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Consistency of a Counterexample to Naimark's Problem

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Consistency of a Counterexample to Naimark's Problem 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 ʈa␭xa␭ Ϫ f(x)a␭ʈ 3 for 0 all x ʦ b(xTg)b 1 (computing the norm in the crossed product), which A (ref.
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