aa r X i v : . [ m a t h . OA ] O c t TRIVIAL ENDOMORPHISMS OF THE CALKIN ALGEBRA
ANDREA VACCARO
Abstract.
We prove that it is consistent with ZFC that every (unital)endomorphism of the Calkin algebra Q ( H ) is unitarily equivalent to anendomorphism of Q ( H ) which is liftable to a (unital) endomorphism of B ( H ). We use this result to classify all unital endomorphisms of Q ( H )up to unitary equivalence by the Fredholm index of the image of theunilateral shift. Finally, we show that it is consistent with ZFC that theclass of C ∗ -algebras that embed into Q ( H ) is not closed under countableinductive limit nor tensor product. Introduction
Let H be a separable, infinite-dimensional, complex Hilbert space. TheCalkin algebra Q ( H ) is the quotient of B ( H ), the algebra of all linear,bounded operators on H , over the ideal of compact operators K ( H ). Let q : B ( H ) → Q ( H ) be the quotient map.Over the last 15 years, the study of the automorphisms of the Calkin alge-bra has been the setting for some of the most significant applications of settheory to C ∗ -algebras. The original motivation for these investigations is ofC ∗ -algebraic nature, and it dates back to the seminal paper [BDF77]. Oneof the most prominent questions asked in [BDF77] is whether there existsa K-theory reverting automorphism of Q ( H ) or, more concretely, an auto-morphism of Q ( H ) sending the unilateral shift to its adjoint. Since all innerautomorphisms act trivially on the K-theory of a C ∗ -algebra, a preliminaryquestion posed in [BDF77] is whether the Calkin algebra has outer automor-phisms. The answer turned out to be depending on set theoretic axioms.Phillips and Weaver show in [PW07] that outer automorphisms of Q ( H )exist if the Continuum Hypothesis
CH is assumed, while in [Far11b] Farahproves that the
Open Coloring Axiom
OCA (see definition 2.1) implies thatall the automorphisms of Q ( H ) are inner. It is still unknown whether it isconsistent with ZFC that there exists an automorphism of Q ( H ) sendingthe unilateral shift to its adjoint, since the automorphisms built in [PW07]act like inner automorphisms on every separable subalgebra of Q ( H ).In this note we investigate the effects of OCA on the endomorphisms of theCalkin algebra. The main consequence of OCA that we show is a completeclassification of the endomorphisms of Q ( H ), up to unitary equivalence, byessentially the Fredholm index of the image of the unilateral shift. Twoendomorphisms ϕ , ϕ : Q ( H ) → Q ( H ) are unitarily equivalent if thereis a unitary v ∈ Q ( H ) such that Ad( v ) ◦ ϕ = ϕ . Let End( Q ( H )) bethe set of all endomorphisms of Q ( H ) modulo unitary equivalence. Let Key words and phrases.
Calkin algebra, endomorphisms, unilateral shift, Open Color-ing Axiom.
End u ( Q ( H )) be the set of all the classes in End( Q ( H )) corresponding tounital endomorphisms. The operation of direct sum ⊕ naturally induces astructure of semigroup on both End( Q ( H )) and End u ( Q ( H )), as well as theoperation of composition ◦ . We fix an orthonormal basis { ξ k } k ∈ N of H andwe let S be the unilateral shift sending ξ k to ξ k +1 . Theorem 1.1.
Assume OCA. Two endomorphisms ϕ , ϕ : Q ( H ) → Q ( H ) are unitarily equivalent if and only if the following two conditions are satis-fied:(1) There is a unitary w ∈ Q ( H ) such that wϕ (1) w ∗ = ϕ (1) .(2) The (finite) Fredholm indices of ϕ ( q ( S ))+(1 − ϕ (1)) and ϕ ( q ( S ))+(1 − ϕ (1)) are equal.Moreover, the map sending ϕ ∈ End u ( Q ( H )) to − ind ( ϕ ( q ( S ))) is a semi-group isomorphism between ( End u ( Q ( H )) , ⊕ ) and ( N \ { } , +) , as well asbetween ( End u ( Q ( H )) , ◦ ) and ( N \ { } , · ) . An explicit description of (End( Q ( H )) , ⊕ ) and (End( Q ( H )) , ◦ ) underOCA is given in remark 5.1.The second consequence of OCA we prove in this paper is related to therecent works [FHV19], [FKV19] and [Vac19, Chapter 2], where methodsfrom set theory are employed in the study of nonseparable subalgebras of Q ( H ). Let E be the class of all C ∗ -algebras that embed into the Q ( H ).Under CH the Calkin algebra has density character ℵ (the first uncountablecardinal), hence a C ∗ -algebra belongs to E if and only if its density characteris at most ℵ (see [FHV19]). Therefore, when CH holds, the class E is closedunder all operations whose output, when starting from C ∗ -algebras of densitycharacter at most ℵ , is a C ∗ -algebra whose density character is at most ℵ ,such as minimal/maximal tensor product and countable inductive limit. Itis not clear whether CH is necessary to prove these closure properties, butwe prove that they might fail if CH is not assumed, answering [FKV19,Question 5.3]. Theorem 1.2.
Assume OCA.(1) The class E is not closed under minimal/maximal tensor product.Moreover, there exists A ∈ E such that A⊗ γ B / ∈ E for every infinite-dimensional, unital B ∈ E and for every tensor norm γ .(2) The class E is not closed under countable inductive limits.In particular, both (1) and (2) are independent from ZFC. Theorems 1.1 and 1.2 are proved in § ϕ : Q ( H ) → Q ( H ) is trivial if there isa unitary v ∈ Q ( H ) and a strongly continuous (i.e. strong-strong contin-uous) endomorphism Φ : B ( H ) → B ( H ) such that the following diagramcommutes. B ( H ) B ( H ) Q ( H ) Q ( H ) Φ q q Ad( v ) ◦ ϕ RIVIAL ENDOMORPHISMS OF THE CALKIN ALGEBRA 3
With this terminology, the main theorem of [Far11b] says that under OCAall automorphisms of Q ( H ) are trivial (indeed, up to unitary equivalence,they lift to the identity). We extend this result to all endomorphisms of Q ( H ). Theorem 1.3.
Assume OCA. All endomorphisms of the Calkin algebra aretrivial.
The proof of theorem 1.3 occupies both § §
4. Similarly to theorem1.2, theorems 1.1 and 1.3 cannot be proved in ZFC alone, in fact they failunder CH. In § Q ( H )existing when CH is assumed (examples 5.3 and 5.4), which highlight dif-ferent levels of failure of the classification of End( Q ( H )) in theorem 1.1. Inparticular, it is possible to find uncountably many inequivalent automor-phisms of Q ( H ) which send the unilateral shift to a unitary of index − Corollary 1.4.
The existence non-trivial endomorphisms of Q ( H ) is inde-pendent from ZFC. We remark that, unlike the results concerning automorphisms of Q ( H )in [Far11b], the commutative analogue of theorem 1.3 for P ( N ) / Fin doesnot hold. In [Dow14] it is proved that there are non-trivial endomorphismsof P ( N ) / Fin in ZFC. In this scenario, the best one can hope for is theso called weak Extension Principle , a consequence of OCA + MA intro-duced and discussed in [Far00, Chapter 4]. A crucial difference in thenoncommutative context is the presence of partial isometries allowing tocompress/decompress operators into/from infinite-dimensional subspaces of H . These objects play a crucial role in the proof of theorem 4.7.The observations and results exposed in this note are in continuity withthe numerous studies investigating the strong rigidity properties induced bythe Proper Forcing Axiom (of which OCA is a consequence) on the Calkinalgebra and on other nonseparable quotient algebras ([Far11b], [MV18],[Vig18], [BFV18]; see also [Far00], [Vel93]). The Continuum Hypothesis, onthe other hand, grants the opposite effect, allowing to prove the existenceof too many maps on these quotients for all of them to be ‘trivial’ ([PW07][CF14], [FMS13]). Woodin’s Σ -absoluteness theorem gives a deeper meta-mathematical motivation for the efficacy of CH in solving these problems(see [Woo02]).The paper is structured as follows. Section 2 contains preliminaries anddefinitions. Sections 3 and 4 are devoted to the proof of theorem 1.3. Insection 3 we show that locally trivial (see definition 2.2) endomorphisms of Q ( H ) are, up to unitary transformation, locally liftable with ‘nice’ unitariesin B ( H ). In section 4, adapting the main arguments from [Far11b] (see also[Far, § Q ( H ) are locallytrivial, and that all locally trivial endomorphisms are trivial. We remarkthat OCA is only needed in §
4, while the results in § § Q ( H )) underCH and some open questions. ANDREA VACCARO Notation and Preliminaries
The only extra set-theoretic assumption required for our proofs is theOpen Coloring Axiom OCA, which is defined as follows.
Definition 2.1.
Given a set X , let [ X ] be the set of all unordered pairsof elements of X . For a topological space X , a coloring [ X ] = K ⊔ K is open if the set K , when naturally identified with a symmetric subset of X × X , is open in the product topology. For a K ⊆ [ X ] , a subset Y of X is K -homogeneous if [ Y ] ⊆ K . OCA.
Let X be a separable, metric space, and let [ X ] = K ⊔ K be anopen coloring. Then either X has an uncountable K -homogeneous set, orit can be covered by countably many K -homogeneous sets.This statement, which contradicts CH, is independent from ZFC and itwas introduced by Todorcevic in [Tod89].For the rest of the paper, fix { ξ k } k ∈ N an orthonormal basis of H andidentify ℓ ∞ , the C ∗ -algebra of all uniformly bounded sequences of complexnumbers, with the atomic masa of all operators in B ( H ) diagonalized bysuch basis. With this identification, the algebra c = ℓ ∞ ∩ K ( H ) is the setof all sequences converging to zero. Given a set M ⊆ N , P M denotes theorthogonal projection onto the closure of span { ξ k : k ∈ M } . If M = { k } ,we simply write P k . Throughout this paper, all the partitions ~E of N arealways implicitly assumed to be composed by consecutive, finite intervals.Given such a partition ~E = { E n } n ∈ N , D [ ~E ] is the von Neumann algebraof all operators in B ( H ) for which each span { ξ k : k ∈ E n } is invariant.Equivalently, D [ ~E ] is the set of all operators which commute with P E n forevery n ∈ N . Given a subset X ⊆ N , D X [ ~E ] denotes the C ∗ -algebra of allthe operators in D [ ~E ] which act as zero on span { ξ k : k ∈ E n } for every n / ∈ X Given a unital ∗ -homomorphism Φ : D [ ~E ] → B ( H ) such that Φ[ D [ ~E ] ∩K ( H )] ⊆ K ( H ), ~n Φ denotes the sequence { rk(Φ( P k )) } k ∈ N of the (finite)ranks of the projections Φ( P k ). Notice that ~n Φ only depends on how Φ actson ℓ ∞ and that n i = n j whenever i, j ∈ E n for some n ∈ N . For a partition ~E = { E n } n ∈ N of N , ~E even is the partition { E n ∪ E n +1 } n ∈ N and ~E odd is thepartition { E } ∪ { E n +1 ∪ E n +2 } n ∈ N .The strong topology on B ( H ) (and on any subalgebra of B ( H )) is the oneinduced by the pointwise norm convergence on H , hence a sequence { T n } n ∈ N of operators in B ( H ) strongly converges to T iff T n ξ → T ξ in norm for every ξ ∈ H .For every partial isometry v in a C ∗ -algebra A , Ad( v ) is the endomorphismsending a to vav ∗ for every a ∈ A .For every subalgebra A of B ( H ), let A Q be the quotient A / ( K ( H ) ∩ A ).Given a map ϕ : A Q → Q ( H ), the function Φ : A → B ( H ) lifts (or is a liftof ) ϕ if the following diagram commutes: A B ( H ) A Q Q ( H ) Φ q qϕ RIVIAL ENDOMORPHISMS OF THE CALKIN ALGEBRA 5
Definition 2.2.
Given a C ∗ -algebra A ⊆ B ( H ), we say that an embedding(i.e. an injective ∗ -homomorphism) ϕ : A Q → Q ( H ) is trivial if thereexists a unitary v ∈ Q ( H ) and a strongly continuous (i.e. strong-strongcontinuous), ∗ -homomorphism Φ : A → B ( H ) such that Φ lifts Ad( v ) ◦ ϕ .An endomorphism ϕ : Q ( H ) → Q ( H ) is locally trivial if, for every partition ~E , the restriction ϕ ↾ D [ ~E ] Q is trivial.Given two operators T, S ∈ B ( H ), we use the notation T ∼ K ( H ) S toabbreviate T − S ∈ K ( H ). Analogously, for a C ∗ -algebra A and two functionsΦ , Φ : A → B ( H ), Φ ∼ K ( H ) Φ abbreviates Φ ( a ) ∼ K ( H ) Φ ( a ) for all a ∈ A .Given a C ∗ -algebra A ⊆ B ( H ) ( A ⊆ Q ( H )), the commutant A ′ ∩ B ( H )( A ′ ∩ Q ( H )) is the set of all the operators in B ( H ) ( Q ( H )) commuting withall elements of A .The ( Fredholm ) index of an operator T ∈ B ( H ) is the integer dim(ker( T )) − codim( T [ H ]). An element a ∈ Q ( H ) is invertible if and only if it can belifted to an operator of finite index ([Mur90, Theorem 1.4.16]), which canbe assumed to be a partial isometry if a is a unitary. Remark . Strongly continuous, unital endomorphisms of B ( H ) have anextremely rigid structure. Indeed, strong continuity implies that such mapsare uniquely determined by how they behave on the projections whose rangeis 1-dimensional. Since all these projections are Murray-von Neumann equiv-alent, the same is true for their images, which therefore all have the samerank. For every m ∈ N , let Φ m : B ( H ) → B ( H ⊗ C m ) be the map sending T to T ⊗ m . As B ( H ) and B ( H ⊗ C m ) are isomorphic, with an abuse of nota-tion we consider Φ m as a map from B ( H ) into B ( H ). If Φ : B ( H ) → B ( H )is a unital, strongly continuous endomorphism sending compact operatorsinto compact operators, by the previous observation it is possible tofind an m ∈ N \ { } and a unitary U ∈ B ( H ) such that Φ = Ad( U ) ◦ Φ m . Ourclassification of End( Q ( H )) and End u ( Q ( H )) in theorem 1.1 will be basedon this simple observation. Since the commutant of the image of ℓ ∞ via Φ m is isomorphic to ℓ ∞ ( M m ( C )) (the C ∗ -algebra of all norm-bounded sequencesof m × m matrices with complex entries), the same is true for the commutantof the image of ℓ ∞ via Φ.Every unital endomorphism ϕ : Q ( H ) → Q ( H ) is uniquely determinedby its restrictions { ϕ ↾ D [ ~E ] Q } as ~E varies among all partitions of N . Thisis a consequence of the following standard fact (see [Far11b, Lemma 1.2] orthe proof of [Ell74, Theorem 3.1]). Proposition 2.4.
For every countable set { T n } n ∈ N in B ( H ) there exists apartition ~E of N such that for every n ∈ N there are T n ∈ D [ ~E even ] and T n ∈ D [ ~E odd ] such that T n ∼ K ( H ) T n + T n . Locally Trivial Endomorphisms
Given a unital, locally trivial endomorphism ϕ : Q ( H ) → Q ( H ), through-out this section we fix, for every partition ~E of N , a partial isometry of finiteindex v ~E and a strongly continuous, ∗ -homomorphism Φ ~E : D [ ~E ] → B ( H )such that Ad( v ~E ) ◦ Φ ~E lifts the restriction of ϕ to D [ ~E ] Q . In this part we ANDREA VACCARO show that, up to considering Ad( v ) ◦ ϕ for some unitary v ∈ Q ( H ), we canassume that Φ ~E is Φ m (as defined in remark 2.3) and v ~E is a unitary inthe commutant of Φ m [ ℓ ∞ ], for every partition ~E . We remark that no extraset-theoretic axiom is required in the present section. Remark . Notice that for a unital, locally trivial endomorphism ϕ : Q ( H ) → Q ( H ), for each partition ~E the projection Φ ~E (1) is a compactperturbation of the identity, hence its range has finite codimension r . There-fore, by multiplying v ~E by a suitable partial isometry of index − r , we canalways assume that Φ ~E is unital, and we will always implicitly do so. Lemma 3.2.
Let Φ : ℓ ∞ → B ( H ) and Φ : ℓ ∞ → B ( H ) be two strongly con-tinuous, unital ∗ -homomorpisms such that Φ [ c ] , Φ [ c ] ⊆ K ( H ) . Supposethere exist two partial isometries of finite index v , v such that Ad ( v ) ◦ Φ ∼ K ( H ) Ad ( v ) ◦ Φ . Then the sequences ~n Φ = { rk (Φ ( P k )) } k ∈ N and ~n Φ = { rk (Φ ( P k )) } k ∈ N are eventually equal.Proof. Since q ( v ) and q ( v ) are unitaries in Q ( H ), we can assume that v is the identity, and we denote v by v . Let ~n Φ = { n k } k ∈ N , ~n Φ = { m k } k ∈ N and suppose there is an infinite X ⊆ N such that n k > m k for every k ∈ X .We inductively define a set Y ⊆ X as follows.Let y be the minimum of X and let Y = { y } . Since rk(Φ ( P y )) > rk(Φ ( P y )) ≥ rk(Ad( v )(Φ ( P y ))), there is a norm-one vector ξ in theimage of Φ ( P y ) which also belongs to the kernel of Ad( v )(Φ ( P y )). Thisis the case since the codimension of ker(Ad( v )(Φ ( P y ))) is strictly smallerthan rk(Φ ( P y )).Suppose Y k = { y < · · · < y k } ⊆ X and that, for every h ≤ k , there isa norm-one vector ξ h such that Φ ( P y h ) ξ h = ξ h and k Ad( v )(Φ ( P Y k )) ξ h k < /
2. Let y k +1 be the smallest element in X greater than y k such that(1) k Ad( v )(Φ ( P Y k ))Φ ( P y k +1 ) k < / k Ad( v )(Φ ( P Y k ∪{ y k +1 } )) ξ h k < / h ≤ k .Such number y k +1 exists since all Ad( v )(Φ ( P Y k )) and the projection ontospan { v ∗ ξ h : h ≤ k } have finite rank and, by strong continuity, the se-quences { Φ ( P k ) } k ∈ N and { Φ ( P k ) } k ∈ N strongly converge to zero. Define Y k +1 = Y k ∪ { y k +1 } . We have to verify that Y k +1 satisfies the inductivehypothesis, namely that for every h ≤ k + 1 there is a norm-one vector ξ h such that Φ ( P y h ) ξ h = ξ h and k Ad( v )(Φ ( P Y k +1 )) ξ h k < /
2. For h ≤ k ,pick the ξ h given by the inductive hypothesis, and the inequality follows byitem (2). Since y k +1 ∈ X , it follows that rk(Φ ( P y k +1 )) > rk(Φ ( P y k +1 )) ≥ rk(Ad( v )(Φ ( P y k +1 ))). There exists thus a norm-one vector ξ k +1 in the im-age of Φ ( P y k +1 ) which also belongs to the kernel of Ad( v )(Φ ( P y k +1 )). Thisis the case since the codimension of ker(Ad( v )(Φ ( P y k +1 ))) is strictly smallerthan rk(Φ ( P y k +1 )). Because of this and item (1): k Ad( v )(Φ ( P Y k +1 )) ξ k +1 k = k Ad( v )(Φ ( P Y k )) ξ k +1 k ≤≤ k Ad( v )(Φ ( P Y k ))Φ ( P y k +1 ) k < / . Let Y = ∪ k ∈ N Y k . We show that, for every k ∈ N , the following holds k (Φ ( P Y ) − Ad( v )(Φ ( P Y ))) ξ k k ≥ / , We denote the restriction of Φ m to D [ ~E ] by Φ m . RIVIAL ENDOMORPHISMS OF THE CALKIN ALGEBRA 7 which contradicts Φ ( P Y ) ∼ K ( H ) Ad( v )(Φ ( P Y )). The previous inequalityfollows since, for every k ∈ N , by strong continuity of Φ we have thatΦ ( P Y ) ξ k = ξ k , and by strong continuity of Φ we have that k Ad( v )(Φ ( P Y )) ξ k k ≤ / . (cid:3) Lemma 3.3.
Let ϕ : Q ( H ) → Q ( H ) be a unital, locally trivial endomor-phism. There exists m ∈ N such that, for every ~E , the sequence ~n Φ ~E = { rk (Φ ~E ( P k )) } k ∈ N is, up to a finite number of entries, constantly equal to m .Proof. By lemma 3.2, it is enough to show that there exists a partition ~E such that ~n Φ ~E is eventually constant. Let ~E be the partition composed bythe intervals { k, k +1 } and ~E be the partition composed by { k +1 , k +2 } ,as k varies in N . Let ~n Φ ~E = { n k } k ∈ N and ~n Φ ~E = { m k } k ∈ N . On the onehand we have that n k = n k +1 and m k +1 = m k +2 for all k ∈ N , since thesecouple of numbers belong to the same intervals in ~E and ~E respectively.On the other hand, by lemma 3.2, there is j ∈ N such that n i = m i for all i ≥ j . It follows that there is m ∈ N such that n i = m i = m for all i ≥ j . (cid:3) Let Φ : D [ ~E ] → B ( H ) be a strongly continuous, unital ∗ -homomorphismsuch that ~n Φ is eventually constant with value m . This is not enough to inferthat, even up to a unitary transformation of H , Φ is a compact perturbationof Φ m . For instance, the map Ψ : ℓ ∞ → B ( H ) sending ( a , a , a , . . . ) ( a , a , a , a , . . . ) is not a compact perturbation of the identity, since it is acompact perturbation of Ad( S ), being S the unilateral shift. Nevertheless,by suitably ‘shifting’ Φ it is possible to obtain a compact perturbation ofΦ m , as shown in the following lemma. Lemma 3.4.
Let ~E be a partition of N and let Φ : D [ ~E ] → B ( H ) be astrongly continuous, unital ∗ -homomorphism such that Φ[ D [ ~E ] ∩ K ( H )] ⊆K ( H ) and such that ~n Φ is eventually constant with value m ∈ N \ { } . Thereexists a partial isometry w of finite index such that Ad ( w ) ◦ Φ ∼ K ( H ) Φ m .Proof. If two strongly continuous, unital embeddings Φ , Φ : D [ ~E ] → B ( H )are such that ~n Φ = ~n Φ , then there is a unitary u ∈ B ( H ) sending Φ ( P k ) H to Φ ( P k ) H for every k ∈ N such that Ad( u ) ◦ Φ = Φ . Thus, it is enoughto show that there is a partial isometry w of finite index and a stronglycontinuous, unital ∗ -homomorphism ˜Φ : D [ ~E ] → B ( H ) such that ˜Φ ∼ K ( H ) Ad w ◦ Φ and such that ~n ˜Φ is constantly equal to m . Let ~n Φ = { n k } k ∈ N and let h ∈ N be such that n k = m for all k ∈ E j and all j ≥ h , whichexists by lemma 3.3. Let k be the minimum of E h , let n = P k 0) or the orthogonal projection onto { ζ i } i ≥− r (if r ≤ 0) and − r ≤ n . Let ˜Φ := (Ψ ◦ Ad( Q )) ⊕ (Ad( S r Φ(1 − Q )) ◦ Φ), where Ψ is a unital ANDREA VACCARO embedding between Q B ( H ) Q ( ∼ = M k ( C )) and P B ( H ) P ( ∼ = M km ( C )). Themap ˜Φ is a strongly continuous, unital ∗ -homomorphism (multiplicativityfollows since S − r S r ≥ Φ(1 − Q ) and Q commutes with every element in D [ ~E ]) with ~n ˜Φ constantly equal to m and such that ˜Φ ∼ K ( H ) Ad( S r ) ◦ Φ. (cid:3) The following lemma, an analogue of [Far11b, Lemma 1.4], shows thatunital, locally trivial embeddings which lift to Φ m on ℓ ∞ /c have nice andregular lifts also on the other D [ ~E ] Q ’s. Lemma 3.5. Let ϕ : Q ( H ) → Q ( H ) be a unital, locally trivial endomor-phism such that Φ m lifts ϕ on ℓ ∞ /c . Then, for every partition ~E , thereexists a unitary u ~E in Φ m [ ℓ ∞ ] ′ ∩ B ( H ) ∼ = ℓ ∞ ( M m ( C )) such that Ad ( u ~E ) ◦ Φ m lifts ϕ ~E on D [ ~E ] Q .Proof. Fix a partition ~E , let v ~E be a partial isometry of finite index and letΦ ~E : D [ ~E ] → B ( H ) be a strongly continuous, unital ∗ -homomorphism suchthat Ad( v ~E ) ◦ Φ ~E lifts ϕ on D [ ~E ]. By assumption we have that Ad( v ~E ) ◦ Φ ~E ∼ K ( H ) Φ m on ℓ ∞ . By lemmas 3.3 and 3.4 there is a finite index isometry w such that Ad( v ~E ) ◦ Φ ~E ∼ K ( H ) Ad( w ) ◦ Φ m on D [ ~E ], hence the latter alsolifts ϕ on D [ ~E ] Q . We have therefore that Ad( w ) ◦ Φ m ∼ K ( H ) Φ m on ℓ ∞ ,which entails that w commutes, up to compact operators, with the elementsin Φ m [ ℓ ∞ ]. The commutant of Φ m [ ℓ ∞ ] is (isomorphic to) ℓ ∞ ( M m ( C )) and by[JP72, Theorem 2.1] we have that w is a compact perturbation of an element u in ℓ ∞ ( M m ( C )). Being an element of ℓ ∞ ( M m ( C )), the operator u hasFredholm index zero, moreover u ∼ K ( H ) w entails that its class q ( u ) in Q ( H )is a unitary. Therefore, the polar decomposition of u in ℓ ∞ ( M m ( C )) providesa unitary u ~E in the commutant of Φ m [ ℓ ∞ ] such that u ~E ∼ K ( H ) w . (cid:3) All Endomorphisms are Trivial We split the proof of theorem 1.3 in two steps. We first prove that allunital, locally trivial endomorphisms of Q ( H ) are trivial, then we showthat all unital endomorphisms of Q ( H ) are locally trivial. We use OCA inboth proofs. The non-unital case follows from the unital one, since everyendomorphism ϕ : Q ( H ) → Q ( H ) can be thought as a unital endomorphismwith codomain Q ( ϕ (1) H ).4.1. Locally trivial endomorphisms are trivial.Theorem 4.1. Assume OCA . Every unital, locally trivial endomorphism ϕ : Q ( H ) → Q ( H ) is trivial. Fix a unital endomorphism ϕ : Q ( H ) → Q ( H ). The endomorphism ϕ istrivial if and only if Ad( v ) ◦ ϕ is trivial for some unitary v ∈ Q ( H ). Thus,by lemmas 3.4 and 3.5, we can assume that there is m ∈ N such that ϕ liftsto Φ m when restricted to ℓ ∞ /c , and that for every partition ~E there is aunitary u ~E in ℓ ∞ ( M m ( C )) such that Ad( u ~E ) ◦ Φ m lifts ϕ on D [ ~E ] Q . Givena unitary u ∈ ℓ ∞ ( M m ( C )) we denote Ad( u ) ◦ Φ m by Φ u (the endomorphism ϕ , and therefore the integer m , will be always fixed through this section,hence we omit m in this notation). RIVIAL ENDOMORPHISMS OF THE CALKIN ALGEBRA 9 The proof of theorem 4.1 is inspired to [Far11b, Section 3], where theorem4.1 is proved for an automorphism, hence in case m = 1. The idea is toglue together the various u ~E in a coherent way in order to define a unitary u ∈ ℓ ∞ ( M m ( C )) such that Φ u lifts ϕ globally. We identify the unitaries in ℓ ∞ ( M m ( C )) with elements in ( U ( M m ( C ))) N , being U ( M m ( C )) the unitarygroup of M m ( C ).For u = ( u ( i )) i ∈ N , v = ( v ( i )) i ∈ N ∈ U ( ℓ ∞ ( M m ( C ))) and I ⊆ N , define∆ I ( u, v ) := sup i,j ∈ I k u ( i ) u ∗ ( j ) − v ( i ) v ∗ ( j ) k . Lemma 4.2. For all I ⊆ N and u, v ∈ U ( ℓ ∞ ( M m ( C ))) :(1) ∆ I ( u, v ) ≤ i ∈ I k u ( i ) − v ( i ) k .(2) ∆ I ( u, v ) ≥ sup j ∈ I k u ( j ) − v ( j ) k − inf i ∈ I k u ( i ) − v ( i ) k . In particular,if u ( k ) = v ( k ) for some k ∈ I , then ∆ I ( u, v ) ≥ sup j ∈ I k u ( j ) − v ( j ) k .(3) If w ∈ U ( M m ( C )) then ∆ I ( u, v ) = ∆ I ( u, vw ) .(4) If I ∩ J = ∅ , then ∆ I ∪ J ( u, v ) ≤ ∆ I ( u, v ) + ∆ J ( u, v ) .(5) inf w ∈U ( M m ( C )) sup i ∈ I k u ( i ) − v ( i ) w k ≤ ∆ I ( u, v ) ≤ w ∈U ( M m ( C )) sup i ∈ I k u ( i ) − v ( i ) w k .Proof. The proof of this lemma can be easily inferred from the proof of[Far11b, Lemma 1.5]. We have that k u ( i ) u ∗ ( j ) − v ( i ) v ∗ ( j ) k = k v ∗ ( i )( u ( i ) u ∗ ( j ) − v ( i ) v ∗ ( j )) u ( j ) k = k v ∗ ( i ) u ( i ) − − v ∗ ( j ) u ( j ) k = k v ∗ ( i )( u ( i ) − v ( i )) − v ∗ ( j )( u ( j ) − v ( j )) k . This entails |k u ( i ) − v ( i ) k − k u ( j ) − v ( j ) k| ≤ k u ( i ) u ∗ ( j ) − v ( i ) v ∗ ( j ) k ≤≤ k u ( i ) − v ( i ) k + k u ( j ) − v ( j ) k , from which both item (1) and (2) follow. Item (3) is straightforward to checksince v ( i ) ww ∗ v ∗ ( j ) = v ( i ) v ∗ ( j ). Notice that, unlikely the 1-dimensionalcase, it is important to consider vw rather than wv . Item (4) follows by thetriangular inequality, since k u ( i ) u ∗ ( j ) − v ( i ) v ∗ ( j ) k is equal to k v ∗ ( i ) u ( i ) − v ∗ ( j ) u ( j ) k . Item (5) follows by item (3) plus items (1) and (2). (cid:3) Lemma 4.3. Let u = ( u ( i )) i ∈ N , v = ( v ( i )) i ∈ N ∈ U ( ℓ ∞ ( M m ( C ))) .(1) If lim i →∞ k u ( i ) − v ( i ) k = 0 then Φ u ∼ K ( H ) Φ v .(2) Φ u ∼ K ( H ) Φ v on D [ ~E ] if and only if lim sup n ∆ E n ( u, v ) = 0 .Proof. This lemma (and its proof) is an adapted version of [Far11b, Lemma1.6] for endomorphisms. If lim i →∞ k u ( i ) − v ( i ) k is zero, it means that u ∼ K ( H ) v , hence Φ u ∼ K ( H ) Φ v .In order to prove item (2), suppose first that lim sup n ∆ E n ( u, v ) = 0. Forevery n ∈ N , let k n be min( E n ). Let w = ( w ( i )) i ∈ N ∈ ℓ ∞ ( M m ( C )) be theunitary defined, for i ∈ E n , as w ( i ) := v ( i ) v ∗ ( k n ) u ( k n ) . The unitary P P E n ⊗ v ∗ ( k n ) u ( k n ) belongs to the commutant of Φ m [ D [ ~E ]],hence Φ w = Φ v on D [ ~E ]. On the other hand, by items (2)-(3) of lemma 4.2 we have that, for i ∈ E n , k w ( i ) − u ( i ) k ≤ ∆ E n ( u, w ) = ∆ E n ( u, v ). Thuslim i →∞ k w ( i ) − u ( i ) k = 0 and, by item (1) of this lemma, Φ u ∼ K ( H ) Φ w = Φ v .To prove the other direction, suppose there is ǫ > { n k } k ∈ N such that ∆ E nk ( u, v ) > ǫ . Fix two sequences i k , j k ∈ E n k such that k u ( j k ) u ∗ ( i k ) − v ( j k ) v ∗ ( i k ) k > ǫ for every k ∈ N . Let η k ∈ C m be a norm-onevector witnessing the previous inequality. Let V be the partial isometry in D [ ~E ] moving ξ i k to ξ j k (from the orthonormal basis of H we fixed at thebeginning of § 2) for every k ∈ N and sending all other vectors in { ξ n } n ∈ N tozero. We have that, if ζ ∈ Φ m ( P i k ),Φ u ( V )( ζ ) = u Φ m ( V ) u ∗ ( i k )( ζ ) = u ( j k ) u ∗ ( i k )( ζ ) , Φ v ( V )( ζ ) = v Φ m ( V ) v ∗ ( i k )( ζ ) = v ( j k ) v ∗ ( i k )( ζ ) . Thus, for the vector η k we fixed before (or rather for 0 ⊕ · · · ⊕ ⊕ η k ⊕ . . . ,where the non-zero coordinate appears in the i k -th position), we have k (Φ u ( V ) − Φ v ( V ))( η k ) k = k ( u ( j k ) u ∗ ( i k ) − v ( j k ) v ∗ ( i k ))( η k ) k > ǫ. Since this holds for every k ∈ N , it follows that the difference Φ u ( V ) − Φ v ( V )is not compact. (cid:3) Given a function f ∈ N N we define E fn := [ f ( n ) , f ( n + 1)) ,F fn := [ f + ( n ) , f + ( n + 1)) ,E f, even n := [ f (2 n ) , f (2 n + 2)) ,E f, odd n := [ f (2 n + 1) , f (2 n + 3)) . The corresponding partitions are ~E f , ~F f , ~E f, even and ~E f, odd respectively.We shall denote ~E f + , even and ~E f + , odd by ~F f, even and ~F f, odd . Lemma 4.4. Let ϕ : Q ( H ) → Q ( H ) be a unital, locally trivial endomor-phism which can be lifted to Φ m on ℓ ∞ /c for some m ∈ N . For every f ∈ N N there is a unitary w ∈ ℓ ∞ ( M m ( C )) such that Φ w lifts ϕ on both D [ ~E f, even ] and D [ ~E f, odd ] .Proof. This proof follows the one of [Far11b, Lemma 3.5]. By assumptionthere are two unitaries u, v ∈ ℓ ∞ ( M m ( C )) such that Φ u and Φ v lift ϕ on D [ ~E f, even ] and D [ ~E f, odd ] respectively. We define inductively two unitaries u ′ , v ′ ∈ ℓ ∞ ( M m ( C )) as follows. For i ∈ [ f (0) , f (2)), let u ′ ( i ) = u ( i ). If u ′ ( i )has been defined for i < f (2 n ), for i ∈ [ f (2 n − , f (2 n + 1)) let v ′ ( i ) = v ( i ) v ∗ ( f (2 n − u ′ ( f (2 n − . If v ′ ( i ) has been defined for i < f (2 n + 1), for i ∈ [ f (2 n ) , f (2 n + 2)) let u ′ ( i ) = u ( i ) u ∗ ( f (2 n )) v ′ ( f (2 n )) . We have that v ′ ( f ( n )) = u ′ ( f ( n )), that Φ u = Φ u ′ on D [ ~E f, even ] and thatΦ v = Φ v ′ on D [ ~E f, odd ]. This implies that, by item (2) of lemma 4.2,sup i ∈ E fn k u ′ ( i ) − v ′ ( i ) k ≤ ∆ E fn ( u ′ , v ′ ) . On the other hand we have that Φ u ′ = Φ u ∼ K ( H ) Ψ v = Ψ v ′ on D [ ~E f ] byhypothesis (remember that D [ ~E f ] ⊆ D [ ~E f, even ] ∩ D [ ~E f, odd ]), therefore by RIVIAL ENDOMORPHISMS OF THE CALKIN ALGEBRA 11 item (2) of lemma 4.3 it follows that lim n ∈∞ ∆ E fn ( u ′ , v ′ ) = 0. By item (1) oflemma 4.3, we infer that Φ u ′ and Ψ v ′ agree on B ( H ) up to compact operator.In conclusion, Φ u ′ lifts ϕ on both D [ ~E f, even ] and D [ ~E f, odd ]. (cid:3) Given f, g ∈ N N , we write g ≤ ∗ f if g ( n ) ≤ f ( n ) for all but finitely many n ∈ N . A subset F ⊆ N N is ≤ ∗ -cofinal if for every g ∈ N N there is f ∈ F such that g ≤ ∗ f . Lemma 4.5 ([Far11b, Lemma 3.3]) . Assume F ⊆ N N is ≤ ∗ -cofinal.(1) If F is partitioned into countably many pieces, then at least one is ≤ ∗ -cofinal.(2) ( ∃ ∞ n )( ∃ i )( ∀ k ≥ n )( ∃ f ∈ F )( f ( i ) ≤ n and f ( i + 1) ≥ k ) .(3) { f + : f ∈ F } is ≤ ∗ -cofinal. Lemma 4.6 ([Far11b, Lemma 3.4]) . Let f, g ∈ N N be such that g ≤ ∗ f . Forall but finitely many n ∈ N there is i such that f + ( i ) ≤ g ( n ) < g ( n + 1) ≤ f + ( i + 2) . If f ( m ) ≥ g ( m ) for all m ∈ N , then the previous statement holdsfor every n ∈ N . Lemma 4.6 entails that if g ≤ ∗ f then for all but finitely many n ∈ N there is i n ∈ N such that E gn ⊆ F fi n ∪ F fi n +1 . In particular, if f ( m ) ≥ g ( m ) for all m ∈ N , then D [ ~E g ] is contained in the algebra generated by D [ ~F f, even ] ∪ D [ ~F f, odd ]. Proof of theorem 4.1. We can assume that ϕ : Q ( H ) → Q ( H ) is locallyrepresented on D [ ~E ] by Φ u ~E , where u ~E is a unitary in ℓ ∞ ( M m ( C )) (see theparagraph after the statement of theorem 4.1). Let X ⊂ N N × U ( M m ( C )) N be the set of all pairs ( f, u ) such that Φ u lifts ϕ on both D [ ~F f, even ] and D [ ~F f, odd ]. By lemma 4.4, for every f ∈ N N there is u such that ( f, u ) ∈ X .Fix ǫ > X ] = K ǫ ⊔ K ǫ , where the pair( f, u ), ( g, v ) has color K ǫ if there are m, n ∈ N such that ∆ F fn ∩ F gm ( u, v ) > ǫ .We consider N N with the Baire space topology, induced by the metric d ( f, g ) = 2 − min { n : f ( n ) = g ( n ) } . This is a complete separable metric. We consider U ( M m ( C )) N with theproduct of the strong operator topology on U ( M m ( C )) and X with theproduct topology. In this setting, it is straightforward to check that K ǫ isopen. Claim 4.6.1. Assume OCA. For every ǫ > there are no uncountable K ǫ -homogeneous subsets of X .Proof. Fix ǫ > H be an uncountable K ǫ -homogeneous subset of X . Let F = { g + : ∃ u ( g, u ) ∈ H} . We can assume that H , and thus F , has size ℵ . By OCA and [Tod89,Theorems 3.4 and 8.5] there is f ∈ N N which is an upper bound for F .Using the pigeonhole principle, we can assume that there is n ∈ N such that f ( m ) ≥ g + ( m ) for all g + ∈ F and all m ≥ n , and moreover that g + ( i ) = h + ( i ) for all g + , h + ∈ F and all i ≤ n . By increasing f by f ( n ) we can alsoassume that f ( m ) ≥ g + ( m ) for all g + ∈ F and m ∈ N . By lemma 4.6 this entails that for every n ∈ N there is i ∈ N such that E g + n = F gn ⊆ F fi ∪ F fi +1 .Let u be a unitary in ℓ ∞ ( M m ( C )) such that Φ u lifts ϕ on D [ ~F f, even ] and D [ ~F f, odd ]. Since, by the previous observations, for every g + ∈ F we havethat D [ ~F g ] is contained in the algebra generated by D [ ~F f, even ] ∪ D [ ~F f, odd ],it follows that Φ u lifts ϕ also on D [ ~F g ] and therefore, by item (2) of lemma4.3, we have that lim n →∞ ∆ F gn ( u, v ) = 0 for every ( g, v ) ∈ H . By taking anuncountable subset of F if necessary, we can assume that there is k suchthat ∆ F gm ( u, v ) < ǫ/ m ≥ k and all ( g, v ) ∈ H . By separability of U ( M m ( C )) N there are ( g, v ), ( h, w ) ∈ H such that g + ( i ) = h + ( i ) for all i ≤ k and k w i − v i k < ǫ/ i ≤ g + ( k ). This entails that if n, m ∈ N are suchthat F gn ∩ F hm = ∅ , then either both m, n ≤ k or m, n ≥ k . In the formercase it follows that ∆ F gn ∩ F hm ( v, w ) < ǫ by item (1) of lemma 4.2. If m, n ≥ k then we have∆ F gn ∩ F hm ( w, v ) ≤ ∆ F gn ∩ F hm ( u, v ) + ∆ F gn ∩ F hm ( w, u ) < ǫ. This is a contradiction since ( g, v ), ( h, w ) ∈ H . (cid:3) By OCA, for every ǫ > X into countably many K ǫ -homogeneous sets. Let ǫ n = 2 − n . Repeatedly using item (1) of lemma4.5, find sequences X ⊇ X ⊇ · · · ⊇ X n ⊇ . . . and 0 = m (0) < m (1) < · · · < m ( n ) < . . . such that X n is K ǫ n -homogeneous and such that the set { f : ( ∃ u )( f, u ) ∈ X n } is ≤ ∗ -cofinal. Let m ( n ) be the natural number given byitem (2) of lemma 4.5 for X n . For each n ∈ N fix a sequence { ( f n,i , u n,i ) } i ∈ N in X n such that, for some j i ∈ N (4.6.1) f + n,i ( j i ) ≤ m ( n ) < m ( n + i ) ≤ f + n,i ( j i + 1) . By compactness of U ( M m ( C )) N , we can assume that each sequence { u n,i } i ∈ N converges to some u n . Claim 4.6.2. There is a subsequence { u n k } k ∈ N such that sup i ∈ [ m ( n h ) , ∞ ) k u n k ( i ) − u n h ( i ) k ≤ ǫ k for all h ≥ k .Proof. We start by showing that ∆ [ m ( n ) , ∞ ) ( u h , u n ) ≤ ǫ h for all h < n . Sup-pose this is not the case and let m ( n ) ≤ i < i be such that k u h ( i ) u ∗ h ( i ) − u n ( i ) u ∗ n ( i ) k > ǫ h . There is j ∈ N such that k u h,j ( i ) u ∗ h,j ( i ) − u n,j ( i ) u ∗ n,j ( i ) k > ǫ h and by (4.6.1) there are k , k ∈ N such that f + n,j ( k ) ≤ m ( n ) < i < f + n,j ( k + 1) ,f + h,j ( k ) ≤ m ( h ) < m ( n ) < i < f + h,j ( k + 1) . In particular, this entails that ∆ F fh,jk ∩ F fn,jk ( u h,j , u n,j ) > ǫ h , which is a con-tradiction since ( f h,j , u h,j ) and ( f n,j , u n,j ) both belong to X h , which is K ǫ h -homogeneous. RIVIAL ENDOMORPHISMS OF THE CALKIN ALGEBRA 13 By item (5) of lemma 4.2, for every h < n there is w h,n ∈ U ( M m ( C )) suchthat sup i ≥ m ( n ) k u n ( i ) − u h ( i ) w h,n k ≤ ǫ h . The unitary w h,n exists by compactness of U ( M m ( C )). Given h < n < k ∈ N we have that, for i ≥ m ( k ) u h ( i ) w h,n ≈ ǫ h u n ( i ) ≈ ǫ h u k ( i ) w ∗ n,k ≈ ǫ h u h ( i ) w h,k w ∗ n,k , hence(4.6.2) k w h,n − w h,k w ∗ n,k k ≤ ǫ h . This can be used to show that there is an infinite Y = { n k } k ∈ N ⊆ N suchthat, for i < j ∈ N , then k − w n i ,n j k ≤ ǫ n i . To see this, define a coloring M ⊔ M on the triples of elements in N , by saying that the triple i < j < k is in M if and only if k − w j,k k ≤ ǫ i . Suppose there is an infinite M -homogeneous set Y . Let h be the minimumof Y . By compactness of the unit ball of M m ( C ) there is n ∈ Y big enoughso that, for some j < k < n all in Y we have that k w j,n − w k,n k < ǫ h . Itfollows that k w j,k − k ≤ k w j,k − w j,n w ∗ k,n k + k − w j,n w ∗ k,n k (4.6.2) ≤ ǫ h , which is a contradiction, since the triple ( h, j, k ) is supposed to be in M . ByRamsey’s theorem there is an infinite M -homogeneous set Y = { n k } k ∈ N .We have therefore, for j > i ≥ k − w n i ,n j k ≤ ǫ n i − . Without loss of generality we can assume that n ≥ 4, hence that, for every i ≥ ǫ n i − ≤ ǫ i − / ǫ i +1 . Summarizing, we have that for every k < h ∈ N sup i ≥ m ( n h ) k u n k ( i ) − u n h ( i ) k ≤ sup i ≥ m ( n h ) k u n k ( i ) w n k ,n h − u n h ( i ) k + k − w n k ,n h k≤ ǫ n k + ǫ k +1 ≤ ǫ k . (cid:3) Let { u n k } k ∈ N be the subsequence given by the previous claim and let v ∈ U ( M m ( C )) N be defined as v ( i ) = u n k ( i ) for all i ∈ [ m ( n k ) , m ( n k +1 )) and v ( i ) = u n ( i ) for all i ≤ m ( n ). It follows then that k v ( i ) − u n k ( i ) k < ǫ k forall i ≥ m ( n k ). Given j ∈ N and ( g, w ) ∈ X n j , we claim that for all i ∈ N we have ∆ F gi \ m nj ( v, w ) ≤ ǫ j . This is the case since for every i ∈ N there is h ∈ N such that [ f + n j ,h ( j h ) , f + n j ,h ( j h + 1)) ⊇ F gi \ m n j . Hence, since both g and f n j ,h belong to X n j , we have that ∆ F gi \ m nj ( w, u n j ,h ) <ǫ n j . By continuity, we also have ∆ F gi \ m nj ( w, u n j ) ≤ ǫ n j . Thus, in conclusion∆ F gi \ m nj ( v, w ) ≤ ∆ F gi \ m nj ( v, u n j ) + ∆ F gi \ m nj ( u n j , w ) ≤ h ∈ F gi \ m nj k v ( h ) − u n k ( h ) k + ǫ n j ≤ ǫ j . (4.6.3)We conclude by showing that Φ v lifts ϕ ~E for every partition ~E . Let g ∈ N N be such that ~E = ~E g and find u ∈ U ( M m ( C )) N such that Φ u lifts ϕ on D [ ~E g ].By item (2) of lemma 4.3, it is enough to show that lim n →∞ ∆ E gn ( u, v ) = 0.Fix k ∈ N , and let ( f, w ) ∈ X n k be such that f ≥ ∗ g . For all but finitelymany n ∈ N there is i ∈ N such that E gn ⊆ F fi n ∪ F fi n +1 . This implies thatlim n →∞ ∆ E gn ( w, u ) = 0, which, by item (4) of lemma 4.2 in turn entailslim n →∞ ∆ E gn ( u, v ) ≤ lim n →∞ ∆ E gn ( u, w ) + ∆ E gn ( w, v ) = lim n →∞ ∆ E gn ( w, v ) ≤ lim n →∞ ∆ F fin ∪ F fin +1 ( w, v ) ≤ lim n →∞ ∆ F fin ( w, v ) + ∆ F fin +1 ( w, v ) (4.6.3) ≤ ǫ k . The inequality above holds for every k ∈ N , thus lim n →∞ ∆ E gn ( u, v ) is zero. (cid:3) All endomorphisms are locally trivial.Theorem 4.7. Assume OCA. Every unital endomorphism ϕ : Q ( H ) →Q ( H ) is locally trivial.Proof. Given a partition ~E , we want to find a finite index isometry v ~E and a strongly continuous, unital ∗ -homomorphism Φ ~E : D [ ~E ] → B ( H )such that Ad( v ~E ) ◦ Φ ~E lifts ϕ on D [ ~E ] Q . Without loss of generality, wetake a partition ~E which is composed of intervals whose length is strictlyincreasing. First we need a fact following from OCA which is proved in[Far11b, § § 7] (see also [Far, § ∗ -homomorphism Ψ : D [ ~E ] → B ( H ) which lifts ϕ on D X [ ~E ], for some infinite X ⊆ N . The paper [Far11b] focuses onautomorphisms Q ( H ), but these proofs also work for unital endomorphisms.More specifically, [Far11b, Lemma 7.2] and [Far11b, Proposition 7.7] can beused, as shown in the proof of [Far11b, Proposition 7.1], to find an infinite Y ⊆ N such that ϕ has a C -measurable (see [Far11b, § ′ on D Y [ ~E ]. The proof of [Far11b, Theorem 6.3] (which does not requireOCA) shows how to find an infinite X ⊆ Y and refine Ψ ′ to a stronglycontinuous ∗ -homomorphism Ψ : D [ ~E ] → B ( H ) such that Ψ lifts ϕ on D X [ ~E ]. Alternatively, it is possible to use [MV18, Theorem 8.4] to directlyobtain X and Ψ. This result however, being a more general statementabout corona algebras rather than only about the Calkin algebra, requiresthe stronger assumption OCA ∞ + MA ℵ (see [MV18, § RIVIAL ENDOMORPHISMS OF THE CALKIN ALGEBRA 15 Ψ and X ⊆ N , the idea now is to exploit the abundance of partial isometriesin Q ( H ) to obtain a global unital lift in D [ ~E ]. Concretely, we ‘compress’elements of D [ ~E ] into D X [ ~E ], apply Ψ and finally we ‘decompress’ theirimage in B ( H ) (see also [Far11b, Lemma 4.1]).Let v ∈ B ( H ) be a partial isometry such that v ∗ v = 1, P := vv ∗ ≤ P X belongs to D X [ ~E ] and such that v D [ ~E ] v ∗ ⊆ D X [ ~E ]. Such partial isome-try exists since, by assumption, the length of the intervals in ~E is strictlyincreasing. Let Q be the image of P via Ψ. Since P ∈ D X [ ~E ] we have that(4.7.1) q ( Q ) = ϕ ( q ( P )) . Let w be a partial isometry lifting ϕ ( q ( v )), hence we have(4.7.2) q ( w ) = ϕ ( q ( v )) , ∼ K ( H ) w ∗ w and Q ∼ K ( H ) ww ∗ . Claim 4.7.1. Up to a compact perturbation, we can assume that w satisfiesthe following porperties.(1) ww ∗ ≤ Q .(2) w ∗ w ≥ Q .Proof. We start by proving item (1). Let w ′ be a lift of ϕ ( q ( v ∗ )). Then w ′ Q is also a lift of ϕ ( q ( v ∗ )), since q ( w ′ Q ) = ϕ ( q ( v ∗ )) ϕ ( q ( P )) = ϕ ( q ( v ∗ P )) = ϕ ( q ( v ∗ )) . Let w ′ = u | w ′ Q | be the polar decomposition of w ′ . Since | w ′ Q | is a com-pact perturbation of the identity, we have that u , whose kernel is equal toker( w ′ Q ) ⊆ ker( Q ), is also a lift of ϕ ( q ( v ∗ )) such that u ∗ u ≤ Q . Let w be u ∗ . Summarizing, we can assume that both 1 − w ∗ w and Q − ww ∗ are finiterank projections.In order to prove item (2), first notice that the space K := Q ⊥ H ∩ w ∗ wH is infinite dimensional, as the former is infinite dimensional and the latterhas finite codimension. Let n be the rank of 1 − w ∗ w and fix a set of linearlyindependent vectors { ζ k } k We are finally ready to prove theorems 1.1 and 1.2. Proof of theorem 1.1. We start by showing the following claim, which doesnot require OCA. Claim 5.0.1. The index of the image of the unilateral shift S via a trivial,unital endomorphism ϕ is finite and negative. RIVIAL ENDOMORPHISMS OF THE CALKIN ALGEBRA 17 Proof. By definition of trivial endomorphism, there is a unitary u ∈ Q ( H )such that Ad( u ) ◦ ϕ is induced by a strongly continuous endomorphism Φ of B ( H ), which can assumed to be unital (see remark 3.1). This means thatthere exists m ∈ N \ { } such that, up to unitary transformation, ϕ lifts tothe map Φ m : B ( H ) → B ( H ⊗ C m ) sending T to T ⊗ m (see remark 2.3).In particular, the index of ϕ ( q ( S )) is − m . (cid:3) The forward direction of the equivalence is straightforward. Suppose thusthat ϕ , ϕ are two endomorphisms of Q ( H ) that satisfy conditions (1) and(2) of the statement. By condition (1) we can assume that ϕ (1) = ϕ (1) = p . If there is a unitary v ∈ Q ( H ) such that Ad( v ) ◦ ϕ = ϕ , then vpv ∗ = p ,hence v and p commute, which means that v is a direct sum of a unitaryin p Q ( H ) p and a unitary in (1 − p ) Q ( H )(1 − p ). The only part of v actingnon-trivially on ϕ [ Q ( H )] is the one in p Q ( H ) p . Hence, without loss ofgenerality, we can assume that p = ϕ (1) = ϕ (1) = 1. By theorem 1.3 both ϕ and ϕ are trivial. Therefore, by condition (2) and claim 5.0.1, we havethat ϕ and ϕ , modulo unitary equivalence, lift to the same endomorphismon B ( H ), hence they are equal.The final sentence of the theorem follows from claim 5.0.1, since Φ m ⊕ Φ n =Φ m + n and Φ m ◦ Φ n = Φ mn for every m, n ∈ N \ { } . (cid:3) Remark . Every non-unital ϕ ∈ End( Q ( H )) can be written as a di-rect sum ϕ ⊕ ϕ is unital and 0 is the zero endomorphism of Q ( H ). Therefore, by theorem 1.3, every non-unital endomorphism ϕ is, upto unitary equivalence, equal to Φ m ⊕ m ∈ N . Consider theset N := ( N × { , } ) \ { (0 , } and, for ϕ ∈ End( Q ( H )), define ind( ϕ ) := − ind( ϕ ( q ( S )) + (1 − ϕ (1))), where S is the unilateral shift. Consider themap: Θ : End( Q ( H )) → N ϕ (0 , 0) if ϕ (1) = 0(ind( ϕ ) , 1) if ϕ (1) = 1(ind( ϕ ) , 0) if 0 < ϕ (1) < p ∈ Q ( H ) such that 0 < p < Q ( H ). For ϕ , ϕ ∈ End( Q ( H )) we have that ϕ ⊕ ϕ (and ϕ ◦ ϕ )is non-unital if and only if at least one between ϕ and ϕ is non-unital.Therefore, the map Θ is a semigroup isomorphism between (End( Q ( H )) , ⊕ )and ( N , +), where the addition on N is defined as ( n, i ) + ( m, j ) = ( n + m, i · j ). Analogously, Θ is an isomorphism between (End( Q ( H )) , ◦ ) and ( N , · ),where ( n, i ) · ( m, j ) = ( n · m, i · j ). Proof of theorem 1.2. (1) Let ϕ : Q ( H ) → Q ( H ) be a trivial, unital endo-morphism. Up to unitary transformation there is m ∈ N such that ϕ isinduced by the map Φ m : B ( H ) → B ( H ⊗ C m ) sending T to T ⊗ m (seeremark 2.3). The commutant of Φ m [ B ( H )] in B ( H ⊗ C m ) is isomorphicto M m ( C ). By [JP72, Lemma 3.2] also the commutant of ϕ [ Q ( H )] in thecodomain Q ( H ) is isomorphic to M m ( C ). Thus, the commutant of the im-age of a trivial, unital endomorphism of Q ( H ) is always finite dimensional. Let B ∈ E be unital and infinite-dimensional. Consider the algebraic tensorproduct Q ( H ) ⊗ alg B and suppose ψ : Q ( H ) ⊗ alg B → Q ( H ) is an embedding.The element ψ (1) is a non-zero projection. Since ψ (1) Q ( H ) ψ (1) ∼ = Q ( H ),we can assume that ψ is unital. On the one hand, by theorem 1.3 the re-striction of ψ to Q ( H ) is unital and trivial, on the other hand ψ sends B into the commutant of ψ [ Q ( H )], which is a contradiction.(2) Let {A n } n ∈ N be an increasing sequence of finite-dimensional, unitalC ∗ -algebra such that A := S n ∈ N A n is an infinite-dimensional, unital AF-algebra. We have that Q ( H ) ⊗ A n ∈ E for all n ∈ N , but, by the proof ofitem (1), Q ( H ) ⊗ A / ∈ E . (cid:3) By the results in [Far11b] we know that it is consistent with ZFC thatthere is no automorphism of Q ( H ) sending the unilateral shift S to itsadjoint. Combining claim 5.0.1 with theorem 1.3 and with the equalityind( q ( S ∗ )) = 1, we can generalize this statement to endomorphisms of Q ( H ). Corollary 5.2. Assume OCA. There is no endomorphism ϕ : Q ( H ) →Q ( H ) sending the unilateral shift to its adjoint or to any unitary of indexzero. The following two examples witness the failure of theorems 1.1 and 1.3when CH holds, suggesting a rather complicated picture of End( Q ( H )) inthat case. In particular, while End( Q ( H )) is countable under OCA (theorem1.1), CH implies that End( Q ( H )) has size 2 ℵ , as shown in the followingexample. Example 5.3. An automorphism is trivial if and only if it is inner. Henceall the inequivalent 2 ℵ outer automorphisms produced in [PW07] are exam-ples of non-trivial endomorphisms of Q ( H ) (outer automorphisms of Q ( H )can also be built using the weak Continuum Hypothesis , see [FS14] and [Far, § ℵ inequivalentautomorphisms which all send the unilateral shift to an element of index-1. Hence this invariant is not enough to classify End u ( Q ( H )), and theo-rem 1.1 fails under CH. It would be interesting to investigate whether CHimplies that for every endomorphism ϕ of Q ( H ) there are 2 ℵ inequivalentendomorphisms of Q ( H ) with the same invariant as ϕ . Example 5.4. In [FHV19] it is proved that all C ∗ -algebras of density char-acter ℵ can be embedded into Q ( H ) with a map whose restriction to theseparable subalgebras is a trivial extension . Under CH the density charac-ter of Q ( H ) is ℵ , thus there is a unital endomorphism ϕ : Q ( H ) → Q ( H )that sends the unilateral shift S to a unitary in Q ( H ) which lifts to a unitaryin B ( H ), which has therefore index zero. By claim 5.0.1, this map cannotbe a trivial endomorphism. With this example we see that without OCAthe range of values assumed by ind( ϕ ( S )), as ϕ varies in End u ( Q ( H )), can Given a C ∗ -algebra A and an embedding θ : A → Q ( H ), the map θ is a trivialextension of A iff there is a ∗ -homomorphism Θ : A → B ( H ) such that θ = q ◦ Θ. Thisnotion of trivial maps, albeit more common, is different from the one we used throughoutthis paper in in definition 2.2. RIVIAL ENDOMORPHISMS OF THE CALKIN ALGEBRA 19 be strictly larger than the negative numbers. Notice that the existence ofan automorphism sending S to S ∗ would give an example where the indexof the image of the shift is positive.In [Far11a] the author extends his results in [Far11b] to all Calkin algebrason nonseparable Hilbert spaces, showing that the Proper Forcing Axiomimplies that all automorphisms of the Calkin algebra on a nonseparableHilbert space are inner. It would be interesting to known whether thosetechniques can be generalized to study the semigroup of endomorphisms ofthe Calkin algebra on a nonseparable Hilbert space.In conclusion, we remark that an interesting consequence of the simplestructure of End u ( Q ( H )) under OCA is that the monoid of (End u ( Q ( H )) , ◦ )is commutative (theorem 1.1). To our knowledge, surprisingly, it is notknown whether commutativity fails when OCA is not assumed. Question 5.5. 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Vaccaro, C ∗ -algebras and the Uncountable: a systematic study of the com-binatorics of the uncountable in the noncommutative framework , Ph.D. thesis,Univeristy of Pisa & York University, 2019.[Vel93] B. Veliˇckovi´c, OCA and automorphisms of P ( ω ) / fin, Topology Appl. (1993),no. 1, 1–13. MR 1202874[Vig18] A. Vignati, Rigidity conjectures , arXiv preprint arXiv:1812.01306 (2018).[Woo02] W. H. Woodin, Beyond Σ absoluteness , Proceedings of the International Con-gress of Mathematicians, Vol. I (Beijing, 2002), Higher Ed. Press, Beijing, 2002,pp. 515–524. MR 1989202(A. Vaccaro) Department of Mathematics, Ben-Gurion University of the Negev,P.O.B. 653, Be’er Sheva 84105, Israel E-mail address : [email protected] URL ::