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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
History and the case of groups Previous results The methods
Outline
History and the case of groups
Previous results
The methods
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.
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.
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.
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.
History and the case of groups Previous results The methods
Setup
• R is a unital commutative ring.
History and the case of groups Previous results The methods
Setup
• R is a unital commutative ring.
• G is a Hausdorff ample groupoid.
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).
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).
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.
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 ).
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.
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
′).
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.
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.
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 ?
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.
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.
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.
History and the case of groups Previous results The methods
How about groups?
• Groups are just one-object ample groupoids.
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.
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.
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?
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).
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.
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.
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.
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.
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).
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.
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.
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)×.
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.
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.
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.
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 .
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.
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.
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.
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?
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.
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.
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.
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.
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.
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.
History and the case of groups Previous results The methods
Zero divisors
• Let R be an integral domain.
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.
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.
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.
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.
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.
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.
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.
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.
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).
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.
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.
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).
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.
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.
History and the case of groups Previous results The methods
Groupoids: previous results
• In the following results R is an integral domain.
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.
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.
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.
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.
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.
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.
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).
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.
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.
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.
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.
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.
History and the case of groups Previous results The methods
Cartan subalgebras
• G is effective iff DR(G ) is a maximal commutativesubring of RG .
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.
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.
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.
History and the case of groups Previous results The methods
Carlsen and Rout
• The best prior result is due to Carlsen and Rout.
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.
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.
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.
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.
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).
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.
History and the case of groups Previous results The methods
Our goals
• We aim to improve on Carlsen and Rout in several ways.
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.
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.
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.
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.
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.
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.
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 ′.
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.
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.
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).
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×.
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.
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.
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.
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.
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.
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.
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.
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 ).
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 .
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.
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 ).
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}.
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)×?
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 )}.
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.
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}.
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 ).
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.
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 )).
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.
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.
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.
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.
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:
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);
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.
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.
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.
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.
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.
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.
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).
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.
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 .
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.
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).
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.
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.
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.
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.
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.
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.
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.
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 ′.
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.
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 .
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 ).
History and the case of groups Previous results The methods
The interior of the isotropy bundle
• Let H = Int(Iso(G )).
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.
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 .
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 ).
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.
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.
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.
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).
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
History and the case of groups Previous results The methods
The end
Thank you for your attention!
History and the case of groupsPrevious resultsThe methods