Purely infinite C ∗ -algebras associated to étale
groupoids
Jonathan Brown
Kansas State University
Joint with L. Clark and A. Sierakowski
Nebraska-Iowa Functional Analysis Seminar
April 2014
Jonathan Brown (KSU)
Purely infinite groupoids
(K)NIFAS
1 / 15
Purely infinite simple
Let A be a simple C ∗ -algebra.
For a ∈ Mn (A), b ∈ Mm (A) be positive, we say a is Cuntz below b,
a - b, if there exist xk ∈ Mm,n (A) such that xk∗ bxk → a in norm.
a ∈ A+ is infinite if there exists b ∈ A+ a ⊕ b - a.
A projection p ∈ A is infinite if and only if it is Murray-von Neumann
equivalent to a proper subprojection of itself. (KR00 Lemma 3.1)
A is purely infinite if every a ∈ A+ − {0} is infinite. (KR00
Theorem 4.16)
Theorem (Kirchberg Phillips)
Let A and B be separable nuclear, purely infinite simple C ∗ -algebras
satisfying the Universal Coefficient Theorem (UCT). Assume A and B are
both unital or both nonunital. If there exists a graded isomorphism
α : K∗ (A) → K∗ (B), which (in the unital case) satisfies α([1A ]) = [1B ],
then there exists an isomorphism φ : A → B
Jonathan Brown (KSU)
Purely infinite groupoids
(K)NIFAS
2 / 15
Graph C ∗ -algebras
Let E = (E 0 , E 1 , r , s) be a directed graph.
E is row-finite if r −1 (v ) < ∞
E has no sources if r −1 (v ) 6= ∅
∀v ∈ E 0 .
∀v ∈ E 0 .
α = α1 α2 · · · is a path if s(αi ) = r (αi+1 ): r (α) = r (α1 ).
E ∗ is the set of finite paths, E ∞ is the set of infinite paths.
For α ∈ E ∗
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|α| is the length α;
s(α) = s(α|α| );
α is a return path if r (α) = s(α).
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A return path α has an entrance if there exists i and
e ∈ r −1 (r (αi )) − {αi }.
Jonathan Brown (KSU)
Purely infinite groupoids
(K)NIFAS
3 / 15
Purely infinite graph algebras
KPRR 1997 construct a C ∗ -algebra C ∗ (E ) from E .
Theorem (KPR 1998)
C ∗ (E ) is purely infinite simple if and only if
1
There exists a return path in E ,
2
Every return path in E has an entrance, and
3
∀v ∈ E 0 , x = x1 x2 · · · ∈ E ∞ ∃α ∈ E ∗ , i ∈ N such that
r (α) = v , s(α) = r (xi ). (cofinal)
C ∗ (E ) simple iff items (2) and (3) above hold.
That (1)-(3) are sufficient is a result about groupoids from
Anantharaman-Delaroch 97.
To show they are necessary, the authors show that if E satisfies (2)
and (3) but not (1) then C ∗ (E ) is AF.
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This dichotomy is particular for graphs.
If we try to generalize to k-graphs, A-D 97 still gives a sufficient condition,
but no necessary condition is known.
Jonathan Brown (KSU)
Purely infinite groupoids
(K)NIFAS
4 / 15
Groupoids
A groupoid G can be defined as a small category in which every morphism
is invertible.
We identify the objects of the category with the identity morphisms
and denote both by G (0) ; G (0) is called the unit space of G .
Denote the range of a morphism γ by r (γ) and its source by s(γ);
r , s : G → G (0) .
We say a pair of morphisms (γ, η) is composable if and only if
s(γ) = r (η) and denote the composition by γη.
G acts on G (0) by γ · s(γ) = r (γ); for C ⊂ G (0) we denote
G · C := {r (γ) : s(γ) ∈ C }.
Jonathan Brown (KSU)
Purely infinite groupoids
(K)NIFAS
5 / 15
Topological groupoids
We say G is a topological groupoid if G has a topology in which
composition and inversion of morphisms are continuous.
This implies r and s are continuous.
We assume this topology is second countable locally compact and
Hausdorff.
G is étale if it is a topological groupoid such that r and s are local
homeomorphisms.
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G étale implies G (0) is open and closed in G .
we call open sets B such that r , s are homeomorphisms on B bisections.
G is topologically principal if {u ∈ G (0) : r −1 (u) ∩ s −1 (u) = {u}} is
dense in G (0) .
G is minimal if G · U ⊂ U and U open implies U ∈ {∅, G (0) }.
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Minimal implies: for x ∈ G (0) , U open there exists an open bisection B
such that x ∈ B · U.
Jonathan Brown (KSU)
Purely infinite groupoids
(K)NIFAS
6 / 15
The groupoid of a directed graph
Let be a row-finite directed graph with no sources. Define:
GE := {(x, k, y ) ∈ E ∞ × Z × E ∞ : ∃N with xi+k = yi for i ≥ N}.
If α = x1 x2 · · · xN+k , β = y1 y2 · · · yN and z = yN+1 yN+2 · · · , then
(x, k, y ) = (αz, |α| − |β|, βz).
GE is a groupoid with unit space E ∞ :
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(x, k, y ) is a morphism from y to x,
composition is given by (x, k, y )(y , `, z) = (x, k + `, z),
(x, k, y )−1 = (y , −k, x).
The sets
Z (α, β) := {(αx, |α| − |β|, βx) : r (x) = s(α) = s(β)}
for α, β ∈ E ∗
form a basis for a locally compact Hausdorff topology on GE :
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Z (α, β) is compact.
GE topologically principal iff every return path in E has an entrance
GE minimal iff E cofinal.
Jonathan Brown (KSU)
Purely infinite groupoids
(K)NIFAS
7 / 15
Groupoid C ∗ -algebras
For an étale groupoid G we define a convolution algebra structure on
Cc (G ) by
X
f ∗ g (γ) =
f (η)g (η −1 γ) f ∗ (γ) = f (γ −1 ).
r (η)=r (γ)
We define the regular representation of Cc (G ) at u ∈ G (0) on `2 (Gu) by
X
πu (f )δγ =
f (η)δγη
s(η)=r (γ)
Cr∗ (G ) is the completion of Cc (G ) in the norm
kf kr = sup kπu (f )k.
u∈G (0)
Cr∗ (G ) is simple if G is topologically principal and minimal (Renault 1980).
Jonathan Brown (KSU)
Purely infinite groupoids
(K)NIFAS
8 / 15
Conditional expectation
Let G be an étale groupoid.
G (0) is open and closed in G .
f ∈ Cc (G (0) ) the f ∈ Cc (G ) (extend by 0)
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This extends to an embedding of C0 (G (0) ) into Cr∗ (G ).
f ∈ Cc (G ) =⇒ f |G (0) ∈ C0 (G (0) ).
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f 7→ f |G (0) extends to a faithful conditional expectation
E : Cr∗ (G ) → C0 (G (0) ).
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That is:
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E (ba) = bE (a) ∀ b ∈ C0 (G (0) ), a ∈ Cr∗ (G )
E (a) ∈ C0 (G (0) )+ ∀ a ∈ Cr∗ (G )+
E (a∗ a) = 0 =⇒ a = 0.
Jonathan Brown (KSU)
Purely infinite groupoids
(K)NIFAS
9 / 15
Purely infinite simple étale groupoids
Given a ∈ Cr∗ (G )+ , E (a) ∈ C0 (G (0) )+ .
Take
h“ = ” max{E (a/kak) − 1/2, 0}
Then h - a by Lemma 2.2 of Kirchberg Rørdam 2000.
So if h is infinite then so is a.
Thus if every element in C0 (G (0) )+ is infinite then every element
Cr∗ (G )+ is.
Theorem (B., Clark, Sierakowski)
If G is a locally compact étale groupoid that is topologically principal and
minimal then Cr∗ (G ) is purely infinite if and only if every element of
C0 (G (0) )+ is infinite (in Cr∗ (G )).
Jonathan Brown (KSU)
Purely infinite groupoids
(K)NIFAS
10 / 15
Purely infinite ample groupoids
A groupoid is ample if it has a basis of compact open bisections.
Ample groupoids are étale and locally compact.
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So the previous theorem applies to them.
Ample groupoids have lots of projections:
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If U compact open bisection then χU is continuous.
For U, V compact open bisections χU ∗ χV = χUV so if U ⊂ G (0) we
have
χU ∈ C0 (G (0) ) and χ2U = χU .
If h ∈ C0 (G (0) )+ , then there exists U and s ∈ R>0 such that h|U ≥ s.
Therefore h ≥ sχU and so χU - h.
Thus if χU is infinite then h is infinite.
Theorem (B., Clark, Sierakowski)
If G is a locally compact ample groupoid that is topological principal and
minimal and B is a basis of compact open sets for G (0) then Cr∗ (G ) is
purely infinite if and only if χU is infinite for all U ∈ B.
Jonathan Brown (KSU)
Purely infinite groupoids
(K)NIFAS
11 / 15
More fun with ample groupoids
Theorem (B., Clark, Sierakowski)
If G is a locally compact ample groupoid that is topological principal and
minimal and B is a basis of compact open sets for G (0) then Cr∗ (G ) is
purely infinite if and only if χU is infinite for all U ∈ B.
Now G is minimal.
So if x ∈ G (0) and U ∈ B there exists a compact open bisection such
that x ∈ r (B) and s(B) ⊂ U.
Since χ∗B χB ≤ χU and χB χ∗B = χr (B) , we have χU infinite if χr (B)
infinite.
Thus
Corollary (B., Clark, Sierakowski)
C ∗ (G ) is purely infinite if and only if χV is infinite for all V ∈ N where N
is a neighborhood basis at some point x ∈ G (0) of compact open sets.
Jonathan Brown (KSU)
Purely infinite groupoids
(K)NIFAS
12 / 15
k-graphs
A k-graph Λ is a generalization of a graph where paths have “shape” given
by elements of Nk .
C ∗ (Λ) is the universal C ∗ -algebra generated by a Cuntz-Krieger
Λ-family {sλ : λ ∈ Λ};
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in particular sλ∗ sλ = ss(λ) .
Define
GΛ := {(x, n, y ) ∈ Λ∞ × Zk × Λ∞ : σ l (x) = σ m (y ), n = l − m}
where σ is the shift map and x and y are infinite paths.
The unit space of GΛ is Λ∞ the set of infinite paths.
For λ, µ ∈ Λ with s(λ) = s(µ), the sets
Z (λ, µ) := {(λz, d(λ) − d(µ), µz) : z ∈ Λ∞ (s(λ))}.
give a basis of a second countable locally compact Hausdorff topology of
compact open sets.
φ : C ∗ (Λ) → C ∗ (GΛ ) sλ 7→ χZ (λ,s(λ)) is an isomorphism.
Jonathan Brown (KSU)
Purely infinite groupoids
(K)NIFAS
13 / 15
Purely infinite k-graphs
For x ∈ Λ∞ the sets Z (λ, λ) where λ ranges over initial segments of
X is a neighborhood basis at x.
sλ sλ∗ is infinite iff φ(sλ sλ∗ ) = χZ (λ,λ) is infinite.
ss(λ) = sλ sλ∗ is equivalent to sλ sλ∗ .
So sλ sλ∗ is infinite if and only if ss(λ) is.
Theorem (B., Clark, Sierakowski)
Let Λ be a row-finite k-graph with no sources then C ∗ (Λ) is purely infinite
if and only if there exists x ∈ Λ∞ such that sv is infinite for all vertices v
in x.
Jonathan Brown (KSU)
Purely infinite groupoids
(K)NIFAS
14 / 15
GPOTS 2014
Kansas State University
May 27-31
Jonathan Brown (KSU)
Purely infinite groupoids
(K)NIFAS
15 / 15