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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!