Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Étale Groupoids Étale Groupoid Algebras Étale groupoid algebras Benjamin Steinberg, City College of New York August 1, 2014 Groups, Rings and Group Rings Representation Theory Étale Groupoids Étale Groupoid Algebras Outline Étale Groupoids Étale Groupoid Algebras Representation Theory Representation Theory Étale Groupoids Étale Groupoid Algebras Representation Theory Background • Groupoid C ∗ -algebras have been a fruitful and unifying notion in the theory of operator algebras since J. Renault’s influential monograph (1980). Étale Groupoids Étale Groupoid Algebras Representation Theory Background Groupoid C ∗ -algebras have been a fruitful and unifying notion in the theory of operator algebras since J. Renault’s influential monograph (1980). • They simultaneously generalize: • Étale Groupoids Étale Groupoid Algebras Representation Theory Background Groupoid C ∗ -algebras have been a fruitful and unifying notion in the theory of operator algebras since J. Renault’s influential monograph (1980). • They simultaneously generalize: • • commutative C ∗ -algebras Étale Groupoids Étale Groupoid Algebras Representation Theory Background Groupoid C ∗ -algebras have been a fruitful and unifying notion in the theory of operator algebras since J. Renault’s influential monograph (1980). • They simultaneously generalize: • • commutative C ∗ -algebras • group C ∗ -algebras Étale Groupoids Étale Groupoid Algebras Representation Theory Background Groupoid C ∗ -algebras have been a fruitful and unifying notion in the theory of operator algebras since J. Renault’s influential monograph (1980). • They simultaneously generalize: • • commutative C ∗ -algebras • group C ∗ -algebras • group action C ∗ -algebras Étale Groupoids Étale Groupoid Algebras Representation Theory Background Groupoid C ∗ -algebras have been a fruitful and unifying notion in the theory of operator algebras since J. Renault’s influential monograph (1980). • They simultaneously generalize: • • • • • commutative C ∗ -algebras group C ∗ -algebras group action C ∗ -algebras Cuntz-Krieger/graph C ∗ -algebras Étale Groupoids Étale Groupoid Algebras Representation Theory Background Groupoid C ∗ -algebras have been a fruitful and unifying notion in the theory of operator algebras since J. Renault’s influential monograph (1980). • They simultaneously generalize: • • • • • • commutative C ∗ -algebras group C ∗ -algebras group action C ∗ -algebras Cuntz-Krieger/graph C ∗ -algebras inverse semigroup C ∗ -algebras. Étale Groupoids Étale Groupoid Algebras Representation Theory Background Groupoid C ∗ -algebras have been a fruitful and unifying notion in the theory of operator algebras since J. Renault’s influential monograph (1980). • They simultaneously generalize: • • • • • • • commutative C ∗ -algebras group C ∗ -algebras group action C ∗ -algebras Cuntz-Krieger/graph C ∗ -algebras inverse semigroup C ∗ -algebras. Algebraic properties can often be seen from the groupoid. Étale Groupoids Étale Groupoid Algebras Representation Theory Background Groupoid C ∗ -algebras have been a fruitful and unifying notion in the theory of operator algebras since J. Renault’s influential monograph (1980). • They simultaneously generalize: • • • • • • commutative C ∗ -algebras group C ∗ -algebras group action C ∗ -algebras Cuntz-Krieger/graph C ∗ -algebras inverse semigroup C ∗ -algebras. Algebraic properties can often be seen from the groupoid. • Morita equivalence of groupoid algebras is often explained by a Morita equivalence of the groupoids. • Étale Groupoids Étale Groupoid Algebras Representation Theory Background Groupoid C ∗ -algebras have been a fruitful and unifying notion in the theory of operator algebras since J. Renault’s influential monograph (1980). • They simultaneously generalize: • • • • • • commutative C ∗ -algebras group C ∗ -algebras group action C ∗ -algebras Cuntz-Krieger/graph C ∗ -algebras inverse semigroup C ∗ -algebras. Algebraic properties can often be seen from the groupoid. • Morita equivalence of groupoid algebras is often explained by a Morita equivalence of the groupoids. • Groupoids are espoused by Connes as noncommutative models of spaces. • Étale Groupoids Étale Groupoid Algebras Representation Theory Algebraic analogues • Many groupoid C ∗ -algebras have analogues in the context of associative algebras. Étale Groupoids Étale Groupoid Algebras Representation Theory Algebraic analogues Many groupoid C ∗ -algebras have analogues in the context of associative algebras. • Group algebras and inverse semigroup algebras are obvious examples. • Étale Groupoids Étale Groupoid Algebras Representation Theory Algebraic analogues Many groupoid C ∗ -algebras have analogues in the context of associative algebras. • Group algebras and inverse semigroup algebras are obvious examples. • Leavitt path algebras are algebraic analogues of Cuntz-Krieger C ∗ -algebras. • Étale Groupoids Étale Groupoid Algebras Representation Theory Algebraic analogues Many groupoid C ∗ -algebras have analogues in the context of associative algebras. • Group algebras and inverse semigroup algebras are obvious examples. • Leavitt path algebras are algebraic analogues of Cuntz-Krieger C ∗ -algebras. • The ring of k-valued continuous functions with compact support on a totally disconnected space is an algebraic analogue of a commutative C ∗ -algebra. • Étale Groupoids Étale Groupoid Algebras Representation Theory Algebraic analogues • • • • • Many groupoid C ∗ -algebras have analogues in the context of associative algebras. Group algebras and inverse semigroup algebras are obvious examples. Leavitt path algebras are algebraic analogues of Cuntz-Krieger C ∗ -algebras. The ring of k-valued continuous functions with compact support on a totally disconnected space is an algebraic analogue of a commutative C ∗ -algebra. Over a field, these are precisely the idempotent-generated commutative algebras. Étale Groupoids Étale Groupoid Algebras Representation Theory Algebraic analogues • • • • • • Many groupoid C ∗ -algebras have analogues in the context of associative algebras. Group algebras and inverse semigroup algebras are obvious examples. Leavitt path algebras are algebraic analogues of Cuntz-Krieger C ∗ -algebras. The ring of k-valued continuous functions with compact support on a totally disconnected space is an algebraic analogue of a commutative C ∗ -algebra. Over a field, these are precisely the idempotent-generated commutative algebras. Surprising similarities between operator algebras and their algebraic analogues have been known for some time. Étale Groupoids Étale Groupoid Algebras Representation Theory History • In 2009 [Adv. Math. ’10], I introduced an algebraic analogue of groupoid C ∗ -algebras for a class of étale groupoids. Étale Groupoids Étale Groupoid Algebras Representation Theory History In 2009 [Adv. Math. ’10], I introduced an algebraic analogue of groupoid C ∗ -algebras for a class of étale groupoids. • My hope was that it would explain many of the similarities between the algebraic and analytic setting, especially for Leavitt path algebras. • Étale Groupoids Étale Groupoid Algebras Representation Theory History In 2009 [Adv. Math. ’10], I introduced an algebraic analogue of groupoid C ∗ -algebras for a class of étale groupoids. • My hope was that it would explain many of the similarities between the algebraic and analytic setting, especially for Leavitt path algebras. • Initially, I focused on applications to inverse semigroup algebras. • Étale Groupoids Étale Groupoid Algebras Representation Theory History In 2009 [Adv. Math. ’10], I introduced an algebraic analogue of groupoid C ∗ -algebras for a class of étale groupoids. • My hope was that it would explain many of the similarities between the algebraic and analytic setting, especially for Leavitt path algebras. • Initially, I focused on applications to inverse semigroup algebras. • Groupoid algebras over C were rediscovered later by L. O. Clark, C. Farthing, A. Sims and M. Tomforde, who have kindly dubbed them “Steinberg algebras.” • Étale Groupoids Étale Groupoid Algebras Representation Theory History • • • • • In 2009 [Adv. Math. ’10], I introduced an algebraic analogue of groupoid C ∗ -algebras for a class of étale groupoids. My hope was that it would explain many of the similarities between the algebraic and analytic setting, especially for Leavitt path algebras. Initially, I focused on applications to inverse semigroup algebras. Groupoid algebras over C were rediscovered later by L. O. Clark, C. Farthing, A. Sims and M. Tomforde, who have kindly dubbed them “Steinberg algebras.” Many of my hopes have since been borne out by J. Brown, L. O. Clark, C. Farthing, A. Sims and M. Tomforde, who seem to produce new results faster than I can keep up with. Étale Groupoids Étale Groupoid Algebras Representation Theory Groupoids • A groupoid G is a small category in which all morphisms are isomorphisms. Étale Groupoids Étale Groupoid Algebras Representation Theory Groupoids A groupoid G is a small category in which all morphisms are isomorphisms. • G (0) is the object set. • Étale Groupoids Étale Groupoid Algebras Representation Theory Groupoids A groupoid G is a small category in which all morphisms are isomorphisms. • G (0) is the object set. • G (1) is the arrow set. • Étale Groupoids Étale Groupoid Algebras Representation Theory Groupoids A groupoid G is a small category in which all morphisms are isomorphisms. • G (0) is the object set. • G (1) is the arrow set. • d, r : G (1) → G (0) are the domain and range maps. • Étale Groupoids Étale Groupoid Algebras Representation Theory Groupoids • • • • • A groupoid G is a small category in which all morphisms are isomorphisms. G (0) is the object set. G (1) is the arrow set. d, r : G (1) → G (0) are the domain and range maps. m : G(2) → G(1) is the multiplication map. Étale Groupoids Étale Groupoid Algebras Representation Theory Groupoids • • • • • • A groupoid G is a small category in which all morphisms are isomorphisms. G (0) is the object set. G (1) is the arrow set. d, r : G (1) → G (0) are the domain and range maps. m : G(2) → G(1) is the multiplication map. ι : G(1) → G(1) is the inversion map. Étale Groupoids Étale Groupoid Algebras Representation Theory Groupoids • • • • • • • A groupoid G is a small category in which all morphisms are isomorphisms. G (0) is the object set. G (1) is the arrow set. d, r : G (1) → G (0) are the domain and range maps. m : G(2) → G(1) is the multiplication map. ι : G(1) → G(1) is the inversion map. We view G (0) ⊆ G (1) by identifying objects and identity arrows. Étale Groupoids Étale Groupoid Algebras Representation Theory Groupoids • • • • • • • • A groupoid G is a small category in which all morphisms are isomorphisms. G (0) is the object set. G (1) is the arrow set. d, r : G (1) → G (0) are the domain and range maps. m : G(2) → G(1) is the multiplication map. ι : G(1) → G(1) is the inversion map. We view G (0) ⊆ G (1) by identifying objects and identity arrows. Often G (0) is called the unit space of G. Étale Groupoids Étale Groupoid Algebras Representation Theory Étale groupoids • A topological groupoid is a groupoid G with a topology on G (1) such that if G (0) is given the induced topology, then all the structure maps are continuous. Étale Groupoids Étale Groupoid Algebras Representation Theory Étale groupoids A topological groupoid is a groupoid G with a topology on G (1) such that if G (0) is given the induced topology, then all the structure maps are continuous. • Étale groupoids form one of the most important classes of topological groupoids. • Étale Groupoids Étale Groupoid Algebras Representation Theory Étale groupoids A topological groupoid is a groupoid G with a topology on G (1) such that if G (0) is given the induced topology, then all the structure maps are continuous. • Étale groupoids form one of the most important classes of topological groupoids. • G is étale if d : G (1) → G (0) is a local homeomorphism. • Étale Groupoids Étale Groupoid Algebras Representation Theory Étale groupoids A topological groupoid is a groupoid G with a topology on G (1) such that if G (0) is given the induced topology, then all the structure maps are continuous. • Étale groupoids form one of the most important classes of topological groupoids. • G is étale if d : G (1) → G (0) is a local homeomorphism. • This forces all the structure maps to be local homeomorphisms and G (0) to be open. • Étale Groupoids Étale Groupoid Algebras Representation Theory Étale groupoids • • • • • A topological groupoid is a groupoid G with a topology on G (1) such that if G (0) is given the induced topology, then all the structure maps are continuous. Étale groupoids form one of the most important classes of topological groupoids. G is étale if d : G (1) → G (0) is a local homeomorphism. This forces all the structure maps to be local homeomorphisms and G (0) to be open. We assume G (0) is locally compact Hausdorff, but G (1) need not be Hausdorff. Étale Groupoids Étale Groupoid Algebras Representation Theory Étale groupoids • • • • • • A topological groupoid is a groupoid G with a topology on G (1) such that if G (0) is given the induced topology, then all the structure maps are continuous. Étale groupoids form one of the most important classes of topological groupoids. G is étale if d : G (1) → G (0) is a local homeomorphism. This forces all the structure maps to be local homeomorphisms and G (0) to be open. We assume G (0) is locally compact Hausdorff, but G (1) need not be Hausdorff. Renault calls an étale groupoid G ample if G (0) has a basis of compact open sets. Étale Groupoids Étale Groupoid Algebras Representation Theory Étale groupoids • • • • • • • A topological groupoid is a groupoid G with a topology on G (1) such that if G (0) is given the induced topology, then all the structure maps are continuous. Étale groupoids form one of the most important classes of topological groupoids. G is étale if d : G (1) → G (0) is a local homeomorphism. This forces all the structure maps to be local homeomorphisms and G (0) to be open. We assume G (0) is locally compact Hausdorff, but G (1) need not be Hausdorff. Renault calls an étale groupoid G ample if G (0) has a basis of compact open sets. Many important C ∗ -algebras come from ample groupoids: group algebras, Cuntz-Krieger algebras and inverse semigroup algebras. Étale Groupoids Étale Groupoid Algebras Representation Theory Discrete groups • A discrete group is the same thing as a one-object étale groupoid. Étale Groupoids Étale Groupoid Algebras Representation Theory Discrete groups A discrete group is the same thing as a one-object étale groupoid. • Indeed, if G (0) is discrete, then because d is a local homeomorphism also G (1) is discrete. • Étale Groupoids Étale Groupoid Algebras Representation Theory Discrete groups A discrete group is the same thing as a one-object étale groupoid. • Indeed, if G (0) is discrete, then because d is a local homeomorphism also G (1) is discrete. • More generally, any discrete groupoid is an étale groupoid. • Étale Groupoids Étale Groupoid Algebras Representation Theory Discrete groups A discrete group is the same thing as a one-object étale groupoid. • Indeed, if G (0) is discrete, then because d is a local homeomorphism also G (1) is discrete. • More generally, any discrete groupoid is an étale groupoid. • In fact, discrete groups and groupoids are ample. • Étale Groupoids Étale Groupoid Algebras Spaces • Let X be a locally compact Hausdorff space. Representation Theory Étale Groupoids Étale Groupoid Algebras Representation Theory Spaces Let X be a locally compact Hausdorff space. • Then X is an étale groupoid in which all arrows are identities. • Étale Groupoids Étale Groupoid Algebras Representation Theory Spaces Let X be a locally compact Hausdorff space. • Then X is an étale groupoid in which all arrows are identities. • That is, G (0) = X = G (1) . • Étale Groupoids Étale Groupoid Algebras Representation Theory Spaces Let X be a locally compact Hausdorff space. • Then X is an étale groupoid in which all arrows are identities. • That is, G (0) = X = G (1) . • X is an ample groupoid if and only if it has a basis of compact open sets. • Étale Groupoids Étale Groupoid Algebras Representation Theory Spaces • • • • • Let X be a locally compact Hausdorff space. Then X is an étale groupoid in which all arrows are identities. That is, G (0) = X = G (1) . X is an ample groupoid if and only if it has a basis of compact open sets. This guarantees X has enough continuous maps into any non-trivial discrete ring to separate points. Étale Groupoids Étale Groupoid Algebras Representation Theory Group actions • Let G be a discrete group acting on a locally compact Hausdorff space X. Étale Groupoids Étale Groupoid Algebras Representation Theory Group actions Let G be a discrete group acting on a locally compact Hausdorff space X. • The semidirect product G ⋉ X has object space X and arrow space G × X. • Étale Groupoids Étale Groupoid Algebras Representation Theory Group actions Let G be a discrete group acting on a locally compact Hausdorff space X. • The semidirect product G ⋉ X has object space X and arrow space G × X. • Think g (g, x) = gx ←−− x. • Étale Groupoids Étale Groupoid Algebras Representation Theory Group actions Let G be a discrete group acting on a locally compact Hausdorff space X. • The semidirect product G ⋉ X has object space X and arrow space G × X. • Think g (g, x) = gx ←−− x. • • Composition is gh ghx g hx h x Étale Groupoids Étale Groupoid Algebras Representation Theory Group actions Let G be a discrete group acting on a locally compact Hausdorff space X. • The semidirect product G ⋉ X has object space X and arrow space G × X. • Think g (g, x) = gx ←−− x. • • Composition is gh ghx • g Inversion is gx h hx g g −1 x x Étale Groupoids Étale Groupoid Algebras Deaconu-Renault groupoid • Let Q = (Q(0) , Q(1) ) be a finite quiver. Representation Theory Étale Groupoids Étale Groupoid Algebras Deaconu-Renault groupoid Let Q = (Q(0) , Q(1) ) be a finite quiver. • Qω is the space of infinite paths in Q. • Representation Theory Étale Groupoids Étale Groupoid Algebras Deaconu-Renault groupoid Let Q = (Q(0) , Q(1) ) be a finite quiver. • Qω is the space of infinite paths in Q. • Let T : Qω → Qω be the shift map • T (e0 e1 e2 · · · ) = e1 e2 · · · . Representation Theory Étale Groupoids Étale Groupoid Algebras Representation Theory Deaconu-Renault groupoid Let Q = (Q(0) , Q(1) ) be a finite quiver. • Qω is the space of infinite paths in Q. • Let T : Qω → Qω be the shift map • T (e0 e1 e2 · · · ) = e1 e2 · · · . • The Deaconu-Renault groupoid GQ has object space Qω and arrow space m−n {x −−−→ y | T m (x) = T n (y)} ⊆ Qω × Z × Qω . Étale Groupoids Étale Groupoid Algebras Representation Theory Deaconu-Renault groupoid Let Q = (Q(0) , Q(1) ) be a finite quiver. • Qω is the space of infinite paths in Q. • Let T : Qω → Qω be the shift map • T (e0 e1 e2 · · · ) = e1 e2 · · · . • The Deaconu-Renault groupoid GQ has object space Qω and arrow space m−n {x −−−→ y | T m (x) = T n (y)} ⊆ Qω × Z × Qω . • Composition is x m−n y m′ + n′ z = x (m + m′ ) − (n + n′ ) z Étale Groupoids Étale Groupoid Algebras Representation Theory Deaconu-Renault groupoid Let Q = (Q(0) , Q(1) ) be a finite quiver. • Qω is the space of infinite paths in Q. • Let T : Qω → Qω be the shift map • T (e0 e1 e2 · · · ) = e1 e2 · · · . • The Deaconu-Renault groupoid GQ has object space Qω and arrow space m−n {x −−−→ y | T m (x) = T n (y)} ⊆ Qω × Z × Qω . • Composition is x • m−n m−n y m′ + n′ n−m z = (x −−−→ y)−1 = y −−−→ x. x (m + m′ ) − (n + n′ ) z Étale Groupoids Étale Groupoid Algebras Representation Theory Deaconu-Renault groupoid Let Q = (Q(0) , Q(1) ) be a finite quiver. • Qω is the space of infinite paths in Q. • Let T : Qω → Qω be the shift map • T (e0 e1 e2 · · · ) = e1 e2 · · · . • The Deaconu-Renault groupoid GQ has object space Qω and arrow space m−n {x −−−→ y | T m (x) = T n (y)} ⊆ Qω × Z × Qω . • Composition is x m−n m−n y m′ + n′ n−m z = x (m + m′ ) − (n + n′ ) (x −−−→ y)−1 = y −−−→ x. • This ample groupoid gives rise to Cuntz-Krieger C ∗ -algebras and Leavitt path algebras. • z Étale Groupoids Étale Groupoid Algebras Representation Theory Inverse semigroups • An inverse semigroup is a semigroup S such that, for all s ∈ S, there exists unique s∗ ∈ S such that ss∗ s = s and s∗ ss∗ = s∗ . Étale Groupoids Étale Groupoid Algebras Representation Theory Inverse semigroups • An inverse semigroup is a semigroup S such that, for all s ∈ S, there exists unique s∗ ∈ S such that ss∗ s = s and s∗ ss∗ = s∗ . • Note that s∗ s, ss∗ are idempotents. Étale Groupoids Étale Groupoid Algebras Representation Theory Inverse semigroups • An inverse semigroup is a semigroup S such that, for all s ∈ S, there exists unique s∗ ∈ S such that ss∗ s = s and s∗ ss∗ = s∗ . Note that s∗ s, ss∗ are idempotents. • Groups are inverse semigroups. • Étale Groupoids Étale Groupoid Algebras Representation Theory Inverse semigroups • An inverse semigroup is a semigroup S such that, for all s ∈ S, there exists unique s∗ ∈ S such that ss∗ s = s and s∗ ss∗ = s∗ . Note that s∗ s, ss∗ are idempotents. • Groups are inverse semigroups. • The idempotents E(S) commute and form a subsemigroup. • Étale Groupoids Étale Groupoid Algebras Representation Theory Inverse semigroups • An inverse semigroup is a semigroup S such that, for all s ∈ S, there exists unique s∗ ∈ S such that ss∗ s = s and s∗ ss∗ = s∗ . Note that s∗ s, ss∗ are idempotents. • Groups are inverse semigroups. • The idempotents E(S) commute and form a subsemigroup. • E(S) is a semilattice with e ∧ f = ef . • Étale Groupoids Étale Groupoid Algebras Representation Theory Inverse semigroups • An inverse semigroup is a semigroup S such that, for all s ∈ S, there exists unique s∗ ∈ S such that ss∗ s = s and s∗ ss∗ = s∗ . • • • • • Note that s∗ s, ss∗ are idempotents. Groups are inverse semigroups. The idempotents E(S) commute and form a subsemigroup. E(S) is a semilattice with e ∧ f = ef . If G is a group acting on a semilattice E, then the semidirect product E ⋊ G is an inverse semigroup. Étale Groupoids Étale Groupoid Algebras Representation Theory Pseudogroups of partial homeomorphisms • Let X be a topological space. Étale Groupoids Étale Groupoid Algebras Representation Theory Pseudogroups of partial homeomorphisms Let X be a topological space. • IX is the inverse semigroup of all homeomorphisms between open subsets of X. • Étale Groupoids Étale Groupoid Algebras Representation Theory Pseudogroups of partial homeomorphisms Let X be a topological space. • IX is the inverse semigroup of all homeomorphisms between open subsets of X. • Composition is defined where it makes sense: • Étale Groupoids Étale Groupoid Algebras Representation Theory Pseudogroups of partial homeomorphisms Let X be a topological space. • IX is the inverse semigroup of all homeomorphisms between open subsets of X. • Composition is defined where it makes sense: • f g g◦f Étale Groupoids Étale Groupoid Algebras Representation Theory Pseudogroups of partial homeomorphisms Let X be a topological space. • IX is the inverse semigroup of all homeomorphisms between open subsets of X. • Composition is defined where it makes sense: • f g g◦f • The empty partial function is a zero element of IX . Étale Groupoids Étale Groupoid Algebras Representation Theory Pseudogroups of partial homeomorphisms Let X be a topological space. • IX is the inverse semigroup of all homeomorphisms between open subsets of X. • Composition is defined where it makes sense: • f g g◦f The empty partial function is a zero element of IX . • If f : U → V , then f ∗ = f −1 : V → U. • Étale Groupoids Étale Groupoid Algebras Representation Theory Pseudogroups of partial homeomorphisms Let X be a topological space. • IX is the inverse semigroup of all homeomorphisms between open subsets of X. • Composition is defined where it makes sense: • f g g◦f The empty partial function is a zero element of IX . • If f : U → V , then f ∗ = f −1 : V → U. • Inverse semigroups abstract pseudogroups of partial homeomorphisms from differential geometry. • Étale Groupoids Étale Groupoid Algebras Actions of inverse semigroups • S is an inverse semigroup. Representation Theory Étale Groupoids Étale Groupoid Algebras Actions of inverse semigroups S is an inverse semigroup. • X is a locally compact Hausdorff space. • Representation Theory Étale Groupoids Étale Groupoid Algebras Representation Theory Actions of inverse semigroups S is an inverse semigroup. • X is a locally compact Hausdorff space. • An action S y X is a homomorphism θ : S → IX with [ dom(θ(e)) = X (1) • e∈E(S) Étale Groupoids Étale Groupoid Algebras Representation Theory Actions of inverse semigroups S is an inverse semigroup. • X is a locally compact Hausdorff space. • An action S y X is a homomorphism θ : S → IX with [ dom(θ(e)) = X (1) • e∈E(S) • Since dom(θ(s)) = dom(θ(s∗ s)), (1) is equivalent to X being the union of the domains of all elements of S. Étale Groupoids Étale Groupoid Algebras Representation Theory Actions of inverse semigroups S is an inverse semigroup. • X is a locally compact Hausdorff space. • An action S y X is a homomorphism θ : S → IX with [ dom(θ(e)) = X (1) • e∈E(S) Since dom(θ(s)) = dom(θ(s∗ s)), (1) is equivalent to X being the union of the domains of all elements of S. • If S is a group, (1) implies the action is by homeomorphisms. • Étale Groupoids Étale Groupoid Algebras Representation Theory Actions of inverse semigroups S is an inverse semigroup. • X is a locally compact Hausdorff space. • An action S y X is a homomorphism θ : S → IX with [ dom(θ(e)) = X (1) • e∈E(S) Since dom(θ(s)) = dom(θ(s∗ s)), (1) is equivalent to X being the union of the domains of all elements of S. • If S is a group, (1) implies the action is by homeomorphisms. • If s ∈ S and e ∈ E(S), then θ(se) is the restriction of θ(s) to dom(θ(e)). • Étale Groupoids Étale Groupoid Algebras Representation Theory Groupoids of germs • Suppose S y X is an action of an inverse semigroup. Étale Groupoids Étale Groupoid Algebras Representation Theory Groupoids of germs Suppose S y X is an action of an inverse semigroup. • We can form a groupoid of germs S ⋉ X. • Étale Groupoids Étale Groupoid Algebras Representation Theory Groupoids of germs Suppose S y X is an action of an inverse semigroup. • We can form a groupoid of germs S ⋉ X. • X is the object space and (S × X)/∼ is the arrow space. • Étale Groupoids Étale Groupoid Algebras Representation Theory Groupoids of germs Suppose S y X is an action of an inverse semigroup. • We can form a groupoid of germs S ⋉ X. • X is the object space and (S × X)/∼ is the arrow space. • Think s [s, x] = sx ←−− x. • Étale Groupoids Étale Groupoid Algebras Representation Theory Groupoids of germs Suppose S y X is an action of an inverse semigroup. • We can form a groupoid of germs S ⋉ X. • X is the object space and (S × X)/∼ is the arrow space. • Think s [s, x] = sx ←−− x. • • Composition and inversion are like in the group action case. Étale Groupoids Étale Groupoid Algebras Representation Theory Groupoids of germs Suppose S y X is an action of an inverse semigroup. • We can form a groupoid of germs S ⋉ X. • X is the object space and (S × X)/∼ is the arrow space. • Think s [s, x] = sx ←−− x. • Composition and inversion are like in the group action case. • (s, x) ∼ (t, y) ⇐⇒ x = y and ∃e ∈ E(S) with x ∈ dom(e), se = te. • Étale Groupoids Étale Groupoid Algebras Representation Theory Groupoids of germs Suppose S y X is an action of an inverse semigroup. • We can form a groupoid of germs S ⋉ X. • X is the object space and (S × X)/∼ is the arrow space. • Think s [s, x] = sx ←−− x. • Composition and inversion are like in the group action case. • (s, x) ∼ (t, y) ⇐⇒ x = y and ∃e ∈ E(S) with x ∈ dom(e), se = te. • S ⋉ X is not Hausdorff in general. • Étale Groupoids Étale Groupoid Algebras Representation Theory Groupoids of germs Suppose S y X is an action of an inverse semigroup. • We can form a groupoid of germs S ⋉ X. • X is the object space and (S × X)/∼ is the arrow space. • Think s [s, x] = sx ←−− x. • Composition and inversion are like in the group action case. • (s, x) ∼ (t, y) ⇐⇒ x = y and ∃e ∈ E(S) with x ∈ dom(e), se = te. • S ⋉ X is not Hausdorff in general. • It is ample if X has a basis of compact open sets and the domains of elements of S are compact open. • Étale Groupoids Étale Groupoid Algebras Paterson’s universal groupoid • [ ⊆ {0, 1}E(S) be the space of non-zero Let E(S) homomorphisms (characters) χ : E(S) → {0, 1}. Representation Theory Étale Groupoids Étale Groupoid Algebras Paterson’s universal groupoid [ ⊆ {0, 1}E(S) be the space of non-zero Let E(S) homomorphisms (characters) χ : E(S) → {0, 1}. [ by s · χ(e) = χ(s∗ es). • S acts on E(S) • Representation Theory Étale Groupoids Étale Groupoid Algebras Paterson’s universal groupoid [ ⊆ {0, 1}E(S) be the space of non-zero Let E(S) homomorphisms (characters) χ : E(S) → {0, 1}. [ by s · χ(e) = χ(s∗ es). • S acts on E(S) • The domain of s consists of those characters with s · χ 6= 0. • Representation Theory Étale Groupoids Étale Groupoid Algebras Paterson’s universal groupoid [ ⊆ {0, 1}E(S) be the space of non-zero Let E(S) homomorphisms (characters) χ : E(S) → {0, 1}. [ by s · χ(e) = χ(s∗ es). • S acts on E(S) • The domain of s consists of those characters with s · χ 6= 0. [ • The universal groupoid G(S) is S ⋉ E(S). • Representation Theory Étale Groupoids Étale Groupoid Algebras Paterson’s universal groupoid [ ⊆ {0, 1}E(S) be the space of non-zero Let E(S) homomorphisms (characters) χ : E(S) → {0, 1}. [ by s · χ(e) = χ(s∗ es). • S acts on E(S) • The domain of s consists of those characters with s · χ 6= 0. [ • The universal groupoid G(S) is S ⋉ E(S). • Paterson proved C ∗ (G(S)) ∼ = C ∗ (S). • Representation Theory Étale Groupoids Étale Groupoid Algebras Representation Theory Groupoid algebras • Let k be a unital comm. ring with the discrete topology. Étale Groupoids Étale Groupoid Algebras Representation Theory Groupoid algebras Let k be a unital comm. ring with the discrete topology. • Let G be an ample groupoid. • Étale Groupoids Étale Groupoid Algebras Representation Theory Groupoid algebras Let k be a unital comm. ring with the discrete topology. • Let G be an ample groupoid. • The groupoid algebra kG is defined as follows. • Étale Groupoids Étale Groupoid Algebras Representation Theory Groupoid algebras Let k be a unital comm. ring with the discrete topology. • Let G be an ample groupoid. • The groupoid algebra kG is defined as follows. • The underlying vector space consists of the continuous k-valued functions on G with compact support. • Étale Groupoids Étale Groupoid Algebras Representation Theory Groupoid algebras Let k be a unital comm. ring with the discrete topology. • Let G be an ample groupoid. • The groupoid algebra kG is defined as follows. • The underlying vector space consists of the continuous k-valued functions on G with compact support. • Warning This definition needs adjustment for the non-Hausdorff case. Étale Groupoids Étale Groupoid Algebras Representation Theory Groupoid algebras • • • • • Let k be a unital comm. ring with the discrete topology. Let G be an ample groupoid. The groupoid algebra kG is defined as follows. The underlying vector space consists of the continuous k-valued functions on G with compact support. Ample implies there are many such functions. Étale Groupoids Étale Groupoid Algebras Representation Theory Groupoid algebras • • • • • • Let k be a unital comm. ring with the discrete topology. Let G be an ample groupoid. The groupoid algebra kG is defined as follows. The underlying vector space consists of the continuous k-valued functions on G with compact support. Ample implies there are many such functions. The product is convolution: X f ∗ g(x) = f (xz −1 )g(z). d(x)=d(z) Étale Groupoids Étale Groupoid Algebras Representation Theory Groupoid algebras • • • • • • Let k be a unital comm. ring with the discrete topology. Let G be an ample groupoid. The groupoid algebra kG is defined as follows. The underlying vector space consists of the continuous k-valued functions on G with compact support. Ample implies there are many such functions. The product is convolution: X f ∗ g(x) = f (xz −1 )g(z). d(x)=d(z) • The sum is finite because fibers of d are closed and discrete and f, g have compact support. Étale Groupoids Étale Groupoid Algebras Representation Theory Examples • If G is a discrete group, kG is the usual group algebra and similarly for discrete groupoids. Étale Groupoids Étale Groupoid Algebras Representation Theory Examples If G is a discrete group, kG is the usual group algebra and similarly for discrete groupoids. • If X is a space, kX is the ring of continuous k-valued functions with compact support. • Étale Groupoids Étale Groupoid Algebras Representation Theory Examples If G is a discrete group, kG is the usual group algebra and similarly for discrete groupoids. • If X is a space, kX is the ring of continuous k-valued functions with compact support. • The Leavitt path algebras are the algebras of Deaconu-Renault groupoids. • Étale Groupoids Étale Groupoid Algebras Representation Theory Examples If G is a discrete group, kG is the usual group algebra and similarly for discrete groupoids. • If X is a space, kX is the ring of continuous k-valued functions with compact support. • The Leavitt path algebras are the algebras of Deaconu-Renault groupoids. • If G y X is a group action, k(G ⋉ X) is the cross product kX ⋊ G. • Étale Groupoids Étale Groupoid Algebras Representation Theory Examples If G is a discrete group, kG is the usual group algebra and similarly for discrete groupoids. • If X is a space, kX is the ring of continuous k-valued functions with compact support. • The Leavitt path algebras are the algebras of Deaconu-Renault groupoids. • If G y X is a group action, k(G ⋉ X) is the cross product kX ⋊ G. • Theorem Let S be Then kS (BS) an inverse semigroup and k a comm. ring with 1. ∼ = kG(S). Étale Groupoids Étale Groupoid Algebras Representation Theory First properties • C ∗ (G) is the completion of CG (Stone-Weierstrass). Étale Groupoids Étale Groupoid Algebras Representation Theory First properties C ∗ (G) is the completion of CG (Stone-Weierstrass). • This lets us recover C ∗ (S) ∼ = C ∗ (G(S)) for an inverse semigroup S. • Étale Groupoids Étale Groupoid Algebras Representation Theory First properties C ∗ (G) is the completion of CG (Stone-Weierstrass). • This lets us recover C ∗ (S) ∼ = C ∗ (G(S)) for an inverse semigroup S. • kG is unital iff G0 is compact. • Étale Groupoids Étale Groupoid Algebras Representation Theory First properties C ∗ (G) is the completion of CG (Stone-Weierstrass). • This lets us recover C ∗ (S) ∼ = C ∗ (G(S)) for an inverse semigroup S. • kG is unital iff G0 is compact. • kG has local units (is a directed union of unital subrings). • Étale Groupoids Étale Groupoid Algebras Representation Theory First properties • • • • • C ∗ (G) is the completion of CG (Stone-Weierstrass). This lets us recover C ∗ (S) ∼ = C ∗ (G(S)) for an inverse semigroup S. kG is unital iff G0 is compact. kG has local units (is a directed union of unital subrings). Abrams, Ahn and Marki developed a successful Morita theory for rings with local units. Étale Groupoids Étale Groupoid Algebras Representation Theory First properties • • • • • • C ∗ (G) is the completion of CG (Stone-Weierstrass). This lets us recover C ∗ (S) ∼ = C ∗ (G(S)) for an inverse semigroup S. kG is unital iff G0 is compact. kG has local units (is a directed union of unital subrings). Abrams, Ahn and Marki developed a successful Morita theory for rings with local units. kG carries an involution f ∗ (g) = f (g −1). Étale Groupoids Étale Groupoid Algebras Representation Theory Isotropy • The isotropy group Gx of x ∈ G (0) consists of all loops g x −→ x. Étale Groupoids Étale Groupoid Algebras Representation Theory Isotropy The isotropy group Gx of x ∈ G (0) consists of all loops g x −→ x. S • The isotropy bundle Giso = x∈X Gx . • Étale Groupoids Étale Groupoid Algebras Representation Theory Isotropy The isotropy group Gx of x ∈ G (0) consists of all loops g x −→ x. S • The isotropy bundle Giso = x∈X Gx . • • G is effective if Int(Giso ) = G (0) . Étale Groupoids Étale Groupoid Algebras Representation Theory Isotropy The isotropy group Gx of x ∈ G (0) consists of all loops g x −→ x. S • The isotropy bundle Giso = x∈X Gx . • G is effective if Int(Giso ) = G (0) . • The Deaconu-Renault groupoid is effective iff the Leavitt path algebra satisfies condition (L). • Étale Groupoids Étale Groupoid Algebras Representation Theory Isotropy The isotropy group Gx of x ∈ G (0) consists of all loops g x −→ x. S • The isotropy bundle Giso = x∈X Gx . • G is effective if Int(Giso ) = G (0) . • The Deaconu-Renault groupoid is effective iff the Leavitt path algebra satisfies condition (L). • G ⋉ X is effective iff G y X is topologically free. • Étale Groupoids Étale Groupoid Algebras Representation Theory Isotropy The isotropy group Gx of x ∈ G (0) consists of all loops g x −→ x. S • The isotropy bundle Giso = x∈X Gx . • G is effective if Int(Giso ) = G (0) . • The Deaconu-Renault groupoid is effective iff the Leavitt path algebra satisfies condition (L). • G ⋉ X is effective iff G y X is topologically free. • f ∈ kG is a class function if: • Étale Groupoids Étale Groupoid Algebras Representation Theory Isotropy The isotropy group Gx of x ∈ G (0) consists of all loops g x −→ x. S • The isotropy bundle Giso = x∈X Gx . • G is effective if Int(Giso ) = G (0) . • The Deaconu-Renault groupoid is effective iff the Leavitt path algebra satisfies condition (L). • G ⋉ X is effective iff G y X is topologically free. • f ∈ kG is a class function if: • 1. supp(f ) ⊆ Giso Étale Groupoids Étale Groupoid Algebras Representation Theory Isotropy The isotropy group Gx of x ∈ G (0) consists of all loops g x −→ x. S • The isotropy bundle Giso = x∈X Gx . • G is effective if Int(Giso ) = G (0) . • The Deaconu-Renault groupoid is effective iff the Leavitt path algebra satisfies condition (L). • G ⋉ X is effective iff G y X is topologically free. • f ∈ kG is a class function if: • 1. supp(f ) ⊆ Giso h g 2. implies f (ghg −1 ) = f (h): Étale Groupoids Étale Groupoid Algebras Representation Theory Isotropy The isotropy group Gx of x ∈ G (0) consists of all loops g x −→ x. S • The isotropy bundle Giso = x∈X Gx . • G is effective if Int(Giso ) = G (0) . • The Deaconu-Renault groupoid is effective iff the Leavitt path algebra satisfies condition (L). • G ⋉ X is effective iff G y X is topologically free. • f ∈ kG is a class function if: • 1. supp(f ) ⊆ Giso h g 2. implies f (ghg −1 ) = f (h): Theorem (BS) The class functions form the center of kG. Étale Groupoids Étale Groupoid Algebras Representation Theory Orbits and minimality • The orbit Ox of x ∈ G (0) consists of all y such that there g exists x −→ y. Étale Groupoids Étale Groupoid Algebras Representation Theory Orbits and minimality The orbit Ox of x ∈ G (0) consists of all y such that there g exists x −→ y. • If G y X is a group action, then the orbits of G ⋉ X are the orbits of G on X. • Étale Groupoids Étale Groupoid Algebras Representation Theory Orbits and minimality The orbit Ox of x ∈ G (0) consists of all y such that there g exists x −→ y. • If G y X is a group action, then the orbits of G ⋉ X are the orbits of G on X. • G is minimal if every orbit is dense. • Étale Groupoids Étale Groupoid Algebras Representation Theory Orbits and minimality The orbit Ox of x ∈ G (0) consists of all y such that there g exists x −→ y. • If G y X is a group action, then the orbits of G ⋉ X are the orbits of G on X. • G is minimal if every orbit is dense. • Theorem (BS) Let G be an effective, Hausdorff ample groupoid with a dense orbit. Then ( k, if G (0) is compact Z(kG) = 0, else. Étale Groupoids Étale Groupoid Algebras Representation Theory Orbits and minimality The orbit Ox of x ∈ G (0) consists of all y such that there g exists x −→ y. • If G y X is a group action, then the orbits of G ⋉ X are the orbits of G on X. • G is minimal if every orbit is dense. • Theorem (BS) Let G be an effective, Hausdorff ample groupoid with a dense orbit. Then ( k, if G (0) is compact Z(kG) = 0, else. • This generalizes an earlier result of L. O. Clark and C. Edie-Michelle for the minimal case. Étale Groupoids Étale Groupoid Algebras Representation Theory Simplicity Theorem (L. O. Clark, C. Edie-Michelle) Let G be a Hausdorff ample groupoid and k a field. Then kG is simple if and only if G is effective and minimal. Étale Groupoids Étale Groupoid Algebras Representation Theory Simplicity Theorem (L. O. Clark, C. Edie-Michelle) Let G be a Hausdorff ample groupoid and k a field. Then kG is simple if and only if G is effective and minimal. • First proved by J. H. Brown, L. O. Clark, C. Farthing and A. Sims over C. Étale Groupoids Étale Groupoid Algebras Representation Theory Simplicity Theorem (L. O. Clark, C. Edie-Michelle) Let G be a Hausdorff ample groupoid and k a field. Then kG is simple if and only if G is effective and minimal. First proved by J. H. Brown, L. O. Clark, C. Farthing and A. Sims over C. • Translates into simplicity criterion for Leavitt path algebras. • Étale Groupoids Étale Groupoid Algebras Representation Theory Simplicity Theorem (L. O. Clark, C. Edie-Michelle) Let G be a Hausdorff ample groupoid and k a field. Then kG is simple if and only if G is effective and minimal. First proved by J. H. Brown, L. O. Clark, C. Farthing and A. Sims over C. • Translates into simplicity criterion for Leavitt path algebras. • I can use to characterize inverse semigroups S with simple contracted semigroup algebras provided G(S) is Hausdorff. • Étale Groupoids Étale Groupoid Algebras Representation Theory Simplicity Theorem (L. O. Clark, C. Edie-Michelle) Let G be a Hausdorff ample groupoid and k a field. Then kG is simple if and only if G is effective and minimal. First proved by J. H. Brown, L. O. Clark, C. Farthing and A. Sims over C. • Translates into simplicity criterion for Leavitt path algebras. • I can use to characterize inverse semigroups S with simple contracted semigroup algebras provided G(S) is Hausdorff. • This answers in part a question of Munn. • Étale Groupoids Étale Groupoid Algebras Representation Theory Simplicity Theorem (L. O. Clark, C. Edie-Michelle) Let G be a Hausdorff ample groupoid and k a field. Then kG is simple if and only if G is effective and minimal. • • • • • First proved by J. H. Brown, L. O. Clark, C. Farthing and A. Sims over C. Translates into simplicity criterion for Leavitt path algebras. I can use to characterize inverse semigroups S with simple contracted semigroup algebras provided G(S) is Hausdorff. This answers in part a question of Munn. A sufficient condition for Hausdorff is being 0-E-unitary: Étale Groupoids Étale Groupoid Algebras Representation Theory Simplicity Theorem (L. O. Clark, C. Edie-Michelle) Let G be a Hausdorff ample groupoid and k a field. Then kG is simple if and only if G is effective and minimal. • • • • • First proved by J. H. Brown, L. O. Clark, C. Farthing and A. Sims over C. Translates into simplicity criterion for Leavitt path algebras. I can use to characterize inverse semigroups S with simple contracted semigroup algebras provided G(S) is Hausdorff. This answers in part a question of Munn. A sufficient condition for Hausdorff is being 0-E-unitary: • se, e ∈ E(S) \ {0} =⇒ s ∈ E(S). Étale Groupoids Étale Groupoid Algebras Representation Theory Primitivity • Recall: a ring is primitive if it has a faithful simple module. Étale Groupoids Étale Groupoid Algebras Representation Theory Primitivity Recall: a ring is primitive if it has a faithful simple module. • Let G be a Hausdorff ample groupoid and k a field. • Étale Groupoids Étale Groupoid Algebras Representation Theory Primitivity Recall: a ring is primitive if it has a faithful simple module. • Let G be a Hausdorff ample groupoid and k a field. • Theorem (BS) If G is effective, then kG is primitive iff G has a dense orbit. Étale Groupoids Étale Groupoid Algebras Representation Theory Primitivity Recall: a ring is primitive if it has a faithful simple module. • Let G be a Hausdorff ample groupoid and k a field. • Theorem (BS) If G is effective, then kG is primitive iff G has a dense orbit. Theorem (BS) Suppose that G has a dense orbit Ox with kGx primitive. Then kG is primitive. The converse holds if x is isolated. Étale Groupoids Étale Groupoid Algebras Representation Theory Primitivity Recall: a ring is primitive if it has a faithful simple module. • Let G be a Hausdorff ample groupoid and k a field. • Theorem (BS) If G is effective, then kG is primitive iff G has a dense orbit. Theorem (BS) Suppose that G has a dense orbit Ox with kGx primitive. Then kG is primitive. The converse holds if x is isolated. • Recall that Gx is the isotropy group at x. Étale Groupoids Étale Groupoid Algebras Representation Theory Primitivity Recall: a ring is primitive if it has a faithful simple module. • Let G be a Hausdorff ample groupoid and k a field. • Theorem (BS) If G is effective, then kG is primitive iff G has a dense orbit. Theorem (BS) Suppose that G has a dense orbit Ox with kGx primitive. Then kG is primitive. The converse holds if x is isolated. Recall that Gx is the isotropy group at x. • A dense orbit is always necessary for primitivity. • Étale Groupoids Étale Groupoid Algebras Representation Theory Primitivity Recall: a ring is primitive if it has a faithful simple module. • Let G be a Hausdorff ample groupoid and k a field. • Theorem (BS) If G is effective, then kG is primitive iff G has a dense orbit. Theorem (BS) Suppose that G has a dense orbit Ox with kGx primitive. Then kG is primitive. The converse holds if x is isolated. Recall that Gx is the isotropy group at x. • A dense orbit is always necessary for primitivity. • There is a non-Hausdorff version. • Étale Groupoids Étale Groupoid Algebras Semiprimitivity: the effective case • Recall: a ring is semiprimitive if it has a faithful semisimple module. Representation Theory Étale Groupoids Étale Groupoid Algebras Representation Theory Semiprimitivity: the effective case • Recall: a ring is semiprimitive if it has a faithful semisimple module. Theorem (BS) Let G be a Hausdorff effective ample groupoid and k a semiprimitive comm. ring with 1. Then kG is semiprimitive. Étale Groupoids Étale Groupoid Algebras Representation Theory Semiprimitivity Theorem (BS) Let G be a Hausdorff ample groupoid and k a semiprimitive comm. ring with 1. If kGx is semiprimitive for all x in some dense subset X ⊆ G (0) , then kG is semiprimitive. The converse holds if X consists of isolated points. Étale Groupoids Étale Groupoid Algebras Representation Theory Semiprimitivity Theorem (BS) Let G be a Hausdorff ample groupoid and k a semiprimitive comm. ring with 1. If kGx is semiprimitive for all x in some dense subset X ⊆ G (0) , then kG is semiprimitive. The converse holds if X consists of isolated points. • A non-Hausdorff version holds. Étale Groupoids Étale Groupoid Algebras Representation Theory Semiprimitivity Theorem (BS) Let G be a Hausdorff ample groupoid and k a semiprimitive comm. ring with 1. If kGx is semiprimitive for all x in some dense subset X ⊆ G (0) , then kG is semiprimitive. The converse holds if X consists of isolated points. A non-Hausdorff version holds. • Isotropy groups for Deaconu-Renault groupoids are trivial or Z, recovering semiprimitivity of Leavitt path algebras. • Étale Groupoids Étale Groupoid Algebras Representation Theory Semiprimitivity Theorem (BS) Let G be a Hausdorff ample groupoid and k a semiprimitive comm. ring with 1. If kGx is semiprimitive for all x in some dense subset X ⊆ G (0) , then kG is semiprimitive. The converse holds if X consists of isolated points. A non-Hausdorff version holds. • Isotropy groups for Deaconu-Renault groupoids are trivial or Z, recovering semiprimitivity of Leavitt path algebras. • Corollary Let G be an ample groupoid and k be a field of characteristic 0 such that k/Q is not algebraic. Then kG is semiprimitive. Étale Groupoids Étale Groupoid Algebras Representation Theory Semiprimitivity Theorem (BS) Let G be a Hausdorff ample groupoid and k a semiprimitive comm. ring with 1. If kGx is semiprimitive for all x in some dense subset X ⊆ G (0) , then kG is semiprimitive. The converse holds if X consists of isolated points. A non-Hausdorff version holds. • Isotropy groups for Deaconu-Renault groupoids are trivial or Z, recovering semiprimitivity of Leavitt path algebras. • Corollary Let G be an ample groupoid and k be a field of characteristic 0 such that k/Q is not algebraic. Then kG is semiprimitive. Proof. Apply Amitsur ’59 with X = G (0) . Étale Groupoids Étale Groupoid Algebras Representation Theory Applications to inverse semigroup algebras Corollary Let k be a semiprimitive comm. ring with 1 and S an inverse semigroup. If kGe is semiprimitive for every maximal subgroup Ge of S, then kS is semiprimitve. The converse holds if E(S) is pseudofinite. Étale Groupoids Étale Groupoid Algebras Representation Theory Applications to inverse semigroup algebras Corollary Let k be a semiprimitive comm. ring with 1 and S an inverse semigroup. If kGe is semiprimitive for every maximal subgroup Ge of S, then kS is semiprimitve. The converse holds if E(S) is pseudofinite. • Sufficiency was proved by Domanov (1976). Étale Groupoids Étale Groupoid Algebras Representation Theory Applications to inverse semigroup algebras Corollary Let k be a semiprimitive comm. ring with 1 and S an inverse semigroup. If kGe is semiprimitive for every maximal subgroup Ge of S, then kS is semiprimitve. The converse holds if E(S) is pseudofinite. Sufficiency was proved by Domanov (1976). • The converse was proved by Munn (1987). • Étale Groupoids Étale Groupoid Algebras Representation Theory Applications to inverse semigroup algebras Corollary Let k be a semiprimitive comm. ring with 1 and S an inverse semigroup. If kGe is semiprimitive for every maximal subgroup Ge of S, then kS is semiprimitve. The converse holds if E(S) is pseudofinite. Sufficiency was proved by Domanov (1976). • The converse was proved by Munn (1987). • Ge is the group of units of eSe for e idempotent. • Étale Groupoids Étale Groupoid Algebras Representation Theory Applications to inverse semigroup algebras Corollary Let k be a semiprimitive comm. ring with 1 and S an inverse semigroup. If kGe is semiprimitive for every maximal subgroup Ge of S, then kS is semiprimitve. The converse holds if E(S) is pseudofinite. Sufficiency was proved by Domanov (1976). • The converse was proved by Munn (1987). • Ge is the group of units of eSe for e idempotent. • E(S) is pseudofinite if the characteristic functions of [ principal filters are discrete in E(S). • Étale Groupoids Étale Groupoid Algebras Representation Theory Applications to inverse semigroup algebras Corollary Let k be a semiprimitive comm. ring with 1 and S an inverse semigroup. If kGe is semiprimitive for every maximal subgroup Ge of S, then kS is semiprimitve. The converse holds if E(S) is pseudofinite. • • • • • Sufficiency was proved by Domanov (1976). The converse was proved by Munn (1987). Ge is the group of units of eSe for e idempotent. E(S) is pseudofinite if the characteristic functions of [ principal filters are discrete in E(S). This is a topological reformulation of the original definition. Étale Groupoids Étale Groupoid Algebras Unitary modules • Let R be a ring with local units. Representation Theory Étale Groupoids Étale Groupoid Algebras Unitary modules Let R be a ring with local units. • An R-module M is unitary if RM = M. • Representation Theory Étale Groupoids Étale Groupoid Algebras Representation Theory Unitary modules Let R be a ring with local units. • An R-module M is unitary if RM = M. • The category of unitary R-modules is denoted R-mod. • Étale Groupoids Étale Groupoid Algebras Representation Theory Unitary modules Let R be a ring with local units. • An R-module M is unitary if RM = M. • The category of unitary R-modules is denoted R-mod. • R and S are Morita equivalent if R-mod is equivalent to S-mod. • Étale Groupoids Étale Groupoid Algebras Representation Theory Unitary modules • • • • • Let R be a ring with local units. An R-module M is unitary if RM = M. The category of unitary R-modules is denoted R-mod. R and S are Morita equivalent if R-mod is equivalent to S-mod. Morita equivalence can also be formulated in terms of Morita contexts. Étale Groupoids Étale Groupoid Algebras Representation Theory Sheaves of G-modules • Let G be an ample groupoid and k a comm. ring with 1. Étale Groupoids Étale Groupoid Algebras Representation Theory Sheaves of G-modules Let G be an ample groupoid and k a comm. ring with 1. • A G-sheaf of k-modules is a sheaf of k-modules over G (0) with an action of G . • Étale Groupoids Étale Groupoid Algebras Representation Theory Sheaves of G-modules Let G be an ample groupoid and k a comm. ring with 1. • A G-sheaf of k-modules is a sheaf of k-modules over G (0) with an action of G . g • Roughly, this means that if x −→ y is in G (1) , then g takes the stalk over x to the stalk over y by a k-module isomorphism. • Étale Groupoids Étale Groupoid Algebras Representation Theory Sheaves of G-modules Let G be an ample groupoid and k a comm. ring with 1. • A G-sheaf of k-modules is a sheaf of k-modules over G (0) with an action of G . g • Roughly, this means that if x −→ y is in G (1) , then g takes the stalk over x to the stalk over y by a k-module isomorphism. • There are also some continuity conditions. • Étale Groupoids Étale Groupoid Algebras Representation Theory Sheaves of G-modules • • • • • Let G be an ample groupoid and k a comm. ring with 1. A G-sheaf of k-modules is a sheaf of k-modules over G (0) with an action of G . g Roughly, this means that if x −→ y is in G (1) , then g takes the stalk over x to the stalk over y by a k-module isomorphism. There are also some continuity conditions. If G is discrete, this amounts to a functor G → k-mod. Étale Groupoids Étale Groupoid Algebras Representation Theory Sheaves of G-modules • • • • • Let G be an ample groupoid and k a comm. ring with 1. A G-sheaf of k-modules is a sheaf of k-modules over G (0) with an action of G . g Roughly, this means that if x −→ y is in G (1) , then g takes the stalk over x to the stalk over y by a k-module isomorphism. There are also some continuity conditions. If G is discrete, this amounts to a functor G → k-mod. Theorem (BS) kG-mod is equivalent to the category of G-sheaves of k-modules. Étale Groupoids Étale Groupoid Algebras Representation Theory Morita equivalence • If Z is a locally compact space and f : Z → G (0) is continuous, there is a pullback groupoid G[Z]. Étale Groupoids Étale Groupoid Algebras Representation Theory Morita equivalence If Z is a locally compact space and f : Z → G (0) is continuous, there is a pullback groupoid G[Z]. g g • G[Z](0) = Z, G[Z](1) = {z −→ z ′ | f (z) −→ f (z ′ )}. • Étale Groupoids Étale Groupoid Algebras Representation Theory Morita equivalence If Z is a locally compact space and f : Z → G (0) is continuous, there is a pullback groupoid G[Z]. g g • G[Z](0) = Z, G[Z](1) = {z −→ z ′ | f (z) −→ f (z ′ )}. • Groupoids G and H are Morita equivalent if there is a locally compact space Z and continuous open surjections p : Z → G (0) and q : Z → H(0) such that G[Z] ∼ = H[Z]. • Étale Groupoids Étale Groupoid Algebras Representation Theory Morita equivalence If Z is a locally compact space and f : Z → G (0) is continuous, there is a pullback groupoid G[Z]. g g • G[Z](0) = Z, G[Z](1) = {z −→ z ′ | f (z) −→ f (z ′ )}. • Groupoids G and H are Morita equivalent if there is a locally compact space Z and continuous open surjections p : Z → G (0) and q : Z → H(0) such that G[Z] ∼ = H[Z]. • Theorem (L. O. Clark, A. Sims) If G and H are Morita equivalent ample groupoids, then kG is Morita equivalent to kH. Étale Groupoids Étale Groupoid Algebras Representation Theory Morita equivalence If Z is a locally compact space and f : Z → G (0) is continuous, there is a pullback groupoid G[Z]. g g • G[Z](0) = Z, G[Z](1) = {z −→ z ′ | f (z) −→ f (z ′ )}. • Groupoids G and H are Morita equivalent if there is a locally compact space Z and continuous open surjections p : Z → G (0) and q : Z → H(0) such that G[Z] ∼ = H[Z]. • Theorem (L. O. Clark, A. Sims) If G and H are Morita equivalent ample groupoids, then kG is Morita equivalent to kH. • Gives a uniform explanation for Morita equivalences of graph C ∗ -algebras and Leavitt path algebras. Étale Groupoids Étale Groupoid Algebras Representation Theory Morita equivalence If Z is a locally compact space and f : Z → G (0) is continuous, there is a pullback groupoid G[Z]. g g • G[Z](0) = Z, G[Z](1) = {z −→ z ′ | f (z) −→ f (z ′ )}. • Groupoids G and H are Morita equivalent if there is a locally compact space Z and continuous open surjections p : Z → G (0) and q : Z → H(0) such that G[Z] ∼ = H[Z]. • Theorem (L. O. Clark, A. Sims) If G and H are Morita equivalent ample groupoids, then kG is Morita equivalent to kH. Gives a uniform explanation for Morita equivalences of graph C ∗ -algebras and Leavitt path algebras. • The original proof constructed a Morita context. • Étale Groupoids Étale Groupoid Algebras Representation Theory Morita equivalence If Z is a locally compact space and f : Z → G (0) is continuous, there is a pullback groupoid G[Z]. g g • G[Z](0) = Z, G[Z](1) = {z −→ z ′ | f (z) −→ f (z ′ )}. • Groupoids G and H are Morita equivalent if there is a locally compact space Z and continuous open surjections p : Z → G (0) and q : Z → H(0) such that G[Z] ∼ = H[Z]. • Theorem (L. O. Clark, A. Sims) If G and H are Morita equivalent ample groupoids, then kG is Morita equivalent to kH. Gives a uniform explanation for Morita equivalences of graph C ∗ -algebras and Leavitt path algebras. • The original proof constructed a Morita context. • A result of Moerdijk implies if G, H are Morita equivalent, then so are their categories of sheaves. • Étale Groupoids Étale Groupoid Algebras Schützenberger representations • Fix x ∈ G (0) . Representation Theory Étale Groupoids Étale Groupoid Algebras Representation Theory Schützenberger representations Fix x ∈ G (0) . • Let Lx = d−1 (x) be the arrows emanating from x. • Étale Groupoids Representation Theory Étale Groupoid Algebras Schützenberger representations Fix x ∈ G (0) . • Let Lx = d−1 (x) be the arrows emanating from x. • The isotropy group Gx acts freely on the right of Lx . • Gx Lx x Étale Groupoids Representation Theory Étale Groupoid Algebras Schützenberger representations Fix x ∈ G (0) . • Let Lx = d−1 (x) be the arrows emanating from x. • The isotropy group Gx acts freely on the right of Lx . • Gx Lx • So kLx is a free kGx -module. x Étale Groupoids Representation Theory Étale Groupoid Algebras Schützenberger representations Fix x ∈ G (0) . • Let Lx = d−1 (x) be the arrows emanating from x. • The isotropy group Gx acts freely on the right of Lx . • Gx Lx So kLx is a free kGx -module. • It is in fact a kG-kGx -bimodule. • x Étale Groupoids Representation Theory Étale Groupoid Algebras Schützenberger representations Fix x ∈ G (0) . • Let Lx = d−1 (x) be the arrows emanating from x. • The isotropy group Gx acts freely on the right of Lx . • Gx Lx x So kLx is a free kGx -module. • It is in fact a kG-kGx -bimodule. • If f ∈ kG and t ∈ Lx , then X f ·t= f (s)st. • d(s)=r(t) Étale Groupoids Étale Groupoid Algebras Representation Theory Induction and restriction functors • There is an exact functor Indx : kGx -mod → kG-mod given by M 7−→ kLx ⊗kGx M. Étale Groupoids Étale Groupoid Algebras Representation Theory Induction and restriction functors • There is an exact functor Indx : kGx -mod → kG-mod given by M 7−→ kLx ⊗kGx M. • It has a right adjoint Resx : kG-mod → kGx -mod. Étale Groupoids Étale Groupoid Algebras Representation Theory Induction and restriction functors • There is an exact functor Indx : kGx -mod → kG-mod given by M 7−→ kLx ⊗kGx M. It has a right adjoint Resx : kG-mod → kGx -mod. • Indx preserves simple modules. • Étale Groupoids Étale Groupoid Algebras Representation Theory Induction and restriction functors • There is an exact functor Indx : kGx -mod → kG-mod given by M 7−→ kLx ⊗kGx M. It has a right adjoint Resx : kG-mod → kGx -mod. • Indx preserves simple modules. • Resx sends simple modules to simple modules or 0. • Étale Groupoids Étale Groupoid Algebras Representation Theory Induction and restriction functors • There is an exact functor Indx : kGx -mod → kG-mod given by M 7−→ kLx ⊗kGx M. It has a right adjoint Resx : kG-mod → kGx -mod. • Indx preserves simple modules. • Resx sends simple modules to simple modules or 0. • Theorem (BS) Let k be a field and G an ample groupoid. Then the finite dimensional simple kG-modules are the Indx (M) with |Ox | < ∞ and M a finite dimensional simple kGx -module. Étale Groupoids Étale Groupoid Algebras Representation Theory A question of Exel Question (Exel) Let k be a field and G a Hausdorff ample groupoid. Is every primitive ideal of kG a kernel of an induced module Indx (M) with M a simple kGx -module? Étale Groupoids Étale Groupoid Algebras Representation Theory A question of Exel Question (Exel) Let k be a field and G a Hausdorff ample groupoid. Is every primitive ideal of kG a kernel of an induced module Indx (M) with M a simple kGx -module? • A C ∗ -algebraic analogue is true if G is amenable. Étale Groupoids Étale Groupoid Algebras Representation Theory A question of Exel Question (Exel) Let k be a field and G a Hausdorff ample groupoid. Is every primitive ideal of kG a kernel of an induced module Indx (M) with M a simple kGx -module? A C ∗ -algebraic analogue is true if G is amenable. • It is true for Leavitt path algebras. • Étale Groupoids Étale Groupoid Algebras Representation Theory Further work • There are still many ample groupoid C ∗ -algebras whose algebraic analogues have not been studied. Étale Groupoids Étale Groupoid Algebras Representation Theory Further work There are still many ample groupoid C ∗ -algebras whose algebraic analogues have not been studied. • Algebras associated to tilings. • Étale Groupoids Étale Groupoid Algebras Representation Theory Further work There are still many ample groupoid C ∗ -algebras whose algebraic analogues have not been studied. • Algebras associated to tilings. • Algebras associated to self-similar/automaton groups. • Étale Groupoids Étale Groupoid Algebras Representation Theory Further work There are still many ample groupoid C ∗ -algebras whose algebraic analogues have not been studied. • Algebras associated to tilings. • Algebras associated to self-similar/automaton groups. • Algebras associated to Cantor dynamical systems. • Étale Groupoids Étale Groupoid Algebras Representation Theory Further work • • • • • There are still many ample groupoid C ∗ -algebras whose algebraic analogues have not been studied. Algebras associated to tilings. Algebras associated to self-similar/automaton groups. Algebras associated to Cantor dynamical systems. Algebras associated to higher rank graphs. Étale Groupoids Étale Groupoid Algebras Representation Theory Further work • • • • • • There are still many ample groupoid C ∗ -algebras whose algebraic analogues have not been studied. Algebras associated to tilings. Algebras associated to self-similar/automaton groups. Algebras associated to Cantor dynamical systems. Algebras associated to higher rank graphs. Algebras associated to partial group actions. Étale Groupoids Étale Groupoid Algebras Representation Theory The end Thank you for your attention!