Consistency of a counterexample to Naimark’s problem
Charles Akemann† and Nik Weaver‡§
†Department of Mathematics, University of California, Santa Barbara, CA 93106; and ‡Department of Mathematics, Washington University, St. Louis, MO 63130
Edited by Vaughan F. Jones, University of California, Berkeley, CA, and approved March 29, 2004 (received for review March 2, 2004) We construct a C*-algebra that has only one irreducible represen- example to Naimark’s problem. Our construction is not carried tation up to unitary equivalence but is not isomorphic to the out in ZFC (Zermelo–Frankel set theory with the axion of algebra of compact operators on any Hilbert space. This answers an choice): it requires Jensen’s ‘‘diamond’’ principle, which follows old question of Naimark. Our construction uses a combinatorial from Go¨del’s axiom of constructibility and hence is relatively statement called the diamond principle, which is known to be consistent with ZFC, i.e., if ZFC is consistent then so is ZFC plus consistent with but not provable from the standard axioms of set diamond (see refs. 13 and 14 for general background on set theory (assuming that these axioms are consistent). We prove that theory). The diamond principle has been used to prove a variety the statement ‘‘there exists a counterexample to Naimark’s prob- of consistency results in mainstream mathematics (see, e.g., refs. lem which is generated by Ꭽ1 elements’’ is undecidable in standard 15–17). set theory. Presumably, the existence of a counterexample to Naimark’s problem is independent of ZFC, but we have not yet been able to show this. We can prove the relative consistency of the et K(H) denote the C*-algebra of compact operators on a Ꭽ complex, not necessarily separable, Hilbert space H. In ref. 1 assertion ‘‘no C*-algebra generated by 1 elements is a coun- L terexample to Naimark’s problem’’; indeed, this follows easily Naimark observed that every irreducible representation (irrep) Ꭽ from ref. 10. Since our counterexample is generated by 1 of K(H) is unitarily equivalent to the identity representation, so Ꭽ that each of these algebras has only one irrep up to equivalence, elements, it follows that the existence of an 1-generated coun- and in ref. 2 he asked whether this property characterizes the terexample is independent of ZFC (see Corollary 7). algebras K(H). In other words: if A is a C*-algebra with only one The basic idea of our construction is to create a nested A␣ ␣ Ͻ Ꭽ irrep up to unitary equivalence, is A isomorphic to some K(H)? transfinite sequence of separable C*-algebras , for 1, f␣ We call any algebra A that satisfies the premise of this question each equipped with a distinguished pure state , such that for any ␣ Ͻ , f is the unique extension of f␣ to a state on A.At but not its conclusion a counterexample to Naimark’s problem. the same time, for each ␣ (or at least ‘‘enough’’ ␣) we want to The problem was quickly settled in the separable case. Build- select a pure state g␣ on A␣ that is not equivalent to f␣, such that ing on results in ref. 2, Rosenberg (3) showed that there are no g␣ has a unique state extension gЈ␣ to A␣ϩ , and gЈ␣ is equivalent separable (indeed, no separably acting) counterexamples to 1 to f␣ϩ . Thus, we build up a continually expanding pool of Naimark’s problem. Around the same time, Kaplansky (4) 1 equivalent pure states in an attempt to ensure that all pure states introduced the so-called ‘‘type I’’ (or ‘‘GCR’’) C*-algebras and on A ϭ ഫA␣ will be equivalent. began developing their representation theory. (See refs. 5–7 for There are two significant difficulties with this approach, which general background on type I C*-algebras.) This development Ꭽ incidentally is essentially the only way an 1-generated counterex- was carried further by Fell and Dixmier, who showed in partic- ample to Naimark’s problem could be constructed. First, there is ular that any two irreps of a type I C*-algebra with the same the technical problem of finding a suitable algebra A␣ϩ that kernel are unitarily equivalent (8) and conversely, any separable 1 contains A␣ and such that f␣ and g␣ extend uniquely to A␣ϩ1 and C*-algebra that is not type I has inequivalent irreps with the become equivalent. We accomplish this via a crossed product same kernel (9). As it is easy to see that no type I C*-algebra can construction which takes advantage of a powerful recent result be a counterexample to Naimark’s problem, the latter result of Kishimoto, Ozawa, and Sakai that, together with earlier work (partially) recovers Rosenberg’s theorem, doing so, moreover, in of Futamura, Kataoka, and Kishimoto, ensures the existence of the context of a general theory. automorphisms on separable C*-algebras that relate inequivalent Next, in a celebrated paper Glimm (10) gave several charac- pure states in a certain manner. The second fundamental challenge terizations of separable type I C*-algebras, showing in particular is to choose the states g␣ in such a way that all pure states are that every separable C*-algebra which is not type I has uncount- eventually made equivalent to one another. This is especially ably many inequivalent irreps. Since counterexamples to Naima- troublesome because pure states can be expected to proliferate rk’s problem cannot be type I, this reestablished the nonexist- exponentially as ␣ increases (every pure state on A␣ extends to at ence of separable counterexamples in an especially dramatic way. least one, but probably many, pure states on A␣ϩ1) and we can only Some of Glimm’s results were initially proven without assum- make a single pair of states equivalent at each step. We handle this ing separability, and subsequent work by Sakai (11, 12) went issue by using the diamond principle, one version of which states further in this direction. However, separability was never re- that it is possible to select a single vertex from each level of the moved from the reverse implication of the equivalence ‘‘type I standard tree of height and width Ꭽ , such that every path down the N 1 irreps with the same kernel are equivalent,’’ and it was tree contains in some sense ‘‘many’’ selected vertices. In our recognized that Naimark’s problem potentially represented a application the vertices of the tree at level ␣ model the states on A␣ fundamental obstruction to a nonseparable generalization of this and diamond informs us how to choose g␣. Then every pure state result. However, it was reasonable to expect that there were no counterexamples to Naimark’s problem because it seemed likely that the separability assumption could be dropped in Glimm’s This paper was submitted directly (Track II) to the PNAS office. theorem that non-type I C*-algebras have uncountably many Abbreviations: irrep, irreducible representation; ZFC, Zermelo–Frankel set theory with the inequivalent irreps, but this was never achieved. axiom of choice. That is the background for the present investigation. We §To whom correspondence should be addressed. E-mail: [email protected]. construct a (necessarily nonseparable and not type I) counter- © 2004 by The National Academy of Sciences of the USA
7522–7525 ͉ PNAS ͉ May 18, 2004 ͉ vol. 101 ͉ no. 20 www.pnas.org͞cgi͞doi͞10.1073͞pnas.0401489101 Downloaded by guest on October 1, 2021 on A induces a path down the model tree, and hence its restriction Proof: (f) Suppose that for some g ʦ G not the identity, f is ؠ to some A␣ equals g␣, so that all pure states are indeed taken care equivalent to f g; we must show that f does not extend uniquely of at some point in our construction. to the crossed product. Let u ʦ A be a unitary such that f ϭ u* ؠ 3 (f g)u (ref. 6, proposition 3.13.4). Let : A B(Hf)bethe ʦ Unique Extension of Pure States to Crossed Products GNS representation associated to f, and let Hf be the image In this section we consider the problem: if f and g are inequiva- of the unit of A, so that f(y) ϭ͗(y),͘ for all y ʦ A. Let b ʦ lent pure states on a C*-algebra A, is it possible to find a A be any positive, norm-one element such that f(b) ϭ 1; then ϭ C*-algebra B that contains A such that (i) f and g have unique (b)( ) . Furthermore, g(u*bu) is also a positive, norm-one ϭ ϭ state extensions fЈ and gЈ to B and (ii) fЈ and gЈ are equivalent? element such that f( g(u*bu)) f(b) 1, so the same is true of ϭ We find that the answer is yes if A is simple, separable, and unital, this element, and it follows that (b g(u*bu))( ) . Hence ʈ ʈ ϭ and moreover we can ensure that B is also simple, separable, and b g(u*bu) 1. ϭ ʦ unital, with the same unit as A. Now let x g(u*). Let b A be any positive, norm-one A net (a) of positive, norm-one elements of a C*-algebra A element such that f(b) ϭ 1 and let y ʦ A. We will show that ʈy Ϫ 2 ʈ Ն is said to excise a state f on A if ʈaxa Ϫ f(x)aʈ 3 for 0 all x ʦ b(xTg)b 1 (computing the norm in the crossed product), which A (ref. 18, p. 1239). By ref. 18, proposition 2.2, for each pure state implies nonunique extension by ref. 19, theorem 3.2. Observe f of a C*-algebra A, there is a decreasing net (a) that excises f. first that As described in ref. 18, p. 1240 (see also ref. 6, p. 87), the support ͑ ͒ ϭ ͑ ͒ ϭ ͑ ͒ Ϫ ͑ ͒ ϭ projection p of f is a rank 1 projection in A**. Further, as shown b xTg b bx g b Tg b g u*bu g 1 u Tg cTg 3 in the last 7 lines of the proof of ref. 18, proposition 2.3, a ʦ where c A has norm 1. [It is the product of b g(u*bu), which p in the strong operator topology on A**. we showed above has norm 1, with the unitary gϪ1(u).] Now to Ϫ ϭ Ϫ estimate the norm of y b(xTg)b y cTg, consider the element Lemma 1. Suppose f and g are inequivalent pure states on a ␦ 2 ␦ ϭ ␦ ϭ e of l (G; A) that satisfies e(e) 1A (the unit of A) and e(h) C*-algebra A,(a) excises f, and (b) excises g. Also assume that Ϫ ␦ ϭ 0 for all h e. Since g e, we have (y cTg)( e) where (b) is decreasing. Then ʈaxbʈ 3 0 for all x ʦ A. ϭ ͑ ϭϪ Ϫ ϭ (e) y, g) g 1(c), and (h) 0 for all other h. The norm Proof: Note that we are assuming (a) and (b) have the same 2 1/2 1/2 of ␦ in l (G; A)isʈ⌺ ␦ (h)*␦ (h)ʈ ϭʈ1Aʈ ϭ 1 and the norm index set. This is easy to arrange by replacing two different index e h e e A A of is sets with their product. Let q denote the rank one projection in 1/2 1/2 A** that supports the pure state g. Suppose the lemma is false ʈ ͑ ͒ ͑ ͒ʈ ϭ ʈ ϩ Ϫ ͑ ͒ Ϫ ͑ ͒ʈ Ն h * h y*y 1 c * 1 c 1 for some x ʦ A. Then, multiplying x by a large enough positive A g g A h number and passing to a subnet, we can assume that ʈaxbʈ Ͼ 1 for all . ʈ Ϫ ͑ ʈϭʈ ʈϭ Ϫ n n since g 1 c) c 1. Thus the norm of y cTg is at least 1, as We first show that f(xbx*) 0. In A**, xbx* xqx*. Let y we needed to show. denote the central cover of q in A**. Since g is pure, yA** is d ؠ ( ) Suppose f is not equivalent to f g for any g e. To isomorphic to some B(H). Thus, f must be 0 on yA** lest it be verify that f has a unique extension to the crossed product, equivalent to g. Therefore, according to (ref. 19, theorem 3.2), we must, for every element z of the crossed product and every Ͼ 0, find x ʦ A and a f͑xbx*͒n f͑xqx*͒ ϭ f͑xyqx*͒ ϭ f͑y͑xqx*͒͒ ϭ 0. positive norm-one element b ʦ A such that f(b) ϭ 1 and ʈx Ϫ Ͻ ͞ bzbʈ Յ. It is sufficient to accomplish this only for a dense set Fix 0 so that f(xb x*) 1 2, and note that because (a) excises 0 Ն ʈ Ϫ ϭ ⌺ f, there exists 0 such that 0 implies axb x*a of elements z, so let z gxgTg where each xg is an element 2 0 f(xb x*)aʈ Ͻ 1͞2. Choose so that Ն and Ն . Using of A and xg 0 for only finitely many g. (Such sums are clearly 0 0 0 ϫ b Յ b , we have dense in A r G.) 0 Let (a) be a decreasing net in A that excises f and satisfies 2 1/2 2 ϭ ʈ 2 ʈaxbʈ Յ ʈaxb ʈ ϭ ʈaxbx*aʈ f(a) 1 for all (ref. 18, proposition 2.2). We claim that f(xe)a Ϫ azaʈ 3 0, which will complete the proof. Observe first that ʈ 2 Ϫ ʈ 3 Յ ʈaxb x*aʈ f(xe)a axea 0 since (a) excises f. Since z is a finite sum 0 ϭ ʈ ʈ 3 and xe xeTe we now need only show that a(xgTg)a 0 for ϭ ϭ Յ ʈ Ϫ ͑ ͒ 2ʈ ϩ ͑ ͒ʈ 2ʈ each g e. But a(xgTg)a (axgb)Tg where b g(a) and (b) axb x*a f xb x* a f xb x* a ؠ ؠ 0 0 0 is a decreasing net that excises f gϪ1. By hypothesis, f and f gϪ1 ʈ ʈ 3 are inequivalent, so Lemma 1 now implies axgb 0. Thus ʈ ʈ 3 Ͻ ͞ ϩ ͞ ϭ (axgb)Tg 0, as desired. 1 2 1 2 1, MATHEMATICS Note that in the proof of the reverse direction of Theorem 2, ϭ ⌺ a contradiction. the fact that z xgTg can be ‘‘compressed’’ to xe implies that Ј Ј ϭ Let be an action of a discrete group G on a unital C*-algebra the unique extension f of f satisfies f (z) f(xe). ϫ A. The reduced crossed product A r G can be defined as the By ref. 20, the pure state space of any simple, separable C*-algebra that acts on the right Hilbert module l2(G; A) and is C*-algebra A is homogeneous: given any two inequivalent pure ʦ generated by (i) for each x A, the multiplication operator Mx states f and g, there is an automorphism of A such that f ϭ ϭ ʦ defined by Mx (g) gϪ1(x) (g) and (ii) for each h G, the g ؠ . In the following corollary we need an even more powerful ϭ Ϫ1 translation operator Th defined by Th (g) (h g). Observe ‘‘strong transitivity’’ result that was proven in ref. 21, theorem ϫ ۋ that the map x Mx isomorphically embeds A in A r G,sowe 7.5 for a certain class of simple, separable C*-algebras. By ϫ can regard A as a subalgebra of A r G by identifying x with Mx. combining the methods of the two papers, that result can be Denote the identity of G by e. achieved for all simple, separable C*-algebras (A. Kishimoto, personal communication). Theorem 2. Let A be a unital C*-algebra, let f be a pure state on A, The result we need states the following: Suppose A is a simple, and let be an action of a discrete group G on A. Embed A in A separable C*-algebra and ( n) and ( n) are sequences of irreducible ϫ r G in the manner just described. Then f has a unique state representations such that the n are mutually inequivalent, as are ϫ ؠ extension to A r G if and only if f is inequivalent to f g for all the n. Then there is an automorphism of A such that n is ؠ g e. equivalent to n for all n. It can be proven by first replacing
Akemann and Weaver PNAS ͉ May 18, 2004 ͉ vol. 101 ͉ no. 20 ͉ 7523 Downloaded by guest on October 1, 2021 ͉ ϭ ϩ ͞ every result in ref. 20 involving a single pure state f (or a pair of f A␣ (f1 f2) 2, where f1 and f2 are distinct states on A␣. Now ഫ ␣ ʦ pure states f and g) with a corresponding result involving a finite ʦS A is dense in A␣ by continuity (unless S0, when this 0  ʦ ͉ ͉ set of mutually inequivalent pure states fi (or a pair of such sets, is true vacuously). Thus there exists S0 such that f1 A f2 A. ͉ ϭ ͉ ϩ ͉ ͞ ͉ with, for each i, fi and gi related in the same way that f and g were). But then f A (f1 A f2 A) 2 contradicts the fact that f A is ͉ The only difference in the proofs will be that every application pure. We conclude that f A␣ is pure and hence that S is closed. of Kadison’s transitivity theorem will now require the n-fold Next observe that there is a sequence (xn)inA such that ͉ ͉͞ʈ ʈ 3 ʚ ␣ Ͻ Ꭽ version for a finite family of mutually inequivalent pure states f(xn) xn 1. Then (xn) A␣ for some 1, so that the (ref. 7, theorem 1.21.16). This achieves a proof of ref. 21, restriction of f to A has norm one, and hence is a state, for all theorem 7.3, for any simple separable C*-algebra, which can be  Ն ␣. Thus, without loss of generality, in proving unbounded- converted into a proof of the result we need in the same way that ness we can assume the restriction of f to any A␣ is a state. ␣ Ͻ Ꭽ ʦ this is done in ref. 21, theorem 7.5. Let 1. We first claim that for any x A␣, for sufficiently large  Ն ␣ we have Corollary 3. Let A be a simple, separable, unital C*-algebra and let ͉ ϭ ͑ ϩ ͒͞ f ͑ ͒ ϭ ͑ ͒ f and g be inequivalent pure states on A. Then there is a simple, f A f1 f2 2 f1 x f2 x separable, unital C*-algebra B that unitally contains A such that f ʦ and g have unique state extensions to B and these extensions are whenever f1 and f2 are states on A. That is, for each x A␣ there ␣Ј Ն ␣  Ն ␣Ј equivalent. exists such that the above holds for all .Ifthe Proof: Observe first that if f and f are equivalent pure states displayed condition holds for all states f1 and f2 on A, then we 1 2 ͉ on A and f has a unique state extension to B, then so does f . say that f A is pure on x. 1 2 ʦ Ͼ Indeed, if u ʦ A is a unitary such that f ϭ u*f u then u also Suppose the claim fails for some x A␣. Then there exist 1 2 ʚ Ꭽ   belongs to B and conjugates the set of extensions of f1 with the 0 and an unbounded set T 1 together with states f1 and f2 on A  ʦ set of extensions of f2. for all T, such that ء It follows from the result quoted before this corollary that ͉ ϭ ϩ  ͞ ͉ ͑ ͒ Ϫ ͑ ͉͒ Ն ʦ f A (f1 f2 ) 2 and f1 x f x . [ ] given any sequence ( n)(n Z) of inequivalent irreps, there is an automorphism of A such that ؠ is equivalent to ϩ n n 1 (If no such and T existed, then for each n ʦ N we could find for all n. Now, since A is simple and separable and has inequiva-   ␣ Ͻ Ꭽ such that for any  Ն ␣ ,nostatesf and f on A satisfy lent pure states it cannot be type I, and therefore it has n 1 n 1 2 with ϭ 1͞n. Then f͉A would be pure on x for all  Ն sup ␣ , uncountably many inequivalent irreps (ref. 6, corollary 6.8.5). *  n contradicting our assumption that the claim fails for x.) Now let Let ( n) be any sequence of mutually inequivalent irreps such  ʦ  Ͼ  U be an ultrafilter on T that contains the set { T : 0} that 1 and 2 are the GNS irreps arising from f and g, and find  Ͻ Ꭽ ϭ ؠ for each 0 1, and define states g1 and g2 on A by g1 limu an automorphism as above. Then g is equivalent to f , and  ϭ  f1 and g2 limu f2, where the limits are taken pointwise on neither f nor g is equivalent to itself composed with any nonzero ϭ ϩ ͞ ͉ Ϫ ͉ Ն elements of A. Then f (g1 g2) 2 and g1(x) f(x) hence power of . Define :Z3 Aut(A)by ϭ n and let B be the crossed f g1, contradicting purity of f. (This argument could also be n carried out by using universal nets.) This establishes the claim. product of A by this action of Z. By Theorem 2, f and g extend ␣ Ͻ Ꭽ Now for any 1, since A␣ is separable we can find a dense uniquely to B, and therefore so do all pure states equivalent to ʚ ؠ sequence (xn) A␣, and the claim implies that for sufficiently g, in particular f , by the comment at the start of this proof.  ͉ ͉ Now if fЈ and (f ؠ )Ј are the unique extensions of f and f ؠ to large , f A is pure on every xn. By density, f A is then pure on every x ʦ A␣. Let ␣* be the least ordinal larger than ␣ such that B, then  Ն ␣ ͉ ʦ * implies that f A is pure on every x A␣. ␣ Ͻ Ꭽ ␣ ϭ ␣ ,* fЈ͑T ͑ x T ͒TϪ ͒ ϭ fЈ͑͑x ͒T ͒ ϭ f͑͑x ͒͒ ϭ ͑f ؠ ͒Ј͑x T ͒ Finally, to see that S is unbounded, fix 1. Let 1 1 n n 1 n n 0 n n ␣ ϭ ␣  ϭ ␣ ͉ ʦ 2 **, etc., and let sup n. Then f A is pure on every x ⌺ ഫ ␣ ͉  for any finite sum xnTn using the comment following Theorem A n, and hence by continuity f A is a pure state on A . This Ј ϭ ؠ Ј Ј  ʦ 2. This shows that T1f TϪ1 (f ) , and hence f is equivalent shows that S, as desired. Ꭽ to (f ؠ )Ј. Using the first paragraph again, we see that the unique A subset of 1 is stationary if it intersects every closed Ꭽ extensions of f and g to B are equivalent. unbounded subset of 1. We require the following version of the B is clearly separable and unital, and it unitally contains A.It diamond principle ({): there exists a transfinite sequence of ␣ 3 Ꭽ ␣ Ͻ Ꭽ Ꭽ is simple by ref. 22, theorem 3.1. functions h␣ : 1 ( 1) such that for any function h : 1 3 Ꭽ ␣ ͉ ϭ We thank A. Kishimoto for a simplification in the above proof. 1 the set { : h ␣ h␣} is stationary (see ref. 13, 22.20, or ref. 14, exercise II.51). The Counterexample Let S(A) denote the set of states on a C*-algebra A. Ꭽ ʚ A subset S of 1 is said to be closed if for every countable S0 ʦ ␣ ʦ Ꭽ { S we have sup S0 S. It is unbounded if for every 1 there Theorem 5. Assume . Then there is a counterexample to Naimark’s Ꭽ exists  ʦ S such that  Ͼ ␣. problem that is generated by 1 elements. We call a nested transfinite sequence of C*-algebras (A␣) Proof: Let (h␣) be a transfinite sequence of functions which { ␣ Ͻ Ꭽ continuous if for every limit ordinal ␣ we have A␣ ϭ ഫϽ␣A. verifies in the form given above. For 1 we recursively construct a continuous nested transfinite sequence of simple ␣ Ͻ Ꭽ Lemma 4. Let (A␣), 1, be a continuous nested transfinite separable unital C*-algebras A␣, all with the same unit; pure sequence of separable C*-algebras and set A ϭ ഫA␣. Then A is a states f␣ on A␣ with the property that ␣ Ͻ  implies f is the C*-algebra, and if f is a pure state on A then {␣ : f restricts to a pure unique state extension of f␣; and injective functions ␣ : S(A␣) 3 Ꭽ state on A␣} is closed and unbounded. 1 as follows. Let A0 be any simple, separable, infinite Proof: We observe first that A is automatically complete. For dimensional, unital C*-algebra and let f0 be any pure state on A0. ʚ ␣ ʦ { any sequence (xn) A we can find indices n such that xn A␣ , Since implies the continuum hypothesis (see ref. 13 or 14), ʚ ␣ ϭ ␣ n Ꭽ and then (xn) A␣ for sup n. Thus if (xn) is Cauchy, its limit S(A0) has cardinality at most 1, so there exists an injective Ꭽ belongs to A␣ and hence to A. This shows that A is a C*-algebra. function from S(A0) into 1; let 0 be any such function. Let f be a pure state on A and let S ϭ {␣ : f restricts to a pure To proceed from stage ␣ of the construction to stage ␣ ϩ 1 ʚ ␣ state on A␣}. First we verify that S is closed. Suppose S0 S is when is a limit ordinal, first check whether there is a pure state ␣ ϭ ͉  ϭ ͉ countable and let sup S0.Iff A␣ is not pure then we can write g␣ on A␣, not equivalent to f␣, such that h␣( ) (g␣ A) for all
7524 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0401489101 Akemann and Weaver Downloaded by guest on October 1, 2021  Ͻ ␣. (By injectivity of all , there is at most one such g␣.) If Proof: Since A cannot be type I, it follows that there is a ϭ so, let A␣ϩ1 be the C*-algebra B given by Corollary 3 with f subalgebra B of A and a surjective *-homomorphism of B onto ϭ ϭ Rϱ f␣ and g g␣. Let f␣ϩ1 be the unique extension of f␣ to A␣ϩ1, and the Fermion algebra F 1 M2(C) (ref. 6, corollary 6.7.4). Let Ꭽ as above let ␣ϩ1 be any injective function from S(A␣ϩ1) into 1. f1 and f2 be the pure states on M2(C) defined by If there is no such state g␣, and whenever ␣ is a successor ordinal ␣ ϭ ϭ ϭ ϭ a a a a (or 0), let A␣ϩ1 A␣, f␣ϩ1 f␣, and ␣ϩ1 ␣. At limit ͩͫ 11 12ͬͪ ϭ ͩͫ 11 12ͬͪ ϭ f1 a11 and f2 a22 . ordinals ␣, let A␣ ϭ ഫϽ␣ A (it is standard that A␣ will be a21 a22 a21 a22 simple, given that each A is simple), define f␣ by requiring ʦ Then for any sequence of indices in {1, 2} the product state f␣͉A ϭ f for all  Ͻ ␣ (f␣ will be pure, by the argument in the  f ϭ R f is a pure state on F. Moreover, ʈf Ϫ fЈʈ ϭ 2 for any two second paragraph of the proof of Lemma 4), and as before let n in Ꭽ distinct states of this form. Lifting to B and extending to A,we ␣ be any injective function from S(A␣) into 1. Ꭽ ϭ ഫ ʦ ͉ ϭ obtain a family of 2 0 pure states on A, the distance between any Let A ␣ϽᎭ A␣ and define f S(A)byf A␣ f␣. Here f is 1 ␣ Ͻ  two of which is 2. As A is a counterexample to Naimark’s well defined since f is an extension of f␣ whenever . Also, Ꭽ problem, there exists a family of 2 0 unitaries in the unitization f is a pure state, again by the reasoning in the second paragraph ˜ A of A that conjugate this family of pure states to a single of the proof of Lemma 4 coupled with the fact that each f␣ is pure. We claim that every pure state g on A is equivalent to f. To (arbitrary) pure state g on A. Uniform separation of the pure states implies uniform separation of the conjugating unitaries: if see this, it is enough to verify that g͉A is equivalent to f␣ for some ␣ ϭ * ϭ * ʦ A ␣; then since f␣ extends uniquely to A the same must be true of g u1 f1u1 u2 f2u2, then for all x ͉ g A␣ (see the first paragraph of the proof of Corollary 3), and the ͉ ͑ ͒ Ϫ ͑ ͉͒ ϭ ͉ ͑ Ϫ ͉͒ ͉ f1 x f2 x f1 x u*xu unitary in A␣ that implements the equivalence of f␣ and g A␣ must then also implement an equivalence between f and g. Now define Յ ͉f ͑x Ϫ u*x͉͒ ϩ ͉f ͑u*x Ϫ u*xu͉͒ Ꭽ 3 Ꭽ ␣ ϭ ͉ 1 1 h : 1 1 by setting h( ) ␣(g A␣). Let S be the set of limit ␣ ͉ Յ ʈ Ϫ ʈʈ ʈ ordinals such that g A␣ is pure. According to Lemma 4, S is 2 1A u x , closed and unbounded. (The intersection of any two closed ϭ ʈ Ϫ ʈ Յ ʈ Ϫ ʈϭ ʈ Ϫ ʈ ˜ unbounded sets is always closed and unbounded.) Therefore, by where u u*2u1,so f1 f2 2 1A u 2 u1 u2 . Thus A, Ꭽ { ␣ ͉ ϭ 0 there exists a limit ordinal such that g A␣ is pure and h␣ and hence A, cannot contain a dense set with fewer than 2 h͉␣, i.e., elements, and consequently it cannot be generated by fewer than Ꭽ 2 0 elements. ͑͒ ϭ ͑ ͉ ͒ h␣  g A Corollary 7. If the continuum hypothesis fails, then no C*-algebra  Ͻ ␣ ͉ for all .Ifg A␣ is equivalent to f␣ then we are done, and generated by Ꭽ elements is a counterexample to Naimark’s prob- ␣ 1 otherwise the construction at stage guarantees that f␣ϩ1 is lem. The existence of a counterexample to Naimark’s problem ͉ equivalent to the unique extension of g A␣ to A␣ϩ1, which must which is generated by Ꭽ elements is independent of ZFC. ͉ 1 be g A␣ϩ1. This completes the proof that f and g are equivalent. The first assertion of Corollary 7 follows immediately from Finally, A is infinite dimensional and unital, so it cannot be Proposition 6, and the second assertion follows from Theorem isomorphic to any K(H). 5 coupled with the first assertion, together with the fact that both We conclude by observing that if the continuum hypothesis diamond and the negation of the continuum hypothesis are fails then there is no counterexample to Naimark’s problem that consistent with ZFC (assuming ZFC is consistent). Ꭽ is generated by 1 elements. N.W. thanks Gert Pedersen for motivating his work on this problem. Proposition 6. Let A be a counterexample to Naimark’s problem. N.W. was partially supported by National Science Foundation Grant Ꭽ Then A cannot be generated by fewer than 2 0 elements. DMS-0070634.
1. Naimark, M. A. (1948) Usp. Mat. Nauk 3, 52–145 (Russian). 13. Jech, T. (1978) Set Theory (Academic, New York). 2. Naimark, M. A. (1951) Usp. Mat. Nauk 6, 160–164 (Russian). 14. Kunen, K. (1980) Set Theory: An Introduction to Independence Proofs (North- 3. Rosenberg, A. (1953) Am. J. Math. 75, 523–530. Holland, Amsterdam). 4. Kaplansky, I. (1951) Trans. Am. Math. Soc. 70, 219–255. 15. Devlin, K. J. (1977) The Axiom of Constructibility: A Guide for the Mathema- 5. Arveson, W. (1976) An Invitation to C*-Algebras (Springer, New York). tician (Springer, New York). 6. Pedersen, G. K. (1979) C*-Algebras and Their Automorphism Groups (Aca- 16. Shelah, S. (1974) Isr. J. Math. 18, 243–256. demic, New York). 17. Shelah, S. (1985) Isr. J. Math. 51, 273–297. 7. Sakai, S. (1971) C*-Algebras and W*-Algebras (Springer, New York). 18. Akemann, C., Anderson, J. & Pedersen, G. K. (1986) Can. J. Math. 38,
8. Fell, J. M. G. (1960) Ill. J. Math. 4, 221–230. 1239–1260. MATHEMATICS 9. Dixmier, J. (1960) Bull. Soc. Math. France 88, 95–112. 19. Anderson, J. (1979) Trans. Am. Math. Soc. 249, 303–329. 10. Glimm, J. (1961) Ann. Math. 73, 572–612. 20. Kishimoto, A., Ozawa, N. & Sakai, S. (2003) Can. Math. Bull. 46, 365–372. 11. Sakai, S. (1966) Bull. Am. Math. Soc. 72, 508–512. 21. Futamura, H., Kataoka, N. & Kishimoto, A. (2001) Int. J. Math. 12, 813–845. 12. Sakai, S. (1967) Proc. Am. Math. Soc. 18, 861–863. 22. Kishimoto, A. (1981) Commun. Math. Phys. 81, 429–435.
Akemann and Weaver PNAS ͉ May 18, 2004 ͉ vol. 101 ͉ no. 20 ͉ 7525 Downloaded by guest on October 1, 2021