reconstructing étale groupoids from their algebras · • groupoid reconstruction asks the...

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


Outline

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.


Overview


• Renault showed that a diagonal-preserving isomorphism ofreduced C∗-algebras induces an isomorphism of groupoidsfor topologically principal étale groupoids.


Overview



• A number of authors have studied the analogous questionfor ample groupoid algebras over rings.


Overview



• 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.


Setup

• R is a unital commutative ring.


Setup


• G is a Hausdorff ample groupoid.


Setup



• As an R-module RG = Cc(G(1), R).


Setup




• The product is convolution:

fg(γ) =∑

d(γ)=d(α)

f(γα−1)g(α).


Setup





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.


Setup





fg(γ) =∑

d(γ)=d(α)

f(γα−1)g(α).



• We call it the diagonal subalgebra DR(G ).


Setup





fg(γ) =∑

d(γ)=d(α)

f(γα−1)g(α).



• We call it the diagonal subalgebra DR(G ).

• DR(G ) is commutative.


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.




′).





• 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.


How about groups?

• Groups are just one-object ample groupoids.


How about groups?


• RG, for G a group, is the usual group algebra.


How about groups?



• R is the diagonal subalgebra.


How about groups?




• So we are asking: when does RG ∼= RH as R-algebrasimply G ∼= H?


How about groups?




• 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).


Negative results

• Any two finite abelian groups of the same order haveisomorphic complex algebras by Fourier Analysis.


Negative results


• So a group cannot be recovered from its group algebra ingeneral.


Negative results



• Clearly C is too big.


Negative results




• It is more hopeful ZG1 ∼= ZG2 =⇒ G1 ∼= G2.


Negative results





• Hertweck found two non-isomorphic groups of order221 · 9728 with isomorphic integral group rings (Annals2001).


Negative results





• 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.


Positive results

• There seems to be only one general method to show thata group algebra determines the group.


Positive results


• If r ∈ R× and g ∈ G, then rg ∈ (RG)×.


Positive results



• These are call trivial units.


Positive results




• RG has no non-trivial units if every unit is trivial.


Positive results




• 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.


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.






• 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?


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.







• 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.


Zero divisors

• Let R be an integral domain.


Zero divisors


• If G has torsion, RG has zero divisors.


Zero divisors



• If G is torsion-free, then RG having no non-trivial unitsimplies RG has no zero divisors.


Zero divisors




• So no non-trivial units + no zero divisors = no non-trivialunits + torsion-free.


Zero divisors





• Kaplansky’s unit conjecture says if G is torsion-free, thenRG has no non-trivial units.


Zero divisors






• Kaplansky’s unit conjecture implies Kaplansky’s zerodivisor conjecture.


Zero divisors






• Kaplansky’s unit conjecture implies Kaplansky’s zerodivisor conjecture.

• Note that zero divisors are irrelevant to the group ringisomorphism problem.


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.









• 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.


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.







• 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.


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.








• 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.


Cartan subalgebras

• G is effective iff DR(G ) is a maximal commutativesubring of RG .


Cartan subalgebras


• If G is effective and there is a diagonal-preservingisomorphism RG → RG ′, then G ′ is effective.


Cartan subalgebras



• Thus if there was a version of Ara et al. for effectivegroupoids, then effective groupoids would be determinedby their algebra and diagonal subalgebra.


Cartan subalgebras



• 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.


Carlsen and Rout

• The best prior result is due to Carlsen and Rout.


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.


Carlsen and Rout



• They prove that a diagonal-preserving isomorphismbetween algebras of groupoids in this class implies agroupoid isomorphism.


Carlsen and Rout




• This recovers Ara et al. since the group ring of the trivialgroup has no zero divisors and no non-trivial units.


Carlsen and Rout





• It also recovers the Leavitt path algebra result of Brownet al. without the ∗-condition.


Carlsen and Rout






• Isotropy groups in path groupoids are either trivial or Z(hence orderable).


Carlsen and Rout






• 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.


Our goals

• We aim to improve on Carlsen and Rout in several ways.


Our goals


• We want our groupoids to be determined up toisomorphism by their algebras and diagonal subalgebrasamongst all groupoids, not just a special subclass.


Our goals



• We want to drop the no zero divisor condition since it isnot used in the group ring case.


Our goals




• We want to work with the interior isotropy groups (theisotropy groups of the interior of the isotropy bundle) inorder to handle the effective case.


Our goals





• We want to allow more general coefficient rings.


Our goals






• We accomplish this by generalizing the property of agroup ring having no non-trivial units to groupoid rings.


Our goals






• 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.


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 ′.


A concrete theorem

Theorem (BS)



1. G ∼= G ′.


• This implies Carlsen-Rout because having no non-trivialunits passes to subgroups.


A concrete theorem

Theorem (BS)



1. G ∼= G ′.



• It covers effective groupoids because the interior isotropygroups are all trivial.


A concrete theorem

Theorem (BS)



1. G ∼= G ′.



• 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).


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.






• A groupoid doesn’t “live” inside its algebra like a groupdoes.

• We work with inverse semigroups instead.


Local bisections

• A local bisection U of G is an open subset of G (1) withd |U , r |U injective.


Local bisections


• The compact local bisections form an inverse semigroupΓc(G ) under setwise multiplication.


Local bisections



• It has been observed by Exel, Lawson and Lenz andprobably others that G ∼= G ′ iff Γc(G ) ∼= Γc(G

′) forample groupoids.


Local bisections





• G is the tight (or ultrafilter) groupoid of Γc(G ).


Local bisections






• Γc(G ) embeds in RG via U 7→ χU .


Local bisections







• Γc(G ) spans RG but is not a basis.


Local bisections








• We aim to recover Γc(G ).


Local bisections









• If G is a group, Γc(G) = G ∪ {0}.


Local bisections









• If G is a group, Γc(G) = G ∪ {0}.

• What inverse semigroup replaces (RG)×?


The normalizer of the diagonal

• Let N = {f ∈ RG | ∃f ′ ∈ RG , ff ′f =f, fDR(G )f

′ ∪ f ′DR(G )f ⊆ DR(G )}.




′ ∪ f ′DR(G )f ⊆ DR(G )}.

• N is the normalizer of the diagonal subalgebra.




′ ∪ f ′DR(G )f ⊆ DR(G )}.


• If G is a group, N = (RG)× ∪ {0}.




′ ∪ f ′DR(G )f ⊆ DR(G )}.



• N is an inverse semigroup containing Γc(G ).




′ ∪ f ′DR(G )f ⊆ DR(G )}.




• All the idempotents of N are in DR(G ) and hencecommute.




′ ∪ f ′DR(G )f ⊆ DR(G )}.





• If R is indecomposable, E(DR(G )) = E(N) = E(Γc(G )).




′ ∪ f ′DR(G )f ⊆ DR(G )}.






• Previous papers prove these facts by explicitly describingthe elements of N using their full hypotheses.




′ ∪ f ′DR(G )f ⊆ DR(G )}.






• Previous papers prove these facts by explicitly describingthe elements of N using their full hypotheses.

• Our proof is direct and elementary.


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:





• If S is an inverse semigroup, K ≤ S is a normal inversesubsemigroup if:1. E(K) = E(S) (K is full);





• 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.







• Normal subsemigroups generalize normal subgroups.

• Idempotent-separating congruences are determined byappropriate normal subsemigroups.


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.








• Since D(N) is commutative it satisfies the abovecondition.

• So there is an idempotent-separating quotientπ : N → N/D(N).


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.





Γc(G )

ψ

N

π

N/D(N)

with ψ injective.• For groups, ψ is an isomorphism iff RG has no non-trivialunits.





Γc(G )

ψ

N

π

N/D(N)


• This is equivalent to saying each unit has singletonsupport, i.e., is a local bisection.





Γc(G )

ψ

N

π

N/D(N)



• We say G satisfies the local bisection hypothesis if thesupport of each element of N is a local bisection.





Γc(G )

ψ

N

π

N/D(N)



• 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.


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 .







• 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 ).


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.







Theorem (BS)


• So a groupoid is determined by its algebra and itsdiagonal subalgebra if the interior of its isotropy bundlesatisfies the local bisection hypothesis.







Theorem (BS)



• I’m not convinced that this condition has a simplerreformulation.







Theorem (BS)



• 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).


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.






• 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.


Graded groupoids

• One can consider more generally an ample groupoid Gwith a cocycle c : G → G.


Graded groupoids


• Then RG is G-graded.


Graded groupoids



• The question is whether the pair (RG , DR(G )), taken upto graded isomorphism determines G and the cocycle.


Graded groupoids




• We show this is the case if the homogeneous componentof 1 satisfies the local bisection condition.


Graded groupoids





• The proof is mostly the same but we work with gradedinverse semigroups.


Graded groupoids





• The proof is mostly the same but we work with gradedinverse semigroups.

• Carlsen and Rout and Ara et al. also worked in the gradedsetting.


The end

Thank you for your attention!

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