trivial embeddings of the calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · trivial embeddings...

Trivial Embeddings of the Calkin algebra

Andrea Vaccaro

Università di Pisa & York University

Texas A&M University, GPOTS 2019

Embeddings into the Calkin algebra

Let H be a separable, infinite-dimensional Hilbert space. TheCalkin algebra is Q(H) := B(H)/K(H).

QuestionWhat C∗-algebras embed into the Calkin algebra?

QuestionWhat can we say about the closure properties of the classE := {A : A ↪→ Q(H)}? For instance:

1 if A,B ∈ E , does A⊗γ B ∈ E?2 if {An}n∈N is an increasing sequence of C∗-algebras such thatAn ∈ E ∀n ∈ N, does A =

⋃n∈NAn ∈ E?

QuestionWhat can we say about the closure properties of the classE := {A : A ↪→ Q(H)}?

For instance:1 if A,B ∈ E , does A⊗γ B ∈ E?2 if {An}n∈N is an increasing sequence of C∗-algebras such thatAn ∈ E ∀n ∈ N, does A =

⋃n∈NAn ∈ E?

1 if A,B ∈ E , does A⊗γ B ∈ E?

2 if {An}n∈N is an increasing sequence of C∗-algebras such thatAn ∈ E ∀n ∈ N, does A =

⋃n∈NAn ∈ E?

The simplest answer

Theorem (Farah-Hirshberg-Vignati, 2018)Every C∗-algebras A of density character ℵ1 embeds into Q(H).

Moreover, the embedding is such that its restriction to eachseparable subalgebra of A can be lifted to an embedding intoB(H).

The Calkin algebra has density character (and size) c := 2ℵ0 , hencelarger C∗-algebras do not embed into it.If the Conitnuum Hypothesis (CH) holds, i.e. c = ℵ1, then:

1 all C∗-algebras whose size is less or equal than c embed intoQ(H), i.e. the Calkin algebra is universal,

2 the class E satisfies all ‘reasonable’ closure properties. Forinstance

⊗α<cQ(H) ∈ E .

The simplest answer

Theorem (Farah-Hirshberg-Vignati, 2018)Every C∗-algebras A of density character ℵ1 embeds into Q(H).Moreover, the embedding is such that its restriction to eachseparable subalgebra of A can be lifted to an embedding intoB(H).

⊗α<cQ(H) ∈ E .

The simplest answer

The Calkin algebra has density character (and size) c := 2ℵ0 , hencelarger C∗-algebras do not embed into it.

If the Conitnuum Hypothesis (CH) holds, i.e. c = ℵ1, then:1 all C∗-algebras whose size is less or equal than c embed intoQ(H), i.e. the Calkin algebra is universal,

⊗α<cQ(H) ∈ E .

The simplest answer

The Calkin algebra has density character (and size) c := 2ℵ0 , hencelarger C∗-algebras do not embed into it.If the Conitnuum Hypothesis (CH) holds, i.e. c = ℵ1,

then:1 all C∗-algebras whose size is less or equal than c embed intoQ(H), i.e. the Calkin algebra is universal,

⊗α<cQ(H) ∈ E .

The simplest answer

⊗α<cQ(H) ∈ E .

The simplest answer

⊗α<cQ(H) ∈ E .

CH is easy, real life is hard

Under PFA there is an abelian C∗-algebra of density character cwhich does not embed into Q(H). (Farah-Hirshberg-Vignati, 2018)

Theorem (Farah-Katsimpas-Vaccaro, 2018)The statement ‘Every C∗-algebra of density character (strictly) lessthan c embeds into the Calkin algebra’ is independent from ZFC.In particular, MA implies that such statement is true.

What about the closure properties of E when CH fails?

Trivial embeddings of Q(H)

A unital embedding φ : Q(H)→ Q(H) is trivial if there is a unitaryv ∈ Q(H) such that Ad(v) ◦ φ is liftable to a strong-strongcontinuous, unital ∗-homomorphism Φ of B(H) into B(H).

B(H) B(H)

Q(H) Q(H)

Adv◦φ

All strong-strong continuous, unital ∗-homomorphisms of B(H)into B(H) sending K(H) to K(H) have the form Ad(u) ◦ Φm

Φm : B(H)→ B(H ⊗ Cm)Ad(u)� B(H)

T 7→ T ⊗ Idm

B(H) B(H)

Q(H) Q(H)

Adv◦φ

T 7→ T ⊗ Idm

B(H) B(H)

Q(H) Q(H)

Adv◦φ

T 7→ T ⊗ Idm

Trivial embeddings are trivial

1 The set of all trivial embeddings Triv(Q(H)) can be classifiedup to unitary equivalence (in Q(H)) by N \ {0}.

Fix an orthonormal basis {ξk}k∈N of H and let S be theunilateral shift sending ξk to ξk+1.

Ind : Triv(Q(H))→ N \ {0}φ 7→ −ind(φ(S))

2 An automorphism of Q(H) is trivial if and only if it is inner.

Are there non-trivial embeddings?

1 The set of all trivial embeddings Triv(Q(H)) can be classifiedup to unitary equivalence (in Q(H)) by N \ {0}.Fix an orthonormal basis {ξk}k∈N of H and let S be theunilateral shift sending ξk to ξk+1.

Yes...

Theorem (Phillips-Weaver, 2007)Assume CH. There is an outer automorphism of the Calkin algebra.

The automorphism α built by Phillips and Weaver is such that forevery a ∈ Q(H) there is a unitary v ∈ Q(H) such thatα(a) = Ad(v)a. In particular Ind(α) = −ind(α(S)) = 1.

Under CH there is a unital embedding ψ : Q(H)→ Q(H) suchthat ind(ψ(S)) = 0.

Yes...

...and no

Theorem (Farah, 2011)Assume OCA. All automorphisms of the Calkin algebra are inner.

TheoremAssume OCA. All unital embeddings of the Calkin algebra intoitself are trivial.

Under OCA all unital embeddings of Q(H) into Q(H) can beclassified up to unitary equivalence by N \ {0}.

...and no

What does this mean?TheoremAssume OCA. All unital embeddings of the Calkin algebra intoitself are trivial.

Let φ : Q(H)→ Q(H) be a trivial embedding. What isφ[Q(H)]′ ∩Q(H)? There exists a unitary v ∈ Q(H) such thatAd(v) ◦ φ is lifted by Φm.

Φm : B(H)→ B(H ⊗ Cm) � B(H)T 7→ T ⊗ Idm

The commutant Φ′m[B(H)] ∩ B(H) is isomorphic to Mm(C).Theorem (Johnson-Parrott, 1972)If T ∈ B(H) is such that TR − RT ∈ K(H) for all R ∈ Φm[B(H)],then there is Q ∈ Φ′m[B(H)] ∩ B(H) such that T − Q ∈ K(H).

CorollaryThe commutant of the image of a trivial embedding is isomorphicto Mm(C) for some m ∈ N \ {0}.

What does this mean?TheoremAssume OCA. All unital embeddings of the Calkin algebra intoitself are trivial.Let φ : Q(H)→ Q(H) be a trivial embedding. What isφ[Q(H)]′ ∩Q(H)?

There exists a unitary v ∈ Q(H) such thatAd(v) ◦ φ is lifted by Φm.

What does this mean?TheoremAssume OCA. All unital embeddings of the Calkin algebra intoitself are trivial.Let φ : Q(H)→ Q(H) be a trivial embedding. What isφ[Q(H)]′ ∩Q(H)? There exists a unitary v ∈ Q(H) such thatAd(v) ◦ φ is lifted by Φm.

The commutant Φ′m[B(H)] ∩ B(H) is isomorphic to Mm(C).

Theorem (Johnson-Parrott, 1972)If T ∈ B(H) is such that TR − RT ∈ K(H) for all R ∈ Φm[B(H)],then there is Q ∈ Φ′m[B(H)] ∩ B(H) such that T − Q ∈ K(H).

Conclusions

CorollaryAssume OCA. The class E is not closed under (any) tensor product.

Proof:If ψ : Q(H)⊗γ A → Q(H) is a unital embedding, then A ismapped into the commutant of ψ[Q(H)], hence it must befinite-dimensional.Q(H) ∈ E and A ∈ E for all infinite-dimensional, unital A, butQ(H)⊗γ A < E .

RemarkUnder CH

⊗α<cQ(H) ∈ E .

Conclusions

Proof:If ψ : Q(H)⊗γ A → Q(H) is a unital embedding, then A ismapped into the commutant of ψ[Q(H)], hence it must befinite-dimensional.

Q(H) ∈ E and A ∈ E for all infinite-dimensional, unital A, butQ(H)⊗γ A < E .

RemarkUnder CH

⊗α<cQ(H) ∈ E .

Conclusions

RemarkUnder CH

⊗α<cQ(H) ∈ E .

Conclusions

RemarkUnder CH

⊗α<cQ(H) ∈ E .

Conclusions II

CorollaryAssume OCA. The class E is not closed under inductive limits.

Proof:Let A an infinite-dimensional, unital AF algebra and {An}n∈N afamily of finite dimensional C∗-aglebras such that A =

⋃n∈NAn.

Q(H)⊗An ∈ E for all n ∈ N , but

Q(H)⊗A =⋃

n∈NQ(H)⊗An

does not belong to E , since A is infinite-dimensional.

Thank you!

Conclusions II

⋃n∈NAn.

Q(H)⊗A =⋃

n∈NQ(H)⊗An

Thank you!

Conclusions II

⋃n∈NAn.

Q(H)⊗An ∈ E for all n ∈ N

Q(H)⊗A =⋃

n∈NQ(H)⊗An

Thank you!

Conclusions II

⋃n∈NAn.

Q(H)⊗A =⋃

n∈NQ(H)⊗An

Thank you!

Conclusions II

⋃n∈NAn.

Q(H)⊗A =⋃

n∈NQ(H)⊗An

Thank you!

trivial embeddings of the calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · trivial embeddings...

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