The Calkin algebra is not countably homogeneous
aa r X i v : . [ m a t h . OA ] F e b THE CALKIN ALGEBRA IS NOT COUNTABLY HOMOGENEOUS
ILIJAS FARAH AND ILAN HIRSHBERG
Abstract.
We show that the Calkin algebra is not countably homogeneous, inthe sense of continuous model theory. We furthermore show that the connectedcomponent of the unitary group of the Calkin algebra is not countably homogeneous.
Motivated by their study of extensions of C ∗ -algebras, Brown, Douglas and Fill-more asked whether the Calkin algebra has a K -theory reversing automorphism andwhether it has outer automorphisms at all ([4, Remark 1.6 (ii)]). By [16] and [8]the answer to the latter question is independent from ZFC. In particular, since innerautomorphisms fix K -theory, a negative answer to the former question is relativelyconsistent with ZFC. It is not known whether the existence of a K -theory reversingautomorphism of the Calkin algebra is relatively consistent with ZFC. All knownautomorphisms of the Calkin algebra ([16] and [8, § K -theory,as they are implemented by a unitary on every separable subalgebra of the Calkinalgebra.A scenario for using Continuum Hypothesis to construct a K -theory reversing auto-morphism of the Calkin algebra on separable Hilbert space, denoted Q , was sketchedin [11, § § Theorem 1.
The Calkin algebra Q is not countably homogeneous, and this is wit-nessed by a quantifier-free type. Theorem 2.
The group U ( Q ) of Fredholm index zero unitaries in Q is not countablyhomogeneous. Our theorems give negative answers to [10, Questions 5.2 and 5.7] and a novelobstruction to countable saturation of Q . In [10, Question 5.1] it was asked whetherall obstructions to (quantifier-free) countable saturation of Q are of K -theoretic na-ture. The obstruction given in our results is finer than the Fredholm index, but it is K -homological and therefore ultimately K -theoretical. In addition, the obstructiongiven in Theorem 1 is quantifier-free and one given in Theorem 2 appears to havelittle to do with the Fredholm index. It should be noted that one of the key ideas,using Ext( M ∞ ), is due to N.C. Phillips, and it was already used in the proof of [10,Proposition 4.2]. Date : May 11, 2018.
Model theory of C ∗ -algebras and their unitary groups is based on [2] and describedin [11, § § Formulas in logic of metric structures aredefined recursively. In case of C ∗ -algebras, atomic formulas are expressions of theform k t (¯ x ) k where t is a noncommutative *-polynomial in variables ¯ x = ( x , . . . , x n ).The set of all formulas is the smallest set F containing all atomic formulas such that (i)for every n , all continuous f : [0 , ∞ ) n → [0 , ∞ ) and all φ , . . . , φ n in F the expression f ( φ , . . . , φ n ) belongs to F and (ii) if φ ∈ F , m ≥
1, and x is a variable symbolthan both sup k x k≤ m φ and inf k x k≤ m φ belong to F (see [11, § φ ( x , . . . , x n ) is aformula, A is a C ∗ -algebra, and a , . . . , a n are elements of A , then the interpretation φ ( a , . . . , a n ) A is obviously defined by recursion. A condition is any expression of theform φ ≤ r for formula φ and r ≥ type is a set of conditions ([11, § φ = r is equivalent to the condition max( φ, r ) ≤ r andevery expression of the form φ ≥ r is equivalent to the condition min(0 , r − φ ) ≤ n ≥
1, the n -type is a typesuch that free variables occurring in its conditions are included in { x , . . . , x n } . Itis important that each free variable x is associated with a domain of quantification,which in our case reduces to asserting that k x k ≤ m for some fixed m .Given a C ∗ -algebra A and sequence ¯ a = ( a j : n ∈ N ) in A , the type of ¯ a in A is theset of all conditions φ ( x , . . . , x m ) ≤ r such that φ ( a , . . . , a m ) A ≤ r . A structure C is said to be countably homogeneous if for every two sequences ¯ a = ( a n : n ∈ N ) and¯ b = ( b n : n ∈ N ) with the same type and every c ∈ C there exists d ∈ C such that(¯ a, c ) and (¯ b, d ) have the same type. Our proof of the failure of countable homogeneityin Q will show that sequences ¯ a and ¯ b can be chosen to be finite.We recall the definitions the semigroups Ext( A ) and Ext w ( A ). If A is a unitalC ∗ -algebra, we consider injective unital *-homomorphisms π : A → Q (such a *-homomorphism is the Busby invariant of an extension of A by the compacts). Byslight abuse of notation, we call such a *-homomorphism an extension. Two extensions π j : A → Q , for j = 1 , u inthe Calkin algebra such that π = Ad u ◦ π . If u above is furthermore required tohave Fredholm index zero then we say that these extensions are equivalent. The setof such *-homomorphisms is equipped with the direct sum operation (using implicitlythe fact that M ( Q ) ∼ = Q ), and the set of equivalence relations forms a semigroup,denoted Ext w ( A ) or Ext( A ), respectively. They correspond to semigroups Ext u ∗ ( A, K )where ∗ = s, w and K denotes the algebra of compact operators on separable Hilbertspace as defined in [3, Definition 15.6.3, Proposition 15.6.2 and § A is a simple unital C ∗ -subalgebra of Q and p ∈ A ′ ∩ Q is a nonzero projection,then a pap is an injective unital *-homomorphism from A into pQp ∼ = Q . Theisomorphism between pQp and Q used here is chosen by picking an isometry v suchthat vv ∗ = p , and the map Q → pQp is given by x vxv ∗ . The choice of v is uniqueup to multiplication by a unitary, and therefore it does not affect the Ext w class. (The HE CALKIN ALGEBRA IS NOT COUNTABLY HOMOGENEOUS 3 choice of v can affect the Ext class, and therefore the choice of p only determines theweak equivalence class.) Therefore projections in A ′ ∩ Q determine Ext w -classses ofunital extensions of A , after identifying pQp with Q in the manner we described.A subalgebra A of Q is split if there is a unital *-homomorphism from Φ : A → B ( H )such that π ◦ Φ = id A . The following lemma is related to [12, Lemma 5.1.2 andLemma 5.1.2]. Lemma 3.
Let A be a simple separable unital subalgebra of Q and let p and q beprojections in A ′ ∩ Q . Then p and q are Murray-von Neumann equivalent in A ′ ∩ Q if and only if the extensions of A corresponding to p and q are weakly equivalent.Proof. The direct implication is trivially true because of our convention that fornonzero p ∈ A ′ ∩ Q we identify pQp with Q and p with unital extension a pap of A . We now prove the converse implication. If the extensions corresponding to p and q are weakly equivalent, then there exists a partial isometry v in Q such that v ∗ v = p , vv ∗ = q , and vpapv ∗ = qaq for all a ∈ A . It will suffice to check that v ∈ A ′ ∩ Q . Fix a ∈ A . We have vav ∗ = vv ∗ avv ∗ , and since vv ∗ ∈ A ′ ∩ Q , we have vv ∗ avv ∗ = avv ∗ and therefore vav ∗ = avv ∗ . Multiplying by v on the right hand sideand using v ∗ v ∈ A ′ ∩ Q we have vv ∗ va = avv ∗ v . But since v is a partial isometry wehave vv ∗ v = v , thus showing that va = av . (cid:3) If A is a separable, unital and nuclear C ∗ -algebra then Ext w ( A ) is a group ([7, p.586]). This implies that every extension of A corresponds to some p ∈ A ′ ∩ Q . To seethat, if π : A → Q is any given extension, then there exists an extension π : A → Q such that π ⊕ π is weakly equivalent to id A . The extension π corresponds to theprojection (cid:18) (cid:19) ∈ ( π ⊕ π )( A ) ′ ∩ M ( Q ). When we identify π ⊕ π with id A via a unitary and an isomorphism M ( Q ) ∼ = Q , the above matrix is identified with aprojection p ∈ A ′ ∩ Q as required. Lemma 4.
Let A be a separable unital subalgebra of Q such that Ext w ( A ) is a group.Then the Cuntz algebra O unitally embeds into A ′ ∩ Q if and only if A is split.Proof. Assume first that A is split. Recall that by Voiculescu’s theorem ([18], [1,Section 4]), all trivial extensions of A are equivalent. In particular, id A is equivalentto id A ⊕ id A : A → Q ⊗ M . Thus, if A is split then there is a projection p ∈ A ′ ∩ Q such that both pAp and (1 − p ) A (1 − p ) are split in pQp and (1 − p ) Q (1 − p ),respectively. Lemma 3 implies that 1, p and 1 − p are Murray–von Neumann equivalentand therefore O embeds unitally into A ′ ∩ Q .Now assume O unitally embeds into A ′ ∩ Q . Then the extension of A correspondingto 1 is an idempotent in Ext( A ). Since the identity is the only idempotent in agroup, A is split. (cid:3) Let A denote the CAR algebra, M ∞ . A is singly generated by [13]. Fix a generator g for A . Since A is nuclear, Ext w ( A ) is a group. ILIJAS FARAH AND ILAN HIRSHBERG
Lemma 5.
For every unital extension π of A , the type of π ( g ) in Q is the same asthe type of π ( g ) corresponding to the trivial extension π of A .Proof. Represent A as a direct limit of M n ( C ) and choose a n ∈ M n ( C ) such thatlim n a n = g . Fix a unital extension π of A . For n ∈ N the group Ext w ( M n ( C )) istrivial, and therefore the type of π ( a n ) in Q is the same as the type of π ( a n ) in Q .Fix a formula φ ( x ). Since the interpretation b φ ( b ) Q is continuous, we have φ ( π ( g )) Q = lim n φ ( π ( a n )) Q = lim n φ ( π ( a n )) Q = φ ( π ( g )) Q . Since φ was arbitrary, the conclusion follows. (cid:3) Proof of Theorem 1.
Fix a unital *-homomorphism π of A into Q and consider the2-type in x , x consisting of conditions x j π ( g ) = π ( g ) x j , x ∗ j x j = 1 , x x ∗ + x x ∗ = 1for j = 1 ,
2. By Lemma 4, this type is realized if and only if π is the trivial extension.Since A has both trivial and nontrivial extensions (as a matter of fact, Ext w ( A ) isuncountable by [5, Proposition 3] or [17]) and the type of π ( g ) does not depend onthe choice of the extension π by Lemma 5, Q is not (countably) homogeneous. (cid:3) The salient point in our proof of Theorem 2 is the fact that the presence of O in A ′ ∩ Q can be detected from A ′ ∩ U ( Q ). We note that in [14, Theorem 4.6] itwas shown that if B and C are simple C ∗ -algebras such that their unitary groupsare isometrically isomorphic then this isomorphism extends to an isomorphism or ananti-isomorphism of B and C . We were not able, however, to use this result directly.By Voiculescu’s theorem ([18]) for a unital separable C ∗ -subalgebra A of Q one has( A ′ ∩ Q ) ′ = A and Z ( A ′ ∩ Q ) = Z ( A ). We need the following self-strengthening ofthis result. Lemma 6. If A is a unital separable C ∗ -subalgebra of Q then ( A ′ ∩ U ( Q )) ′ = A and Z ( A ′ ∩ U ( Q )) = Z ( A ) ∩ U ( Q ) .Proof. Assume b ∈ Q is such that b / ∈ A . Since A = ( A ′ ∩ Q ) ′ , there exists anelement x ∈ A ′ ∩ Q such that xb = bx . By replacing x by its real or imaginarypart, we may assume that x is self-adjoint, and we may assume that k x k < π . Set u = exp( ix ). Then u ∈ A ′ ∩ U ( Q ) and since x ∈ C ∗ ( u ), we have ub = bu . Therefore b / ∈ ( A ′ ∩ U ( Q )) ′ . Since A = ( A ′ ∩ Q ) ′ and b was arbitrary, this proves ( A ′ ∩ U ( Q )) ′ = A .The second equality is a standard consequence of the first. If b ∈ Z ( A ′ ∩ U ( Q )),then by the above b ∈ A and therefore b ∈ Z ( A ). Since Z ( A ) ⊆ A ′ ∩ Q , the conclusionfollows. (cid:3) Lemma 7.
The Cuntz algebra O is the universal C ∗ -algebra generated by three uni-taries u , v and w satisfying the following relations: (1) u = v = w = 1 . HE CALKIN ALGEBRA IS NOT COUNTABLY HOMOGENEOUS 5 (2) k w − k = 1 . (3) uw u = − w . (4) vw v = e πi/ w .Proof. In [6, Theorem 2.6] Choi proved that every C ∗ -algebra generated by unitaries u and v and projection p satisfying the following conditions is isomorphic to O :(5) u = v = 1.(6) p + upu = 1 and p + vpv + v pv = 1.Denote γ = e πi/ . It is straightforward to check that such u and v , together with w = p + γvpv + γ v pv, satisfy our conditions. It will therefore suffice to prove that our conditions imply(7) w = p + γq + γ r , with projections p, q, r satisfying p + q + r = 1.(8) upu + p = 1 and p + vpv + v pv = 1.Since w = 1 and k w − k = 1 = | e πi/ − | , Sp( w ) is contained in { γ, , γ } , with atleast one of γ or γ belonging to it. By (4), the unitaries w and γ w are conjugateand Sp( w ) = { γ λ : λ ∈ Sp( w ) } . Therefore Sp( w ) = { γ, , γ } , and we can write(9) w = p + γq + γ r for projections p, q, r satisfying p + q + r = 1. By applying (4) to (9) we obtain vpv = q and vqv = r . In particular, p + vpv + v pv = 1.Since uw u = − w , u and w generate a unital copy of M ( C ) and p = ( w + 1) / p + upu = 1.Thus u, v and p = ( w +1) / (cid:3) Lemma 8.
There is a 4-type s (¯ x ) in the language of metric groups such that for aunital C ∗ -algebra A and a closed group G satisfying U ( A ) ⊆ G ⊆ A and Z ( G ) = T the following are equivalent. (1) s is realized in G . (2) A has a unital subalgebra isomorphic to O whose unitary group is includedin G .Proof. Define a type s (¯ x ) consisting of the following conditions:(3) x = x = x = 1.(4) sup y k x yx − y − − k = 0(5) k x − k = 1(6) k x − k = 1.(7) x x x = − x .(8) x x x = x x .Observe that x satisfies condition (4) if and only if x ∈ Z ( G ). Write γ = e πi/ , andnote that γ and γ are the only elements of Z ( A ) = T at the distance exactly 1 fromthe identity. Therefore (5) implies that x = γ · x = γ · ILIJAS FARAH AND ILAN HIRSHBERG If x = γ ·
1, the remaining conditions are satisfied by x , x and x in G if and onlyif they satisfy the assumptions of Lemma 7. If x = γ ·
1, then x , x and x − satisfythose conditions. Either way, we see that if G realizes the type then by Lemma 7there exists a unital copy of O in A .Since every unitary in O is of the form exp( ia ) for a self-adjoint a ([15]), its unitarygroup is connected. Therefore if A and G are as above then A has a unital copy of O if and only if it has a unital copy of O whose unitary group is included in G . By theabove, this is equivalent to s being realized in G . (cid:3) Proof of Theorem 2.
As in the proof of Theorem 1, let A denote the CAR alge-bra M ∞ and let g j , for j ∈ N , be an enumeration of a dense subgroup of the unitarygroup U ( A ). Fix a unital *-homomorphism π : A → Q . Since the unitary group of A is connected, we have π ( U ( A )) ⊆ U ( Q ) and π ( A ) ′ ∩ U ( Q ) = U ( π ( A )) ′ ∩ U ( Q ).As in Theorem 1, the type of ( π ( g j ) : j ∈ ω ) does not depend on the choice of π .Since Z ( A ) = C , Lemma 6 implies Z ( π ( A ) ′ ∩ U ( Q )) = T .Let 4-type s + (¯ x ) consist of s (¯ x ) as in Lemma 8 together with all conditions of theform k g j x k g − j x − k − k = 0for j ∈ N and 1 ≤ k ≤
3. Then s + is realized in U ( Q ) if and only if s is realized in U ( π ( A )) ′ ∩ U ( Q ) = π ( A ) ′ ∩ U ( Q ). Since the assumptions of Lemma 8 are satisfied, s is realized in U ( π ( A )) ′ ∩ U ( Q ) if and only if O unitally embeds into π ( A ) ′ ∩ Q .Since there are π : A → Q and π : A → Q such that π ( A ) ′ ∩ Q has a unital copyof O and π ( A ) ′ ∩ Q does not, our proof is complete. (cid:3) We conclude with a remark on the role of the Continuum Hypothesis in theconstruction of a possible K -theory reversing automorphism of the Calkin algebra.Woodin’s Σ ∼ absoluteness theorem (see [19]) implies that, under a suitable large car-dinal assumption, the following holds. If there exists a forcing extension in which theCalkin algebra has a K -theory reversing automorphism, then every forcing extensionin which the Continuum Hypothesis holds contains a K -theory reversing automor-phism of the Calkin algebra. This means that if the existence of a K -theory reversingautomorphism of the Calkin algebra is consistent with ZFC, then it most likely followsfrom the Continuum Hypothesis. Acknowledgment.
The results of this note were proved during first author’s visitto the Ben Gurion University in May 2015. He would like to thank the Departmentof Mathematics, and the second author in particular, for their warm hospitality. Wewould also like to thank the referee for suggesting several improvements.
References
1. W. Arveson,
Notes on extensions of C ∗ -algebras , Duke Math. J. (1977), no. 2, 329–355. HE CALKIN ALGEBRA IS NOT COUNTABLY HOMOGENEOUS 7
2. I. Ben Yaacov, A. Berenstein, C.W. Henson, and A. Usvyatsov,
Model theory for metric struc-tures , Model Theory with Applications to Algebra and Analysis, Vol. II (Z. Chatzidakis et al.,eds.), London Math. Soc. Lecture Notes Series, no. 350, Cambridge University Press, 2008,pp. 315–427.3. B. Blackadar, K -theory for operator algebras , second ed., Mathematical Sciences Research In-stitute Publications, vol. 5, Cambridge University Press, Cambridge, 1998.4. L. G. Brown, R. G. Douglas, and P. A. Fillmore, Extensions of C ∗ -algebras and K -homology ,Ann. of Math. (2) (1977), no. 2, 265–324.5. S. J. Cho, Strong extensions vs. weak extensions of C ∗ -algebras , Canad. Math. Bull. (1978),no. 2, 143–147.6. M.-D. Choi, A simple C ∗ -algebra generated by two finite-order unitaries , Canad. J. Math. (1979), no. 4, 867–880.7. M.-D. Choi and E.G. Effros, The completely positive lifting problem for c*-algebras , Annals ofMathematics (1976), 585–609.8. I. Farah,
All automorphisms of the Calkin algebra are inner , Ann. of Math. (2) (2011),no. 2, 619–661.9. ,
Logic and operator algebras , Proceedings of the Seoul ICM (S.Y. Jang, Y.R. Kim, D.-W.Lee, and I. Yie, eds.), vol. II, Kyung Moon SA, 2014, pp. 15–40.10. I. Farah and B. Hart,
Countable saturation of corona algebras , C.R. Math. Rep. Acad. Sci.Canada (2013), 35–56.11. I. Farah, B. Hart, and D. Sherman, Model theory of operator algebras II: Model theory , Israel J.Math. (2014), 477–505.12. N. Higson and J. Roe,
Analytic K -homology , Oxford Mathematical Monographs, Oxford Uni-versity Press, Oxford, 2000, Oxford Science Publications.13. C.L. Olsen and W.R. Zame, Some C ∗ -algebras with a single generator , Trans. Amer. Math. Soc. (1976), 205–217.14. A.L.T. Paterson, Harmonic analysis on unitary groups , J. Funct. Anal. (1983), no. 3, 203–223.15. N.C. Phillips, Exponential length and traces , Proc. Roy. Soc. Edinburgh Sect. A (1995),no. 1, 13–29.16. N.C. Phillips and N. Weaver,
The Calkin algebra has outer automorphisms , Duke Math. J. (2007), no. 1, 185–202.17. M. Pimsner and S. Popa,
On the Ext-group of an AF -algebra , Rev. Roumaine Math. Pures Appl. (1978), no. 2, 251–267.18. D. Voiculescu, A non-commutative Weyl-von Neumann theorem , Rev. Roumaine Math. PuresAppl. (1976), no. 1, 97–113.19. W.H. Woodin, Beyond Σ ∼ absoluteness , Proceedings of the International Congress of Mathe-maticians, Vol. I (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 515–524. Department of Mathematics and Statistics, York University, 4700 Keele Street,North York, Ontario, Canada, M3J 1P3
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