Reconstructing étale groupoids from their algebras · • Groupoid reconstruction asks the...
Transcript of Reconstructing étale groupoids from their algebras · • Groupoid reconstruction asks the...
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History and the case of groups Previous results The methods
Reconstructing étale groupoids from their
algebras
Benjamin Steinberg (City College of New York)
December 5, 2017Facets of Irreversibility
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History and the case of groups Previous results The methods
Outline
History and the case of groups
Previous results
The methods
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History and the case of groups Previous results The methods
Overview
• Groupoid reconstruction asks the question: when is anétale groupoid determined by the pair consisting of itsalgebra and its diagonal subalgebra.
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History and the case of groups Previous results The methods
Overview
• Groupoid reconstruction asks the question: when is anétale groupoid determined by the pair consisting of itsalgebra and its diagonal subalgebra.
• Renault showed that a diagonal-preserving isomorphism ofreduced C∗-algebras induces an isomorphism of groupoidsfor topologically principal étale groupoids.
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History and the case of groups Previous results The methods
Overview
• Groupoid reconstruction asks the question: when is anétale groupoid determined by the pair consisting of itsalgebra and its diagonal subalgebra.
• Renault showed that a diagonal-preserving isomorphism ofreduced C∗-algebras induces an isomorphism of groupoidsfor topologically principal étale groupoids.
• A number of authors have studied the analogous questionfor ample groupoid algebras over rings.
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History and the case of groups Previous results The methods
Overview
• Groupoid reconstruction asks the question: when is anétale groupoid determined by the pair consisting of itsalgebra and its diagonal subalgebra.
• Renault showed that a diagonal-preserving isomorphism ofreduced C∗-algebras induces an isomorphism of groupoidsfor topologically principal étale groupoids.
• A number of authors have studied the analogous questionfor ample groupoid algebras over rings.
• We present here what we believe is the best result onecan get from the present methodology.
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History and the case of groups Previous results The methods
Setup
• R is a unital commutative ring.
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History and the case of groups Previous results The methods
Setup
• R is a unital commutative ring.
• G is a Hausdorff ample groupoid.
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History and the case of groups Previous results The methods
Setup
• R is a unital commutative ring.
• G is a Hausdorff ample groupoid.
• As an R-module RG = Cc(G(1), R).
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History and the case of groups Previous results The methods
Setup
• R is a unital commutative ring.
• G is a Hausdorff ample groupoid.
• As an R-module RG = Cc(G(1), R).
• The product is convolution:
fg(γ) =∑
d(γ)=d(α)
f(γα−1)g(α).
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History and the case of groups Previous results The methods
Setup
• R is a unital commutative ring.
• G is a Hausdorff ample groupoid.
• As an R-module RG = Cc(G(1), R).
• The product is convolution:
fg(γ) =∑
d(γ)=d(α)
f(γα−1)g(α).
• Cc(G(0), R) sits inside of RG with convolution restricting
to pointwise multiplication.
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History and the case of groups Previous results The methods
Setup
• R is a unital commutative ring.
• G is a Hausdorff ample groupoid.
• As an R-module RG = Cc(G(1), R).
• The product is convolution:
fg(γ) =∑
d(γ)=d(α)
f(γα−1)g(α).
• Cc(G(0), R) sits inside of RG with convolution restricting
to pointwise multiplication.
• We call it the diagonal subalgebra DR(G ).
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History and the case of groups Previous results The methods
Setup
• R is a unital commutative ring.
• G is a Hausdorff ample groupoid.
• As an R-module RG = Cc(G(1), R).
• The product is convolution:
fg(γ) =∑
d(γ)=d(α)
f(γα−1)g(α).
• Cc(G(0), R) sits inside of RG with convolution restricting
to pointwise multiplication.
• We call it the diagonal subalgebra DR(G ).
• DR(G ) is commutative.
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History and the case of groups Previous results The methods
Diagonal-preserving isomorphisms
• An isomorphism Φ: RG → RG ′ is diagonal-preserving ifΦ(DR(G )) = DR(G
′).
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History and the case of groups Previous results The methods
Diagonal-preserving isomorphisms
• An isomorphism Φ: RG → RG ′ is diagonal-preserving ifΦ(DR(G )) = DR(G
′).
• An isomorphism φ : G → G ′ induces a diagonal-preservingisomorphism.
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History and the case of groups Previous results The methods
Diagonal-preserving isomorphisms
• An isomorphism Φ: RG → RG ′ is diagonal-preserving ifΦ(DR(G )) = DR(G
′).
• An isomorphism φ : G → G ′ induces a diagonal-preservingisomorphism.
• We are interested in when the converse holds.
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History and the case of groups Previous results The methods
Diagonal-preserving isomorphisms
• An isomorphism Φ: RG → RG ′ is diagonal-preserving ifΦ(DR(G )) = DR(G
′).
• An isomorphism φ : G → G ′ induces a diagonal-preservingisomorphism.
• We are interested in when the converse holds.
• So we want to know: when does the pair (RG , DR(G ))determine G ?
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History and the case of groups Previous results The methods
Diagonal-preserving isomorphisms
• An isomorphism Φ: RG → RG ′ is diagonal-preserving ifΦ(DR(G )) = DR(G
′).
• An isomorphism φ : G → G ′ induces a diagonal-preservingisomorphism.
• We are interested in when the converse holds.
• So we want to know: when does the pair (RG , DR(G ))determine G ?
• We work in the category of rings: we just ask Φ to be aring isomorphism.
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History and the case of groups Previous results The methods
Diagonal-preserving isomorphisms
• An isomorphism Φ: RG → RG ′ is diagonal-preserving ifΦ(DR(G )) = DR(G
′).
• An isomorphism φ : G → G ′ induces a diagonal-preservingisomorphism.
• We are interested in when the converse holds.
• So we want to know: when does the pair (RG , DR(G ))determine G ?
• We work in the category of rings: we just ask Φ to be aring isomorphism.
• This is not much of a loss of generality because DR(G )knows a lot about R.
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History and the case of groups Previous results The methods
Diagonal-preserving isomorphisms
• An isomorphism Φ: RG → RG ′ is diagonal-preserving ifΦ(DR(G )) = DR(G
′).
• An isomorphism φ : G → G ′ induces a diagonal-preservingisomorphism.
• We are interested in when the converse holds.
• So we want to know: when does the pair (RG , DR(G ))determine G ?
• We work in the category of rings: we just ask Φ to be aring isomorphism.
• This is not much of a loss of generality because DR(G )knows a lot about R.
• We do not work with the ∗-ring structure.
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History and the case of groups Previous results The methods
How about groups?
• Groups are just one-object ample groupoids.
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History and the case of groups Previous results The methods
How about groups?
• Groups are just one-object ample groupoids.
• RG, for G a group, is the usual group algebra.
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History and the case of groups Previous results The methods
How about groups?
• Groups are just one-object ample groupoids.
• RG, for G a group, is the usual group algebra.
• R is the diagonal subalgebra.
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History and the case of groups Previous results The methods
How about groups?
• Groups are just one-object ample groupoids.
• RG, for G a group, is the usual group algebra.
• R is the diagonal subalgebra.
• So we are asking: when does RG ∼= RH as R-algebrasimply G ∼= H?
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History and the case of groups Previous results The methods
How about groups?
• Groups are just one-object ample groupoids.
• RG, for G a group, is the usual group algebra.
• R is the diagonal subalgebra.
• So we are asking: when does RG ∼= RH as R-algebrasimply G ∼= H?
• This is the classical isomorphism problem for group rings(goes back to 1940s).
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History and the case of groups Previous results The methods
Negative results
• Any two finite abelian groups of the same order haveisomorphic complex algebras by Fourier Analysis.
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History and the case of groups Previous results The methods
Negative results
• Any two finite abelian groups of the same order haveisomorphic complex algebras by Fourier Analysis.
• So a group cannot be recovered from its group algebra ingeneral.
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History and the case of groups Previous results The methods
Negative results
• Any two finite abelian groups of the same order haveisomorphic complex algebras by Fourier Analysis.
• So a group cannot be recovered from its group algebra ingeneral.
• Clearly C is too big.
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History and the case of groups Previous results The methods
Negative results
• Any two finite abelian groups of the same order haveisomorphic complex algebras by Fourier Analysis.
• So a group cannot be recovered from its group algebra ingeneral.
• Clearly C is too big.
• It is more hopeful ZG1 ∼= ZG2 =⇒ G1 ∼= G2.
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History and the case of groups Previous results The methods
Negative results
• Any two finite abelian groups of the same order haveisomorphic complex algebras by Fourier Analysis.
• So a group cannot be recovered from its group algebra ingeneral.
• Clearly C is too big.
• It is more hopeful ZG1 ∼= ZG2 =⇒ G1 ∼= G2.
• Hertweck found two non-isomorphic groups of order221 · 9728 with isomorphic integral group rings (Annals2001).
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History and the case of groups Previous results The methods
Negative results
• Any two finite abelian groups of the same order haveisomorphic complex algebras by Fourier Analysis.
• So a group cannot be recovered from its group algebra ingeneral.
• Clearly C is too big.
• It is more hopeful ZG1 ∼= ZG2 =⇒ G1 ∼= G2.
• Hertweck found two non-isomorphic groups of order221 · 9728 with isomorphic integral group rings (Annals2001).
• These groups cannot be recovered from their group ringsover any base ring.
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History and the case of groups Previous results The methods
Positive results
• There seems to be only one general method to show thata group algebra determines the group.
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History and the case of groups Previous results The methods
Positive results
• There seems to be only one general method to show thata group algebra determines the group.
• If r ∈ R× and g ∈ G, then rg ∈ (RG)×.
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History and the case of groups Previous results The methods
Positive results
• There seems to be only one general method to show thata group algebra determines the group.
• If r ∈ R× and g ∈ G, then rg ∈ (RG)×.
• These are call trivial units.
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History and the case of groups Previous results The methods
Positive results
• There seems to be only one general method to show thata group algebra determines the group.
• If r ∈ R× and g ∈ G, then rg ∈ (RG)×.
• These are call trivial units.
• RG has no non-trivial units if every unit is trivial.
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History and the case of groups Previous results The methods
Positive results
• There seems to be only one general method to show thata group algebra determines the group.
• If r ∈ R× and g ∈ G, then rg ∈ (RG)×.
• These are call trivial units.
• RG has no non-trivial units if every unit is trivial.
• That is ψ in the diagram
G
ψ
(RG)×
(RG)×/R×
is an isomorphism.
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History and the case of groups Previous results The methods
No non-trivial units
• If RG and RH have no non-trivial units and RG ∼= RHas R-algebras, then G ∼= (RG)×/R× ∼= (RH)×/R× ∼= H .
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History and the case of groups Previous results The methods
No non-trivial units
• If RG and RH have no non-trivial units and RG ∼= RHas R-algebras, then G ∼= (RG)×/R× ∼= (RH)×/R× ∼= H .
• In fact, if RG has no non-trivial units and RG ∼= RH asR-algebras, then RH has no non-trivial units.
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History and the case of groups Previous results The methods
No non-trivial units
• If RG and RH have no non-trivial units and RG ∼= RHas R-algebras, then G ∼= (RG)×/R× ∼= (RH)×/R× ∼= H .
• In fact, if RG has no non-trivial units and RG ∼= RH asR-algebras, then RH has no non-trivial units.
• The proof uses that a group is a basis for its group ring.
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History and the case of groups Previous results The methods
No non-trivial units
• If RG and RH have no non-trivial units and RG ∼= RHas R-algebras, then G ∼= (RG)×/R× ∼= (RH)×/R× ∼= H .
• In fact, if RG has no non-trivial units and RG ∼= RH asR-algebras, then RH has no non-trivial units.
• The proof uses that a group is a basis for its group ring.
• So if RG has no non-trivial units, then G is determined byits group ring up to diagonal-preserving isomorphism.
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History and the case of groups Previous results The methods
No non-trivial units
• If RG and RH have no non-trivial units and RG ∼= RHas R-algebras, then G ∼= (RG)×/R× ∼= (RH)×/R× ∼= H .
• In fact, if RG has no non-trivial units and RG ∼= RH asR-algebras, then RH has no non-trivial units.
• The proof uses that a group is a basis for its group ring.
• So if RG has no non-trivial units, then G is determined byits group ring up to diagonal-preserving isomorphism.
• But which group rings have no non-trivial units?
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History and the case of groups Previous results The methods
Group rings with no non-trivial units
• If G has the unique product property (upp), RG has nonon-trivial units for any integral domain R.
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History and the case of groups Previous results The methods
Group rings with no non-trivial units
• If G has the unique product property (upp), RG has nonon-trivial units for any integral domain R.
• Left orderable groups have upp.
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History and the case of groups Previous results The methods
Group rings with no non-trivial units
• If G has the unique product property (upp), RG has nonon-trivial units for any integral domain R.
• Left orderable groups have upp.
• This includes torsion-free abelian groups, free groups andbraid groups.
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History and the case of groups Previous results The methods
Group rings with no non-trivial units
• If G has the unique product property (upp), RG has nonon-trivial units for any integral domain R.
• Left orderable groups have upp.
• This includes torsion-free abelian groups, free groups andbraid groups.
• Upp groups are torsion-free.
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History and the case of groups Previous results The methods
Group rings with no non-trivial units
• If G has the unique product property (upp), RG has nonon-trivial units for any integral domain R.
• Left orderable groups have upp.
• This includes torsion-free abelian groups, free groups andbraid groups.
• Upp groups are torsion-free.
• Higman proved if ZG has no non-trivial units, Z[G× C2]has no non-trivial units.
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History and the case of groups Previous results The methods
Group rings with no non-trivial units
• If G has the unique product property (upp), RG has nonon-trivial units for any integral domain R.
• Left orderable groups have upp.
• This includes torsion-free abelian groups, free groups andbraid groups.
• Upp groups are torsion-free.
• Higman proved if ZG has no non-trivial units, Z[G× C2]has no non-trivial units.
• He also proved that ZG has no non-trivial units if G isfinite abelian of exponent dividing 4 or 6 or if G is aquaternion group.
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History and the case of groups Previous results The methods
Zero divisors
• Let R be an integral domain.
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History and the case of groups Previous results The methods
Zero divisors
• Let R be an integral domain.
• If G has torsion, RG has zero divisors.
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History and the case of groups Previous results The methods
Zero divisors
• Let R be an integral domain.
• If G has torsion, RG has zero divisors.
• If G is torsion-free, then RG having no non-trivial unitsimplies RG has no zero divisors.
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History and the case of groups Previous results The methods
Zero divisors
• Let R be an integral domain.
• If G has torsion, RG has zero divisors.
• If G is torsion-free, then RG having no non-trivial unitsimplies RG has no zero divisors.
• So no non-trivial units + no zero divisors = no non-trivialunits + torsion-free.
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History and the case of groups Previous results The methods
Zero divisors
• Let R be an integral domain.
• If G has torsion, RG has zero divisors.
• If G is torsion-free, then RG having no non-trivial unitsimplies RG has no zero divisors.
• So no non-trivial units + no zero divisors = no non-trivialunits + torsion-free.
• Kaplansky’s unit conjecture says if G is torsion-free, thenRG has no non-trivial units.
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History and the case of groups Previous results The methods
Zero divisors
• Let R be an integral domain.
• If G has torsion, RG has zero divisors.
• If G is torsion-free, then RG having no non-trivial unitsimplies RG has no zero divisors.
• So no non-trivial units + no zero divisors = no non-trivialunits + torsion-free.
• Kaplansky’s unit conjecture says if G is torsion-free, thenRG has no non-trivial units.
• Kaplansky’s unit conjecture implies Kaplansky’s zerodivisor conjecture.
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History and the case of groups Previous results The methods
Zero divisors
• Let R be an integral domain.
• If G has torsion, RG has zero divisors.
• If G is torsion-free, then RG having no non-trivial unitsimplies RG has no zero divisors.
• So no non-trivial units + no zero divisors = no non-trivialunits + torsion-free.
• Kaplansky’s unit conjecture says if G is torsion-free, thenRG has no non-trivial units.
• Kaplansky’s unit conjecture implies Kaplansky’s zerodivisor conjecture.
• Note that zero divisors are irrelevant to the group ringisomorphism problem.
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History and the case of groups Previous results The methods
Beyond integral domains
• If RG has no non-trivial units, it determines G for anyring R.
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History and the case of groups Previous results The methods
Beyond integral domains
• If RG has no non-trivial units, it determines G for anyring R.
• However, if G 6= 1 and RG has no non-trivial units Rmust be indecomposable and reduced.
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History and the case of groups Previous results The methods
Beyond integral domains
• If RG has no non-trivial units, it determines G for anyring R.
• However, if G 6= 1 and RG has no non-trivial units Rmust be indecomposable and reduced.
• R is indecomposable if 0, 1 are its only idempotents (R isnot a direct product).
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History and the case of groups Previous results The methods
Beyond integral domains
• If RG has no non-trivial units, it determines G for anyring R.
• However, if G 6= 1 and RG has no non-trivial units Rmust be indecomposable and reduced.
• R is indecomposable if 0, 1 are its only idempotents (R isnot a direct product).
• R is reduced if 0 is its only nilpotent.
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History and the case of groups Previous results The methods
Beyond integral domains
• If RG has no non-trivial units, it determines G for anyring R.
• However, if G 6= 1 and RG has no non-trivial units Rmust be indecomposable and reduced.
• R is indecomposable if 0, 1 are its only idempotents (R isnot a direct product).
• R is reduced if 0 is its only nilpotent.
• Integral domains are indecomposable and reduced.
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History and the case of groups Previous results The methods
Beyond integral domains
• If RG has no non-trivial units, it determines G for anyring R.
• However, if G 6= 1 and RG has no non-trivial units Rmust be indecomposable and reduced.
• R is indecomposable if 0, 1 are its only idempotents (R isnot a direct product).
• R is reduced if 0 is its only nilpotent.
• Integral domains are indecomposable and reduced.
• So are coordinate rings of connected affine varieties, e.g.,C[x, y]/(xy).
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History and the case of groups Previous results The methods
Beyond integral domains
• If RG has no non-trivial units, it determines G for anyring R.
• However, if G 6= 1 and RG has no non-trivial units Rmust be indecomposable and reduced.
• R is indecomposable if 0, 1 are its only idempotents (R isnot a direct product).
• R is reduced if 0 is its only nilpotent.
• Integral domains are indecomposable and reduced.
• So are coordinate rings of connected affine varieties, e.g.,C[x, y]/(xy).
• It follows from a result of Neher if kG has no non-trivialunits for every field k, then RG has no non-trivial unitsfor every indecomposable reduced ring R.
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History and the case of groups Previous results The methods
Beyond integral domains
• If RG has no non-trivial units, it determines G for anyring R.
• However, if G 6= 1 and RG has no non-trivial units Rmust be indecomposable and reduced.
• R is indecomposable if 0, 1 are its only idempotents (R isnot a direct product).
• R is reduced if 0 is its only nilpotent.
• Integral domains are indecomposable and reduced.
• So are coordinate rings of connected affine varieties, e.g.,C[x, y]/(xy).
• It follows from a result of Neher if kG has no non-trivialunits for every field k, then RG has no non-trivial unitsfor every indecomposable reduced ring R.
• This applies to upp groups and left orderable groups.
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History and the case of groups Previous results The methods
Groupoids: previous results
• In the following results R is an integral domain.
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History and the case of groups Previous results The methods
Groupoids: previous results
• In the following results R is an integral domain.
• Brown, Clark, an Huef proved that a diagonal-preserving∗-ring isomorphism of Leavitt path algebras implies anisomorphism of their path groupoids.
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History and the case of groups Previous results The methods
Groupoids: previous results
• In the following results R is an integral domain.
• Brown, Clark, an Huef proved that a diagonal-preserving∗-ring isomorphism of Leavitt path algebras implies anisomorphism of their path groupoids.
• Leavitt path algebras are the ring theoretic versions ofgraph C∗-algebras.
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History and the case of groups Previous results The methods
Groupoids: previous results
• In the following results R is an integral domain.
• Brown, Clark, an Huef proved that a diagonal-preserving∗-ring isomorphism of Leavitt path algebras implies anisomorphism of their path groupoids.
• Leavitt path algebras are the ring theoretic versions ofgraph C∗-algebras.
• They are defined by the same groupoids as in the C∗-case.
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History and the case of groups Previous results The methods
Groupoids: previous results
• In the following results R is an integral domain.
• Brown, Clark, an Huef proved that a diagonal-preserving∗-ring isomorphism of Leavitt path algebras implies anisomorphism of their path groupoids.
• Leavitt path algebras are the ring theoretic versions ofgraph C∗-algebras.
• They are defined by the same groupoids as in the C∗-case.
• Ara, Bosa, Hazrat, and Sims proved a diagonal-preservingring isomorphism between algebras of topologicallyprincipal groupoids implies an isomorphism of groupoids.
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History and the case of groups Previous results The methods
Groupoids: previous results
• In the following results R is an integral domain.
• Brown, Clark, an Huef proved that a diagonal-preserving∗-ring isomorphism of Leavitt path algebras implies anisomorphism of their path groupoids.
• Leavitt path algebras are the ring theoretic versions ofgraph C∗-algebras.
• They are defined by the same groupoids as in the C∗-case.
• Ara, Bosa, Hazrat, and Sims proved a diagonal-preservingring isomorphism between algebras of topologicallyprincipal groupoids implies an isomorphism of groupoids.
• This is the ring theoretic analogue of Renault’s result.
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History and the case of groups Previous results The methods
Topologically principal versus effective
• G is topologically principal if there is a dense set ofobjects with trivial isotropy.
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History and the case of groups Previous results The methods
Topologically principal versus effective
• G is topologically principal if there is a dense set ofobjects with trivial isotropy.
• G is effective if Int(Iso(G )) = G (0).
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History and the case of groups Previous results The methods
Topologically principal versus effective
• G is topologically principal if there is a dense set ofobjects with trivial isotropy.
• G is effective if Int(Iso(G )) = G (0).
• Topologically principal implies effective.
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History and the case of groups Previous results The methods
Topologically principal versus effective
• G is topologically principal if there is a dense set ofobjects with trivial isotropy.
• G is effective if Int(Iso(G )) = G (0).
• Topologically principal implies effective.
• The converse is true for second countable groupoids.
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History and the case of groups Previous results The methods
Topologically principal versus effective
• G is topologically principal if there is a dense set ofobjects with trivial isotropy.
• G is effective if Int(Iso(G )) = G (0).
• Topologically principal implies effective.
• The converse is true for second countable groupoids.
• Brown, Clark, Farthing and Sims produced an effectivegroupoid with every isotropy group isomorphic to Z.
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History and the case of groups Previous results The methods
Topologically principal versus effective
• G is topologically principal if there is a dense set ofobjects with trivial isotropy.
• G is effective if Int(Iso(G )) = G (0).
• Topologically principal implies effective.
• The converse is true for second countable groupoids.
• Brown, Clark, Farthing and Sims produced an effectivegroupoid with every isotropy group isomorphic to Z.
• Guess: for any group G, there is an effective groupoidwith every isotropy group isomorphic to G.
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History and the case of groups Previous results The methods
Topologically principal versus effective
• G is topologically principal if there is a dense set ofobjects with trivial isotropy.
• G is effective if Int(Iso(G )) = G (0).
• Topologically principal implies effective.
• The converse is true for second countable groupoids.
• Brown, Clark, Farthing and Sims produced an effectivegroupoid with every isotropy group isomorphic to Z.
• Guess: for any group G, there is an effective groupoidwith every isotropy group isomorphic to G.
• For results on simplicity, Cuntz-Krieger uniquenesstheorems, etc., effective is the right notion for groupoidalgebras.
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History and the case of groups Previous results The methods
Cartan subalgebras
• G is effective iff DR(G ) is a maximal commutativesubring of RG .
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History and the case of groups Previous results The methods
Cartan subalgebras
• G is effective iff DR(G ) is a maximal commutativesubring of RG .
• If G is effective and there is a diagonal-preservingisomorphism RG → RG ′, then G ′ is effective.
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History and the case of groups Previous results The methods
Cartan subalgebras
• G is effective iff DR(G ) is a maximal commutativesubring of RG .
• If G is effective and there is a diagonal-preservingisomorphism RG → RG ′, then G ′ is effective.
• Thus if there was a version of Ara et al. for effectivegroupoids, then effective groupoids would be determinedby their algebra and diagonal subalgebra.
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History and the case of groups Previous results The methods
Cartan subalgebras
• G is effective iff DR(G ) is a maximal commutativesubring of RG .
• If G is effective and there is a diagonal-preservingisomorphism RG → RG ′, then G ′ is effective.
• Thus if there was a version of Ara et al. for effectivegroupoids, then effective groupoids would be determinedby their algebra and diagonal subalgebra.
• It is not immediately obvious that being topologicallyprincipal is invariant under diagonal-preservingisomorphism.
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History and the case of groups Previous results The methods
Carlsen and Rout
• The best prior result is due to Carlsen and Rout.
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History and the case of groups Previous results The methods
Carlsen and Rout
• The best prior result is due to Carlsen and Rout.
• They consider groupoids having a dense set of objectswhose isotropy group rings over R have no zero divisorsand no non-trivial units.
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History and the case of groups Previous results The methods
Carlsen and Rout
• The best prior result is due to Carlsen and Rout.
• They consider groupoids having a dense set of objectswhose isotropy group rings over R have no zero divisorsand no non-trivial units.
• They prove that a diagonal-preserving isomorphismbetween algebras of groupoids in this class implies agroupoid isomorphism.
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History and the case of groups Previous results The methods
Carlsen and Rout
• The best prior result is due to Carlsen and Rout.
• They consider groupoids having a dense set of objectswhose isotropy group rings over R have no zero divisorsand no non-trivial units.
• They prove that a diagonal-preserving isomorphismbetween algebras of groupoids in this class implies agroupoid isomorphism.
• This recovers Ara et al. since the group ring of the trivialgroup has no zero divisors and no non-trivial units.
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History and the case of groups Previous results The methods
Carlsen and Rout
• The best prior result is due to Carlsen and Rout.
• They consider groupoids having a dense set of objectswhose isotropy group rings over R have no zero divisorsand no non-trivial units.
• They prove that a diagonal-preserving isomorphismbetween algebras of groupoids in this class implies agroupoid isomorphism.
• This recovers Ara et al. since the group ring of the trivialgroup has no zero divisors and no non-trivial units.
• It also recovers the Leavitt path algebra result of Brownet al. without the ∗-condition.
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History and the case of groups Previous results The methods
Carlsen and Rout
• The best prior result is due to Carlsen and Rout.
• They consider groupoids having a dense set of objectswhose isotropy group rings over R have no zero divisorsand no non-trivial units.
• They prove that a diagonal-preserving isomorphismbetween algebras of groupoids in this class implies agroupoid isomorphism.
• This recovers Ara et al. since the group ring of the trivialgroup has no zero divisors and no non-trivial units.
• It also recovers the Leavitt path algebra result of Brownet al. without the ∗-condition.
• Isotropy groups in path groupoids are either trivial or Z(hence orderable).
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History and the case of groups Previous results The methods
Carlsen and Rout
• The best prior result is due to Carlsen and Rout.
• They consider groupoids having a dense set of objectswhose isotropy group rings over R have no zero divisorsand no non-trivial units.
• They prove that a diagonal-preserving isomorphismbetween algebras of groupoids in this class implies agroupoid isomorphism.
• This recovers Ara et al. since the group ring of the trivialgroup has no zero divisors and no non-trivial units.
• It also recovers the Leavitt path algebra result of Brownet al. without the ∗-condition.
• Isotropy groups in path groupoids are either trivial or Z(hence orderable).
• This result does not cover effective groupoids and groupswith no non-trivial units but with torsion.
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History and the case of groups Previous results The methods
Our goals
• We aim to improve on Carlsen and Rout in several ways.
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History and the case of groups Previous results The methods
Our goals
• We aim to improve on Carlsen and Rout in several ways.
• We want our groupoids to be determined up toisomorphism by their algebras and diagonal subalgebrasamongst all groupoids, not just a special subclass.
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History and the case of groups Previous results The methods
Our goals
• We aim to improve on Carlsen and Rout in several ways.
• We want our groupoids to be determined up toisomorphism by their algebras and diagonal subalgebrasamongst all groupoids, not just a special subclass.
• We want to drop the no zero divisor condition since it isnot used in the group ring case.
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History and the case of groups Previous results The methods
Our goals
• We aim to improve on Carlsen and Rout in several ways.
• We want our groupoids to be determined up toisomorphism by their algebras and diagonal subalgebrasamongst all groupoids, not just a special subclass.
• We want to drop the no zero divisor condition since it isnot used in the group ring case.
• We want to work with the interior isotropy groups (theisotropy groups of the interior of the isotropy bundle) inorder to handle the effective case.
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History and the case of groups Previous results The methods
Our goals
• We aim to improve on Carlsen and Rout in several ways.
• We want our groupoids to be determined up toisomorphism by their algebras and diagonal subalgebrasamongst all groupoids, not just a special subclass.
• We want to drop the no zero divisor condition since it isnot used in the group ring case.
• We want to work with the interior isotropy groups (theisotropy groups of the interior of the isotropy bundle) inorder to handle the effective case.
• We want to allow more general coefficient rings.
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History and the case of groups Previous results The methods
Our goals
• We aim to improve on Carlsen and Rout in several ways.
• We want our groupoids to be determined up toisomorphism by their algebras and diagonal subalgebrasamongst all groupoids, not just a special subclass.
• We want to drop the no zero divisor condition since it isnot used in the group ring case.
• We want to work with the interior isotropy groups (theisotropy groups of the interior of the isotropy bundle) inorder to handle the effective case.
• We want to allow more general coefficient rings.
• We accomplish this by generalizing the property of agroup ring having no non-trivial units to groupoid rings.
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History and the case of groups Previous results The methods
Our goals
• We aim to improve on Carlsen and Rout in several ways.
• We want our groupoids to be determined up toisomorphism by their algebras and diagonal subalgebrasamongst all groupoids, not just a special subclass.
• We want to drop the no zero divisor condition since it isnot used in the group ring case.
• We want to work with the interior isotropy groups (theisotropy groups of the interior of the isotropy bundle) inorder to handle the effective case.
• We want to allow more general coefficient rings.
• We accomplish this by generalizing the property of agroup ring having no non-trivial units to groupoid rings.
• We then give concrete conditions to have this abstractproperty.
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History and the case of groups Previous results The methods
A concrete theorem
Theorem (BS)
Let R be indecomposable, G have a dense set of objectswhose interior isotropy group rings have no non-trivial units
and G ′ be any ample groupoid. The following are equivalent.
1. G ∼= G ′.
2. There is a diagonal-preserving isomorphismΦ: RG → RG ′.
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History and the case of groups Previous results The methods
A concrete theorem
Theorem (BS)
Let R be indecomposable, G have a dense set of objectswhose interior isotropy group rings have no non-trivial units
and G ′ be any ample groupoid. The following are equivalent.
1. G ∼= G ′.
2. There is a diagonal-preserving isomorphismΦ: RG → RG ′.
• This implies Carlsen-Rout because having no non-trivialunits passes to subgroups.
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History and the case of groups Previous results The methods
A concrete theorem
Theorem (BS)
Let R be indecomposable, G have a dense set of objectswhose interior isotropy group rings have no non-trivial units
and G ′ be any ample groupoid. The following are equivalent.
1. G ∼= G ′.
2. There is a diagonal-preserving isomorphismΦ: RG → RG ′.
• This implies Carlsen-Rout because having no non-trivialunits passes to subgroups.
• It covers effective groupoids because the interior isotropygroups are all trivial.
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History and the case of groups Previous results The methods
A concrete theorem
Theorem (BS)
Let R be indecomposable, G have a dense set of objectswhose interior isotropy group rings have no non-trivial units
and G ′ be any ample groupoid. The following are equivalent.
1. G ∼= G ′.
2. There is a diagonal-preserving isomorphismΦ: RG → RG ′.
• This implies Carlsen-Rout because having no non-trivialunits passes to subgroups.
• It covers effective groupoids because the interior isotropygroups are all trivial.
• It covers Leavitt path algebras over indecomposablereduced rings. Just indecomposable is needed undercondition (L).
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History and the case of groups Previous results The methods
Recovering groupoids
• If RG has no non-trivial units, we recover G as(RG)×/R×.
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History and the case of groups Previous results The methods
Recovering groupoids
• If RG has no non-trivial units, we recover G as(RG)×/R×.
• This is the unit group mod the diagonal units.
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History and the case of groups Previous results The methods
Recovering groupoids
• If RG has no non-trivial units, we recover G as(RG)×/R×.
• This is the unit group mod the diagonal units.
• We have to replace this with something else for groupoids.
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History and the case of groups Previous results The methods
Recovering groupoids
• If RG has no non-trivial units, we recover G as(RG)×/R×.
• This is the unit group mod the diagonal units.
• We have to replace this with something else for groupoids.
• A groupoid doesn’t “live” inside its algebra like a groupdoes.
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History and the case of groups Previous results The methods
Recovering groupoids
• If RG has no non-trivial units, we recover G as(RG)×/R×.
• This is the unit group mod the diagonal units.
• We have to replace this with something else for groupoids.
• A groupoid doesn’t “live” inside its algebra like a groupdoes.
• We work with inverse semigroups instead.
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History and the case of groups Previous results The methods
Local bisections
• A local bisection U of G is an open subset of G (1) withd |U , r |U injective.
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History and the case of groups Previous results The methods
Local bisections
• A local bisection U of G is an open subset of G (1) withd |U , r |U injective.
• The compact local bisections form an inverse semigroupΓc(G ) under setwise multiplication.
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History and the case of groups Previous results The methods
Local bisections
• A local bisection U of G is an open subset of G (1) withd |U , r |U injective.
• The compact local bisections form an inverse semigroupΓc(G ) under setwise multiplication.
• It has been observed by Exel, Lawson and Lenz andprobably others that G ∼= G ′ iff Γc(G ) ∼= Γc(G
′) forample groupoids.
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History and the case of groups Previous results The methods
Local bisections
• A local bisection U of G is an open subset of G (1) withd |U , r |U injective.
• The compact local bisections form an inverse semigroupΓc(G ) under setwise multiplication.
• It has been observed by Exel, Lawson and Lenz andprobably others that G ∼= G ′ iff Γc(G ) ∼= Γc(G
′) forample groupoids.
• G is the tight (or ultrafilter) groupoid of Γc(G ).
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History and the case of groups Previous results The methods
Local bisections
• A local bisection U of G is an open subset of G (1) withd |U , r |U injective.
• The compact local bisections form an inverse semigroupΓc(G ) under setwise multiplication.
• It has been observed by Exel, Lawson and Lenz andprobably others that G ∼= G ′ iff Γc(G ) ∼= Γc(G
′) forample groupoids.
• G is the tight (or ultrafilter) groupoid of Γc(G ).
• Γc(G ) embeds in RG via U 7→ χU .
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History and the case of groups Previous results The methods
Local bisections
• A local bisection U of G is an open subset of G (1) withd |U , r |U injective.
• The compact local bisections form an inverse semigroupΓc(G ) under setwise multiplication.
• It has been observed by Exel, Lawson and Lenz andprobably others that G ∼= G ′ iff Γc(G ) ∼= Γc(G
′) forample groupoids.
• G is the tight (or ultrafilter) groupoid of Γc(G ).
• Γc(G ) embeds in RG via U 7→ χU .
• Γc(G ) spans RG but is not a basis.
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History and the case of groups Previous results The methods
Local bisections
• A local bisection U of G is an open subset of G (1) withd |U , r |U injective.
• The compact local bisections form an inverse semigroupΓc(G ) under setwise multiplication.
• It has been observed by Exel, Lawson and Lenz andprobably others that G ∼= G ′ iff Γc(G ) ∼= Γc(G
′) forample groupoids.
• G is the tight (or ultrafilter) groupoid of Γc(G ).
• Γc(G ) embeds in RG via U 7→ χU .
• Γc(G ) spans RG but is not a basis.
• We aim to recover Γc(G ).
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History and the case of groups Previous results The methods
Local bisections
• A local bisection U of G is an open subset of G (1) withd |U , r |U injective.
• The compact local bisections form an inverse semigroupΓc(G ) under setwise multiplication.
• It has been observed by Exel, Lawson and Lenz andprobably others that G ∼= G ′ iff Γc(G ) ∼= Γc(G
′) forample groupoids.
• G is the tight (or ultrafilter) groupoid of Γc(G ).
• Γc(G ) embeds in RG via U 7→ χU .
• Γc(G ) spans RG but is not a basis.
• We aim to recover Γc(G ).
• If G is a group, Γc(G) = G ∪ {0}.
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History and the case of groups Previous results The methods
Local bisections
• A local bisection U of G is an open subset of G (1) withd |U , r |U injective.
• The compact local bisections form an inverse semigroupΓc(G ) under setwise multiplication.
• It has been observed by Exel, Lawson and Lenz andprobably others that G ∼= G ′ iff Γc(G ) ∼= Γc(G
′) forample groupoids.
• G is the tight (or ultrafilter) groupoid of Γc(G ).
• Γc(G ) embeds in RG via U 7→ χU .
• Γc(G ) spans RG but is not a basis.
• We aim to recover Γc(G ).
• If G is a group, Γc(G) = G ∪ {0}.
• What inverse semigroup replaces (RG)×?
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History and the case of groups Previous results The methods
The normalizer of the diagonal
• Let N = {f ∈ RG | ∃f ′ ∈ RG , ff ′f =f, fDR(G )f
′ ∪ f ′DR(G )f ⊆ DR(G )}.
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History and the case of groups Previous results The methods
The normalizer of the diagonal
• Let N = {f ∈ RG | ∃f ′ ∈ RG , ff ′f =f, fDR(G )f
′ ∪ f ′DR(G )f ⊆ DR(G )}.
• N is the normalizer of the diagonal subalgebra.
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History and the case of groups Previous results The methods
The normalizer of the diagonal
• Let N = {f ∈ RG | ∃f ′ ∈ RG , ff ′f =f, fDR(G )f
′ ∪ f ′DR(G )f ⊆ DR(G )}.
• N is the normalizer of the diagonal subalgebra.
• If G is a group, N = (RG)× ∪ {0}.
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History and the case of groups Previous results The methods
The normalizer of the diagonal
• Let N = {f ∈ RG | ∃f ′ ∈ RG , ff ′f =f, fDR(G )f
′ ∪ f ′DR(G )f ⊆ DR(G )}.
• N is the normalizer of the diagonal subalgebra.
• If G is a group, N = (RG)× ∪ {0}.
• N is an inverse semigroup containing Γc(G ).
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History and the case of groups Previous results The methods
The normalizer of the diagonal
• Let N = {f ∈ RG | ∃f ′ ∈ RG , ff ′f =f, fDR(G )f
′ ∪ f ′DR(G )f ⊆ DR(G )}.
• N is the normalizer of the diagonal subalgebra.
• If G is a group, N = (RG)× ∪ {0}.
• N is an inverse semigroup containing Γc(G ).
• All the idempotents of N are in DR(G ) and hencecommute.
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History and the case of groups Previous results The methods
The normalizer of the diagonal
• Let N = {f ∈ RG | ∃f ′ ∈ RG , ff ′f =f, fDR(G )f
′ ∪ f ′DR(G )f ⊆ DR(G )}.
• N is the normalizer of the diagonal subalgebra.
• If G is a group, N = (RG)× ∪ {0}.
• N is an inverse semigroup containing Γc(G ).
• All the idempotents of N are in DR(G ) and hencecommute.
• If R is indecomposable, E(DR(G )) = E(N) = E(Γc(G )).
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History and the case of groups Previous results The methods
The normalizer of the diagonal
• Let N = {f ∈ RG | ∃f ′ ∈ RG , ff ′f =f, fDR(G )f
′ ∪ f ′DR(G )f ⊆ DR(G )}.
• N is the normalizer of the diagonal subalgebra.
• If G is a group, N = (RG)× ∪ {0}.
• N is an inverse semigroup containing Γc(G ).
• All the idempotents of N are in DR(G ) and hencecommute.
• If R is indecomposable, E(DR(G )) = E(N) = E(Γc(G )).
• Previous papers prove these facts by explicitly describingthe elements of N using their full hypotheses.
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History and the case of groups Previous results The methods
The normalizer of the diagonal
• Let N = {f ∈ RG | ∃f ′ ∈ RG , ff ′f =f, fDR(G )f
′ ∪ f ′DR(G )f ⊆ DR(G )}.
• N is the normalizer of the diagonal subalgebra.
• If G is a group, N = (RG)× ∪ {0}.
• N is an inverse semigroup containing Γc(G ).
• All the idempotents of N are in DR(G ) and hencecommute.
• If R is indecomposable, E(DR(G )) = E(N) = E(Γc(G )).
• Previous papers prove these facts by explicitly describingthe elements of N using their full hypotheses.
• Our proof is direct and elementary.
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History and the case of groups Previous results The methods
Normal subsemigroups
• RG has no non-trivial units when G is isomorphic to unitsmod diagonal units.
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History and the case of groups Previous results The methods
Normal subsemigroups
• RG has no non-trivial units when G is isomorphic to unitsmod diagonal units.
• The analogue for us should be normalizers mod diagonalnormalizers.
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History and the case of groups Previous results The methods
Normal subsemigroups
• RG has no non-trivial units when G is isomorphic to unitsmod diagonal units.
• The analogue for us should be normalizers mod diagonalnormalizers.
• If S is an inverse semigroup, K ≤ S is a normal inversesubsemigroup if:
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History and the case of groups Previous results The methods
Normal subsemigroups
• RG has no non-trivial units when G is isomorphic to unitsmod diagonal units.
• The analogue for us should be normalizers mod diagonalnormalizers.
• If S is an inverse semigroup, K ≤ S is a normal inversesubsemigroup if:1. E(K) = E(S) (K is full);
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History and the case of groups Previous results The methods
Normal subsemigroups
• RG has no non-trivial units when G is isomorphic to unitsmod diagonal units.
• The analogue for us should be normalizers mod diagonalnormalizers.
• If S is an inverse semigroup, K ≤ S is a normal inversesubsemigroup if:1. E(K) = E(S) (K is full);2. sKs−1 ⊆ K for all s ∈ S.
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History and the case of groups Previous results The methods
Normal subsemigroups
• RG has no non-trivial units when G is isomorphic to unitsmod diagonal units.
• The analogue for us should be normalizers mod diagonalnormalizers.
• If S is an inverse semigroup, K ≤ S is a normal inversesubsemigroup if:1. E(K) = E(S) (K is full);2. sKs−1 ⊆ K for all s ∈ S.
• If φ : S → T is a homomorphism, ker φ = φ−1(E(T )) is anormal subsemigroup.
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History and the case of groups Previous results The methods
Normal subsemigroups
• RG has no non-trivial units when G is isomorphic to unitsmod diagonal units.
• The analogue for us should be normalizers mod diagonalnormalizers.
• If S is an inverse semigroup, K ≤ S is a normal inversesubsemigroup if:1. E(K) = E(S) (K is full);2. sKs−1 ⊆ K for all s ∈ S.
• If φ : S → T is a homomorphism, ker φ = φ−1(E(T )) is anormal subsemigroup.
• Normal subsemigroups generalize normal subgroups.
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History and the case of groups Previous results The methods
Normal subsemigroups
• RG has no non-trivial units when G is isomorphic to unitsmod diagonal units.
• The analogue for us should be normalizers mod diagonalnormalizers.
• If S is an inverse semigroup, K ≤ S is a normal inversesubsemigroup if:1. E(K) = E(S) (K is full);2. sKs−1 ⊆ K for all s ∈ S.
• If φ : S → T is a homomorphism, ker φ = φ−1(E(T )) is anormal subsemigroup.
• Normal subsemigroups generalize normal subgroups.
• Idempotent-separating congruences are determined byappropriate normal subsemigroups.
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History and the case of groups Previous results The methods
Idempotent-separating congruences
• A congruence or homomorphism on S isidempotent-separating if it is injective on idempotents.
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History and the case of groups Previous results The methods
Idempotent-separating congruences
• A congruence or homomorphism on S isidempotent-separating if it is injective on idempotents.
• An idempotent-separating congruence is uniquelydetermined by its kernel.
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History and the case of groups Previous results The methods
Idempotent-separating congruences
• A congruence or homomorphism on S isidempotent-separating if it is injective on idempotents.
• An idempotent-separating congruence is uniquelydetermined by its kernel.
• In particular, it is injective iff its kernel is E(S)(idempotent-pure).
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History and the case of groups Previous results The methods
Idempotent-separating congruences
• A congruence or homomorphism on S isidempotent-separating if it is injective on idempotents.
• An idempotent-separating congruence is uniquelydetermined by its kernel.
• In particular, it is injective iff its kernel is E(S)(idempotent-pure).
• K ⊳ S is the kernel of an idempotent-separatingcongruence iff aa−1 = a−1a for all a ∈ K.
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History and the case of groups Previous results The methods
Idempotent-separating congruences
• A congruence or homomorphism on S isidempotent-separating if it is injective on idempotents.
• An idempotent-separating congruence is uniquelydetermined by its kernel.
• In particular, it is injective iff its kernel is E(S)(idempotent-pure).
• K ⊳ S is the kernel of an idempotent-separatingcongruence iff aa−1 = a−1a for all a ∈ K.
• D(N) = N ∩DR(G ) is a normal subsemigroup of N .
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History and the case of groups Previous results The methods
Idempotent-separating congruences
• A congruence or homomorphism on S isidempotent-separating if it is injective on idempotents.
• An idempotent-separating congruence is uniquelydetermined by its kernel.
• In particular, it is injective iff its kernel is E(S)(idempotent-pure).
• K ⊳ S is the kernel of an idempotent-separatingcongruence iff aa−1 = a−1a for all a ∈ K.
• D(N) = N ∩DR(G ) is a normal subsemigroup of N .
• Since D(N) is commutative it satisfies the abovecondition.
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History and the case of groups Previous results The methods
Idempotent-separating congruences
• A congruence or homomorphism on S isidempotent-separating if it is injective on idempotents.
• An idempotent-separating congruence is uniquelydetermined by its kernel.
• In particular, it is injective iff its kernel is E(S)(idempotent-pure).
• K ⊳ S is the kernel of an idempotent-separatingcongruence iff aa−1 = a−1a for all a ∈ K.
• D(N) = N ∩DR(G ) is a normal subsemigroup of N .
• Since D(N) is commutative it satisfies the abovecondition.
• So there is an idempotent-separating quotientπ : N → N/D(N).
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History and the case of groups Previous results The methods
The local bisection hypothesis
• The restriction of π to Γc(G ) is injective since thediagonal elements of Γc(G ) are exactly the idempotents.
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History and the case of groups Previous results The methods
The local bisection hypothesis
• The restriction of π to Γc(G ) is injective since thediagonal elements of Γc(G ) are exactly the idempotents.
• So we have a commutative diagram
Γc(G )
ψ
N
π
N/D(N)
with ψ injective.
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History and the case of groups Previous results The methods
The local bisection hypothesis
• The restriction of π to Γc(G ) is injective since thediagonal elements of Γc(G ) are exactly the idempotents.
• So we have a commutative diagram
Γc(G )
ψ
N
π
N/D(N)
with ψ injective.• For groups, ψ is an isomorphism iff RG has no non-trivialunits.
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History and the case of groups Previous results The methods
The local bisection hypothesis
• The restriction of π to Γc(G ) is injective since thediagonal elements of Γc(G ) are exactly the idempotents.
• So we have a commutative diagram
Γc(G )
ψ
N
π
N/D(N)
with ψ injective.• For groups, ψ is an isomorphism iff RG has no non-trivialunits.
• This is equivalent to saying each unit has singletonsupport, i.e., is a local bisection.
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History and the case of groups Previous results The methods
The local bisection hypothesis
• The restriction of π to Γc(G ) is injective since thediagonal elements of Γc(G ) are exactly the idempotents.
• So we have a commutative diagram
Γc(G )
ψ
N
π
N/D(N)
with ψ injective.• For groups, ψ is an isomorphism iff RG has no non-trivialunits.
• This is equivalent to saying each unit has singletonsupport, i.e., is a local bisection.
• We say G satisfies the local bisection hypothesis if thesupport of each element of N is a local bisection.
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History and the case of groups Previous results The methods
The local bisection hypothesis
• The restriction of π to Γc(G ) is injective since thediagonal elements of Γc(G ) are exactly the idempotents.
• So we have a commutative diagram
Γc(G )
ψ
N
π
N/D(N)
with ψ injective.• For groups, ψ is an isomorphism iff RG has no non-trivialunits.
• This is equivalent to saying each unit has singletonsupport, i.e., is a local bisection.
• We say G satisfies the local bisection hypothesis if thesupport of each element of N is a local bisection.
• This occurs iff ψ is an isomorphism.
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History and the case of groups Previous results The methods
The local bisection hypothesis II
• The construction G 7→ N/D(N) is clearly functorial withrespect to diagonal-preserving isomorphisms.
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History and the case of groups Previous results The methods
The local bisection hypothesis II
• The construction G 7→ N/D(N) is clearly functorial withrespect to diagonal-preserving isomorphisms.
• So if G and G ′ satisfy the local bisection hypothesis andtheir algebras are isomorphic via a diagonal-preservingisomorphism, then Γc(G ) ∼= Γc(G
′) and hence G ∼= G ′.
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History and the case of groups Previous results The methods
The local bisection hypothesis II
• The construction G 7→ N/D(N) is clearly functorial withrespect to diagonal-preserving isomorphisms.
• So if G and G ′ satisfy the local bisection hypothesis andtheir algebras are isomorphic via a diagonal-preservingisomorphism, then Γc(G ) ∼= Γc(G
′) and hence G ∼= G ′.
• In fact, diagonal-preserving isomorphisms preserve thelocal bisection hypothesis and so the assumption is onlyneeded for one of the groupoids.
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History and the case of groups Previous results The methods
The local bisection hypothesis II
• The construction G 7→ N/D(N) is clearly functorial withrespect to diagonal-preserving isomorphisms.
• So if G and G ′ satisfy the local bisection hypothesis andtheir algebras are isomorphic via a diagonal-preservingisomorphism, then Γc(G ) ∼= Γc(G
′) and hence G ∼= G ′.
• In fact, diagonal-preserving isomorphisms preserve thelocal bisection hypothesis and so the assumption is onlyneeded for one of the groupoids.
• The proof is a bit like the group ring case but is trickierbecause Γc(G ) is not linearly independent in RG .
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History and the case of groups Previous results The methods
The local bisection hypothesis II
• The construction G 7→ N/D(N) is clearly functorial withrespect to diagonal-preserving isomorphisms.
• So if G and G ′ satisfy the local bisection hypothesis andtheir algebras are isomorphic via a diagonal-preservingisomorphism, then Γc(G ) ∼= Γc(G
′) and hence G ∼= G ′.
• In fact, diagonal-preserving isomorphisms preserve thelocal bisection hypothesis and so the assumption is onlyneeded for one of the groupoids.
• The proof is a bit like the group ring case but is trickierbecause Γc(G ) is not linearly independent in RG .
• We use instead the order structure of Γc(G ).
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History and the case of groups Previous results The methods
The interior of the isotropy bundle
• Let H = Int(Iso(G )).
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History and the case of groups Previous results The methods
The interior of the isotropy bundle
• Let H = Int(Iso(G )).
• Then G (0) ⊆ H ⊆ G and these are open subgroupoids.
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History and the case of groups Previous results The methods
The interior of the isotropy bundle
• Let H = Int(Iso(G )).
• Then G (0) ⊆ H ⊆ G and these are open subgroupoids.
• Thus DR(G ) ≤ RH ≤ RG .
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History and the case of groups Previous results The methods
The interior of the isotropy bundle
• Let H = Int(Iso(G )).
• Then G (0) ⊆ H ⊆ G and these are open subgroupoids.
• Thus DR(G ) ≤ RH ≤ RG .
• Moreover, DR(G ) = DR(H ).
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History and the case of groups Previous results The methods
The interior of the isotropy bundle
• Let H = Int(Iso(G )).
• Then G (0) ⊆ H ⊆ G and these are open subgroupoids.
• Thus DR(G ) ≤ RH ≤ RG .
• Moreover, DR(G ) = DR(H ).
Theorem (BS)
G satisfies the local bisection hypothesis iff H does.
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History and the case of groups Previous results The methods
The interior of the isotropy bundle
• Let H = Int(Iso(G )).
• Then G (0) ⊆ H ⊆ G and these are open subgroupoids.
• Thus DR(G ) ≤ RH ≤ RG .
• Moreover, DR(G ) = DR(H ).
Theorem (BS)
G satisfies the local bisection hypothesis iff H does.
• So a groupoid is determined by its algebra and itsdiagonal subalgebra if the interior of its isotropy bundlesatisfies the local bisection hypothesis.
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History and the case of groups Previous results The methods
The interior of the isotropy bundle
• Let H = Int(Iso(G )).
• Then G (0) ⊆ H ⊆ G and these are open subgroupoids.
• Thus DR(G ) ≤ RH ≤ RG .
• Moreover, DR(G ) = DR(H ).
Theorem (BS)
G satisfies the local bisection hypothesis iff H does.
• So a groupoid is determined by its algebra and itsdiagonal subalgebra if the interior of its isotropy bundlesatisfies the local bisection hypothesis.
• I’m not convinced that this condition has a simplerreformulation.
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History and the case of groups Previous results The methods
The interior of the isotropy bundle
• Let H = Int(Iso(G )).
• Then G (0) ⊆ H ⊆ G and these are open subgroupoids.
• Thus DR(G ) ≤ RH ≤ RG .
• Moreover, DR(G ) = DR(H ).
Theorem (BS)
G satisfies the local bisection hypothesis iff H does.
• So a groupoid is determined by its algebra and itsdiagonal subalgebra if the interior of its isotropy bundlesatisfies the local bisection hypothesis.
• I’m not convinced that this condition has a simplerreformulation.
• The key point is f ∈ N and α, β ∈ supp(f) impliesd(α) = d(β) ⇐⇒ r(α) = r(β).
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History and the case of groups Previous results The methods
Some sufficient conditions
• If there is a dense set of objects such that the interiorisotropy group rings have no non-trivial units, it is easy toshow that H satisfies the local bisection hypothesis.
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History and the case of groups Previous results The methods
Some sufficient conditions
• If there is a dense set of objects such that the interiorisotropy group rings have no non-trivial units, it is easy toshow that H satisfies the local bisection hypothesis.
• If G (0) has a dense set X of isolated points, then Gsatisfies the local bisection hypothesis iff the isotropygroup ring at each x ∈ X has no non-trivial units.
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History and the case of groups Previous results The methods
Some sufficient conditions
• If there is a dense set of objects such that the interiorisotropy group rings have no non-trivial units, it is easy toshow that H satisfies the local bisection hypothesis.
• If G (0) has a dense set X of isolated points, then Gsatisfies the local bisection hypothesis iff the isotropygroup ring at each x ∈ X has no non-trivial units.
• A number of inverse semigroup universal groupoids havethis property.
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History and the case of groups Previous results The methods
Some sufficient conditions
• If there is a dense set of objects such that the interiorisotropy group rings have no non-trivial units, it is easy toshow that H satisfies the local bisection hypothesis.
• If G (0) has a dense set X of isolated points, then Gsatisfies the local bisection hypothesis iff the isotropygroup ring at each x ∈ X has no non-trivial units.
• A number of inverse semigroup universal groupoids havethis property.
• An element of RH belongs to N iff its restriction toeach isotropy group is either 0 or a unit and its support isa local bisection iff each restriction is either 0 or a trivialunit.
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History and the case of groups Previous results The methods
Some sufficient conditions
• If there is a dense set of objects such that the interiorisotropy group rings have no non-trivial units, it is easy toshow that H satisfies the local bisection hypothesis.
• If G (0) has a dense set X of isolated points, then Gsatisfies the local bisection hypothesis iff the isotropygroup ring at each x ∈ X has no non-trivial units.
• A number of inverse semigroup universal groupoids havethis property.
• An element of RH belongs to N iff its restriction toeach isotropy group is either 0 or a unit and its support isa local bisection iff each restriction is either 0 or a trivialunit.
• So for group bundles, the local bisection hypothesis isquite similar to the no non-trivial unit hypothesis forgroup rings.
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History and the case of groups Previous results The methods
Graded groupoids
• One can consider more generally an ample groupoid Gwith a cocycle c : G → G.
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History and the case of groups Previous results The methods
Graded groupoids
• One can consider more generally an ample groupoid Gwith a cocycle c : G → G.
• Then RG is G-graded.
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History and the case of groups Previous results The methods
Graded groupoids
• One can consider more generally an ample groupoid Gwith a cocycle c : G → G.
• Then RG is G-graded.
• The question is whether the pair (RG , DR(G )), taken upto graded isomorphism determines G and the cocycle.
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History and the case of groups Previous results The methods
Graded groupoids
• One can consider more generally an ample groupoid Gwith a cocycle c : G → G.
• Then RG is G-graded.
• The question is whether the pair (RG , DR(G )), taken upto graded isomorphism determines G and the cocycle.
• We show this is the case if the homogeneous componentof 1 satisfies the local bisection condition.
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History and the case of groups Previous results The methods
Graded groupoids
• One can consider more generally an ample groupoid Gwith a cocycle c : G → G.
• Then RG is G-graded.
• The question is whether the pair (RG , DR(G )), taken upto graded isomorphism determines G and the cocycle.
• We show this is the case if the homogeneous componentof 1 satisfies the local bisection condition.
• The proof is mostly the same but we work with gradedinverse semigroups.
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History and the case of groups Previous results The methods
Graded groupoids
• One can consider more generally an ample groupoid Gwith a cocycle c : G → G.
• Then RG is G-graded.
• The question is whether the pair (RG , DR(G )), taken upto graded isomorphism determines G and the cocycle.
• We show this is the case if the homogeneous componentof 1 satisfies the local bisection condition.
• The proof is mostly the same but we work with gradedinverse semigroups.
• Carlsen and Rout and Ara et al. also worked in the gradedsetting.
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History and the case of groups Previous results The methods
The end
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History and the case of groupsPrevious resultsThe methods