Download Groupoid C*-algebras with Hausdorff Spectrum

GPOTS 2013
Berkeley, CA
Groupoid C ∗ -algebras with Hausdorff Spectrum
Geoff Goehle
Western Carolina University
Cullowhee, NC
GPOTS 2013
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum
GPOTS 2013
Berkeley, CA
Transformation Group Case
Suppose an abelian, second countable, locally compact Hausdorff
group H acts on a second countable, locally compact Hausdorff
space X . Let H n X be the transformation groupoid and
C ∗ (H n X ) the groupoid C ∗ -algebra.
Theorem ([Williams; 82])
Suppose that the maps of H/Hx onto H · x are homeomorphisms
for each x ∈ X . The spectrum C ∗ (H n X )∧ is Hausdorff if and
only if the map x 7→ Hx is continuous with respect to the Fell
topology and X /H is Hausdorff.
Note: We can use [Orloff Clark; 07] and [Ramsay; 90] to show that
if either X /H or C ∗ (H n X )∧ is T0 then the maps from H/Hx to
H · x are automatically homeomorphisms.
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum
GPOTS 2013
Berkeley, CA
Transformation Group Case
Suppose an abelian, second countable, locally compact Hausdorff
group H acts on a second countable, locally compact Hausdorff
space X . Let H n X be the transformation groupoid and
C ∗ (H n X ) the groupoid C ∗ -algebra.
Theorem ([Williams; 82])
Suppose that the maps of H/Hx onto H · x are homeomorphisms
for each x ∈ X . The spectrum C ∗ (H n X )∧ is Hausdorff if and
only if the map x 7→ Hx is continuous with respect to the Fell
topology and X /H is Hausdorff.
Note: We can use [Orloff Clark; 07] and [Ramsay; 90] to show that
if either X /H or C ∗ (H n X )∧ is T0 then the maps from H/Hx to
H · x are automatically homeomorphisms.
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum
GPOTS 2013
Berkeley, CA
Transformation Group Case
Suppose an abelian, second countable, locally compact Hausdorff
group H acts on a second countable, locally compact Hausdorff
space X . Let H n X be the transformation groupoid and
C ∗ (H n X ) the groupoid C ∗ -algebra.
Theorem ([Williams; 82])
Suppose that the maps of H/Hx onto H · x are homeomorphisms
for each x ∈ X . The spectrum C ∗ (H n X )∧ is Hausdorff if and
only if the map x 7→ Hx is continuous with respect to the Fell
topology and X /H is Hausdorff.
Note: We can use [Orloff Clark; 07] and [Ramsay; 90] to show that
if either X /H or C ∗ (H n X )∧ is T0 then the maps from H/Hx to
H · x are automatically homeomorphisms.
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum
GPOTS 2013
Berkeley, CA
Groupoid Generalization
I
Let G be a second countable, locally compact Hausdorff
groupoid with a Haar system and abelian stabilizer subgroups,
which we will denote by Sx .
I
The naive generalization of the previous theorem is that
C ∗ (G ) will have Hausdorff spectrum if and only if the map
x 7→ Sx is continuous, and the orbit space G (0) /G is
Hausdorff.
b be the dual
Let S be the stabilizer subgroupoid and S
stabilizer groupoid.
I
I
I
b are the Pontryagin duals of the fibers of S.
The fibers of S
b via conjugation:
There is an action of G on S
γ · χ(s) = χ(γ −1 sγ)
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum
GPOTS 2013
Berkeley, CA
Groupoid Generalization
I
Let G be a second countable, locally compact Hausdorff
groupoid with a Haar system and abelian stabilizer subgroups,
which we will denote by Sx .
I
The naive generalization of the previous theorem is that
C ∗ (G ) will have Hausdorff spectrum if and only if the map
x 7→ Sx is continuous, and the orbit space G (0) /G is
Hausdorff.
b be the dual
Let S be the stabilizer subgroupoid and S
stabilizer groupoid.
I
I
I
b are the Pontryagin duals of the fibers of S.
The fibers of S
b via conjugation:
There is an action of G on S
γ · χ(s) = χ(γ −1 sγ)
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum
GPOTS 2013
Berkeley, CA
Groupoid Generalization
I
Let G be a second countable, locally compact Hausdorff
groupoid with a Haar system and abelian stabilizer subgroups,
which we will denote by Sx .
I
The naive generalization of the previous theorem is that
C ∗ (G ) will have Hausdorff spectrum if and only if the map
x 7→ Sx is continuous, and the orbit space G (0) /G is
Hausdorff.
b be the dual
Let S be the stabilizer subgroupoid and S
stabilizer groupoid.
I
I
I
b are the Pontryagin duals of the fibers of S.
The fibers of S
b via conjugation:
There is an action of G on S
γ · χ(s) = χ(γ −1 sγ)
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum
GPOTS 2013
Berkeley, CA
Forward Direction
The forward direction of the naive generalization holds:
I
It follows from [Orloff Clark; 07] that, since C ∗ (G )∧ is
Hausdorff and hence T0 , G (0) /G is T0 .
I
If C ∗ (G )∧ is Hausdorff then [Muhly, Renault, Williams; 96]
shows that the stabilizers vary continuously.
I
The main result of [G; 12] implies that in this case C ∗ (G )∧ is
b .
homeomorphic to S/G
I
Since G (0) /G is easily seen to be homeomorphic to its image
b , it follows that G (0) /G is Hausdorff.
in S/G
However! The converse fails. There are groupoids G such that the
stabilizers vary continuously and G (0) /G is Hausdorff but C ∗ (G )∧
is not Hausdorff.
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum
GPOTS 2013
Berkeley, CA
Forward Direction
The forward direction of the naive generalization holds:
I
It follows from [Orloff Clark; 07] that, since C ∗ (G )∧ is
Hausdorff and hence T0 , G (0) /G is T0 .
I
If C ∗ (G )∧ is Hausdorff then [Muhly, Renault, Williams; 96]
shows that the stabilizers vary continuously.
I
The main result of [G; 12] implies that in this case C ∗ (G )∧ is
b .
homeomorphic to S/G
I
Since G (0) /G is easily seen to be homeomorphic to its image
b , it follows that G (0) /G is Hausdorff.
in S/G
However! The converse fails. There are groupoids G such that the
stabilizers vary continuously and G (0) /G is Hausdorff but C ∗ (G )∧
is not Hausdorff.
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum
GPOTS 2013
Berkeley, CA
Forward Direction
The forward direction of the naive generalization holds:
I
It follows from [Orloff Clark; 07] that, since C ∗ (G )∧ is
Hausdorff and hence T0 , G (0) /G is T0 .
I
If C ∗ (G )∧ is Hausdorff then [Muhly, Renault, Williams; 96]
shows that the stabilizers vary continuously.
I
The main result of [G; 12] implies that in this case C ∗ (G )∧ is
b .
homeomorphic to S/G
I
Since G (0) /G is easily seen to be homeomorphic to its image
b , it follows that G (0) /G is Hausdorff.
in S/G
However! The converse fails. There are groupoids G such that the
stabilizers vary continuously and G (0) /G is Hausdorff but C ∗ (G )∧
is not Hausdorff.
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum
GPOTS 2013
Berkeley, CA
Forward Direction
The forward direction of the naive generalization holds:
I
It follows from [Orloff Clark; 07] that, since C ∗ (G )∧ is
Hausdorff and hence T0 , G (0) /G is T0 .
I
If C ∗ (G )∧ is Hausdorff then [Muhly, Renault, Williams; 96]
shows that the stabilizers vary continuously.
I
The main result of [G; 12] implies that in this case C ∗ (G )∧ is
b .
homeomorphic to S/G
I
Since G (0) /G is easily seen to be homeomorphic to its image
b , it follows that G (0) /G is Hausdorff.
in S/G
However! The converse fails. There are groupoids G such that the
stabilizers vary continuously and G (0) /G is Hausdorff but C ∗ (G )∧
is not Hausdorff.
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum
GPOTS 2013
Berkeley, CA
Forward Direction
The forward direction of the naive generalization holds:
I
It follows from [Orloff Clark; 07] that, since C ∗ (G )∧ is
Hausdorff and hence T0 , G (0) /G is T0 .
I
If C ∗ (G )∧ is Hausdorff then [Muhly, Renault, Williams; 96]
shows that the stabilizers vary continuously.
I
The main result of [G; 12] implies that in this case C ∗ (G )∧ is
b .
homeomorphic to S/G
I
Since G (0) /G is easily seen to be homeomorphic to its image
b , it follows that G (0) /G is Hausdorff.
in S/G
However! The converse fails. There are groupoids G such that the
stabilizers vary continuously and G (0) /G is Hausdorff but C ∗ (G )∧
is not Hausdorff.
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum
GPOTS 2013
Berkeley, CA
Counter Example
I
If G (0) /G is Hausdorff and the stabilizers vary continuously
b .
then [G; 12] implies that C ∗ (G )∧ is homeomorphic to S/G
I
Using Green’s famous example of a free group action that is
not proper we can construct a non-abelian group H which
acts on a space X so that the stabilizers are abelian and vary
b
continuously and the orbit space is Hausdorff but S/G
fails to
be Hausdorff.
I
This tells us that, even though it is not obvious what other
commutativity requirements one could make regarding
groupoids, the assumption that the stabilizers are abelian
cannot always stand in for abelian transformation groups.
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum
GPOTS 2013
Berkeley, CA
Counter Example
I
If G (0) /G is Hausdorff and the stabilizers vary continuously
b .
then [G; 12] implies that C ∗ (G )∧ is homeomorphic to S/G
I
Using Green’s famous example of a free group action that is
not proper we can construct a non-abelian group H which
acts on a space X so that the stabilizers are abelian and vary
b
continuously and the orbit space is Hausdorff but S/G
fails to
be Hausdorff.
I
This tells us that, even though it is not obvious what other
commutativity requirements one could make regarding
groupoids, the assumption that the stabilizers are abelian
cannot always stand in for abelian transformation groups.
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum
GPOTS 2013
Berkeley, CA
Counter Example
I
If G (0) /G is Hausdorff and the stabilizers vary continuously
b .
then [G; 12] implies that C ∗ (G )∧ is homeomorphic to S/G
I
Using Green’s famous example of a free group action that is
not proper we can construct a non-abelian group H which
acts on a space X so that the stabilizers are abelian and vary
b
continuously and the orbit space is Hausdorff but S/G
fails to
be Hausdorff.
I
This tells us that, even though it is not obvious what other
commutativity requirements one could make regarding
groupoids, the assumption that the stabilizers are abelian
cannot always stand in for abelian transformation groups.
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum
GPOTS 2013
Berkeley, CA
Correct Characterization
In this case the correct generalization requires an additional
topological condition.
Theorem ([G; 13])
Suppose G is a second countable, locally compact Hausdorff
groupoid with a Haar system and abelian stabilizers. Then C ∗ (G )∧
is Hausdorff if and only if:
I
the stabilizers vary continuously,
I
the orbit space G (0) /G is Hausdorff, and
b and {γi } ⊂ G such that γi can act
given sequences {χi } ⊂ S
on χi for each i, if χi → χ and γi · χi → ω with χ and ω in
the same fiber then χ = ω.
I
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum
GPOTS 2013
Berkeley, CA
Correct Characterization
In this case the correct generalization requires an additional
topological condition.
Theorem ([G; 13])
Suppose G is a second countable, locally compact Hausdorff
groupoid with a Haar system and abelian stabilizers. Then C ∗ (G )∧
is Hausdorff if and only if:
I
the stabilizers vary continuously,
I
the orbit space G (0) /G is Hausdorff, and
b and {γi } ⊂ G such that γi can act
given sequences {χi } ⊂ S
on χi for each i, if χi → χ and γi · χi → ω with χ and ω in
the same fiber then χ = ω.
I
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum
GPOTS 2013
Berkeley, CA
Correct Characterization
In this case the correct generalization requires an additional
topological condition.
Theorem ([G; 13])
Suppose G is a second countable, locally compact Hausdorff
groupoid with a Haar system and abelian stabilizers. Then C ∗ (G )∧
is Hausdorff if and only if:
I
the stabilizers vary continuously,
I
the orbit space G (0) /G is Hausdorff, and
b and {γi } ⊂ G such that γi can act
given sequences {χi } ⊂ S
on χi for each i, if χi → χ and γi · χi → ω with χ and ω in
the same fiber then χ = ω.
I
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum
GPOTS 2013
Berkeley, CA
Examples
If G has abelian and continuously varying stabilizers then the third
condition of the theorem will be automatically satisfied if G is one
of the following classes of groupoid:
I
groupoids constructed from abelian transformation groups,
I
principal groupoids,
I
proper groupoids,
I
transitive groupoids,
I
graph groupoids,
I
Cartan groupoids.
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum
GPOTS 2013
Berkeley, CA
Graph Example
Suppose E is a row finite directed graph with no sources. We can
build the graph groupoid G as in [Kumjain, Pask, Raeburn,
Renault; 97] and the graph C ∗ -algebra will be isomorphic to
C ∗ (G ). We can provide conditions for the graph algebra to have
Hausdorff spectrum.
I
The stabilizers of G will vary continuously if and only if no
cycle in the graph has an entry.
I
The orbit space will be Hausdorff if and only if given non-shift
equivalent paths x and y there exist vertices u and v such that
there is a path from a vertex on x to u, a path from a vertex
on y to v , and no vertex w which has a path to both u and v .
I
The third condition is automatically satisfied.
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum
GPOTS 2013
Berkeley, CA
Graph Example
Suppose E is a row finite directed graph with no sources. We can
build the graph groupoid G as in [Kumjain, Pask, Raeburn,
Renault; 97] and the graph C ∗ -algebra will be isomorphic to
C ∗ (G ). We can provide conditions for the graph algebra to have
Hausdorff spectrum.
I
The stabilizers of G will vary continuously if and only if no
cycle in the graph has an entry.
I
The orbit space will be Hausdorff if and only if given non-shift
equivalent paths x and y there exist vertices u and v such that
there is a path from a vertex on x to u, a path from a vertex
on y to v , and no vertex w which has a path to both u and v .
I
The third condition is automatically satisfied.
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum
GPOTS 2013
Berkeley, CA
Graph Example
Suppose E is a row finite directed graph with no sources. We can
build the graph groupoid G as in [Kumjain, Pask, Raeburn,
Renault; 97] and the graph C ∗ -algebra will be isomorphic to
C ∗ (G ). We can provide conditions for the graph algebra to have
Hausdorff spectrum.
I
The stabilizers of G will vary continuously if and only if no
cycle in the graph has an entry.
I
The orbit space will be Hausdorff if and only if given non-shift
equivalent paths x and y there exist vertices u and v such that
there is a path from a vertex on x to u, a path from a vertex
on y to v , and no vertex w which has a path to both u and v .
I
The third condition is automatically satisfied.
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum
GPOTS 2013
Berkeley, CA
Graph Example
Suppose E is a row finite directed graph with no sources. We can
build the graph groupoid G as in [Kumjain, Pask, Raeburn,
Renault; 97] and the graph C ∗ -algebra will be isomorphic to
C ∗ (G ). We can provide conditions for the graph algebra to have
Hausdorff spectrum.
I
The stabilizers of G will vary continuously if and only if no
cycle in the graph has an entry.
I
The orbit space will be Hausdorff if and only if given non-shift
equivalent paths x and y there exist vertices u and v such that
there is a path from a vertex on x to u, a path from a vertex
on y to v , and no vertex w which has a path to both u and v .
I
The third condition is automatically satisfied.
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum
GPOTS 2013
Berkeley, CA
The T1 Case
Interestingly, the T1 (and T0 ) case behave very differently from the
T2 case.
I
I
Suppose the groupoid G has continuously varying, abelian
stabilizers. Then, as before, as long as either G (0) /G or
C ∗ (G )∧ is at least T0 we will be able to identify C ∗ (G )∧ with
b .
S/G
b
Point set arguments show that either G (0) /G , S/G and S/G
will all be T1 , or none of them will be T1 . Consequently
C ∗ (G )∧ will be T1 if and only if G (0) /G is T1 .
I
As we saw before there are topological obstructions to
generalizing this argument to the Hausdorff case.
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum
GPOTS 2013
Berkeley, CA
The T1 Case
Interestingly, the T1 (and T0 ) case behave very differently from the
T2 case.
I
I
Suppose the groupoid G has continuously varying, abelian
stabilizers. Then, as before, as long as either G (0) /G or
C ∗ (G )∧ is at least T0 we will be able to identify C ∗ (G )∧ with
b .
S/G
b
Point set arguments show that either G (0) /G , S/G and S/G
will all be T1 , or none of them will be T1 . Consequently
C ∗ (G )∧ will be T1 if and only if G (0) /G is T1 .
I
As we saw before there are topological obstructions to
generalizing this argument to the Hausdorff case.
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum
GPOTS 2013
Berkeley, CA
The T1 Case
Interestingly, the T1 (and T0 ) case behave very differently from the
T2 case.
I
I
Suppose the groupoid G has continuously varying, abelian
stabilizers. Then, as before, as long as either G (0) /G or
C ∗ (G )∧ is at least T0 we will be able to identify C ∗ (G )∧ with
b .
S/G
b
Point set arguments show that either G (0) /G , S/G and S/G
will all be T1 , or none of them will be T1 . Consequently
C ∗ (G )∧ will be T1 if and only if G (0) /G is T1 .
I
As we saw before there are topological obstructions to
generalizing this argument to the Hausdorff case.
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum
GPOTS 2013
Berkeley, CA
The T1 Case
Interestingly, the T1 (and T0 ) case behave very differently from the
T2 case.
I
I
Suppose the groupoid G has continuously varying, abelian
stabilizers. Then, as before, as long as either G (0) /G or
C ∗ (G )∧ is at least T0 we will be able to identify C ∗ (G )∧ with
b .
S/G
b
Point set arguments show that either G (0) /G , S/G and S/G
will all be T1 , or none of them will be T1 . Consequently
C ∗ (G )∧ will be T1 if and only if G (0) /G is T1 .
I
As we saw before there are topological obstructions to
generalizing this argument to the Hausdorff case.
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum
References
Lisa Orloff Clark, CCR and GCR groupoid C ∗ -algebras, Indiana
University Mathematics Journal 56 (2007), no. 5, 2087–2110.
Geoff Goehle, The Mackey machine for regular groupoid crossed
products. II, Rocky Mountain Journal of Mathematics 42 (2012),
no. 3, 1–28.
, Groupoid C ∗ -algebras with Hausdorff spectrum, to appear,
2013.
Jean N. Renault Paul S. Muhly and Dana P. Williams, Continuous
trace groupoid C ∗ -algebras, III, Transactions of the American
Mathematical Society 348 (1996), no. 9, 3621–3641.
Dana P. Williams, Transformation group C ∗ -algebras with Hausdorff
spectrum, Illinois Journal of Mathematics 26 (1982), no. 2, 317–321.
Geoff Goehle
Groupoid C ∗ -algebras with Hausdorff Spectrum