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Decomposition rank of UHF-absorbing C -algebras Joint work with - - PowerPoint PPT Presentation

Dec 11, 2023 305 likes •547 views

Decomposition rank of UHF-absorbing C -algebras Decomposition rank of UHF-absorbing C -algebras Joint work with Hiroki Matui 12, Mar., 2013. Sde Boker . . . . . . Decomposition rank of UHF-absorbing C -algebras Murray-von

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Decomposition rank of UHF-absorbing C∗-algebras

Decomposition rank of UHF-absorbing C∗-algebras

Joint work with Hiroki Matui 12, Mar., 2013. Sde Boker

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Decomposition rank of UHF-absorbing C∗-algebras Murray-von Neumann equivalence

Murray-von Neumann equivalence

M: a finite von Neumann algebra, p, q: two projections in M.

If τ(p) = τ(q), for any tracial state τ of M, then there exists v ∈ M such that v ∗v = p, vv ∗ = q.

This condition plays an essential role in the classification theorem of injective factors.

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Decomposition rank of UHF-absorbing C∗-algebras Murray-von Neumann equivalence

Murray-von Neumann equivalence and AFD

  • A. Connes proved that any injective factor with a

separable predual is approximately finite dimensional (AFD), by using his deep study of automorphisms. And he classified injective factors of type II and type IIIλ, λ ̸= 1.

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Decomposition rank of UHF-absorbing C∗-algebras Murray-von Neumann equivalence

Murray-von Neumann equivalence and AFD

  • A. Connes proved that any injective factor with a

separable predual is approximately finite dimensional (AFD), by using his deep study of automorphisms. And he classified injective factors of type II and type IIIλ, λ ̸= 1.

  • U. Haagerup gave an alternative proof (injectivity⇒

AFD) without using automorphisms, and classified the injective factor of thpe III1.

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Decomposition rank of UHF-absorbing C∗-algebras Murray-von Neumann equivalence

Murray-von Neumann equivalence and AFD

  • A. Connes proved that any injective factor with a

separable predual is approximately finite dimensional (AFD), by using his deep study of automorphisms. And he classified injective factors of type II and type IIIλ, λ ̸= 1.

  • U. Haagerup gave an alternative proof (injectivity⇒

AFD) without using automorphisms, and classified the injective factor of thpe III1.

  • S. Popa also gave another short proof by using excisions
  • f amenable traces.

In Connes and Haagerup’s argument, they showed AFD by using a partial isometry v which induces the Murray-von Neumann equivalence.

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Decomposition rank of UHF-absorbing C∗-algebras Murray-von Neumann equivalence

Murray-von Neumann equivalence for C∗-algebras

Theorem（1980. Cuntz-Pedersen.） Let A be a C ∗-algebra, p, q projections in A. If τ(p) = τ(q) for any tracial state τ of A. Then ∃vi ∈ A, i = 1, 2, ..., N, such that ∑ v∗

i vi = p,

∑ viv∗

i = q.

（ ）

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Decomposition rank of UHF-absorbing C∗-algebras Murray-von Neumann equivalence

Murray-von Neumann equivalence for C∗-algebras

Theorem（1980. Cuntz-Pedersen.） Let A be a C ∗-algebra, p, q projections in A. If τ(p) = τ(q) for any tracial state τ of A. Then ∃vi ∈ A, i = 1, 2, ..., N, such that ∑ v∗

i vi = p,

∑ viv∗

i = q.

Lemma（2013. Matui-Sato） Let A be a C ∗-algebra with strict comparison for projections, p, q projections in A. If τ(p) = τ(q) for any tracial state τ of

  • A. Then ∃vi ∈ A⊗Mn, i = 1, 2, such that

∑ v∗

i vi ≈4/n p ⊗ 1n,

∑ viv∗

i ≈4/n q ⊗ 1n.

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Decomposition rank of UHF-absorbing C∗-algebras Murray-von Neumann equivalence

Main theorem

Theorem （2013. H. Matui - Y. Sato） Let A be a unital separable, simple, C∗-algebra with a unique tracial state. Then A is nuclear, with strict comparison, and is quasidiagonal ⇐ ⇒ dr(A) < ∞, (in particular dr(A) ≤ 3).

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Decomposition rank of UHF-absorbing C∗-algebras Murray-von Neumann equivalence

Main theorem

Theorem （2013. H. Matui - Y. Sato） Let A be a unital separable, simple, C∗-algebra with a unique tracial state. Then A is nuclear, with strict comparison, and is quasidiagonal ⇐ ⇒ dr(A) < ∞, (in particular dr(A) ≤ 3). If A is in the above theorem, A has strict comparison ⇐ ⇒ A ⊗ Z ∼ = A.

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Decomposition rank of UHF-absorbing C∗-algebras Murray-von Neumann equivalence

Main theorem

Theorem （2013. H. Matui - Y. Sato） Let A be a unital separable, simple, C∗-algebra with a unique tracial state. Then A is nuclear, with strict comparison, and is quasidiagonal ⇐ ⇒ dr(A) < ∞, (in particular dr(A) ≤ 3). If A is in the above theorem, A has strict comparison ⇐ ⇒ A ⊗ Z ∼ = A. A is quasidiagonal ⇐ ⇒ A ֒ → ∏ Mkn/ ⊕ Mkn,

  • D. Voiculescu.
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Decomposition rank of UHF-absorbing C∗-algebras Decomposition rank

Decomposition rank

Definition (2002. E. Kirchberg - W. Winter.) Let A be a separable C∗-algebra. A has decomposition rank at most N, dr(A) ≤ N, if ∃φn : A − → ⊕N

i=0 Mki,n : c.p.c,

∃ψi,n : Mki,n − → A : order zero (disjointness preserving), c.p.c such that ∑ ψi,n is also contractive, ∥(∑ ψi,n) ◦ φn(a) − a∥ → 0, ∀a ∈ A,

where we simply write (∑ ψi,n)(⊕ xi) := ∑ ψi,n(xi).

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Decomposition rank of UHF-absorbing C∗-algebras Decomposition rank

Decomposition rank

Definition (2002. E. Kirchberg - W. Winter.) Let A be a separable C∗-algebra. A has decomposition rank at most N, dr(A) ≤ N, if ∃φn : A − → ⊕N

i=0 Mki,n : c.p.c,

∃ψi,n : Mki,n − → A : order zero (disjointness preserving), c.p.c such that ∑ ψi,n is also contractive, ∥(∑ ψi,n) ◦ φn(a) − a∥ → 0, ∀a ∈ A,

where we simply write (∑ ψi,n)(⊕ xi) := ∑ ψi,n(xi).

dr(A) < ∞ = ⇒ A is quasidiagonal, Kirchberg-Winter.

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Decomposition rank of UHF-absorbing C∗-algebras Decomposition rank

Decomposition rank

Definition (2002. E. Kirchberg - W. Winter.) Let A be a separable C∗-algebra. A has decomposition rank at most N, dr(A) ≤ N, if ∃φn : A − → ⊕N

i=0 Mki,n : c.p.c,

∃ψi,n : Mki,n − → A : order zero (disjointness preserving), c.p.c such that ∑ ψi,n is also contractive, ∥(∑ ψi,n) ◦ φn(a) − a∥ → 0, ∀a ∈ A,

where we simply write (∑ ψi,n)(⊕ xi) := ∑ ψi,n(xi).

dr(A) < ∞ = ⇒ A is quasidiagonal, Kirchberg-Winter. dr(A) < ∞ = ⇒ A ⊗ Z ∼ = A, 2007 Winter.

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Decomposition rank of UHF-absorbing C∗-algebras Decomposition rank

Decomposition rank

Definition (2002. E. Kirchberg - W. Winter.) Let A be a separable C∗-algebra. A has decomposition rank at most N, dr(A) ≤ N, if ∃φn : A − → ⊕N

i=0 Mki,n : c.p.c,

∃ψi,n : Mki,n − → A : order zero (disjointness preserving), c.p.c such that ∑ ψi,n is also contractive, ∥(∑ ψi,n) ◦ φn(a) − a∥ → 0, ∀a ∈ A,

where we simply write (∑ ψi,n)(⊕ xi) := ∑ ψi,n(xi).

dr(A) < ∞ = ⇒ A is quasidiagonal, Kirchberg-Winter. dr(A) < ∞ = ⇒ A ⊗ Z ∼ = A, 2007 Winter. A⊗Z ∼ = A = ⇒ A has strict comparison, 2001 M. Rørdam.

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Decomposition rank of UHF-absorbing C∗-algebras Decomposition rank

Main theorem

Theorem （2013. H. Matui - Y. Sato） Let A be a unital separable, simple, C∗-algebra with a unique tracial state. Then A is nuclear, with strict comparison, and is quasidiagonal ⇐ ⇒ dr(A) < ∞, (in particular dr(A) ≤ 3). If A is in the above theorem, A has strict comparison ⇐ ⇒ A ⊗ Z ∼ = A. A is quasidiagonal ⇐ ⇒ A ֒ → ∏ Mkn/ ⊕ Mkn,

  • D. Voiculescu.
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Decomposition rank of UHF-absorbing C∗-algebras Decomposition rank

Toms - Winter conjecture

Conjecture （2009. A. Toms, W. Winter.） Let A be a unital separable simple nuclear finite C∗-algebra with infinite-dimension. Then the following are equivalent.

(i) A ⊗ Z ∼ = A.

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Decomposition rank of UHF-absorbing C∗-algebras Decomposition rank

Toms - Winter conjecture

Conjecture （2009. A. Toms, W. Winter.） Let A be a unital separable simple nuclear finite C∗-algebra with infinite-dimension. Then the following are equivalent.

(i) A ⊗ Z ∼ = A. (ii) A has the strict comparison.

(i)⇒ (ii) 2001. M. Rørdam.

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Decomposition rank of UHF-absorbing C∗-algebras Decomposition rank

Toms - Winter conjecture

Conjecture （2009. A. Toms, W. Winter.） Let A be a unital separable simple nuclear finite C∗-algebra with infinite-dimension. Then the following are equivalent.

(i) A ⊗ Z ∼ = A. (ii) A has the strict comparison. (iii) dr(A) < ∞

(i)⇒ (ii) 2001. M. Rørdam. (iii)⇒ (i) 2009. W. Winter. Corollary of the main theorem Suppose that A is a unital separable simple nuclear C∗-algebra. Assume that A is quasidiagonal and with a unique tracial state. Then (ii)= ⇒ (iii).

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Decomposition rank of UHF-absorbing C∗-algebras Decomposition rank

Without Q.D.

Here we assume that the T. W. conjecture has been completely proved without Q.D. For a unital separable simple nuclear C∗-algebra A it follows that A ⊗ Z also absorbs Z ((i) in the T. W. conjecture.),

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Decomposition rank of UHF-absorbing C∗-algebras Decomposition rank

Without Q.D.

Here we assume that the T. W. conjecture has been completely proved without Q.D. For a unital separable simple nuclear C∗-algebra A it follows that A ⊗ Z also absorbs Z ((i) in the T. W. conjecture.), = ⇒ dr(A ⊗ Z) < ∞ ((iii) in T.W. conjecture). = ⇒ A ⊗ Z is quasidiagonal.

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Decomposition rank of UHF-absorbing C∗-algebras Decomposition rank

Without Q.D.

Here we assume that the T. W. conjecture has been completely proved without Q.D. For a unital separable simple nuclear C∗-algebra A it follows that A ⊗ Z also absorbs Z ((i) in the T. W. conjecture.), = ⇒ dr(A ⊗ Z) < ∞ ((iii) in T.W. conjecture). = ⇒ A ⊗ Z is quasidiagonal. = ⇒ A is quasidiagonal. ∴ stably-finite and nuclearity ⇝ quasidiagonality.

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Decomposition rank of UHF-absorbing C∗-algebras Decomposition rank

Without Q.D.

Here we assume that the T. W. conjecture has been completely proved without Q.D. For a unital separable simple nuclear C∗-algebra A it follows that A ⊗ Z also absorbs Z ((i) in the T. W. conjecture.), = ⇒ dr(A ⊗ Z) < ∞ ((iii) in T.W. conjecture). = ⇒ A ⊗ Z is quasidiagonal. = ⇒ A is quasidiagonal. ∴ stably-finite and nuclearity ⇝ quasidiagonality. Problem (in Blackadar-Kirchberg) Is any stably-finite C∗-algebra quasidiagonal?

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Decomposition rank of UHF-absorbing C∗-algebras Decomposition rank

Main theorem

Theorem （2013. H. Matui - Y. Sato） Let A be a unital separable, simple, C∗-algebra with a unique tracial state. Then A is nuclear, with strict comparison, and is quasidiagonal ⇐ ⇒ dr(A) < ∞, (in particular dr(A) ≤ 3). If A is in the above theorem, A has strict comparison ⇐ ⇒ A ⊗ Z ∼ = A. A is quasidiagonal ⇐ ⇒ A ֒ → ∏ Mkn/ ⊕ Mkn,

  • D. Voiculescu.