [17, p. 504]). This coding is sufficiently absolute,so that a transitive model of ZFC - P that does not include the set of all realnumbers can still contains codes for some Borel sets and be correct abouttheir properties (the proof of [17, Lemma 25.46] applies to show this).2.8. Forcing. A forcing notion is a partially ordered set P . The elementsof P are also called conditions , and if p ≤ q then p is said to extend q . Twoconditions are compatible if a single condition extends both of them. A subset D of P is called open if it contains all extensions of all of its elements. Asubset D of P is called dense if it contains some extension of every condition Purists may prefer working with transitive structures closed under the rudimentary func-tions, see e.g., [17, Definition 27.2]. AN YOU TAKE AKEMANN–WEAVER’S ♦ AWAY? 5 in P . A subset G of P is a filter if it satisfies the following two conditions:(i) p ∈ G and p ≤ q implies q ∈ G and (ii) every two elements of G have acommon extension in G . If D is a family of dense open subsets of P , then afilter G is called D -generic if it intersects every element of D non-trivially.If M is a transitive model of ZFC - P , P is a forcing notion in M , and afilter G ⊆ P intersects all dense open subsets of P that belong to M , then G is said to be M -generic. In this situation one can define the forcing (orgeneric) extension M [ G ] which is a transitive model of ZFC - P that includes M and contains G . The model M is usually referred to as the ground model .3. Forcing an approximately inner automorphism Let A be a simple and unital C ∗ -algebra. Given ¯ ϕ and ¯ ψ , two elements of P m ( A ) , we will define a forcing notion E A ◦ ( ¯ ϕ, ¯ ψ ) whose generic object is anapproximately inner automorphism Φ G of A such that ¯ ϕ ◦ Φ G = ¯ ψ .Besides ♦ , the Akemann–Weaver construction uses a modification of adeep 2001 result due to Kishimoto, Ozawa, and Sakai (see [20], also [9,§5.6]) that implies that all pure states of a separable, simple, unital C ∗ -algebra are conjugate by an approximately (and even asymptotically) innerautomorphism. A crucial lemma in the proof of the Kishimoto–Ozawa–Sakai theorem (see [20, Lemma 2.2], also [9, Lemma 5.6.7]) together with[14, Property 7] serves as a motivation for the following. Lemma 3.1. Let A be a simple, unital, and infinite-dimensional C ∗ -algebra.For all m ≥ , ¯ ϕ ∈ P m ( A ) , F ⋐ A , and ε > the following holds.(1) The weak ∗ -closure of the set (cid:26) ¯ ϕ ◦ Ad u : u ∈ U ( A ) & max b ∈ F k ub − bu k < ε (cid:27) contains a weak ∗ -open neighbourhood of ¯ ϕ in P m ( A ) .(2) For every l ≥ and ¯ ρ ∈ P l ( A ) such that ¯ ϕ ⌢ ¯ ρ ∈ P m + l ( A ) there exists ¯ ρ ′ ∈ P l ( A ) such that ¯ ρ ′ ∼ ¯ ρ and the weak ∗ -closure of the set (cid:26) ( ¯ ϕ ⌢ ¯ ρ ′ ) ◦ Ad u : u ∈ U ( A ) & max b ∈ F k ub − bu k < ε (cid:27) contains a weak ∗ -open neighbourhood of ¯ ϕ ⌢ ¯ ρ ′ in P m ( A ) .Proof. Using the structure of weak ∗ -open neighbourhoods (see §2.5), (1) isequivalent to the following.For all m ≥ , ¯ ϕ ∈ P m ( A ) , F ⋐ A , and ε > there exists a pair ( G, δ ) with G ⋐ A and δ > such that the set (cid:26) ¯ ϕ ◦ Ad u : u ∈ U ( A ) & max b ∈ F k ub − bu k < ε (cid:27) is weak ∗ -dense in B ( ¯ ϕ, G, δ ) . This is [9, Lemma 5.6.7].Analogously to (1), (2) is equivalent to the following. DANIEL CALDERÓN AND ILIJAS FARAH For all m ≥ , ¯ ϕ ∈ P m ( A ) , F ⋐ A , and ε > , there exists a pair ( G, δ ) with G ⋐ A and δ > such that for all l ≥ , and for all ¯ ρ ∈ P l ( A ) satisfying ¯ ϕ ⌢ ¯ ρ ∈ P m + l ( A ) , there exists ¯ ρ ′ ∈ P l ( A ) with ¯ ρ ′ ∼ ¯ ρ and such that the set (cid:26) ( ¯ ϕ ⌢ ¯ ρ ′ ) ◦ Ad u : u ∈ U ( A ) & max b ∈ F k ub − bu k < ε (cid:27) is weak ∗ -dense in B ( ¯ ϕ ⌢ ¯ ρ ′ , G, δ ) .The proof of [9, Lemma 5.6.7], with some minor modifications, gives thisstronger statement as follows (we assume that the reader has a copy of [9]handy, which is a great idea anyway):First note that [9, Lemma 5.6.3] can be modified by weakening the con-dition P k A pair ( G, δ ) such that the closure of the set in (2) ofLemma 3.1 contains B ( ¯ ϕ ⌢ ¯ ρ ′ , G, δ ) is called a ( ¯ ϕ, F, ε ) -good pair .Fix a simple, unital C ∗ -algebra A and tuples ¯ ϕ and ¯ ψ in P m ( A ) . Wealso fix a norm-dense Q + i Q -subalgebra A ◦ of A such that U ( A ) ∩ A ◦ isnorm-dense in U ( A ) . When A is separable, we will take A ◦ to be countable.In Definition 3.3 we introduce a forcing notion E A ◦ ( ¯ ϕ, ¯ ψ ) which adds ageneric automorphism Φ G of A such that ¯ ϕ ◦ Φ G = ¯ ψ . More precisely, it addstwo nets of unitaries, v p and w p , for p ∈ G , such that each of the nets of Ad v p AN YOU TAKE AKEMANN–WEAVER’S ♦ AWAY? 7 and Ad w ∗ p indexed by p ∈ G converges pointwise to an automorphism of A (this is assured by condition (c)). The automorphism Φ G is the compositionof these two automorphisms. Definition 3.3. Let E A ◦ ( ¯ ϕ, ¯ ψ ) be the set of tuples q = ( F q , G q , ε q , δ q , v q , w q ) such that:(1) F q and G q are finite subsets of A ◦ ,(2) ε q and δ q are positive real numbers,(3) v q and w q are unitaries of A in A ◦ ,(4) ( G q , δ q ) is a ( ¯ ϕ ◦ Ad v q , F q ∪ Ad v ∗ q [ F q ] , ε q / -good pair, and(5) ¯ ψ ◦ Ad w q ∈ B ( ¯ ϕ ◦ Ad v q , G q , δ q ) .We order E A ◦ ( ¯ ϕ, ¯ ψ ) by p ≤ q if:(a) F p ⊇ F q , G p ⊇ G q ,(b) ε p ≤ ε q , δ p ≤ δ q , and(c) for all b ∈ F q max (cid:8) k Ad v p ( b ) − Ad v q ( b ) k , (cid:13)(cid:13) Ad v ∗ p ( b ) − Ad v ∗ q ( b ) (cid:13)(cid:13)(cid:9) ≤ ε q − ε p , and max (cid:8) k Ad w p ( b ) − Ad w q ( b ) k , (cid:13)(cid:13) Ad w ∗ p ( b ) − Ad w ∗ q ( b ) (cid:13)(cid:13)(cid:9) ≤ ε q − ε p . Remark . The bound ε q − ε p in (c) of Definition 3.3 is used to assure thatthe relation ≤ is transitive on E A ◦ ( ¯ ϕ, ¯ ψ ) . Lemma 3.5. For all finite subsets F and G of A ◦ and all positive realnumbers ε and δ , the set D ε δF G of conditions p such that F ⊆ F p , G ⊆ G p , ε p ≤ ε , and δ p ≤ δ is a dense and open subset of E A ◦ ( ¯ ϕ, ¯ ψ ) .Proof. We need to prove that an extension of every condition in D ε δF G belongsto D ε δF G ; this is obvious. Second, we need to prove that every condition hasan extension that belongs to D ε δF G . Let q ∈ E A ◦ ( ¯ ϕ, ¯ ψ ) . By Lemma 3.1,there exists a ( ¯ ψ ◦ Ad w q , F q ∪ Ad w ∗ q [ F q ] , ε q ) -good pair; denote it ( G , δ ) .Since q is a condition, ( G q , δ q ) is ( ¯ ϕ ◦ Ad v q , F q ∪ Ad v ∗ q [ F q ] , ε q ) -good and ¯ ψ ◦ Ad w q ∈ B ( ¯ ϕ ◦ Ad v q , G q , δ q ) .Using the goodness of ( G q , δ q ) , choose v ∈ U ( A ) ∩ A ◦ to be some unitarysuch that ¯ ϕ ◦ Ad v q v ∈ B ( ¯ ψ ◦ Ad w q , G , δ ) and k b − Ad v ( b ) k < ε q / for all b ∈ F q ∪ Ad v ∗ q [ F q ] . Set ε p := min { ε, ε q / } , F p := F q ∪ F , and v p := v q v .By Lemma 3.1, there is a ( ¯ ϕ ◦ Ad v p , F p ∪ Ad v ∗ p [ F p ] , ε p ) -good pair, denoted ( G , δ ) . Also, let G p := G ∪ G q ∪ G and δ p := min { δ, δ q , δ } .Using now the goodness of ( G , δ ) , let w ∈ U ( A ) ∩ A ◦ be such that ¯ ψ ◦ Ad w q w ∈ B ( ¯ ϕ ◦ Ad v p , G p , δ p ) and k b − Ad w ( b ) k < ε q / for every b ∈ F q ∪ Ad w ∗ q [ F q ] . Define w p as w q w and set p to be ( F p , G p , ε p , δ p , v p , w p ) . See §2.8. DANIEL CALDERÓN AND ILIJAS FARAH Clearly F ⊆ F p , G ⊆ G p , ε p ≤ ε and δ p ≤ δ . Also, since ( G p , δ p ) is ( ¯ ϕ ◦ Ad v p , F p ∪ Ad v ∗ p [ F p ] , ε p ) -good, p ∈ E A ◦ ( ¯ ϕ, ¯ ψ ) . Finally, if b ∈ F q then k Ad v ∗ q ( b ) − Ad v ∗ p ( b ) k = k Ad v (Ad v ∗ q ( b )) − Ad v ∗ q ( b ) k < ε q / < ε q − ε q / ≤ ε q − ε p . Also, k Ad v q ( b ) − Ad v p ( b ) k = k b − Ad v ( b ) k < ε q − ε p . The calculations for w p and w q are analogous and therefore p ≤ q . (cid:3) An attentive reader may have noticed that in the proof of Lemma 3.5 wedid not use the full strength of condition (4) in Definition 3.3. Rest assuredthat it will come handy later on (tip: see the proof of the second part ofLemma 4.1). Theorem 3.6. Let A be a simple and unital C ∗ -algebra, m ≥ and let ¯ ϕ ,and ¯ ψ be elements of P m ( A ) . Then(1) Forcing with E A ◦ ( ¯ ϕ, ¯ ψ ) adds an approximately inner automorphism Φ G of A such that ¯ ϕ ◦ Φ G = ¯ ψ .(2) If A is separable, then E A ◦ ( ¯ ϕ, ¯ ψ ) is equivalent to the Cohen forcing.Proof. (1) Let M be a countable transitive model of a large enough fragment ZFC such that E A ◦ ( ¯ ϕ, ¯ ψ ) is an element of M and let G be an M -generic filteron E A ◦ ( ¯ ϕ, ¯ ψ ) . By Lemma 3.5, for any F ⋐ A and for all ε > , there exists q ∈ E A ◦ ( ¯ ϕ, ¯ ψ ) such that if p ≤ q then for all b ∈ F , k Ad v p ( b ) − Ad v q ( b ) k < ε and k Ad v ∗ p ( b ) − Ad v ∗ q ( b ) k < ε . Therefore, the nets Ad v p and Ad v ∗ p , for p ∈ G , are Cauchy with respect to the point-norm topology in Aut ( A ) . By[9, Lemma 2.6.3], Φ L ∈ Aut ( A ) defined pointwise as Φ L ( a ) := lim G Ad v p ( a ) is an endomorphism of A , and its inverse is given by Φ − L ( a ) = lim G Ad v ∗ p ( a ) for each a ∈ A . An analogous argument shows that Φ R ( a ) := lim G Ad w p ( a ) for each a ∈ A is an approximately inner automorphism of A .Let now a ∈ A be arbitrary and let ε > . Again using Lemma 3.5,choose p ∈ G such that some b ∈ F p ∩ G p satisfies k a − b k < ε/ and that max { ε p , δ p } < ε/ . Then | ¯ ϕ ◦ Φ L ( a ) − ¯ ψ ◦ Φ R ( a ) | < ε . Set Φ G := Φ L ◦ Φ − R .Since ε was arbitrary, ¯ ϕ ◦ Φ G = ¯ ψ .(2) Since A is separable, A ◦ is countable. Also, the conditions such thatboth ε p and δ p are rational, comprise a dense subset of E A ◦ ( ¯ ϕ, ¯ ψ ) . This setis countable, and by [18, Proposition 10.20], E A ◦ ( ¯ ϕ, ¯ ψ ) is equivalent to theCohen forcing. (cid:3) The idea of restricting to a countable dense set in order to assure thecountable chain condition of a poset was first used in the context of operatoralgebras in [29].4. The Unique Extension Property of pure states The most remarkable property of the generic automorphism introducedby E A ◦ ( ¯ ϕ, ¯ ψ ) is the genericity of its action on P ( A ) (see Theorem 4.2). AN YOU TAKE AKEMANN–WEAVER’S ♦ AWAY? 9 Let A be a simple, unital, and non-type I C ∗ -algebra. We will see lateron (see Proposition 5.4) that after forcing with E A ◦ ( ¯ ϕ, ¯ ψ ) , new equivalenceclasses of pure states of A will appear. The aim of this section is to studythe ground-model pure states of A that have a unique pure state extensionto A ⋊ Φ G Z in M [ G ] . In order to assure that pure states of A in a prescribedset have unique pure extensions to the crossed product, we use the toolsintroduced in [1] and presented in a gory detail in [9, §5.4].By Theorem 3.1 in [19], if A is simple and Φ is outer, then A ⋊ Φ Z issimple as well. By Theorem 2 in [1] (see also [9, Proposition 5.4.7]), a purestate ϕ on A has a unique extension to a pure state on A ⋊ Φ Z if and only if ϕ is not equivalent to ϕ ◦ Φ n for all n ≥ . Since the set of all extensions of ϕ is a face in S ( A ⋊ Φ Z ) (see [9, Lemma 5.4.1]), ϕ has a unique extension ifand only if it has a unique pure state extension. If two pure states ϕ and ψ on A have unique extensions ˜ ϕ and ˜ ψ to A ⋊ Φ Z , then these extensions areequivalent if and only if ϕ is equivalent to ψ ◦ Φ n for some n ∈ Z (this is thecase Γ = Z of [9, Theorem 5.4.8]).In the following, we use the notation established in Theorem 3.6. Notethat for a fixed A and A ◦ , the conditions in all forcings E A ◦ ( ¯ ϕ, ¯ ψ ) have thesame format and that ¯ ϕ and ¯ ψ behave as side-conditions. We will now relatethese forcing notions. Lemma 4.1. Suppose A is a simple and unital C ∗ -algebra, m ≥ , and ¯ ϕ and ¯ ψ are in P m ( A ) . Also suppose that l ≥ , ¯ ρ and ¯ σ belong to P l ( A ) , andmoreover ¯ ϕ ⌢ ¯ ρ and ¯ ψ ⌢ ¯ σ belong to P m + l ( A ) . Let us write P := E A ◦ ( ¯ ϕ, ¯ ψ ) , P := E A ◦ ( ¯ ϕ ⌢ ¯ ρ, ¯ ψ ⌢ ¯ σ ) , and ≤ j for the ordering on P j for j < . Then(1) Every condition in P is a condition in P . Moreover, if p and q arein P then p ≤ q if and only if p ≤ q .(2) For every q ∈ P there are ¯ ρ ′ ∼ ¯ ρ and ¯ σ ′ ∼ ¯ σ such that there exists p ≤ q which belongs to P ′ := E A ◦ ( ¯ ϕ ⌢ ¯ ρ ′ , ¯ ψ ⌢ ¯ σ ′ ) . In short, P is a subordering of P and the union of all P ′ as in (2) is densein P . This formulation is dangerously misleading, since P is typically nota regular subordering of P . Proof. To see that the first part of (1) holds, fix q ∈ P . Conditions (1)–(3)of Definition 3.3 do not depend on the tuples of pure states, while (4) and(5) are weakened as one passes to sub-tuples of pure states. Since conditions(a)–(c) do not refer to the pure states, the second part of (1) follows.(2)Fix q ∈ P . Since ( G q , δ q ) is ( ¯ ϕ ◦ Ad v q , F q ∪ Ad v ∗ q [ F q ] , ε q / -good, thereexists ¯ ρ ′ ∼ ¯ ρ such that the set (cid:8)(cid:0) ¯ ϕ ⌢ ¯ ρ ′ (cid:1) ◦ Ad v q u : (cid:0) ∀ b ∈ F q ∪ Ad v ∗ q [ F q ] (cid:1) ( k ub − bu k < ε q / (cid:9) is weak ∗ -dense in B (( ¯ ϕ ⌢ ¯ ρ ′ ) ◦ Ad v q , G q , δ q ) . By Glimm’s lemma (see [9, Lem-ma 5.2.6]) there exists a pure state ¯ σ ′ ∼ ¯ σ such that ¯ ψ ⌢ ¯ σ ′ ∈ P m + l ( A ) and ( ¯ ψ ⌢ ¯ σ ′ ) ◦ Ad w q ∈ B (( ¯ ϕ ⌢ ¯ ρ ′ ) ◦ Ad v q , G q , δ q ) .An argument analogous to that in Lemma 3.5 (using the full strength ofthe condition (4) from Definition 3.3) provides a condition p ≤ q in P such that ( ¯ ψ ⌢ ¯ σ ′ ) ◦ Ad w p ∈ B (( ¯ ϕ ⌢ ¯ ρ ′ ) ◦ Ad v p , G p , δ p ) , and such that ( G p , δ p ) is a (( ¯ ϕ ⌢ ¯ ρ ′ ) ◦ Ad v p , F p ∪ Ad v ∗ p [ F p ] , ε p / -good pair.Therefore, p belongs to P ′ , as required. (cid:3) Theorem 4.2. Suppose A is a simple, unital, and non-type I C ∗ -algebra,and let ¯ ϕ and ¯ ψ be elements of P m ( A ) . If ρ is a pure state on A , then somecondition in E A ◦ ( ¯ ϕ, ¯ ψ ) forces that ρ ◦ Φ n G is equivalent to a ground-model purestate σ for some n ≥ if and only if n = 1 and there exists i < m such that ρ ∼ ϕ i and σ ∼ ψ i .Proof. The converse implication is the conclusion of Theorem 3.6.If the direct implication is false, there are ground-model pure states ρ and σ of A such that ¯ ϕ ⌢ ρ ∈ P m +1 ( A ) and some condition q ∈ E A ◦ ( ¯ ϕ, ¯ ψ ) forces that ρ ◦ Φ n G ∼ σ for some n ≥ . Since the ground-model U ( A ) isdense in U ( A ) of the generic extension, by extending q , we may assume thatthere exists u ∈ U ( A ) ∩ A ◦ such that q (cid:13) k ρ ◦ Φ n G − σ ◦ Ad u k < / . Let ¯ ρ := ( ρ, ρ , . . . , ρ n − ) and let η ≁ σ be such that if ¯ η := ( ρ , . . . , ρ n − , η ) ,then both ¯ ϕ ⌢ ¯ ρ and ¯ ψ ⌢ ¯ η are elements of P m + n ( A ) . By Lemma 4.1, let p ≤ q , ¯ ρ ′ ∼ ¯ ρ , and ¯ η ′ ∼ ¯ η be such that p ∈ E A ◦ ( ¯ ϕ ⌢ ¯ ρ ′ , ¯ ψ ⌢ ¯ η ′ ) .Let H be an M -generic filter on E A ◦ ( ¯ ϕ ⌢ ¯ ρ ′ , ¯ ψ ⌢ ¯ η ′ ) containing p . By The-orem 3.6, in M [ H ] we have ¯ ρ ′ ◦ Φ H = ¯ η ′ . Let v ∈ U ( A ) be such that ¯ ρ = ¯ ρ ′ ◦ Ad v and set v H := Φ − n H ( v ) . Since Ad v ◦ Φ n H = Φ n H ◦ Ad v H , wehave ρ ◦ Φ n H = ρ ′ ◦ Ad v ◦ Φ n H = ρ ′ ◦ Φ n H ◦ Ad v H . Since σ is not equivalent to η ′ n − ◦ Ad v H , we can find a ∈ A M ≤ such that (cid:12)(cid:12)(cid:0) η ′ n − ◦ Ad v H (cid:1) ( a ) − ( σ ◦ Ad u ) ( a ) (cid:12)(cid:12) ≥ / . Since A M is norm-dense in A M [ H ] , let r ∈ E A ◦ ( ¯ ϕ ⌢ ¯ ρ ′ , ¯ ψ ⌢ ¯ η ′ ) , with r ≤ p , besuch that some b ∈ G r satisfies that k b − Ad v H ( a ) k < / , and δ r < / . Byusing the (easy) first part of Lemma 4.1, we have that r ∈ E A ◦ ( ¯ ϕ, ¯ ψ ) and r extends q as an element of E A ◦ ( ¯ ϕ, ¯ ψ ) . However, r forces (in either of theposets) that | ( ρ ◦ Φ n H )( a ) − ( η ′ n − ◦ Ad v H )( a ) | = | ( ρ ′ ◦ Φ n H )(Ad v H ( a )) − η ′ n − (Ad v H ( a )) |≤ | ( ρ ′ ◦ Φ n H )( b ) − η ′ n − ( b ) | + 2 k b − Ad v H ( a ) k < / contradiction. (cid:3) In the following corollary there is no need to explicitly refer to the groundmodel. Corollary 4.3. Suppose that A is a simple, unital, and non-type I C ∗ -algebra. Fix ¯ ϕ and ¯ ψ in P m ( A ) , and fix ρ and σ in P ( A ) . If G is a genericfilter in E A ◦ ( ¯ ϕ, ¯ ψ ) , then the following statements hold in the forcing exten-sion.(1) ρ has multiple pure state extensions to A ⋊ Φ G Z if and only if thereexists i < m such that ρ ∼ ϕ i ∼ ψ i . AN YOU TAKE AKEMANN–WEAVER’S ♦ AWAY? 11 (2) If both ρ and σ have unique pure state extensions to A ⋊ Φ G Z thenthese extensions are equivalent if and only if some i < m satisfies { ρ, σ } = { ϕ i , ψ i } .Proof. By a result from [1] (or see [9, Proposition 5.4.7]), a pure state ζ hasa unique pure state extension to the crossed product if and only if ζ ≁ ζ ◦ Φ n G for all n = 0 . By Theorem 3.6 and Theorem 4.2, this happens if and only if ρ ∼ ϕ i ∼ ψ i for some i < m . This proves the first part.To prove the second part, in the above argument, use [9, Theorem 5.4.8]in place of [9, Proposition 5.4.7]. (cid:3) C ∗ -algebras in generic extensions In this section we state and prove a few straightforward results on C ∗ -algebras in models of ZFC - P .As common in set theory, ‘reals’ are elements of any uncountable, de-finable, Polish space. This is justified by a classical result of Kuratowski,asserting that any two uncountable Polish spaces are Borel-isomorphic. Us-ing the coding for Borel sets, the property of being a Borel-isomorphismbetween two Polish spaces is absolute between transitive models of ZFC - P .A forcing notion P adds a new real if and only if it adds a new elementto some (equivalently, every) non-trivial C ∗ -algebra. Because of this, in aforcing extension M [ G ] we identify A with its completion and pure statesof A with their unique continuous extensions to the completion of A . Ascommon in set theory, by A M we denote the original C ∗ -algebra A in M andby A M [ G ] we denote its completion in M [ G ] . Note that A M is an element of M [ G ] which is an algebra over the field C M (but not a C -algebra, hence nota C ∗ -algebra) dense in A M [ G ] .This section contains straightforward results on the relation between A M and A M [ G ] that we could not find in the literature.A property P of C ∗ -algebras is said to be absolute if for all A , M and M [ G ] as above, A M has P if and only if A M [ G ] has P . Lemma 5.1. Both simplicity and being non-type I are absolute properties ofC ∗ -algebras.Proof. To prove that simplicity is absolute, we first consider the case when A is separable and unital. Fix a countable norm-dense subset D of A . Thensome x ∈ A generates a proper two-sided ideal if and only if for every m ∈ N and all m -tuples ( a j : j < m ) and ( b j : j < m ) of elements of D we have k A − P i Suppose A is a separable, simple, unital, and non-type IC ∗ -algebra. Then E A ◦ ( ¯ ϕ, ¯ ψ ) forces that A ⋊ Φ G Z has all of these properties.Proof. By Lemma 5.1, these properties of A are absolute. The crossed prod-uct is therefore unital and non-type I, and it remains to prove that it issimple. Being non-type I, A has continuum many inequivalent pure statesand Theorem 4.2 implies that Φ G moves a pure state to an inequivalent purestate, and is therefore outer. By [19, Theorem 3.1], this implies that thecrossed product is simple. (cid:3) The following lemma will be used tacitly. Lemma 5.3. With A , M , and M [ G ] as above, we have the following.(1) U ( A M ) is norm-dense in U ( A M [ G ] ) .(2) If ϕ and ψ are pure states of A in M , then their (unique) pure stateextensions to A M [ G ] are equivalent in M [ G ] if and only if ϕ and ψ are equivalent in M .Proof. Note that (1) is a consequence of the continuous functional calculus,as follows. Suppose that ( a n : n ∈ N ) is a sequence of elements of A in M [ G ] AN YOU TAKE AKEMANN–WEAVER’S ♦ AWAY? 13 that converges to a unitary u . Then k a ∗ n a n − k → and k a n a ∗ n − k → as n → ∞ . Therefore, a ∗ n a n is invertible for a large enough n , and hence | a n | := ( a ∗ n a n ) / is invertible for a large enough n . The unitary from thepolar decomposition of a n , u n := a n | a n | − , satisfies k u n − u k → .To see (2), as pointed out in §2.4, ϕ and ψ are equivalent if and only ifthere is a unitary u such that k ϕ ◦ Ad u − ψ k < , hence the conclusionfollows from (1). (cid:3) Every definable (in the model-theoretic sense, see [10, §3]) subset of A has the absoluteness property proved for U ( A ) in Lemma 5.3 by a proofanalogous to that of Lemma 5.3.The following was essentially proved in [1, Proposition 6]. We include itsproof for completeness. Proposition 5.4. Let M be a countable transitive model of ZFC - P , let P bea forcing notion in M , and let A ∈ M be a unital, non-type I C ∗ -algebra.If P adds a new real to M , then it adds a new pure state to A which isinequivalent to any ground-model pure state of A .Proof. The construction is very similar to the one in the proof of Theorem5.5.4 in [9], where additional details can be found. For i < define a linearfunctional δ i on M ( C ) by δ i (cid:18)(cid:18) λ λ λ λ (cid:19)(cid:19) = λ ii . This is a pure state of M ( C ) . For r ∈ N define a linear functional ϕ r on theCAR algebra as follows: on the elementary tensors (note that in N n ∈ N a n wehave a n = 1 for all but finitely many n ) let ϕ r ( ⊗ n ∈ N a n ) = Q n ∈ N δ r ( n ) ( a n ) .The linear extension of ϕ r (still denoted ϕ r ) is a pure state on M ∞ . ByGlimm’s theorem (see [9, Theorem 3.7.2]), A includes some separable C ∗ -subalgebra B which has the CAR algebra as a quotient. The compositionof ϕ r with the quotient map is a pure state of B , and this pure state canbe extended to a pure state ψ r of A . Clearly r can be recovered from ψ r byevaluation.Suppose that ψ r is equivalent to a ground-model pure state. Since U ( A ) M is norm-dense in U ( A ) M [ G ] , there exists a ground-model pure state σ of A such that k ψ r − σ k < . Then the restriction of σ to B still factors through thequotient map to a state of the CAR algebra and r can be recovered from thisstate. But this implies that r belongs to the ground model; contradiction. (cid:3) Since the poset E A ◦ ( ¯ ϕ, ¯ ψ ) is countable when A is separable, the space of all(characteristic functions of) filters in E A ◦ ( ¯ ϕ, ¯ ψ ) (see §2.8) is easily checked tobe a G δ subset of the power-set of this countable set, and therefore a Polishspace. The tricky condition is the requirement that for all p and q in a filter G , there is r ∈ G that extends both p and q . In the case of the Cohen forcing any two compatible conditionshave the largest lower bound and the set of filters is closed. . . but we do not need this fact. Suppose M is a transitive model of ZFC - P and X is a Polish space with acode in M . By a result of Solovay, an element r of X is Cohen-generic over M if and only if it belongs to every dense open subset of X coded in M (see[17, Lemma 26.24]). Thus, there exists a Cohen-generic element of X over M if and only if the closed nowhere dense subsets of X with codes in M donot cover X .The minimal cardinality of a family of nowhere dense sets that cover thereal line is denoted cov( M ) (see [9, §8.4]). The Baire category theoremimplies that this cardinal is uncountable, and Martin’s Axiom for κ densesets implies that cov( M ) > κ (see [21]). Corollary 5.5. Suppose that A is a separable, simple, unital and non-type IC ∗ -algebra, and that X ⊆ P ( A ) satisfies | X | < cov( M ) . Then for every pairof inequivalent pure states ϕ and ψ on A there exists Φ ∈ Aut ( A ) such that ϕ ◦ Φ = ψ and every ρ ∈ X has a unique pure state extension to A ⋊ Φ Z .Proof. Let M be an elementary submodel of H (2 ℵ ) + such that A ∈ M , X ⊆ M , and | M | < cov( M ) . Let ¯ M be the transitive collapse of M . Thenthe nowhere dense subsets of R coded in ¯ M are too few to cover R . By a resultof Solovay, a real that does not belong to any of these sets is Cohen-genericover ¯ M (see [17, Lemma 26.24]). By Theorem 3.6, there exists an ¯ M -genericfilter G on E A ◦ ( ¯ ϕ, ¯ ψ ) , and by Corollary 4.3, Φ := Φ G is as required. (cid:3) A proof of Theorem A from ♦ Cohen In this section we introduce our weakening of Jensen’s ♦ principle, ♦ Cohen ,and use it to construct the C ∗ -algebra as required in Theorem A. The relativeconsistency of ♦ Cohen will be discussed in §9.A subset of ℵ is closed and unbounded ( club ) if it is unbounded andcontains the supremum of each of its bounded subsets. A subset of ℵ is stationary if it intersects evert club non-trivially. Definition 6.1. A chain ( M α : α < ℵ ) is a ♦ Cohen -chain if: ♦ Cohen (a) Each M α is a (not necessarily countable) transitive model of ZFC - P . ♦ Cohen (b) For every X ⊆ ℵ the set { α < ℵ : X ∩ α ∈ M α } is stationary. ♦ Cohen (c) For every α < ℵ , some real in M α +1 is Cohen-generic over M α .We say that the principle ♦ Cohen holds if there is a ♦ Cohen -chain.Clearly ♦ ℵ implies ♦ Cohen . For a partial converse, note that if each M α in a ♦ Cohen -chain is countable, or if its intersection with α is countable, then ♦ holds. This is a consequence of [21, Theorem III.7.8]. We will discuss therelative consistency of ♦ Cohen in § .Let us discuss the relevance of the condition ♦ Cohen (b).First, observe that it implies that S α< ℵ M α contains all reals. The salient case is when A is a C ∗ -algebra in M , A ◦ is in M , ¯ ϕ and ¯ ψ are tuples ofinequivalent pure states of A , and X is the space of all filters of E A ◦ ( ¯ ϕ, ¯ ψ ) . In other words, it is unbounded and closed in the ordinal topology. AN YOU TAKE AKEMANN–WEAVER’S ♦ AWAY? 15 Second, this condition applies when X is replaced with any object ofcardinality ℵ or with a complete metric spaces of density character ℵ .More specifically, a C ∗ -algebra of density character κ can be coded by asubset of κ (see the introduction to §3 in [11] or [9, §7.1-2]). Similarly, if ¯ ψ isa tuple of states of a C ∗ -algebra B of density character κ , then the structure ( B, ¯ ψ ) can be coded by a subset of κ . Moreover, the version of Löwenheim–Skolem theorem for logic of metric structures stated in [9, Theorem 7.1.4]implies the following. Lemma 6.2. Suppose κ is a regular and uncountable cardinal, A = lim −→ α<κ A α is a C ∗ -algebra such that the density character of each A α is strictly smallerthan κ , A β = lim −→ α<β A α for every limit ordinal β , ¯ ϕ is a tuple of states of A , and X ⊆ κ is a code for ( A, ¯ ϕ ) . Then the set { α < κ : X ∩ α is a code for ( A α , ¯ ϕ ↾ A α ) } includes a club. (cid:3) If M is a transitive model of ZFC - P then we say that a C ∗ -algebra B belongs to M if some code for B belongs to M . The analogous remarkapplies to states of B .Glimm’s dichotomy (see e.g., [9, Corollary 5.5.8]) implies that every C ∗ -algebra A of density character κ < ℵ either has a unique pure state upto unitary equivalence, and in this case A ∼ = K ( ℓ ( κ )) , or has ℵ manyequivalence classes. The conclusion of the following theorem was deducedfrom ♦ in [11], as announced in [8, §8.2]. The special case when m = 1 (using the full ♦ ) is the Akemann–Weaver result.The case when m = 1 of Theorem 6.3 below is Theorem A. In its proofwe adopt the approach to ♦ constructions introduced in [23]. Theorem 6.3. If ♦ Cohen + CH holds, then for every m ≥ there is a simpleC ∗ -algebra of density character ℵ with exactly m pure states up to unitaryequivalence that is not isomorphic to any algebra of compact operators on acomplex Hilbert space.Proof. Let ( M α : α < ℵ ) be a ♦ Cohen -chain. Using the Continuum Hypoth-esis, fix a surjection f : ℵ → H ℵ such that every element of H ℵ is listedcofinally often. By recursion on β < ℵ , we will define an inductive systemof separable, simple, unital, and non-type I C ∗ -algebras, A β .Let A be a separable, simple, unital, non-type I C ∗ -algebra and let ϕ i ,for i < m , be inequivalent pure states of A . At the latter stages of theconstruction we will assure that the following conditions hold for all α < ℵ .(1) If ξ < α then A ξ is a unital C ∗ -subalgebra of A α .(2) With γ ( α ) := min { γ : A α ∈ M γ } , A α +1 belongs to M γ ( α )+1 . This function is well-defined: Since A α is separable, it is coded by a real and thereforebelongs to S α< ℵ M α . (3) Every pure state of A α that belongs to M γ ( α ) has a unique pure stateextension to A α +1 .(4) If f ( α ) is a code for a pair ( A ξ , ψ ) , where ξ < α and ψ is a pure stateof A ξ which has a unique pure state extension to A α , then ψ has aunique pure state extension to A α +1 , and this extension is equivalentto (the unique pure state extension of) some ϕ i , for i < m .To describe the recursive construction, suppose that β is a countable ordinalsuch that A α as required has been defined for all α < β .Consider first the case when β is a successor ordinal, β = α +1 . Suppose fora moment that f ( α ) is a code for a pair ( A ξ , ψ ) with the following properties.(a) ξ < α .(b) ψ is a pure state of A ξ that has a unique extension ˜ ψ to a pure stateof A α .(c) For all i < m , ˜ ψ is inequivalent to the unique extension of ϕ i to A α (still denoted ϕ i ).By the second part of Theorem 3.6, E A ◦ α ( ϕ , ˜ ψ ) is forcing-equivalent tothe poset for adding a single Cohen real. Since M γ ( α )+1 contains a realthat is Cohen-generic over M γ ( α ) , it contains an M γ ( α ) -generic filter G on E A ◦ α ( ϕ , ˜ ψ ) . By the first part of Theorem 3.6, Φ G is an approximately innerautomorphism of A α such that ϕ ◦ Φ G = ˜ ψ . By Corollary 4.3, the C ∗ -algebra A α +1 := A α ⋊ Φ G Z has the property that every pure state of A α that belongs to M γ ( α ) has aunique pure state extension to A α +1 . By the second part of Corollary 4.3,the unique pure state extensions of ϕ i , for i < m , to A α +1 are inequivalent.Also, A α +1 is separable, simple, unital and non-type I by Corollary 5.2.If f ( α ) does not satisfy the conditions (a)–(c), let A α +1 := A α .If β is a limit ordinal, take A β := lim −→ α<β A α .Finally, let A ℵ := lim −→ α< ℵ A α .By the construction, each one of the the pure states ϕ i , for i < m , of A has a unique pure state extension to A ℵ and these pure state extensions areinequivalent.Suppose that ψ is a pure state of A ℵ . In order to prove that it is equivalentto ϕ i for some i < m , fix a code X ⊆ ℵ for the pair ( A ℵ , ψ ) . By [9,Proposition 7.3.10], the set { α < ℵ : ψ ↾ A α is pure } is a club, and byLemma 6.2, the set { α < ℵ : X ∩ α is a code for ( A α , ψ ↾ A α ) } is a club as well. By ♦ Cohen (b), there exists α in the intersection of thesetwo clubs such that X ∩ α ∈ M α . In particular, both A α and ψ ↾ A α belongto M α —i.e., γ ( α ) = α . This implies that ψ ↾ A α has a unique pure stateextension to A β for all β > α . (This is proved by induction on β . Theproof uses properties (2) and (3) at the successor stages. At the limits, note AN YOU TAKE AKEMANN–WEAVER’S ♦ AWAY? 17 that the unique pure state extension of ψ ↾ A α is definable from ψ ↾ A ξ , for α < ξ < β , and therefore belongs to the relevant model.)By the choice of the function f , there exists β < ℵ such that f ( β ) codesthe pair ( A α , ψ ↾ A α ). By the definition of A β +1 , the restrictions of ψ and ϕ i to A β +1 are equivalent.This proves that A ℵ has exactly m inequivalent pure states. Since A ℵ is infinite-dimensional and unital, it is not isomorphic to any algebra ofcompact operators. (cid:3) The proof of Theorem A will be completed in §9. In this section, wewill prove that ♦ Cohen + CH is relatively consistent with the negation of ♦ .Once proven, this will provide a model of ZFC in which both ♦ and Glimm’sdichotomy fail. 7. A proof of Theorem B, part I This section contains finer analysis of the forcing notion E A ◦ ( ¯ ϕ, ¯ ψ ) , cul-minating in Lemma 7.6. The following is an analog of Theorem 4.2. Theorem 7.1. Suppose that Θ is an outer automorphism of a separable,simple, unital, non-type I C ∗ -algebra A , m ≥ , ¯ ϕ and ¯ ψ belong to P m ( A ) ,and ρ is a pure state of A inequivalent to all ϕ i , for i < m . If Θ G is definedas Φ G ◦ Θ ◦ Φ − G then E A ◦ ( ¯ ϕ, ¯ ψ ) forces that ρ ◦ Θ G is inequivalent to anyground-model pure state of A .Proof. Towards obtaining a contradiction, assume that in M [ G ] we have ρ ◦ Θ G ∼ σ for a ground-model pure state σ . Then fix u ∈ U ( A ) ∩ A ◦ and q ∈ E A ◦ ( ¯ ϕ, ¯ ψ ) such that q forces that k ρ ◦ Θ G − σ ◦ Ad u k < / . We obtain k ρ ◦ Φ G ◦ Θ − σ ◦ Ad u ◦ Φ G k < / . We first consider the most difficult case, when ρ is equivalent to σ .Since Θ is outer, by the slight extension of [19, Theorem 2.1] proved in[11, Theorem 2.4], there exists an uncountable set of pure states η of A eachof which satisfies η ◦ Θ ≁ η . We can therefore choose η such that η ◦ Θ isinequivalent to η and each one of η and η ◦ Θ is inequivalent to all ψ i , for i < m . By Lemma 4.1, there are ρ ′ ∼ ρ , η ′ ∼ η , and a condition p ≤ q in E A ◦ ( ¯ ϕ, ¯ ψ ) such that p also belongs to the poset E A ◦ ( ¯ ϕ ⌢ ρ ′ , ¯ ψ ⌢ η ′ ) .The remainder is analogous to the corresponding part of the proof ofTheorem 4.2: if H is an M -generic filter on E A ◦ ( ¯ ϕ ⌢ ρ ′ , ¯ ψ ⌢ η ′ ) containing p then, in M [ H ] , ρ ′ ◦ Φ H = η ′ . Let v ∈ U ( A ) be such that ρ = ρ ′ ◦ Ad v andset v H := Φ − H ( v ) , so that ρ ◦ Φ H = ρ ′ ◦ Ad v ◦ Φ H = ρ ′ ◦ Φ H ◦ Ad v H . Since η ◦ Θ ≁ η and η ∼ η ′ , we have η ′ ◦ Ad v H ◦ Θ ≁ η ′ ◦ Ad v H . Because of this, wecan find a ∈ A M ≤ such that | ( η ′ ◦ Ad v H ◦ Ad u )( a ) − ( η ′ ◦ Ad v H ◦ Θ)( a ) | ≥ / .Since A M is norm-dense in A M [ H ] there is r ∈ E A ◦ ( ¯ ϕ ⌢ ρ ′ , ¯ ψ ⌢ η ′ ) , extending p , such that δ r < / and some b and c in G r satisfy k b − Ad v H u ( a ) k < / and k c − Ad v H (Θ( a )) k < / . By the easy part of Lemma 4.1, r ∈ E A ◦ ( ¯ ϕ, ¯ ψ ) and it is below q in this poset. From the choice of r , we can conclude that,in E A ◦ ( ¯ ϕ, ¯ ψ ) , r forces | ( ρ ◦ Φ H ◦ Ad u )( a ) − ( η ′ ◦ Ad v H ◦ Ad u )( a ) | = | ( ρ ′ ◦ Φ H )(Ad v H u ( a )) − η ′ (Ad v H u ( a )) |≤ | ( ρ ′ ◦ Φ H )( b ) − η ′ ( b ) | + 2 k b − Ad v H u ( a ) k < / , and also r forces | ( ρ ◦ Φ H ◦ Θ)( a ) − ( η ′ ◦ Ad v H ◦ Θ)( a ) | = | ( ρ ′ ◦ Φ H )(Ad v H (Θ( a ))) − η ′ (Ad v H (Θ( a ))) |≤ | ( ρ ′ ◦ Φ H )( c ) − η ′ ( c ) | + 2 k c − Ad v H (Θ( a )) k < / . By the triangle inequality and the choice of a , we obtain / / > / ;contradiction. This concludes the discussion of the case when ρ ∼ σ .Suppose now that ρ ≁ σ . As in the first case, in each of the two subcases ofthis case we will use Lemma 4.1 to define a forcing notion P and a condition p ≤ q in E A ◦ ( ¯ ϕ, ¯ ψ ) that also belongs to P .If σ ∼ ϕ i for some i < m , choose a pure state ζ , that is not equivalent toany of the ψ j and such that in addition ζ ◦ Θ ≁ ψ i . This is possible because A has ℵ inequivalent pure states. By Lemma 4.1, there are ρ ′ ∼ ρ , ζ ′ ∼ ζ ,and a condition p ≤ q in the poset P := E A ◦ ( ¯ ϕ ⌢ ρ ′ , ¯ ψ ⌢ ζ ′ ) .If σ is not equivalent to any of the ϕ i , choose two pure states, ζ and η ,that are not equivalent to any of the ψ j and such that in addition ζ ◦ Θ ≁ η .By Lemma 4.1, there are ρ ′ ∼ ρ , σ ′ ∼ σ , ζ ′ ∼ ζ , η ′ ∼ η , and a condition p ≤ q in the poset P := E A ◦ ( ¯ ϕ ⌢ ρ ′ ⌢ σ ′ , ¯ ψ ⌢ ζ ′ ⌢ η ′ ) .In each of the two cases the proof that the assumptions lead to a contra-diction is analogous to the proof in the case when ρ ∼ σ and is thereforeomitted. (cid:3) Corollary 7.2. Suppose that A is a separable, simple, unital C ∗ -algebra, Θ is an outer automorphism of A of order two, m ≥ , ¯ ϕ and ¯ ψ belong to P m ( A ) , and ρ is a pure state of A inequivalent to all the ϕ i , for i < m . If Θ G := Φ G ◦ Θ ◦ Φ − G then ρ has multiple pure state extensions to A ⋊ Θ G Z / Z if and only if there exists some i < m such that ρ ∼ ϕ i ∼ ψ i .Proof. The proof is analogous to the proof of Corollary 4.3, using Theo-rem 7.1 in place of Theorem 4.2. (cid:3) In order to prove Theorem 8.1, we need to take a closer look at the innerworkings of the GNS construction (see [9, §1.10]).If ϕ is a state on a C ∗ -algebra A , then it defines a sesquilinear form on A by ( a | b ) ϕ := ϕ ( b ∗ a ) . The completion of A with respect to this norm is a Hilbertspace ℓ ( A, ϕ ) (denoted H ϕ in [9]), and the representation π ϕ is defined bythe left multiplication. If A is a C ∗ -subalgebra of B and ˜ ϕ is a state on B that extends ϕ , then ℓ ( A, ϕ ) is naturally identified with a closed subspaceof ℓ ( B, ˜ ϕ ) . AN YOU TAKE AKEMANN–WEAVER’S ♦ AWAY? 19 Lemma 7.3. Let A be a C ∗ -algebra with an outer automorphism Φ of ordertwo, B := A ⋊ Φ Z / Z , ϕ ∈ P ( A ) be such that ϕ = ϕ ◦ Φ , and ψ ∈ S ( B ) bean extension of ϕ . Then(1) There is a unitary u ∈ B such that ψ is uniquely determined by ψ ( u ) ,and it is pure if and only if ψ ( u ) = ± .(2) If ψ ( u ) = ± , then ℓ ( B, ψ ) = ℓ ( A, ϕ ) .Proof. We will prove (1) and (2) simultaneously. Let π ψ : B → B ( H ) be theGNS representation associated with ψ with the cyclic vector ξ . For simplicityof notation we identify B and A with their images under π ψ . Let u be theunitary of B such that uau = Φ( a ) for all a ∈ A (note that Φ = id A implies u = 1 , hence u is a self-adjoint unitary). Claim. For all a ∈ A we have ψ (Φ( a ) u ) = ψ ( au ) .Proof. Since every element of A is a linear combination of four positive el-ements (see [9, Exercise 1.11.16]), by linearity, it suffices to prove this inthe case when a is positive. Since Φ = id A and ϕ ◦ Φ = ϕ , we have ϕ ( a Φ( a )) = ϕ (Φ( a Φ( a )) = ϕ (Φ( a ) a ) and therefore (after expanding andcancelling) ϕ (( a − Φ( a )) ) = 0 . Recall that ψ extends ϕ .By the Cauchy–Schwarz inequality, since Φ( a ) − a is self-adjoint, | ψ (Φ( a ) u ) − ψ ( au ) | = | ψ ((Φ( a ) − a ) u ) | ≤ ψ ((Φ( a ) − a ) ) ψ ( u ∗ u ) = 0 , as required. (cid:3) Let ξ = ( ξ + uξ ) and ξ = ( ξ − uξ ) . Fix a and b in A . Since ϕ ◦ Φ = ϕ ,the claim implies ( auξ | bξ ) = ( b ∗ auξ | ξ ) = ψ ( b ∗ au ) = ψ (Φ( b ∗ a ) u ) = ( ub ∗ aξ | ξ ) = ( aξ | buξ ) . Using this, we have aξ | bξ ) = ( a ( ξ + uξ ) | b ( ξ − uξ ))= ( aξ | bξ ) − ( aξ | buξ ) + ( auξ | bξ ) − ( auξ | buξ ) = 0 . Since uξ = ξ , this implies ( auξ | bξ ) = 0 . Since every element of B is ofthe form a + cu for some a and c in A , we conclude that Bξ and Bξ are(naturally identified with) orthogonal subspaces of H and H = Bξ ⊕ Bξ .We claim that ψ is pure if and only if | ψ ( u ) | = 1 . Consider three cases.(a) Suppose ξ = 0 . Then Bξ = { } and H = Bξ . Also, uξ = ξ , ξ = ξ is the GNS vector and ψ ( au ) = ϕ ( a ) for all a ∈ A . Since ϕ ispure, the image of A is weak operator topology-dense in B ( H ) , andso is B . This implies that ψ is pure. It also implies that ℓ ( B, ψ ) = Bξ = ℓ ( A, ϕ ) , hence (2) holds as well.(b) If ξ = 0 , then uξ = − ξ , ξ = ξ is the GNS vector and ψ ( au ) = − ϕ ( a ) for all a ∈ A . As in the previous case, ψ is pure and ℓ ( B, ψ ) = Bξ = ℓ ( A, ϕ ) , hence (2) holds as well.(c) Finally, suppose that both ξ and ξ are non-zero. Then Bξ is a non-trivial subspace of B ( H ) which is invariant under π ψ [ B ] (this spaceis automatically closed since ϕ is pure). Therefore, ψ is not pure. Itwas already proved that ψ ( u ) is neither +1 nor − in this case. Wehave ψ ( au ) = ( auξ | ξ ) + ( auξ | ξ ) = k ξ k ψ ( au ) + k ξ k ψ − ( au ) ,which shows that ψ is uniquely determined by ψ ( u ) .This concludes the proof. (cid:3) The following remark is not used in the proof of Theorem 8.1. Remark . Lemma 7.3 can be generalized to the case when m ≥ , Φ m =id A , Φ j is outer for all ≤ j < m , and B = A ⋊ Φ Z /m Z . In this case, ϕ has exactly m pure state extensions ϕ λ , where λ ranges over the m throots of unity and satisfies ϕ λ ( u ) = λ . This implies that ϕ λ ( au k ) = ϕ ( a ) λ k by [9, Proposition 1.7.8]. Since in this case B is the linear span of the set { au j : 0 ≤ j < m } , λ determines ϕ λ uniquely. The states of B that extend ϕ are convex combinations of { ϕ λ : λ m = 1 } . A proof of this is analogous tothe proof of Lemma 7.3.Lemma 7.6 below is based on [11, Lemma 2.7]. The key property of M ∞ used in it is extracted in the following lemma implicit in [11]. Lemma 7.5. There are inequivalent pure states ρ j , σ j , η j for j ∈ N on M ∞ and an automorphism Θ of M ∞ of order two such that the following condi-tions hold.(1) σ j = ρ j ◦ Θ and η j = η j ◦ Θ for all j .(2) M ∞ ⋊ Θ Z / Z is isomorphic to M ∞ .Proof. Identify M ∞ with N N A n , where A n ∼ = M n ( N ) . Let ϕ j , for j ∈ N ,be a family of separated product states of M ∞ (see [11, Definition 2.5]).The existence of such family is guaranteed by [11, Lemma 2.6, (1) implies(2)]. Let u n be a self-adjoint adjoint unitary in A n as defined in the proofof [11, Lemma 2.7] so that for every n the projections in A n separating thepure states satisfy the analogues of conditions (6)–(8). Let Θ := N N Ad u n .Then the action of Θ on the distinguished pure states is as required. Tocomplete the proof, note that as in [11, Lemma 2.7], the classification of AFalgebras implies that M ∞ ⋊ Θ Z / Z is isomorphic to M ∞ . (cid:3) Lemma 7.6. Suppose that X , Y , and Z are disjoint finite sets of purestates of M ∞ and F : X → Y is a bijection. Then there are ¯ ϕ and ¯ ψ in P m + l ( M ∞ ) (where m = | X | and l = | Y | ) and Θ ∈ Aut( M ∞ ) such that E M ◦ ∞ ( ¯ ϕ, ¯ ψ ) forces the following.(1) With Θ G := Φ G ◦ Θ ◦ Φ − G we have that B := M ∞ ⋊ Θ G Z / Z isisomorphic to M ∞ .(2) Every η ∈ Z has exactly two pure state extensions, denoted η +1 and η − , to B , and ℓ ( A, η ) = ℓ ( B, η ± ) . AN YOU TAKE AKEMANN–WEAVER’S ♦ AWAY? 21 (3) For every η ∈ X , η and F ( η ) have unique pure state extensions to B ,and these extensions are equivalent.Proof. For convenience we write A := M ∞ . The plan is to match the purestates in X , Y , and Z to those provided by Lemma 7.5 and import Θ fromthere. More specifically, fix ρ j , σ j , η j , and Θ as guaranteed by Lemma 7.5.Enumerate X as ϕ j , for j < m , and let ϕ m + j := F ( ϕ j ) for j < m . Enumerate Y as ϕ m + j , for j < l . Now let ψ j := ρ j and ψ m + j := σ j if j < m , and let ψ m + j := η j if j < k .(1) Since Θ G is conjugate to Θ , we have B ∼ = M ∞ .(2) Fix η ∈ Z . Then η = ϕ i = ψ i for some i , and therefore Lemma 7.3implies that η has exactly two pure state extensions, η ± , to B andthat ℓ ( A, η ) = ℓ ( B, η ± ) .(3) If η ∈ X and ζ := F ( η ) , then ϕ i = η and ψ i = ζ for some i .Theorem 4.2 implies that η and ζ have unique pure state extensionsto B , and Theorem 3.6 implies that they are equivalent.This concludes the proof. (cid:3) A proof of Theorem B, part II In this section we prove that ♦ Cohen + CH implies that the conclusion ofGlimm’s dichotomy fails even for separably represented C ∗ -algebras. Moreprecisely, we prove the following. Theorem 8.1. Assume ♦ Cohen + CH . For every m ≥ and every n ≥ there exists a simple, unital C ∗ -algebra of density character ℵ with exactly m + n unitary equivalence classes of irreducible representations such that m of these representations are on a separable Hilbert space and n of theserepresentations are on a non-separable Hilbert space. A simple C ∗ -algebra with irreducible representations on both separableand non-separable Hilbert spaces can be constructed in ZFC (see [9, Theo-rem 10.4.3]). Both this C ∗ -algebra and the one in Theorem 8.1 are inductivelimits of inductive systems of C ∗ -algebras all of which are isomorphic to theCAR algebra. Proof. With Lemma 7.6 at our disposal, this proof is analogous to that ofTheorem 6.3. Fix a ♦ Cohen -chain ( M α : α < ℵ ) . Using the ContinuumHypothesis, fix a surjection f : ℵ → H ℵ such that every element of H ℵ islisted cofinally often. By recursion on α < ℵ we will define an inductivesystem of separable, simple, unital, non-type I C ∗ -algebras, A α . For each A α we will have a distinguished ( m + n ) -tuple of inequivalent pure states, ( ϕ αi : i < m + n ) .Let A be the CAR algebra with inequivalent pure states ϕ i , for i < m + n .At the latter steps of the construction, we will assure that for all α < ℵ thefollowing conditions hold.(1) If ξ < α then A ξ is a unital C ∗ -subalgebra of A α . (2) With γ ( α ) := min { γ : A α ∈ M γ } , A α +1 belongs to M γ ( α )+1 .(3) Every pure state ψ of A α that belongs to M γ ( α ) , except ϕ αi , for i < m ,has a unique pure state extension to A α +1 .(4) If f ( α ) is a code for a pair ( A ξ , ψ ) , where ξ < α , ψ is a pure stateof A ξ which has a unique pure state extension ˜ ψ to A α , and ˜ ψ isinequivalent to ϕ αi for all i < m + n , then ψ has a unique pure stateextension to A α +1 , and this extension is equivalent to ϕ α +1 m .(5) For all i < m , ϕ α +1 i extends ϕ αi and ℓ ( A α +1 , ϕ α +1 i ) = ℓ ( A α , ϕ αi ) . In order to describe the recursive construction, suppose that β is a countableordinal such that A α as required has been defined for all α < β . As in theproof of Theorem 6.3, the interesting case is when β = α + 1 for some α and f ( α ) is a code for a pair ( A ξ , ψ ) that satisfies the following conditions.(a) ξ < α .(b) ψ is a pure state of A ξ that has a unique extension ˜ ψ to a pure stateof A α .(c) For all i < m + n , ˜ ψ is inequivalent to ϕ αi .By the second part of Theorem 3.6, any forcing notion of the form E A ◦ α (¯ ρ, ¯ σ ) is forcing-equivalent to the poset for adding a single Cohen real. Since M γ ( α )+1 contains a real that is Cohen-generic over M γ ( α ) , it contains an M γ ( α ) -generic filter for any forcing notion of this form. Therefore, Lemma7.6 implies that in M γ ( α )+1 there exists an automorphism Θ G of A α of ordertwo such that the C ∗ -algebra A α +1 := A α ⋊ Θ G Z / Z is isomorphic to the CAR algebra, each ϕ αi for m ≤ i < m + n has a uniquepure state extension to A α +1 , ψ has a unique pure state extension to A α +1 equivalent to ϕ α +1 m , and ϕ αi ◦ Θ G = ϕ αi for i < m . Lemma 7.3 impliesthat, for i < m , ϕ αi has a pure state extension ϕ α +1 i to A α +1 that satisfies ℓ ( A α , ϕ αi ) = ℓ ( A α +1 , ϕ α +1 i ) . By Corollary 7.2, any other pure states of A α that belongs to M γ ( α ) has a unique pure state extension to A α +1 . Also, A α +1 is separable, simple, unital and non-type I by Corollary 5.2.If f ( α ) does not satisfy the conditions (a)–(c), let A α +1 := A α and, foreach i < m + n , let ϕ α +1 i := ϕ αi .If β is a limit ordinal, take A β := lim −→ α<β A α and, for each i < m + n ,define ϕ βi as the unique pure state of A β that extends ϕ αi for all α < β . Since ϕ βi is definable from its restrictions, it belongs to the relevant model.This describes the recursive constructionLet A ℵ := lim −→ α< ℵ A α , and for i < m + n let ϕ i be the unique pure stateof A ℵ that extends ϕ αi for all α < ℵ .By the construction, the pure states ϕ i , for i < m + n , are inequivalent. By(5) and induction, for i < m , we have ℓ ( A ℵ , ϕ i ) = ℓ ( A , ϕ i ) and therefore For the notation, see the discussion preceding Lemma 7.3. Note that n ≥ , hence ϕ αm is well-defined for every α < ℵ . AN YOU TAKE AKEMANN–WEAVER’S ♦ AWAY? 23 the GNS Hilbert space associated with ϕ i is separable. If m ≤ i < m + n ,then ϕ α +1 i is the unique extension of ϕ αi and therefore Lemma 7.6 implies that ℓ ( A α , ϕ αi ) is a proper subspace of ℓ ( A α +1 , ϕ α +1 i ) for all α < ℵ . Therefore,the GNS Hilbert space associated with ϕ i is non-separable.It remains to prove that every pure state of A ℵ is equivalent to some ϕ i .The proof of this is analogous to the corresponding proof in Theorem 6.3and therefore omitted. (cid:3) The reader may wonder whether it is possible to sharpen the conclusion ofTheorem 8.1 and obtain a simple, unital, infinite-dimensional C ∗ -algebra A with at most m ≤ ℵ irreducible representations up to unitary equivalencesuch that every irreducible representation of A is on a separable Hilbertspace. The answer is well-known to be negative in the case when m = 1 (itis Rosenberg’s result that a counterexample to Naimark’s problem cannotbe separably represented). A proof analogous to that of Rosenberg’s resultprovides a negative answer in the general case. Proposition 8.2. Suppose that A is a non-type I C ∗ -algebra all of whoseirreducible representations are on a separable Hilbert space. Then A has atleast ℵ spatially inequivalent irreducible representations.Proof. The assumption on A is used only to prove that it has a self-adjointelement a whose spectrum is a perfect set. In M ∞ there exists a positivecontraction a with this property. To see this, note that the diagonal masa isisomorphic to the algebra of continuous functions on the Cantor space, andlet a correspond to the identity map on the Cantor space via the continuousfunctional calculus. By Glimm’s theorem, A has a subalgebra whose quotientis isomorphic to M ∞ . Let a be a self-adjoint lift of a to A (see [9, §2.5]).Then the spectrum of a includes the spectrum of a , hence a is as required.For every element x of the spectrum of a , fix a pure state ϕ x on A suchthat ϕ x ( a ) = x . We can take ϕ x to be a pure state extension of the point-evaluation at x . Then the cyclic vector is an x -eigenvector of π x ( a ) . Since theeigenvectors corresponding to distinct eigenvalues are orthogonal, in everyirreducible representation π of A the operator π ( a ) has only countably manyeigenvectors. Therefore A has at least ℵ spatial equivalence classes ofirreducible representations. (cid:3) The combinatorial principle ♦ Cohen This section contains only set-theoretic considerations: we prove that ♦ Cohen does not imply ♦ and that it does not decide the cardinality of ℵ .As the attentive reader may have noticed during the proof of Theorem 6.3(or Theorem 8.1) the combinatorial principle ♦ Cohen can be thought as anoracle in which the required tasks at successor steps can be done by the meanof a Cohen real. On the upside, and in opposition to the usual application ofJensen’s ♦ , such tasks can be delayed (this is the job of the book-keeping)and they do not have to be handled at the moment they are captured by the oracle. Our weakening of ♦ is, at the end of the day, a sort of guessing-plus-forcing axiom in which the generic objects exist (in a prescribed extension)only for countable posets that are elements of models whose job is to capturesubsets of ℵ correctly. Lemma 9.1. It is relatively consistent with ZFC that ♦ Cohen + CH + ¬♦ holds.Proof. Let M be a countable transitive model of a large enough fragmentof ZFC + CH in which ♦ fails. Such model was first constructed by Jensen(see [4], also [25, §V]). Let ( P α , ˙ Q α ) α< ℵ be a finite support iteration of non-trivial ccc forcings each of which has cardinality at most ℵ . Let G ⊆ P ℵ be an M -generic filter. By the countable chain condition, ♦ fails in M [ G ] (see [21, Exercise IV.7.57]) and the standard ‘counting of names’ argumentshows that the Continuum Hypothesis holds in M [ G ] .For α < ℵ , M α := M [ G ∩ P ω · α ] (here ω · α is the α th limit ordinal) is theintermediate forcing extension. By the countable chain condition, no realsare added at stages of uncountable cofinality (see [15, Lemma 18.9]), andtherefore every real in M [ G ] belongs to some M α for α < ℵ . Since a finitesupport iteration of non-trivial ccc forcings adds a Cohen real at every limitstage of countable cofinality (see [21, Exercise V.4.25]), for every α < ℵ themodel M α +1 contains a real that is Cohen-generic over M α .Fix a name for a subset X of ℵ . Again, by the countable chain conditionand the standard closing off argument, there is a club C ⊆ ℵ such thatfor every α ∈ C the forcing P α adds X ∩ α . Therefore, X ∩ α ∈ M α forstationary many α and P ℵ forces that ♦ Cohen holds. (cid:3) The following corollary exhibits a substantial difference between the prin-ciples ♦ and ♦ Cohen . Corollary 9.2. The principle ♦ Cohen does not decide the value of ℵ .Proof. If in the proof of Lemma 9.1 we begin with a model of ℵ = κ , then M [ G ] is a model of ♦ Cohen + 2 ℵ = κ . (cid:3) To see that ♦ Cohen is not a consequence of CH , we will show that, unlikethe Continuum Hypothesis, ♦ Cohen implies the existence of a Suslin tree.In [22], Moore, Hrušák and Džamonja introduced a variety of parametrized ♦ principles based on the weak diamond (see [5]) which have a similar relation-ship to ♦ as cardinal invariants of the continuum have to CH . Definition 9.3 ([22]) . The principle ♦ ( non ( M )) holds if for every function F : 2 < ℵ → M such that F ↾ α , for α < ℵ , is Borel there exists some g : ℵ → R such that for all f : ℵ → , the set { α < ℵ : g ( α ) / ∈ F ( f ↾ α ) } is stationary. Proposition 9.4. The principle ♦ ( non ( M )) is a consequence of ♦ Cohen .Proof. Let ( M α : α < ℵ ) be a ♦ Cohen -chain and F : 2 < ℵ → M be suchthat for all α < ℵ the restriction F ↾ α is Borel-measurable. For each α < ℵ let r α ∈ N N be such that F ↾ α is definable from r α and let AN YOU TAKE AKEMANN–WEAVER’S ♦ AWAY? 25 α ≤ φ ( α ) < ℵ be such that r α ∈ M φ ( α ) . Define g : ℵ → R by choosing g ( α ) to be Cohen-generic over M φ ( α ) . Let f : ℵ → be arbitrary. Since { α < ℵ : f ↾ α ∈ M α } is stationary, and g ( α ) is Cohen-generic over a modelcontaining both f ↾ α and r α , then { α < ℵ : g ( α ) / ∈ F ( f ↾ α ) } is stationaryas well. (cid:3) Corollary 9.5. Following the notation above.(1) If ♦ Cohen holds then there is a Suslin tree.(2) The principle ♦ Cohen is not a consequence of CH .Proof. By [22, Theorem 3.1], ♦ ( non ( M )) implies that there is a Suslin treeand therefore (1) follows from Proposition 9.4. (2) follows from (1) andLemma 9.1. (cid:3) One could consider ♦ Random , ♦ Hechler , or diamonds associated to otherSuslin ccc forcings. The countable chain condition of the forcing is used inorder to assure the property ♦ Cohen (b) in Definition 6.1. We are not awareof any applications of these axioms.10. Concluding Remarks Our title was inspired by the title of ground-breaking Shelah’s paper [24],but the answers to the questions posed in these titles are quite different.Solovay’s inaccessible may or may not be taken away depending on whetherone requires the Baire-measurability alone, or the Lebesgue-measurability aswell. In our case, the ♦ is not necessary for the construction. The questionwhether a counterexample to Naimark’s problem can be constructed in ZFC alone, in ZFC + CH , or using ♦ κ for some κ ≥ ℵ , remains open.Around 2010, the senior author conjectured that Naimark’s problem haspositive answer in a model obtained by adding a sufficient number (super-compact cardinal, if need be) of Cohen reals. Theorem A and its proofgive some (inconclusive) support for the negation of this conjecture. Ad-ditional support would be provided by a proof that a forcing notion withthe properties of E A ◦ ( ¯ ϕ, ¯ ψ ) and the countable chain condition be definedfor tuples of inequivalent pure states for every simple and unital (not nec-essarily separable) C ∗ -algebra. The experience suggests that the countablechain condition and non-commutativity do not mix well (see [13] and [7,Lemma 4.1]). Moreover, if A has irreducible representations on both sepa-rable and non-separable Hilbert spaces (see e.g., [9, Theorem 10.4.3]), thenadding an automorphism of A that moves one of the associated pure statesto another, necessarily collapses ℵ . Thus the relevant question is whethersuch forcing can be constructed for C ∗ -algebras that are inductive limits thatappear in the course of the proof of Theorem A.Another possible route towards constructing a counterexample to Naimark’sproblem would be the following. Instead of forcing with E A ◦ ( ¯ ϕ, ¯ ψ ) or a mod-ification thereof, find a separable C ∗ -subalgebra B of A such that the re-strictions ¯ ϕ ′ and ¯ ψ ′ to B of all the pure states involved, uniquely determine their extensions to A . Force with E B ◦ ( ¯ ϕ ′ , ¯ ψ ′ ) to produce a generic automor-phism Φ of B such that ¯ ϕ ′ and ¯ ψ ′ have unique, and equivalent, extensionsto B ⋊ Φ Z . This plan hinges on the answer to the following purely C ∗ -algebraic question. For simplicity, it is stated for single pure states insteadof m -tuples. Question 10.1. Suppose that A is a unital C ∗ -algebra, B is a unital C ∗ -subalgebra of A , both A and B are simple, and ϕ ′ and ψ ′ are pure statesof B with the unique pure state extensions to A . In addition, suppose that Φ is a sufficiently generic automorphism of B such that ϕ ′ and ψ ′ haveunique and unitarily equivalent pure state extensions to B ⋊ Φ Z . Is therean amalgamation C of B ⋊ Φ Z and A such that ϕ and ψ have unique purestate extensions to C ?Such amalgamation would be a ‘partial crossed product’ of sorts of A byan automorphism of B . There is a rich literature on partial crossed products(see [6] and the references thereof), but our situation does not satisfy therequirements imposed on partial dynamical systems in [6, Definition 6.4]. Question 10.2. Let A be a separable, simple, non-type I C ∗ -algebra. Doesthere exist an automorphism Θ of A such that σ ◦ Θ ≁ σ for every pure state σ of A ?If A has an automorphism Θ as in Question 10.2, then for every m ≥ andtuples ¯ ϕ in ¯ ψ in P m ( A ) , it has an automorphism Φ with the same propertythat in addition satisfies ¯ ϕ ◦ Φ = ¯ ψ . Such an automorphism can be obtainedby conjugating Θ by a Kishimoto–Ozawa–Sakai-type automorphism as inTheorem 7.1. Assuming in addition one could assure that all pure states of A have unique pure state extensions to a crossed product associated with Θ ,one would secure the assumptions of the following. Proposition 10.3. Suppose that there exists a class A of separable, simple,unital C ∗ -algebras such that:(1) A is closed under inductive limits, and(2) For every A ∈ A and pure states ϕ and ψ of A there exists anextension B ∈ A of A such that (i) ϕ and ψ have equivalent purestate extensions to B and (ii) every pure state of A has a uniquepure state extension to B .Then CH implies that there is a counterexample to Naimark’s problem.Proof. Suppose that CH holds, and fix X ⊆ ℵ such that the inner model L [ X ] (see [21, Definition II.6.29]) contains all reals. Then ♦ holds in L [ X ] (see [21, Exercise III.7.21]).Working in L [ X ] , modify the construction of a counterexample as in [1](see also [9, Theorem 11.2.2]) as follows. One constructs an inductive system Note however that we may assume B is an elementary submodel of A ; see [9, Appen-dix D]. AN YOU TAKE AKEMANN–WEAVER’S ♦ AWAY? 27 of C ∗ -algebras A α , for α < ℵ , in A so that at every successor step ofthe construction the extension A α +1 of A α is chosen using ♦ and (2). Atlimit stages take inductive limits. The inductive limit A of this system isa counterexample to Naimark’s problem in L [ X ] , by a proof analogous tothose in [1] or [9, Theorem 11.2.2].We claim that A remains a counterexample to Naimark’s problem in theuniverse V . Assume otherwise. Since it is a counterexample to Naimark’sproblem in L [ X ] , there exists a pure state η of A that belongs to V butnot to L [ X ] . The set C := { α < ℵ : the restriction of η to A α is pure } includes a club (see [9, Proposition 7.3.10]). Let α := min( C ) . By inductionon countable ordinals β ≥ α , one proves that η ↾ A β is the unique purestate extension of η ↾ A α to A β in L [ X ] , for every β < ℵ . At the successorstages this is a consequence of the choice of A β +1 , and at the limit stages itis automatic. This provides a definition of η in L [ X ] ; contradiction. (cid:3) The proof of Proposition 10.3 begs the question: is it possible to add anew pure state to a counterexample to Naimark’s without adding new reals?The answer is, at least assuming ♦ , positive (see [9, Exercise 11.4.11]). Onecan see that ♦ Cohen + CH suffices for this construction.The main result of [11] is a construction (using ♦ ) of a nuclear, simple,C ∗ -algebra not isomorphic to its opposite algebra. We do not know whetherthe existence of an algebra with this property follows from ♦ Cohen + CH . Inthe same theorem, a counterexample to Glimm’s dichotomy with exactly ℵ imequivalent pure states was constructed using ♦ . Such construction us-ing ♦ Cohen would require a generalization of the forcing E A ◦ ( ¯ ϕ, ¯ ψ ) to count-able sequences of inequivalent pure states.In [26] it was shown that a counterexample to Naimark’s problem cannotbe a graph C ∗ -algebra (not to be confused with the ‘graph CCR algebras’ of[9, §10]). We conjecture that a sweeping generalization of this result holds:If a C ∗ -algebra A (Γ) is defined from a discrete object (graph, group, semi-group, etc.) Γ in a way that assures that (using the notation of §5) A (Γ) as computed in M is dense in A (Γ) as computed in M [ G ] for all M and all M -generic filter G , then A (Γ) is (provably in ZFC ) not a counterexample toNaimark’s problem.By [19, Corollary 2.3], an automorphism Φ of a separable and simple C ∗ -algebra satisfies ϕ ◦ Φ ∼ ϕ for all pure state ϕ of A if and only if it is inner.Theorem 4.2 and Theorem 7.1 imply that both the generic automorphism Φ G and the conjugate of a ground-model outer automorphism by Φ G send everyground-model pure state to an inequivalent pure state. We conjecture thatthis property is shared by every reduced word in outer automorphisms of A and Φ G in which the latter occurs. This resembles the properties of genericautomorphisms (and anti-automorphisms) of II factors as exhibited in [16,Lemma A.2] and [28], and used there to construct interesting examples of II factors with a separable predual. These lemmas, combined with an iteratedcrossed product construction à la Akemann–Weaver propelled by ♦ was used in [12] to construct a hyperfinite II factor with non-separable predual andnot isomorphic to its opposite. It is not difficult to see that ♦ Cohen + CH inplace of ♦ suffices for this construction. References 1. C. Akemann and N. Weaver, Consistency of a counterexample to Naimark’s problem ,Proc. Natl. Acad. Sci. USA (2004), no. 20, 7522–7525.2. B. Blackadar, Operator algebras , Encyclopaedia of Mathematical Sciences, vol. 122,Springer-Verlag, Berlin, 2006, Theory of C ∗ -algebras and von Neumann algebras, Op-erator Algebras and Non-commutative Geometry, III.3. N. Brown and N. 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(2017), no. 3-4, 479–495.27. A. Vaccaro, Trace spaces of counterexamples to Naimark’s problem , J. Funct. Anal. (2018), no. 10, 2794–2816.28. S. Vaes, Factors of type II without non-trivial finite index subfactors , Trans. Amer.Math. Soc. (2009), no. 5, 2587–2606.29. E. Wofsey, P ( ω ) / fin and projections in the Calkin algebra , Proc. Amer. Math. Soc. (2008), no. 2, 719–726. Department of Mathematics and Statistics, York University, 4700 KeeleStreet, Toronto, Ontario, Canada, M3J 1P3 E-mail address : [email protected] Department of Mathematics and Statistics, York University, 4700 KeeleStreet, Toronto, Ontario, Canada, M3J 1P3Matematički Institut SANU, Kneza Mihaila 36, 11000 Beograd, p.p. 367,Serbia E-mail address ::