I NTERPLAY BETWEEN C ˚ AND VON N EUMANN
ALGEBRAS
Stuart White
University of Glasgow
26 March 2013, British Mathematical Colloquium,
University of Sheffield.
C ˚ AND VON N EUMANN ALGEBRAS
C ˚ -algebras
Banach algebra A
with an involution ˚
von Neuman algebras
Weak operator closed
˚ -subalgebra M Ď BpHq
satisfying the C ˚ -identity
}x ˚ x} “ }x}2 for all x P A.
Weak operator topology:
xi Ñ x ô xpxi ´ xqξ, ηy Ñ 0
for all ξ, η P H
Norm closed ˚ -subalgebra
A Ď BpHq
C ˚ -algebra M which is
isometrically the dual space
of some Banach space.
Abelian C ˚ -algebras of form
C0 pX q for X locally compact Hff
Abelian vNas of form L8 pX , µq
for some measure space pX , µq.
C ˚ AND VON N EUMANN ALGEBRAS
C ˚ -algebras
Banach algebra A
with an involution ˚
von Neuman algebras
Weak operator closed
˚ -subalgebra M Ď BpHq
satisfying the C ˚ -identity
}x ˚ x} “ }x}2 for all x P A.
Weak operator topology:
xi Ñ x ô xpxi ´ xqξ, ηy Ñ 0
for all ξ, η P H
Norm closed ˚ -subalgebra
A Ď BpHq
C ˚ -algebra M which is
isometrically the dual space
of some Banach space.
Abelian C ˚ -algebras of form
C0 pX q for X locally compact Hff
Abelian vNas of form L8 pX , µq
for some measure space pX , µq.
C ˚ AND VON N EUMANN ALGEBRAS
C ˚ -algebras
Banach algebra A
with an involution ˚
von Neuman algebras
Weak operator closed
˚ -subalgebra M Ď BpHq
satisfying the C ˚ -identity
}x ˚ x} “ }x}2 for all x P A.
Weak operator topology:
xi Ñ x ô xpxi ´ xqξ, ηy Ñ 0
for all ξ, η P H
Norm closed ˚ -subalgebra
A Ď BpHq
C ˚ -algebra M which is
isometrically the dual space
of some Banach space.
Abelian C ˚ -algebras of form
C0 pX q for X locally compact Hff
Abelian vNas of form L8 pX , µq
for some measure space pX , µq.
C ˚ AND VON N EUMANN ALGEBRAS
C ˚ -algebras
Banach algebra A
with an involution ˚
von Neuman algebras
Weak operator closed
˚ -subalgebra M Ď BpHq
satisfying the C ˚ -identity
}x ˚ x} “ }x}2 for all x P A.
Weak operator topology:
xi Ñ x ô xpxi ´ xqξ, ηy Ñ 0
for all ξ, η P H
Norm closed ˚ -subalgebra
A Ď BpHq
C ˚ -algebra M which is
isometrically the dual space
of some Banach space.
Abelian C ˚ -algebras of form
C0 pX q for X locally compact Hff
Abelian vNas of form L8 pX , µq
for some measure space pX , µq.
C ˚ AND VON N EUMANN ALGEBRAS
C ˚ -algebras
Banach algebra A
with an involution ˚
von Neuman algebras
Weak operator closed
˚ -subalgebra M Ď BpHq
satisfying the C ˚ -identity
}x ˚ x} “ }x}2 for all x P A.
Weak operator topology:
xi Ñ x ô xpxi ´ xqξ, ηy Ñ 0
for all ξ, η P H
Norm closed ˚ -subalgebra
A Ď BpHq
C ˚ -algebra M which is
isometrically the dual space
of some Banach space.
Abelian C ˚ -algebras of form
C0 pX q for X locally compact Hff
Abelian vNas of form L8 pX , µq
for some measure space pX , µq.
E XAMPLES
Mn , BpHq;
C0 pX q Ă BpL2 pX qq, L8 pX q Ă BpL2 pX qq;
G discrete group
group C ˚ and von Neumann algebras;
H “ `2 pGq, basis tδg : g P Gu. Define unitaries λg : H Ñ H by
λg pδh q “ δgh ;
}
Cr˚ pGq “ Spantλg : g P Gu ; LG “ Spantλg : g P Gu
α : G ñ X Dynamical system
wot
.
CpX q ¸r G, L8 pX q ¸ G.
CpX q ¸r G is a C ˚ -algebra generated by CpX q and copy of G;
Unitaries pλg qgPG with λgh “ λg λh , λg f λ˚g “ f ˝ αg´1 for g, h P G,
f P CpX q.
How much information about G or G ñ X is captured by their
operator algebras?
E XAMPLES
Mn , BpHq;
C0 pX q Ă BpL2 pX qq, L8 pX q Ă BpL2 pX qq;
G discrete group
group C ˚ and von Neumann algebras;
H “ `2 pGq, basis tδg : g P Gu. Define unitaries λg : H Ñ H by
λg pδh q “ δgh ;
}
Cr˚ pGq “ Spantλg : g P Gu ; LG “ Spantλg : g P Gu
α : G ñ X Dynamical system
wot
.
CpX q ¸r G, L8 pX q ¸ G.
CpX q ¸r G is a C ˚ -algebra generated by CpX q and copy of G;
Unitaries pλg qgPG with λgh “ λg λh , λg f λ˚g “ f ˝ αg´1 for g, h P G,
f P CpX q.
How much information about G or G ñ X is captured by their
operator algebras?
A NOTHER EXAMPLE
Consider M2 ãÑ M4 – M2 b M2 ; x ÞÑ x b 1.
This preserves the normalised trace on M2 and M4 . Carry
on and form
M2 ãÑ M2 b M2 ãÑ M2b3 ãÑ ¨ ¨ ¨
Ä
to obtain direct limit ˚ -algebra 8
n“1 M2 .
Perform GNS construction for canonical trace τ and take
the weak operator closure — obtain hyperfinite II1 factor R.
Could also close in the norm topology to get the
CAR-algebra M28 .
I MPORTANT CONSEQUENCE
R – R bR
A NOTHER EXAMPLE
Consider M2 ãÑ M4 – M2 b M2 ; x ÞÑ x b 1.
This preserves the normalised trace on M2 and M4 . Carry
on and form
M2 ãÑ M2 b M2 ãÑ M2b3 ãÑ ¨ ¨ ¨
Ä
to obtain direct limit ˚ -algebra 8
n“1 M2 .
Perform GNS construction for canonical trace τ and take
the weak operator closure — obtain hyperfinite II1 factor R.
Could also close in the norm topology to get the
CAR-algebra M28 .
I MPORTANT CONSEQUENCE
R – R bR
A NOTHER EXAMPLE
Consider M2 ãÑ M4 – M2 b M2 ; x ÞÑ x b 1.
This preserves the normalised trace on M2 and M4 . Carry
on and form
M2 ãÑ M2 b M2 ãÑ M2b3 ãÑ ¨ ¨ ¨
Ä
to obtain direct limit ˚ -algebra 8
n“1 M2 .
Perform GNS construction for canonical trace τ and take
the weak operator closure — obtain hyperfinite II1 factor R.
Could also close in the norm topology to get the
CAR-algebra M28 .
I MPORTANT CONSEQUENCE
R – R bR
II1 FACTORS
D EFINITION
A factor is a von Neumann algebra M with trivial centre:
Z pMq “ C1.
“Every” vNa can be written as a “direct integral” of factors.
Murray and von Neumann classified factors into types.
D EFINITION
A factor M is type II1 if:
it is infinite dimensional;
D tracial state τ : M Ñ C1, i.e. τ pxy q “ τ pyxq, @x, y P M.
R;
LG for countable discrete G with |thgh´1 : h P Gu| “ 8 for
g ‰ e;
LpX q ¸ G for G ñ pX , µq is free, ergodic and probability
measure preserving.
H YPERFINITENESS
D EFINITION (M URRAY AND VON N EUMANN )
A vNa M is hyperfinite if every finite subset of M can be
arbitrarily approximated arbitrarily in the weak˚ -topology by a
finite dimensional subalgebra of M.
Separably acting hyperfinite vnas arise are inductive limits
of finite dimensional vnas.
T HEOREM (M URRAY AND VON N EUMANN )
There is a unique (separably acting) hyperfinite II1 factor.
U NIFORM PERTURBATIONS
Given two operator algebras A, B Ă BpHq, write dpA, Bq for the
Hausdorff distance between the unit balls of A and B in the
operator norm.
For a unitary u, dpA, uAu ˚ q ď 2}u ´ 1H }.
T HEOREM (C HRISTENSEN , J OHNSON , R AEBURN -TAYLOR
’77)
Let M, N Ă BpHq be vNas with M hyperfinite.
dpM, Nq ă 1{101 ùñ D unitary u P W ˚ pM Y Nq s.t. uMu ˚ “ N
and }u ´ 1H } ď 150dpM, Nq.
I DEA
Establish dimension independent perturbation results for
near containments of finite dimensional algebras.
i.e. @ε ą 0, Dδ ą 0 such that if F is finite dim, F Ăδ N, then
D unitary u P W ˚ pF , Nq with uFu ˚ Ď N and }u ´ 1H } ă ε.
U NIFORM PERTURBATIONS
Given two operator algebras A, B Ă BpHq, write dpA, Bq for the
Hausdorff distance between the unit balls of A and B in the
operator norm.
For a unitary u, dpA, uAu ˚ q ď 2}u ´ 1H }.
T HEOREM (C HRISTENSEN , J OHNSON , R AEBURN -TAYLOR
’77)
Let M, N Ă BpHq be vNas with M hyperfinite.
dpM, Nq ă 1{101 ùñ D unitary u P W ˚ pM Y Nq s.t. uMu ˚ “ N
and }u ´ 1H } ď 150dpM, Nq.
I DEA
Establish dimension independent perturbation results for
near containments of finite dimensional algebras.
i.e. @ε ą 0, Dδ ą 0 such that if F is finite dim, F Ăδ N, then
D unitary u P W ˚ pF , Nq with uFu ˚ Ď N and }u ´ 1H } ă ε.
AF ALGEBRAS
D EFINITION
C˚ -algebra A is AF if every finite subset of A can be arbitrarily
approximated in norm by finite dimensional subalgebras of A.
Separable AF algebras are inductive limits of finite
dimensional algebras.
eg M28 . Changing matrix sizes matters: M38 fl M28 .
C LASSIFICATION — E LLIOTT
Separable AF -algebras are classified by K0 .
T HEOREM (C HRISTENSEN 80)
Let A, B Ă BpHq be C ˚ -algebras, with A separable and AF. If
dpA, Bq ă 10´9 , then there exists a unitary u P W ˚ pA, Bq with
uAu ˚ “ B.
Don’t get }u ´ 1H } small, in terms of dpA, Bq but can
control Adpuq ´ ιA in the point-norm topology.
AF ALGEBRAS
D EFINITION
C˚ -algebra A is AF if every finite subset of A can be arbitrarily
approximated in norm by finite dimensional subalgebras of A.
Separable AF algebras are inductive limits of finite
dimensional algebras.
eg M28 . Changing matrix sizes matters: M38 fl M28 .
C LASSIFICATION — E LLIOTT
Separable AF -algebras are classified by K0 .
T HEOREM (C HRISTENSEN 80)
Let A, B Ă BpHq be C ˚ -algebras, with A separable and AF. If
dpA, Bq ă 10´9 , then there exists a unitary u P W ˚ pA, Bq with
uAu ˚ “ B.
Don’t get }u ´ 1H } small, in terms of dpA, Bq but can
control Adpuq ´ ιA in the point-norm topology.
AF ALGEBRAS
D EFINITION
C˚ -algebra A is AF if every finite subset of A can be arbitrarily
approximated in norm by finite dimensional subalgebras of A.
Separable AF algebras are inductive limits of finite
dimensional algebras.
eg M28 . Changing matrix sizes matters: M38 fl M28 .
C LASSIFICATION — E LLIOTT
Separable AF -algebras are classified by K0 .
T HEOREM (C HRISTENSEN 80)
Let A, B Ă BpHq be C ˚ -algebras, with A separable and AF. If
dpA, Bq ă 10´9 , then there exists a unitary u P W ˚ pA, Bq with
uAu ˚ “ B.
Don’t get }u ´ 1H } small, in terms of dpA, Bq but can
control Adpuq ´ ιA in the point-norm topology.
A MENABLE VON N EUMANN ALGEBRAS
T HEOREM (C ONNES 75)
Let M be a (separably acting) von Neumann algebra. Then
(amongst others) the following are equivalent.
1
M is injective (abstract categorical property)
2
M is semidiscrete (finite dimensional approximation
property)
3
M is hyperfinite (inductive limit structure)
Call these factors amenable.
A linear map φ : A Ñ B between C ˚ -algebras is positive if
φpxq ě 0 when x ě 0.
φ is completely positive if each φpnq : Mn pAq Ñ Mn pBq is
positive.
M is semidiscrete if idM can be approximated in the point
weak˚ -topology by completely positive contractions (cpc)
M Ñ F Ñ M, with with F finite dimensional.
A MENABLE VON N EUMANN ALGEBRAS
T HEOREM (C ONNES 75)
Let M be a (separably acting) von Neumann algebra. Then
(amongst others) the following are equivalent.
1
M is injective (abstract categorical property)
2
M is semidiscrete (finite dimensional approximation
property)
3
M is hyperfinite (inductive limit structure)
Call these factors amenable.
A linear map φ : A Ñ B between C ˚ -algebras is positive if
φpxq ě 0 when x ě 0.
φ is completely positive if each φpnq : Mn pAq Ñ Mn pBq is
positive.
M is semidiscrete if idM can be approximated in the point
weak˚ -topology by completely positive contractions (cpc)
M Ñ F Ñ M, with with F finite dimensional.
A MENABLE VON N EUMANN ALGEBRAS
T HEOREM (C ONNES 75)
Let M be a (separably acting) von Neumann algebra. Then
(amongst others) the following are equivalent.
1
M is injective (abstract categorical property)
2
M is semidiscrete (finite dimensional approximation
property)
3
M is hyperfinite (inductive limit structure)
Call these factors amenable.
A linear map φ : A Ñ B between C ˚ -algebras is positive if
φpxq ě 0 when x ě 0.
φ is completely positive if each φpnq : Mn pAq Ñ Mn pBq is
positive.
M is semidiscrete if idM can be approximated in the point
weak˚ -topology by completely positive contractions (cpc)
M Ñ F Ñ M, with with F finite dimensional.
M ORE ON C ONNES THEOREM
C OROLLARY (C ONNES )
There is a unique (separably acting) injective II1 factor.
Connes didn’t classify injective II1 factors: he showed the
abstract property of injectivity implies the existence of an
inductive limit structure.
Factors with this inductive limit structure had already been
classified by Murray and von Neumann.
I MPORTANT STEP IN C ONNES ’
PROOF
First show that an injective II1 factor M tensorially absorbs the
hyperfinite II1 factor, i.e. M – M b R.
II1 factors M with M – M b R where characterised by
McDuff.
M – M b R iff the central sequence algebra M ω X M 1 is
non-abelian iff Mk ãÑ M ω X M 1 for all k .
M ORE ON C ONNES THEOREM
C OROLLARY (C ONNES )
There is a unique (separably acting) injective II1 factor.
Connes didn’t classify injective II1 factors: he showed the
abstract property of injectivity implies the existence of an
inductive limit structure.
Factors with this inductive limit structure had already been
classified by Murray and von Neumann.
I MPORTANT STEP IN C ONNES ’
PROOF
First show that an injective II1 factor M tensorially absorbs the
hyperfinite II1 factor, i.e. M – M b R.
II1 factors M with M – M b R where characterised by
McDuff.
M – M b R iff the central sequence algebra M ω X M 1 is
non-abelian iff Mk ãÑ M ω X M 1 for all k .
A MENABLE C ˚ - ALGEBRAS
D EFINITION
A C ˚ -algebra A is nuclear if for every C ˚ -algebra B there is only
one way of completing the algebraic tensor product A d B to
obtain a C ˚ -algebra.
D EFINITION
A C ˚ -algebra A has the completely positive factorisation
property if idA : A Ñ A can be approximated by completely
positive contractions A Ñ F Ñ A in the point norm topology.
T HEOREM (C ONNES , (K IRCHBERG , C HOI -E FFROS ))
Let A be a C ˚ -algebra. TFAE:
1
A is nuclear;
2
A˚˚ is semidiscrete;
3
A has the completely positive approximation property.
A MENABLE C ˚ - ALGEBRAS
D EFINITION
A C ˚ -algebra A is nuclear if for every C ˚ -algebra B there is only
one way of completing the algebraic tensor product A d B to
obtain a C ˚ -algebra.
D EFINITION
A C ˚ -algebra A has the completely positive factorisation
property if idA : A Ñ A can be approximated by completely
positive contractions A Ñ F Ñ A in the point norm topology.
T HEOREM (C ONNES , (K IRCHBERG , C HOI -E FFROS ))
Let A be a C ˚ -algebra. TFAE:
1
A is nuclear;
2
A˚˚ is semidiscrete;
3
A has the completely positive approximation property.
A MENABLE C ˚ - ALGEBRAS
D EFINITION
A C ˚ -algebra A is nuclear if for every C ˚ -algebra B there is only
one way of completing the algebraic tensor product A d B to
obtain a C ˚ -algebra.
D EFINITION
A C ˚ -algebra A has the completely positive factorisation
property if idA : A Ñ A can be approximated by completely
positive contractions A Ñ F Ñ A in the point norm topology.
T HEOREM (C ONNES , (K IRCHBERG , C HOI -E FFROS ))
Let A be a C ˚ -algebra. TFAE:
1
A is nuclear;
2
A˚˚ is semidiscrete;
3
A has the completely positive approximation property.
A˚˚ semidiscrete
A˚˚ semidiscrete
Hahn Banach argument
ùñ
A has cpap
currently requires Connes
ðù
A has cpap
Due to Connes theorem, can witness semidiscreteness of
φ
A˚˚ with maps A˚˚ Ñ F Ñ A˚˚ with φ a ˚ -homomorphism.
T HEOREM (H IRSHBERG , K IRCHBERG , W.)
Let A be a nuclear C ˚ -algebra. Then idA : A Ñ A can be
φ
approximated by cpc maps A Ñ F Ñ A where φ is a convex
combination of cpc order zero maps.
A could be projectionless: no ˚ -hms F Ñ A.
Order zero maps, next best thing.
φ : F Ñ A is order zero if φ preserves orthogonality, i.e.
e, f ě 0, ef “ 0 ùñ φpeqφpf q “ 0.
All cpc order zero maps φ obtained by compressing
˚ -homomorphism by a positive element which commutes
with φpAq.
A˚˚ semidiscrete
A˚˚ semidiscrete
Hahn Banach argument
ùñ
A has cpap
currently requires Connes
ðù
A has cpap
Due to Connes theorem, can witness semidiscreteness of
φ
A˚˚ with maps A˚˚ Ñ F Ñ A˚˚ with φ a ˚ -homomorphism.
T HEOREM (H IRSHBERG , K IRCHBERG , W.)
Let A be a nuclear C ˚ -algebra. Then idA : A Ñ A can be
φ
approximated by cpc maps A Ñ F Ñ A where φ is a convex
combination of cpc order zero maps.
A could be projectionless: no ˚ -hms F Ñ A.
Order zero maps, next best thing.
φ : F Ñ A is order zero if φ preserves orthogonality, i.e.
e, f ě 0, ef “ 0 ùñ φpeqφpf q “ 0.
All cpc order zero maps φ obtained by compressing
˚ -homomorphism by a positive element which commutes
with φpAq.
A˚˚ semidiscrete
A˚˚ semidiscrete
Hahn Banach argument
ùñ
A has cpap
currently requires Connes
ðù
A has cpap
Due to Connes theorem, can witness semidiscreteness of
φ
A˚˚ with maps A˚˚ Ñ F Ñ A˚˚ with φ a ˚ -homomorphism.
T HEOREM (H IRSHBERG , K IRCHBERG , W.)
Let A be a nuclear C ˚ -algebra. Then idA : A Ñ A can be
φ
approximated by cpc maps A Ñ F Ñ A where φ is a convex
combination of cpc order zero maps.
A could be projectionless: no ˚ -hms F Ñ A.
Order zero maps, next best thing.
φ : F Ñ A is order zero if φ preserves orthogonality, i.e.
e, f ě 0, ef “ 0 ùñ φpeqφpf q “ 0.
All cpc order zero maps φ obtained by compressing
˚ -homomorphism by a positive element which commutes
with φpAq.
T HEOREM (H IRSHBERG , K IRCHBERG , W. 2011)
Let A be a nuclear C ˚ -algebra. Then idA : A Ñ A can be
φ
approximated by cpc maps A Ñ F Ñ A where φ is a convex
combination of cpc order zero maps.
C OROLLARY (HKW, BUILDING ON RESULTS OF
C HRISTENSEN , S INCLAIR , S MITH , W, W INTER )
Let A be separable and nuclear, and suppose that A Ăγ B for
γ ă 1{210000. Then A ãÑ B.
If in addition dpA, Bq small, then D unitary u P W ˚ pA, Bq
with uAu ˚ “ B.
D EFINITION (W INTER , Z ACHARIAS , 2009)
A C ˚ -algebra has nuclear dimension at most n if one can find
contractive
φ
cp factorisations A
Ñ
F Ñ A such that φ is a sum of at
most n ` 1 cpc order zero maps.
dimnuc pCpX qq “ dimpX q; dimnuc pAq “ 0 ô A is AF.
T HEOREM (H IRSHBERG , K IRCHBERG , W. 2011)
Let A be a nuclear C ˚ -algebra. Then idA : A Ñ A can be
φ
approximated by cpc maps A Ñ F Ñ A where φ is a convex
combination of cpc order zero maps.
C OROLLARY (HKW, BUILDING ON RESULTS OF
C HRISTENSEN , S INCLAIR , S MITH , W, W INTER )
Let A be separable and nuclear, and suppose that A Ăγ B for
γ ă 1{210000. Then A ãÑ B.
If in addition dpA, Bq small, then D unitary u P W ˚ pA, Bq
with uAu ˚ “ B.
D EFINITION (W INTER , Z ACHARIAS , 2009)
A C ˚ -algebra has nuclear dimension at most n if one can find
contractive
φ
cp factorisations A
Ñ
F Ñ A such that φ is a sum of at
most n ` 1 cpc order zero maps.
dimnuc pCpX qq “ dimpX q; dimnuc pAq “ 0 ô A is AF.
T HEOREM (H IRSHBERG , K IRCHBERG , W. 2011)
Let A be a nuclear C ˚ -algebra. Then idA : A Ñ A can be
φ
approximated by cpc maps A Ñ F Ñ A where φ is a convex
combination of cpc order zero maps.
C OROLLARY (HKW, BUILDING ON RESULTS OF
C HRISTENSEN , S INCLAIR , S MITH , W, W INTER )
Let A be separable and nuclear, and suppose that A Ăγ B for
γ ă 1{210000. Then A ãÑ B.
If in addition dpA, Bq small, then D unitary u P W ˚ pA, Bq
with uAu ˚ “ B.
D EFINITION (W INTER , Z ACHARIAS , 2009)
A C ˚ -algebra has nuclear dimension at most n if one can find
contractive
φ
cp factorisations A
Ñ
F Ñ A such that φ is a sum of at
most n ` 1 cpc order zero maps.
dimnuc pCpX qq “ dimpX q; dimnuc pAq “ 0 ô A is AF.
C LASSIFICATION PROGRAMME
C ONNES , H AAGERUP
Classification of separably acting injective factors.
The analogous class of C ˚ -algebras are the simple, separable
and unital C ˚ -algebras.
E LLIOTT ’ S CLASSIFICATION PROGRAMME : I NITIAL AIM
Classify all simple, separable, nuclear C ˚ -algebras A by
K -theory: data about homotopy equivalence classes of
projections and unitaries in matrices over A
C LASSIFICATION PROGRAMME
C ONNES , H AAGERUP
Classification of separably acting injective factors.
The analogous class of C ˚ -algebras are the simple, separable
and unital C ˚ -algebras.
E LLIOTT ’ S CLASSIFICATION PROGRAMME : I NITIAL AIM
Classify all simple, separable, nuclear C ˚ -algebras A by
K -theory: data about homotopy equivalence classes of
projections and unitaries in matrices over A
P URELY INFINITE ALGEBRAS
D EFINITION
A simple C ˚ -algebra A is purely infinite if for all x ‰ 0 there
exists a, b P A such that 1 “ axb.
Very strong infiniteness condition. Implies that all non-zero
projections are equivalent;
Equivalently: every hereditary subalgebra has an infinite
projection.
˚
e.g Cuntz algebras On — universal
generated
řn C -algebras
˚
by n isometries s1 , . . . , sn with i“1 si si “ 1.
T HEOREM (K IRCHBERG , K IRCHBERG -P HILIPS )
Purely infinite simple separable nuclear C ˚ -algebras (satisfying
the UCT) are classified by K -theory.
P URELY INFINITE ALGEBRAS
D EFINITION
A simple C ˚ -algebra A is purely infinite if for all x ‰ 0 there
exists a, b P A such that 1 “ axb.
Very strong infiniteness condition. Implies that all non-zero
projections are equivalent;
Equivalently: every hereditary subalgebra has an infinite
projection.
˚
e.g Cuntz algebras On — universal
generated
řn C -algebras
˚
by n isometries s1 , . . . , sn with i“1 si si “ 1.
T HEOREM (K IRCHBERG , K IRCHBERG -P HILIPS )
Purely infinite simple separable nuclear C ˚ -algebras (satisfying
the UCT) are classified by K -theory.
A BSOPTION THEOREMS : KEY STEPS IN
K IRCHBERG -P HILIIPS CLASSIFICATION
T HEOREM (K IRCHBERG )
Let A be simple, separable and nuclear. Then
A is purely infinite ðñ A – A b O8 .
R ECALL
To prove his theorem, Connes needed to show that an injective
II1 factor has M – M b R.
T HEOREM (K IRCHBERG )
Let A be simple, separable unital and nuclear. Then
O2 – A b O2 .
The tensorial absorption O2 – O2 b O2 (due to Elliott) plays a
significant role in this.
T HE FINITE SETTING
Oddly, in the C ˚ -setting the purely infinite case appears to be
easier than the finite case.
R EASONS
Higher dimensional topological phenoma can occur in
simple C ˚ -algebras.
The condition of pure infiniteness prevents this: in a
precise sense these algebras have low topological
dimension.
In the finite case, need traces as well as K -theory.
In late 90’s Jiang-Su constructed Z, an infinite dimensional
C ˚ -algebra with the same Elliott data as the complex
numbers.
For “reasonable” A, A and A b Z have same Elliott data.
Rørdam, Toms used a Chern class obstruction of Villadsen
to produce vast counter examples to Elliott’s conjecture.
T HE FINITE SETTING
Oddly, in the C ˚ -setting the purely infinite case appears to be
easier than the finite case.
R EASONS
Higher dimensional topological phenoma can occur in
simple C ˚ -algebras.
The condition of pure infiniteness prevents this: in a
precise sense these algebras have low topological
dimension.
In the finite case, need traces as well as K -theory.
In late 90’s Jiang-Su constructed Z, an infinite dimensional
C ˚ -algebra with the same Elliott data as the complex
numbers.
For “reasonable” A, A and A b Z have same Elliott data.
Rørdam, Toms used a Chern class obstruction of Villadsen
to produce vast counter examples to Elliott’s conjecture.
T HE FINITE SETTING
Oddly, in the C ˚ -setting the purely infinite case appears to be
easier than the finite case.
R EASONS
Higher dimensional topological phenoma can occur in
simple C ˚ -algebras.
The condition of pure infiniteness prevents this: in a
precise sense these algebras have low topological
dimension.
In the finite case, need traces as well as K -theory.
In late 90’s Jiang-Su constructed Z, an infinite dimensional
C ˚ -algebra with the same Elliott data as the complex
numbers.
For “reasonable” A, A and A b Z have same Elliott data.
Rørdam, Toms used a Chern class obstruction of Villadsen
to produce vast counter examples to Elliott’s conjecture.
E STABLISHING Z - STABILITY
We expect that simple, separable nuclear C ˚ -algebras should
be classifable when they absorb Z tensorially.
T HEOREM (W INTER 2010)
Let A be simple, separable unital and nuclear and have finite
nuclear dimension. Then A – A b Z.
Can view this as obtaining tensorial absoption from a
topological property.
Absorbing Z gives room to maneouver in classification
theorems.
T HEOREM (T OMS , W INTER 2009)
The class C “ tCpX q ¸ Z : X finite dimensional metrisable,
Z ñ X minimal, uniquely ergodicu is classified by K -theory.
E STABLISHING Z - STABILITY
We expect that simple, separable nuclear C ˚ -algebras should
be classifable when they absorb Z tensorially.
T HEOREM (W INTER 2010)
Let A be simple, separable unital and nuclear and have finite
nuclear dimension. Then A – A b Z.
Can view this as obtaining tensorial absoption from a
topological property.
Absorbing Z gives room to maneouver in classification
theorems.
T HEOREM (T OMS , W INTER 2009)
The class C “ tCpX q ¸ Z : X finite dimensional metrisable,
Z ñ X minimal, uniquely ergodicu is classified by K -theory.
Further, Z-stability is analogous to absorbing R in a very
striking way. Recall that a II1 factor M has M – M b R if and
only if Mk ãÑ M ω X M 1 .
For separable A, A – A b Z if and only if the central
sequence algebra Aω X A1 is sufficiently non-commutative.
For separable unital A, A – A b Z if and only if, for some
k ě 2, there exists a “large” cpc order zero map
φ : Mk Ñ Aω X A1 .
Losely, large means that 1A ´ φp1k q is dominated by φpe11 q.
Further, Z-stability is analogous to absorbing R in a very
striking way. Recall that a II1 factor M has M – M b R if and
only if Mk ãÑ M ω X M 1 .
For separable A, A – A b Z if and only if the central
sequence algebra Aω X A1 is sufficiently non-commutative.
For separable unital A, A – A b Z if and only if, for some
k ě 2, there exists a “large” cpc order zero map
φ : Mk Ñ Aω X A1 .
Losely, large means that 1A ´ φp1k q is dominated by φpe11 q.
Further, Z-stability is analogous to absorbing R in a very
striking way. Recall that a II1 factor M has M – M b R if and
only if Mk ãÑ M ω X M 1 .
For separable A, A – A b Z if and only if the central
sequence algebra Aω X A1 is sufficiently non-commutative.
For separable unital A, A – A b Z if and only if, for some
k ě 2, there exists a “large” cpc order zero map
φ : Mk Ñ Aω X A1 .
Losely, large means that 1A ´ φp1k q is dominated by φpe11 q.
T ENSORIAL ABSORPTION FROM ABSTRACT PROPERTIES
G OAL ( PART OF A CONJECTURE OF T OMS AND W INTER )
Let A be simple unital separable and nuclear. Then
A has strict comparison ô A – A b Z.
ð is Rørdam.
Strict comparison is a condition which says that we can
determine the order on positive elements by looking at their
ranks. The analogous condition holds for all II1 factors.
This would be a vast generalisation of Kirchberg’s
O8 -absorption theorem as when A is simple, separable
and traceless Rørdam has shown:
A – A b Z ô A – A b O8
A purely infinite ô A has strict comparson
T ENSORIAL ABSORPTION FROM ABSTRACT PROPERTIES
G OAL ( PART OF A CONJECTURE OF T OMS AND W INTER )
Let A be simple unital separable and nuclear. Then
A has strict comparison ô A – A b Z.
ð is Rørdam.
Strict comparison is a condition which says that we can
determine the order on positive elements by looking at their
ranks. The analogous condition holds for all II1 factors.
This would be a vast generalisation of Kirchberg’s
O8 -absorption theorem as when A is simple, separable
and traceless Rørdam has shown:
A – A b Z ô A – A b O8
A purely infinite ô A has strict comparson
T ENSORIAL ABSORPTION FROM ABSTRACT PROPERTIES
G OAL ( PART OF A CONJECTURE OF T OMS AND W INTER )
Let A be simple unital separable and nuclear. Then
A has strict comparison ô A – A b Z.
ð is Rørdam.
Strict comparison is a condition which says that we can
determine the order on positive elements by looking at their
ranks. The analogous condition holds for all II1 factors.
This would be a vast generalisation of Kirchberg’s
O8 -absorption theorem as when A is simple, separable
and traceless Rørdam has shown:
A – A b Z ô A – A b O8
A purely infinite ô A has strict comparson
C URRENT S TATUS
T HEOREM (K IRCHBERG -RØRDAM , S ATO ,
T OMS -W-W INTER , BUILDING ON M ATUI -S ATO )
Let A be simple separable unital and nuclear and suppose that
T pAq is non-empty, and has a compact extreme boundary of
finite covering dimension. Then
A has strict comparison ô A – A b Z.
Proofs use interplay between von Neumann and C ˚ -algebras
directly:
For each extreme trace τ , the weak closure Mτ of A in the
GNS representation is an injective II1 factor.
By Connes, Mτ – Mτ b R so have Mk ãÑ Mτω X Mτ1 .
Can use the surjectivity of Aω X A1 Ñ Mτω X Mτ1 to obtain
order zero maps Mk Ñ Aω X A1 large wrt τ .
The condition on T pAq is used to glue these together to
obtain one large order zero map Mk Ñ Aω X A1 .