trivial embeddings of the calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · trivial embeddings...
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![Page 1: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/1.jpg)
Trivial Embeddings of the Calkin algebra
Andrea Vaccaro
Università di Pisa & York University
Texas A&M University, GPOTS 2019
![Page 2: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/2.jpg)
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?
![Page 3: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/3.jpg)
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?
![Page 4: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/4.jpg)
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?
![Page 5: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/5.jpg)
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?
![Page 6: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/6.jpg)
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?
![Page 7: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/7.jpg)
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 .
![Page 8: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/8.jpg)
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 .
![Page 9: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/9.jpg)
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 .
![Page 10: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/10.jpg)
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 .
![Page 11: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/11.jpg)
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 .
![Page 12: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/12.jpg)
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 .
![Page 13: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/13.jpg)
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?
![Page 14: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/14.jpg)
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?
![Page 15: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/15.jpg)
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?
![Page 16: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/16.jpg)
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)
Φ
q q
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
![Page 17: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/17.jpg)
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)
Φ
q q
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
![Page 18: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/18.jpg)
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)
Φ
q q
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
![Page 19: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/19.jpg)
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?
![Page 20: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/20.jpg)
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?
![Page 21: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/21.jpg)
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?
![Page 22: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/22.jpg)
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?
![Page 23: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/23.jpg)
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?
![Page 24: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/24.jpg)
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.
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).
Under CH there is a unital embedding ψ : Q(H)→ Q(H) suchthat ind(ψ(S)) = 0.
![Page 25: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/25.jpg)
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.
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).
Under CH there is a unital embedding ψ : Q(H)→ Q(H) suchthat ind(ψ(S)) = 0.
![Page 26: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/26.jpg)
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.
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).
Under CH there is a unital embedding ψ : Q(H)→ Q(H) suchthat ind(ψ(S)) = 0.
![Page 27: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/27.jpg)
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.
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).
Under CH there is a unital embedding ψ : Q(H)→ Q(H) suchthat ind(ψ(S)) = 0.
![Page 28: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/28.jpg)
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.
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).
Under CH there is a unital embedding ψ : Q(H)→ Q(H) suchthat ind(ψ(S)) = 0.
![Page 29: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/29.jpg)
...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}.
![Page 30: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/30.jpg)
...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}.
![Page 31: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/31.jpg)
...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}.
![Page 32: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/32.jpg)
...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}.
![Page 33: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/33.jpg)
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}.
![Page 34: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/34.jpg)
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}.
![Page 35: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/35.jpg)
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}.
![Page 36: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/36.jpg)
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}.
![Page 37: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/37.jpg)
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}.
![Page 38: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/38.jpg)
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 .
![Page 39: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/39.jpg)
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 .
![Page 40: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/40.jpg)
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 .
![Page 41: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/41.jpg)
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 .
![Page 42: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/42.jpg)
Conclusions II
CorollaryAssume OCA. The class E is not closed under (any) tensor product.
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!
![Page 43: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/43.jpg)
Conclusions II
CorollaryAssume OCA. The class E is not closed under (any) tensor product.
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!
![Page 44: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/44.jpg)
Conclusions II
CorollaryAssume OCA. The class E is not closed under (any) tensor product.
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!
![Page 45: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/45.jpg)
Conclusions II
CorollaryAssume OCA. The class E is not closed under (any) tensor product.
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!
![Page 46: Trivial Embeddings of the Calkin algebrapeople.dm.unipi.it/vaccaro/embed.pdf · Trivial Embeddings of the Calkin algebra AndreaVaccaro Università di Pisa & York University TexasA&MUniversity,GPOTS2019](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1869acafc09581d5601e1/html5/thumbnails/46.jpg)
Conclusions II
CorollaryAssume OCA. The class E is not closed under (any) tensor product.
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!