Download On the simplicity of twisted k-graph C*-algebras

On the simplicity of twisted k-graph C∗ -algebras
Preliminary report of work in progress
Alex Kumjian1 ,
1
David Pask2 ,
Aidan Sims2
University of Nevada, Reno
2
University of Wollongong
GPOTS14, Kansas State University,
Manhattan, 27 May 2014
Kumjian, Pask, Sims
On the simplicity of twisted k-graph C∗ -algebras
Introduction
C∗ -algebras of higher-rank graphs (or k-graphs) were introduced in [KP00]
as generalizations of graph C∗ -algebra which model C∗ -algebras arising
from certain group actions on buildings discovered by Robertson and Steger.
If a k-graph Λ satisfies mild hypotheses, there is a path groupoid GΛ such
that C∗ (Λ) ∼
= C∗ (GΛ ).
Using this isomorphism it was shown that C∗ (Λ) is simple iff Λ is aperiodic
and cofinal.
Given a k-graph Λ and a T-valued 2-cocycle c, one may form the twisted
k-graph C∗ -algebra C∗ (Λ, c) (see [KPS]).
Examples include all noncommutative tori and crossed products of Cuntz
algebras by quasifree automorphisms.
Our goal is to characterize the simplicity of C∗ (Λ, c). Sufficient conditions
were given in (see [SWW]).
This talk is based on joint work with David Pask and Aidan Sims of the
University of Wollongong.
Kumjian, Pask, Sims
On the simplicity of twisted k-graph C∗ -algebras
k-graphs
Let k ∈ N := {0, 1, 2, . . . }.
Definition (see [KP00])
Let Λ be a countable small category and let d : Λ → Nk be a functor.
Then (Λ, d) is a k-graph if it satisfies the factorization property:
For every λ ∈ Λ and m, n ∈ Nk such that
d(λ) = m + n
there exist unique µ, ν ∈ Λ such that λ = µν, d(µ) = m and d(ν) = n.
Set Λn := d−1 (n) and identify Λ0 = Obj (Λ), the set of vertices.
An element λ ∈ Λei is called an edge.
We assume throughout that Λ is row-finite and source-free, that is
for all v ∈ Λ0 , n ∈ Nk , vΛn := r−1 (v) ∩ Λn is finite and nonempty.
Kumjian, Pask, Sims
On the simplicity of twisted k-graph C∗ -algebras
Remarks and Examples
Let Λ be a k-graph.
If k = 0, then d is trivial and Λ is just a set.
If k = 1, then Λ is the path category of a directed graph.
If k ≥ 2, think of Λ as generated by k graphs of different colors that
share the same set of vertices Λ0 .
The k-graph Tk := Nk may be regarded as the k-graph analog of a torus.
Let Ωk := {(m, n) ∈ Nk × Nk | m ≤ n}
be the k-graph with structure maps
..
.
..
.
.
..
.
..
Ω2
···
s(m, n) = n
r(m, n) = m
···
d(m, n) = n − m
···
(`, m)(m, n) = (`, n)
Kumjian, Pask, Sims
···
On the simplicity of twisted k-graph C∗ -algebras
The path groupoid
The infinite path space Λ∞ is the set of k-graph morphisms x : Ωk → Λ.
The shift map: for q ∈ Nk define σ q : Λ∞ → Λ∞ by
σ q (x)(m, n) = x(m + q, n + q)
for ∈ Ωk .
We define the path groupoid GΛ ⊂ Λ∞ × Z × Λ∞ by
GΛ := {(x, m − n, y) : σ m (x) = σ n (y) for some m, n ∈ Nk }.
The unit space is identified with Λ∞ via the map x 7→ (x, 0, x).
Λ is cofinal if for every v ∈ Λ0 and x ∈ Λ∞ , there are λ ∈ Λ and n ∈ Nk
such that s(λ) = x(n, n) and r(λ) = v. If Λ is cofinal, GΛ is minimal.
For v ∈ Λ0 the local periodicity group at v PΛ (v) is the set of all m − n ∈ Zk
such that m, n ∈ Nk and σ m (x) = σ n (x) for all x ∈ vΛ∞ .
Λ is aperiodic if PΛ (v) = 0 for all v ∈ Λ0 . If Λ is aperiodic, GΛ is
topologically principal, that is, points with trivial isotropy are dense.
Kumjian, Pask, Sims
On the simplicity of twisted k-graph C∗ -algebras
Categorical Cohomology
∗
The categorical cohomology, Hcat
(Λ, A), is the usual cocycle cohomology
for groupoids (see [R]) extended to small categories.
2
Consider the 2-cocycle c ∈ Zcat
(Λ, A), that is, a map c : Λ ∗ Λ → A such that
for any composable triple (λ1 , λ2 , λ3 ) we have
c(λ1 , λ2 ) + c(λ1 λ2 , λ3 ) = c(λ1 , λ2 λ3 ) + c(λ2 , λ3 )
and c is a 2-coboundary if there is b : Λ → A such that
c(λ1 , λ2 ) = b(λ1 ) − b(λ1 λ2 ) + b(λ2 ).
2
Hcat
(Λ, A) is the quotient group (2-cocycles modulo 2-coboundaries).
2
There is a homomorphism Hcat
(Λ, A) → H 2 (GΛ , A) induced by a map
c 7→ σc (see [KPS, §6]).
Kumjian, Pask, Sims
On the simplicity of twisted k-graph C∗ -algebras
The C∗ -algebra C∗ (Λ, c)
Definition (see [KPS, §5])
2
For c ∈ Zcat
(Λ, T) let C∗ (Λ, c) be the universal C*-algebra generated by the
set {tλ : λ ∈ Λ} satisfying:
1
{tv : v ∈ Λ0 } is a family of orthogonal projections.
2
For λ ∈ Λ, ts(λ) = tλ∗ tλ .
3
If s(λ) = r(µ), then tλ tµ = c(λ, µ)tλµ .
4
For v ∈ Λ0 , n ∈ Nk
tv =
X
tλ tλ∗ .
λ∈vΛn
Remark: If c and c0 are cohomologous, then C∗ (Λ, c) ∼
= C∗ (Λ, c0 ).
Theorem (see [KPS, §7])
2
Let c ∈ Zcat
(Λ, T) and let σc ∈ Z 2 (GΛ , T) be as above. Then
C∗ (Λ, c) ∼
= C∗ (GΛ , σc ).
Kumjian, Pask, Sims
On the simplicity of twisted k-graph C∗ -algebras
Structure of C∗ (Λ, c) when Λ is cofinal
If C∗ (Λ, c) is simple, then Λ is cofinal but not necessarily aperiodic.
Suppose Λ is cofinal. Then PΛ := PΛ (v) does not depend on v ∈ Λ0 and
there is a short exact sequence of étale groupoids
i
Λ∞ × PΛ →
− GΛ → HΛ
where HΛ is minimal and topologically principal (see [KPSS]).
The cohomology class of i∗x (σc ) in H 2 (PΛ , T) is independent of x.
Moreover, there is σ ∈ Z 2 (GΛ , T) and ω ∈ Z 2 (PΛ , T) such that [σ] = [σc ]
and i∗x (σ) = ω for all x ∈ Λ∞ . It follows that
C∗ (Λ∞ × PΛ , i∗ (σ)) ∼
= C0 (Λ∞ ) ⊗ C∗ (PΛ , ω).
Theorem
c
c
over HΛ such that BΛ
|Λ∞ ∼
There is a Fell bundle BΛ
= Λ∞ × C∗ (PΛ , ω) and
c
C∗ (Λ, c) ∼
).
= C∗ (HΛ ; BΛ
Kumjian, Pask, Sims
On the simplicity of twisted k-graph C∗ -algebras
Simplicity itself
Let Zω := {q ∈ PΛ | ω(p, q)ω(q, p) = 1} then by [OPT] we have
Prim C∗ (PΛ , ω) ∼
= Zc
ω.
By [IW, §2] there is an action of HΛ on
Prim C0 (Λ∞ ) ⊗ C∗ (PΛ , ω) = Λ∞ × Zc
ω.
The action is determined by a cocycle c̃ ∈ Z 1 (HΛ , Zc
ω ).
Theorem
Suppose Λ is cofinal. Then C∗ (Λ, c) is simple iff the action of HΛ on
Λ∞ × Zc
ω is minimal.
If Zω = 0, then C∗ (Λ, c) is simple. But this condition is not necessary as the
following example shows.
Kumjian, Pask, Sims
On the simplicity of twisted k-graph C∗ -algebras
An example
In this example Λ is cofinal and ω is trivial, but C∗ (Λ, c) is simple.
Let B2 be the 1-graph with 1 vertex and 2 edges and let T1 = N be the
1-graph with 1 vertex and 1 edge.
Let Λ := B2 × T1 and c((µ, m), (ν, n)) := znd(µ) where z := e2πiθ and θ is
irrational.
a1
Λ
1
a2
Since PΛ ∼
= Z, ω is a coboundary. Thus Zω = PΛ .
We have GΛ ∼
= HΛ × Z and Λ∞ ∼
= B∞
2 = {(x1 , x2 , . . . ) | xi = a1 or a2 }.
Moreover, c̃(x, n, y) = z−n and hence C∗ (Λ, c) is simple.
Kumjian, Pask, Sims
On the simplicity of twisted k-graph C∗ -algebras
References
[CKSS] Carlsen, Kang, Shotwell and Sims,
The primitive ideals of the Cuntz - Krieger algebra of a row-finite
higher-rank graph with no sources, 2014.
[IW] Ionescu and Williams,
Remarks on the ideal structure of Fell bundle C∗ -algebras, 2012.
[KP00] A. Kumjian and D. Pask,
Higher rank graph C∗ -algebras, 2000.
[KPS] A. Kumjian, D. Pask and A. Sims,
On twisted higher-rank graph C∗ -algebras, in press.
[OPT] Olesen, Pedersen and Takesaki,
Ergodic actions of compact abelian groups, 1980.
[R] J. Renault,
A groupoid approach to C*-algebras, 1980.
[SWW] Sims, Whitehead and Whittaker,
Twisted C∗ -algebras assoc. to finitely aligned higher-rank graphs, ppt.
Kumjian, Pask, Sims
On the simplicity of twisted k-graph C∗ -algebras