New York Journal of Mathematics
New York J. Math. 6 (2000) 1{20.
Higher Rank Graph C -Algebras
Alex Kumjian and David Pask
Abstract. Building on recent work of Robertson and Steger, we associate a
C {algebra to a combinatorial object which may be thought of as a higher
rank graph. This C {algebra is shown to be isomorphic to that of the associated path groupoid. Various results in this paper give sucient conditions on
the higher rank graph for the associated C {algebra to be: simple, purely innite and AF. Results concerning the structure of crossed products by certain
natural actions of discrete groups are obtained a technique for constructing
rank 2 graphs from \commuting" rank 1 graphs is given.
Contents
1. Higher rank graph C {algebras
2. The path groupoid
3. The gauge invariant uniqueness theorem
4. Aperiodicity and its consequences
5. Skew products and group actions
6. 2-graphs
References
3
6
9
12
14
17
19
In this paper we shall introduce the notion of a higher rank graph and associate
a C {algebra to it in such a way as to generalise the construction of the C {algebra
of a directed graph as studied in CK, KPRR, KPR] (amongst others). Graph C {
algebras include up to strong Morita equivalence Cuntz{Krieger algebras and AF
algebras. The motivation for the form of our generalisation comes from the recent
work of Robertson and Steger RS1, RS2, RS3]. In RS1] the authors study crossed
product C {algebras arising from certain group actions on A~2 -buildings and show
that they are generated by two families of partial isometries which satisfy certain
relations amongst which are Cuntz{Krieger type relations RS1, Equations (2), (5)]
as well as more intriguing commutation relations RS1, Equation (7)]. In RS2] they
give a more general framework for studying such algebras involving certain families
Received November 8, 1999.
Mathematics Subject Classication. Primary 46L05 Secondary 46L55.
Key words and phrases. Graphs as categories, Graph algebra, Path groupoid, C {algebra.
Research of the rst author partially supported by NSF grant DMS-9706982.
Research of the second author supported by University of Newcastle RMC project grant.
1
ISSN 1076-9803/00
2
Alex Kumjian and David Pask
of commuting 0 ; 1 matrices. In particular the associated C {algebras are simple,
purely innite and generated by a family of Cuntz{Krieger algebras associated to
these matrices. It is this framework which we seek to cast in graphical terms to
include a wider class of examples (including graph C {algebras).
What follows is a brief outline of the paper. In the rst section we introduce the
notion of a higher rank graph as a purely combinatorial object: a small category gifted with a degree map d : ! Nk (called shape in RS2]) playing the role of the
length function. No detailed knowledge of category theory is required to read this
paper. The associated C {algebra C () is dened as the universal C {algebra
generated by a family of partial isometries fs : 2 g satisfying relations similar
to those of KPR]. (Our standing assumption is that our higher rank graphs satisfy
conditions analogous to a directed graph being row{nite and having no sinks.)
We then describe some basic examples and indicate the relationship between our
formalism and that of RS2].
In the second section we introduce the path groupoid G associated to a higher
rank graph (cf. R, D, KPRR]). Once the innite path space 1 is formed (and
a few elementary facts are obtained) the construction is fairly routine. It follows
from the gauge-invariant uniqueness theorem (Theorem 3.4) that C () = C (G ).
By the universal property C () carries a canonical action of Tk dened by
(1)
t (s ) = td() s
called the gauge action. In the third section we prove the gauge{invariant uniqueness theorem, which is the key result for analysing C () (cf. BPRS, aHR], see
also CK, RS2] where similar techniques are used to prove simplicity). It gives
conditions under which a homomorphism with domain C () is faithful: roughly
speaking, if the homomorphism is equivariant for the gauge action and nonzero on
the generators then it is faithful. This theorem has a number of interesting consequences, amongst which are the isomorphism mentioned above and the fact that
the higher rank Cuntz{Krieger algebras of RS2] are isomorphic to C {algebras
associated to suitably chosen higher rank graphs.
In the fourth section we characterise, in terms of an aperiodicity condition on ,
the circumstances under which the groupoid G is essentially free. This aperiodicity
condition allows us to prove a second uniqueness theorem analogous to the original
theorem of CK]. In 4.8 and 4.9 we obtain conditions under which C () is simple
and purely innite respectively which are similar to those in KPR] but with the
aperiodicity condition replacing condition (L).
In the next section we show that, given a functor c : ! G where G is a discrete
group, then as in KP] one may construct a skew product G c which is also a
higher rank graph. If G is abelian then there is a natural action c : Gb ! Aut C ()
such that
(2)
c (s ) = h c()is moreover C () oc Gb = C (G c ). Comparing (1) and (2) we see that the
gauge action is of the form d and as a consequence we may show that the
crossed product of C () by the gauge action is isomorphic to C (Zk d ) this
C {algebra is then shown to be AF. By Takai duality C () is strongly Morita
equivalent to a crossed product of this AF algebra by the dual action of Zk . Hence
C () belongs to the bootstrap class N of C {algebras for which the UCT applies
Higher Rank Graph C -Algebras
3
(see RSc]) and is consequently nuclear. If a discrete group G acts freely on a kgraph , then the quotient object =G inherits the structure of a k{graph moreover
(as a generalisation of GT, Theorem 2.2.2]) there is a functor c : =G ! G such
that = G c (=G) in an equivariant way. This fact allows us to prove that
; C () o G = C (=G) K `2 (G)
where the action of G on C () is induced from that on . Finally in Section 6,
a technique for constructing a 2-graph from \commuting" 1-graphs A B with the
same vertex set is given. The construction depends on the choice of a certain
bijection between pairs of composable edges: : (a b) 7! (b0 a0 ) where a a0 2 A1
and b b0 2 B 1 the resulting 2-graph is denoted A B . It is not hard to show that
every 2-graph is of this form.
Throughout this paper we let N = f0 1 : : : g denote the monoid of natural
numbers under addition. For k 1, regard Nk as an abelian monoid under addition
with identity 0 (it will sometimes be useful to regard Nk as a small category with
one object) and canonical generators ei for i = 1 : : : k. We shall also regard Nk as
the positive cone of Zk under the usual coordinatewise partial order: thus m n
if and only if mi ni for all i, where m = (m1 : : : mk ), and n = (n1 : : : nk ).
(This makes Nk a lattice.)
We wish to thank Guyan Robertson and Tim Steger for providing us with an
early version of their paper RS2] the rst author would also like to thank them for
a number of stimulating conversations and the sta of the Mathematics Department
at Newcastle University for their hospitality during a recent visit.
1.
Higher rank graph C {algebras
In this section we rst introduce what we shall call a higher rank graph as a
purely combinatorial object. (We do not know whether this concept has been
studied before.) Our denition of a higher rank graph is modelled on the path
category of a directed graph (see H], Mu], MacL, xII.7] and Example 1.3). Thus
a higher rank graph will be dened to be a small category gifted with a degree
map (called shape in RS2]) satisfying a certain factorisation property. We then
introduce the associated C {algebra whose denition is modelled on that of the
C {algebra of a graph as well as the denition of RS2].
Denitions 1.1. A k-graph (rank k graph or higher rank graph) ( d) consists
of a countable small category (with range and source maps r and s respectively)
together with a functor d : ! Nk satisfying the factorisation property: for
every 2 and m n 2 Nk with d() = m + n, there are unique elements 2 such that = and d(
) = m, d( ) = n. For n 2 Nk we write n := d;1 (n).
A morphism between k-graphs (1 d1 ) and (2 d2 ) is a functor f : 1 ! 2
compatible with the degree maps.
Remarks 1.2. The factorisation property of 1.1 allows us to identify Obj(), the
objects of with 0. Suppose = in then by the the factorisation property = left cancellation follows similarly. We shall write the objects of as
u v w : : : and the morphisms as greek letters : : : . We shall frequently refer
to as a k-graph without mentioning d explicitly.
4
Alex Kumjian and David Pask
It might be interesting to replace Nk in Denition 1.1 above by a monoid or
perhaps the positive cone of an ordered abelian group.
Recall that 2 are composable if and only if r(
) = s(), and then 2 on the other hand two nite paths in a directed graph may be composed to
give the path provided that r() = s(
) so in 1.3 below we will need to switch
the range and source maps.
Example 1.3. Given a 1-graph , dene E 0 = 0 and E 1 = 1. If we dene
sE () = r() and rE () = s() then the quadruple (E 0 E 1 rE sE ) is a directed
graph in the sense of KPR, KP]. On the other hand, given a directed graph
E = (E 0 E 1 rE sE ), then E = n0 E n , the collection of nite paths, may be
viewed as small category with range and source maps given by s() = rE () and
r() = sE (). If we let d : E ! N be the length function (i.e., d() = n i
2 E n ) then (E d) is a 1-graph.
We shall associate a C {algebra to a k-graph in such a way that for k = 1 the
associated C {algebra is the same as that of the directed graph. We shall consider
other examples later.
Denitions 1.4. The k-graph is row nite if for each m 2 Nk and v 2 0
the set m(v) := f 2 m : r() = vg is nite. Similarly has no sources if
m(v) 6= for all v 2 0 and m 2 Nk .
Clearly if E is a directed graph then E is row nite (resp. has no sinks) if and
only if E is row nite (resp. has no sources). Throughout this paper we will assume
(unless otherwise stated) that any k-graph is row nite and has no sources, that
is
(3)
0 < #n (v) < 1 for every v 2 0 and n 2 Nk :
The Cuntz{Krieger relations CK, p.253] and the relations given in KPR, x1]
may be interpreted as providing a representation of a certain directed graph by
partial isometries and orthogonal projections. This view motivates the denition
of C ().
Denitions 1.5. Let be a k-graph (which satises the standing hypothesis (3)).
Then C () is dened to be the universal C {algebra generated by a family fs :
2 g of partial isometries satisfying:
(i) fsv : v 2 0 g is a family of mutually orthogonal projections,
(ii) s = s s for all 2 such that s() = r(
),
(iii) s s = ss() for all 2 ,
X
s s .
(iv) for all v 2 0 and n 2 Nk we have sv =
2n (v)
For 2 , dene p = s s (note that pv = sv for all v 2 0 ). A family of partial
isometries satisfying (i){(iv) above is called a {representation of .
Remarks 1.6. (i) If ft : 2 g is a {representation of then the map
s 7! t denes a {homomorphism from C () to C (ft : 2 g).
(ii) If E is the 1-graph associated to the directed graph E (see 1.3), then by
restricting a {representation to E 0 and E 1 one obtains a Cuntz{Krieger
family for E in the sense of KPR, x1]. Conversely every Cuntz{Krieger
family for E extends uniquely to a {representation of E .
Higher Rank Graph C -Algebras
5
(iii) In fact we only need the relation (iv) above to be satised for n = ei 2 Nk for
i = 1 : : : k, the relations for all n will then follow (cf. RS2, Lemma 3.2]).
Note that the denition of C () given in 1.5 may be extended to the case
where there are sources by only requiring that relation (iv) hold for n = ei
and then only if ei (v) 6= (cf. KPR, Equation (2)]).
(iv) For 2 if s() 6= s(
) then s s = 0. The converse follows from 2.11.
(v) Increasing nite
sums of pv 's form an approximate identity for C () (if 0
P
is nite then v20 pv is the unit for C ()). It follows from relations (i)
and (iv) above that for any n 2 Nk , fp : d() = ng forms a collection of
orthogonal projections (cf. RS2, 3.3]) likewise increasing nite sums of these
form an approximate identity for C () (see 2.5).
(vi) The above denition is not stated most eciently. Any family of operators
fs : 2 g satisfying the above conditions must consist of partial isometries.
The rst two axioms could also be replaced by:
(
s s = s if s() = r(
)
0
otherwise.
Examples 1.7. (i) If E is a directed graph, then by 1.6 (i) and (ii) we have
C (E ) (see 1.3).
C (E ) =
(ii) For k 1 let = k be the small category with objects Obj ( ) = Nk ,
and morphisms = f(m n) 2 Nk Nk : m ng the range and source
maps are given by r(m n) = m, s(m n) = n. Let d : ! Nk be dened by
d(m n) = n ;; m. It is then straightforward to show that k is a k-graph and
C ( k) = K `2 (Nk ) .
(iii) Let T = Tk be the semigroup Nk viewed as a small category, then if d : T !
Nk is the identity map then (T d) is a k-graph. It is not hard to show that
C (Tk ), where sei for 1 i k are the canonical unitary generators.
C (T ) =
(iv) Let fM1 : : : Mk g be square f0 1g matrices satisfying conditions (H0){(H3)
of RS2] and let A be the associated C -algebra. For m 2 Nk let Wm be the
collection of undecorated words in the nite alphabet A of shape m as dened
in RS2] then let
W =
m2Nk
Wm :
Together with range and source maps r() = o(), s() = t() and product
dened in RS2, Denition 0.1] W is a small category. If we dene d : W ! Nk
by d() = (), then one checks that d satises the factorisation property,
and then from the second part of (H2) we see that (W d) is an irreducible
k-graph in the sense that for all u v 2 W0 there is 2 W such that s() = u
and r() = v.
We claim that the map s 7! ss() for 2 W extends to a *-homomorphism
C (W ) ! A for which s s 7! s (since these generate A this will show
that the map is onto). It suces to verify that fss() : 2 W g constitutes a
{representation of W . Conditions (i) and (iii) are easy to check, (iv) follows
from RS2, 0.1c,3.2] with u = v 2 W 0 . We check condition (ii): if s() = r(
)
Alex Kumjian and David Pask
6
apply RS2, 3.2]
ss() ss() =
X
W d() (s())
s ss() = s ss() = ss()
where the sum simplies using RS2, 3.1, 3.3] . We shall show below that
C (W ) = A.
We may combine higher rank graphs using the following fact, whose proof is
straightforward.
Proposition 1.8. Let (1 d1) and (2 d2 ) be rank k1 , k2 graphs respectively, then
(1 2 d1 d2 ) is a rank k1 + k2 graph where 1 2 is the product category and
d1 d2 : 1 2 ! Nk1 +k2 is given by d1 d2 (1 2 ) = (d1 (1 ) d2 (2 )) 2 Nk1 Nk2
for 1 2 1 and 2 2 2 .
An example of this construction is discussed in RS2, Remark 3.11]. It is clear
that k+` = k ` where k ` > 0.
Denition 1.9. Let f : N` ! Nk be a monoid morphism, then if ( d) is a kgraph we may form the `-graph f () as follows: (the objects of f () may be
identied with those of and) f () = f( n) : d() = f (n)g with d( n) = n,
s( n) = s() and r( n) = r().
Examples 1.10. (i) Let be a k{graph and put ` = 1, then if we dene the
morphism fi (n) = nei for 1 i k, we obtain the coordinate graphs
i := fi () of (these are 1{graphs).
(ii) Suppose E is a directed graph and dene f : N2 ! N by (m1 m2 ) 7! m1 +m2
then the two coordinate graphs of f (E ) are isomorphic to E . We will show
below that C (f (E )) = C (E ) C (T).
(iii) Suppose E and F are directed graphs and dene f : N ! N2 by f (m) =
(m m) then f (E F ) = (E F ) where E F denotes the cartesian
product graph (see KP, Def. 2.1]).
Proposition 1.11. Let be a k-graph and f : N` ! Nk a monoid morphism,
then there is a {homomorphism f : C (f ()) ! C () such that s(n) 7! s moreover if f is surjective, then f is too.
Proof. By 1.6(i) it suces to show that this is a {representation of f (). Properties (i){(iii) are straightforward to verify and property (iv) follows by observing
that for xed n 2 N` and v 2 0 the map f ()n (v) ! f (n)(v) given by ( n) 7! is a bijection. If f is surjective, then it is clear that every generator s of C () is
in the range of f .
Later in 3.5 we will also show that f is injective if f is injective.
2.
The path groupoid
In this section we construct the path groupoid G associated to a higher rank
graph ( d) along the lines of KPRR, x2]. Because some of the details are not
quite the same as those in KPRR, x2] we feel it is useful to sketch the construction.
First we introduce the following analog of an innite path in a higher rank graph:
Higher Rank Graph C -Algebras
7
Denitions 2.1. Let be a k-graph, then
1 = fx : k ! : x is a k-graph morphismg
is the innite path space of . For v 2 0 let 1 (v) = fx 2 1 : x(0) = vg. For
each p 2 Nk dene p : 1 ! 1 by p (x)(m n) = x(m + p n + p) for x 2 1
and (m n) 2 . (Note that p+q = p q ).
By our standing assumption (3) one can show that for every v 2 0 we have
1
(v) =
6 . Our denition of 1 is related to the denition of W1 , the space
of innite words, given in the proof of RS2, Lemma 3.8]. If E is the 1-graph
associated to the directed graph E then (E )1 may be identied with E 1 .
Remarks 2.2. By the factorisation property the values of x(0 m) for m 2 Nk
completely determine x 2 1 . To see this, suppose that x(0 m) is given for all
m 2 Nk then for (m n) 2 , x(m n) is the unique element 2 such that
x(0 n) = x(0 m).
More generally, let fnj : j 0g be an increasing conal sequence in Nk with
n0 = 0 (for example one could take nj = jp where p = (1 : : : 1) 2 Nk ) then
x 2 1 is completely determined by the values of x(0 nj ). Moreover, given a
sequence fj : j 1g in such that s(j ) = r(j+1 ) and d(j ) = nj ; nj;1 there
is a unique x 2 1 such that x(nj;1 nj ) = j . For (m n) 2 we dene x(m n) by
the factorisation property as follows: let j be the smallest index such that n nj .
Then x(m n) is the unique element of degree n ; m such that 1 j = x(m n)
where d(
) = m and d( ) = nj ; n. It is straightforward to show that x has the
desired properties.
We now establish a factorisation property for 1 which is an easy consequence
of the above remarks.
Proposition 2.3. Let be a rank k graph. For all 2 and x 2 1 with
x(0) = s(), there is a unique y 2 1 such that x = d() y and = y(0 d()) we
write y = x. Note that for every x 2 1 and p 2 Nk we have x = x(0 p)p x.
Proof. Fix 2 and x 2 1 with x(0) = s(). The sequence fnj : j 0g
dened by n0 = 0 and nj = (j ; 1)p + d() for j 1 is conal. Set 1 = and
j = x((j ; 2)p (j ; 1)p) for j 2 and let y 2 1 be dened by the method given
in 2.2. Then y has the desired properties.
Next we construct a basis of compact open sets for the topology on 1 indexed
by .
Denitions 2.4. Let be a rank k graph. For 2 dene
Z () = fx 2 1 : s() = x(0)g = fx : x(0 d()) = g:
Remarks 2.5. Note that Z (v) = 1(v) for all v 2 0. For xed n 2 Nk the sets
fZ () : d() = ng form a partition of 1 (see 1.6(v)) moreover for every 2 we have
(4)
Z () =
Z (
):
d()=n
r()=s()
We endow 1 with the topology generated by the collection fZ () : 2 g.
Note that the map given by x 7! x induces a homeomorphism between Z () and
8
Alex Kumjian and David Pask
Z (s()) for all 2 . Hence, for every p 2 Nk the map p : 1 ! 1 is a local
homeomorphism.
Lemma 2.6. For each 2 , Z () is compact.
Proof. By 2.5 it suces to show that Z (v) is compact for all v 2 0. Fix v 2 0
and let fxn gn1 be a sequence in Z (v). For every m, xn (0 m) may take only nitely
many values (by (3)). Hence there is a 2 m such that xn (0 m) = for innitely
many n. We may therefore inductively construct a sequence fj : j 1g in p
such that s(j ) = r(j+1 ) and xn (0 jp) = 1 j for innitely many n (recall
p = (1 : : : 1) 2 Nk ). Choose a subsequence fxnj g such that xnj (0 jp) = 1 j .
Since fjpg is conal, there is a unique y 2 1 (v) such that y((j ; 1)p jp) = j for
j 1 then xnj ! y and hence Z (v) is compact.
Note that 1 is compact if and only if 0 is nite.
Denition 2.7. If is k-graph then let
G = f(x n y) 2 1 Zk 1 : ` x = m y n = ` ; mg:
Dene range and source maps r s : G ! 1 by r(x n y) = x, s(x n y) = y. For
(x n y), (y ` z ) 2 G set (x n y)(y ` z ) = (x n+` z ), and (x n y);1 = (y ;n x)
G is called the path groupoid of (cf. R, D, KPRR]).
One may check that G is a groupoid with 1 = G0 under the identication
x 7! (x 0 x). For , 2 such that s() = s(
) dene
Z ( ) = f(z d() ; d(
) z ) : z 2 1 (s())g:
We collect certain standard facts about G in the following result.
Proposition 2.8. Let be a k{graph. The sets fZ ( ) : 2 s() = s(
)g
form a basis for a locally compact Hausdor topology on G . With this topology
G is a second countable, r{discrete locally compact groupoid in which each Z ( )
is a compact open bisection. The topology on 1 agrees with the relative topology
under the identication of 1 with the subset G0 of G .
Proof. One may check that the sets Z ( ) form a basis for a topology on G .
To see that multiplication is continuous, suppose that (x n y)(y ` z ) = (x n +
` z ) 2 Z ( ). Since (x n y) (y ` z ) are composable in G there are 2 and
t 2 1 such that x = t, y = t and z = t. Hence (x k y) 2 Z ( ) and
(y ` z ) 2 Z ( ) and the product maps the open set G2 \ (Z ( ) Z ( ))
into Z ( ). The remaining parts of the proof are similar to those given in KPRR,
Proposition 2.6].
Note that Z ( ) = Z (s()), via the map (z d() ; d(
) z ) 7! z . Again
we note that in the case k = 1 we have = E for some directed graph E and
the groupoid GE = GE , the graph groupoid of E which is described in detail in
KPRR, x2].
Proposition 2.9. Let be a k-graph and let f : N` ! Nk be a morphism. The
map x 7! f (x) given by f (x)(m n) = (x(f (m) f (n)) n ; m) denes a continuous
surjective map f : 1 ! f ()1 . Moreover, if the image of f is conal (equivalently f (p) is strictly positive in the sense that all of its coordinates are nonzero )
then f is a homeomorphism.
Higher Rank Graph C -Algebras
9
Proof. Given x 2 f ()1 choose a sequence fmig such that nj = Pji=1 mi
is conal in N` . Set n0 = 0 and let j 2 f (mj ) be dened by the condition that x(nj;1 nj ) = (j mj ). We must show that there is an x0 2 1
such that x0 (f (nj;1 ) f (nj )) = j . It suces to show that the the intersection
\j Z (1 j ) =6 . But this follows by the nite intersection property. One checks
that x = f (x0 ). Furthermore the inverse image of Z ( n) is Z () and hence f is
continuous.
Now suppose that the image of f is conal, then the procedure dened above
gives a continuous inverse for f . Given x 2 f ()1 , then since f (nj ) is conal,
the intersection \j Z (1 j ) contains a single point x0 . Note that x0 depends on
x continuously.
For higher rank graphs of the form f () with f surjective (see 1.9), the associated groupoid Gf () decomposes as a direct product as follows.
Proposition 2.10. Let be a k-graph and let f : N` ! Nk be a surjective morphism. Then
Gf () = G Z`;k :
Proof. Since f is surjective, the map f : 1 ! f ()1 is a homeomorphism (see
2.9). The map f extends to a surjective morphism f : Z` ! Zk . Let j : Zk ! Z`
be a section for f and let i : Z`;k ! Z` be an identication of Z`;k with ker f .
Then we get a groupoid isomorphism by the map
((x n y) m) 7! (f x i(m) + j (n) f y)
where ((x n y) m) 2 G Z`;k .
Finally, as in RS2, Lemma 3.8] we demonstrate that there is a nontrivial
{representation of ( d).
Proposition 2.11. Let ( d) be a k-graph. Then there exists a representation
fS : 2 g of on a Hilbert space with all partial isometries S nonzero.
Proof. Let H = `2(1), then for 2 dene S 2 B(H) by
e if s() = y(0)
S ey = 0y otherwise,
where fey : y 2 1 g is the canonical basis for H. Notice that S is nonzero since
1 (s()) 6= one then checks that the family fS : 2 g satises conditions
1.5(i){(iv).
3.
The gauge invariant uniqueness theorem
By the universal property of C () there is a canonical action of the k-torus Tk ,
called the gauge action: : Tk ! Aut C () dened for t = (t1 : : : tk ) 2 Tk
and s 2 C () by
(5)
t (s ) = td() s
mk
k
1
where tm = tm
1 tk for m = (m1 : : : mk ) 2 N . It is straightforward to show
that is strongly continuous. As in CK, Lemma 2.2] and RS2, Lemma 3.6] we
shall need the following.
Alex Kumjian and David Pask
10
Lemma 3.1. Let be a k-graph. Then for 2 and q 2 Nk with d(),
d(
) q we have
s s =
(6)
X
=
d()=q
s s
:
Hence every nonzero word in s s may be written as a nite sum of partial isometries of the form s s
where s() = s( ) their linear span then forms a dense
{subalgebra of C ().
Proof. Applying 1.5(iv) to s() with n = q ; d(), to s(
) with n = q ; d(
) and
using 1.5 (ii) we get
s s =
(7)
=
0
1 0
1
X
X
ps() s s ps() = @
s s A s s @
s
s
A
q
;
d
q
;
d
0
1 0 (s()) 1 (s())
@ X ss A @ X s
s
A :
( )
q;d() (s())
( )
q;d() (s())
By 1.6(iv) if d() = d(
) but 6= , then the range projections p , p
are orthogonal and hence one has s s
= 0. If = then s s
= pv
where v = s() and so s s s
s
= s pv s
= s s
formula (6) then follows from
formula (7). The rest of the proof is now routine.
Following RS2, x4]: for m 2 Nk let Fm denote the C {subalgebra of C ()
generated by the elements s s for 2 m where s() = s(
), and for v 2 0
denote Fm (v) the C {subalgebra generated by s s where s() = v.
Lemma 3.2. For m 2 Nk , v 2 0 there exist isomorphisms
;
Fm(v) = K `2 (f 2 m : s() = vg)
L 0 F (v). Moreover, the C {algebras F m 2 Nk , form a
and Fm =
m
v2 m
directed system under inclusion, and F = Fm is an AF C {algebra.
Proof. Fix v 2 0 and let , , , 2 m be such that s() = s(
) and s() =
s( ), then by 1.6(v) we have
;s s ;s s = s s (8)
so that the map which sends s s 2 Fm (v) to the matrix unit
;
ev 2 K `2 (f 2 m : s() = vg)
for all , 2 m with s() = s(
) = v extends to an isomorphism. The second
isomorphism also follows from (8) (since s(
) 6= s() implies 6= ). We claim
that Fm is contained in Fn whenever m n. To see this we apply 1.5(iv) to give
X
X
(9)
s s = s ps() s =
s s s s =
s s
` (s())
` (s())
where ` = n ; m. Hence the C {algebras Fm m 2 Nk , form a directed system as
required.
Higher Rank Graph C -Algebras
11
Note that F may also be expressed as the closure of 1
j =1 Fjp where p =
(1 : : : 1) 2 Nk .
Clearly for t 2 Tk the gauge automorphism t dened in (5) xes those elements
s s 2 C () with d() = d(
) (since t (s s ) = td();d()s s ) and hence F
is contained in the xed point algebra C () . Consider the linear map on C ()
dened by
"(x) =
Z
t (x) dt
Tk
where dt denotes normalised Haar measure on Tk and note that "(x) 2 C ()
for all x 2 C (). As the proof of the following result is now standard, we omit it
(see CK, Proposition 2.11], RS2, Lemma 3.3], BPRS, Lemma 2.2]).
Lemma 3.3. Let ", F be as described above.
(i) The map " is a faithful conditional expectation from C () onto C () .
(ii) F = C () .
Hence the xed point algebra C () is an AF algebra. This fact is key to the
proof of the gauge{invariant uniqueness theorem for C () (see BPRS, Theorem
2.1], aHR, Theorem 2.3], see also CK, RS2] where a similar technique is used in
the proof of simplicity).
Theorem 3.4. Let B be a C {algebra, : C () ! B be a homomorphism and
let : Tk ! Aut (B ) be an action such that t = t for all t 2 Tk . Then is faithful if and only if (pv ) 6= 0 for all v 2 0 .
Proof. If (pv ) = 0 for some v 2 0 then clearly is not faithful. Conversely,
suppose that is equivariant andS that (pv ) 6= 0 for all v 2 0 . We rst show
that is faithful on C () = j0 Fjp . For any ideal I in C () , we have
S
I = j0 (I \ Fjp ) (see B, Lemma 3.1], ALNR, Lemma 1.3]). Thus it is enough
to prove that is faithful on each Fn . But by 3.2 it suces to show that it is
faithful on Fn (v), for all v 2 0 . Fix v 2 0 and 2 n with s() = s(
) = v
we need only show that (s s ) 6= 0. Since (pv ) 6= 0 we have
0 6= (p2v ) = (s s s s ) = (s )(s s )(s ):
Hence (s s ) 6= 0 and is faithful on C () . Let a 2 C () be a nonzero
positive element then since " is faithful "(a) 6= 0 and as is faithful on C ( )
we have
Z
Z
0 6= ("(a)) = (a) dt =
t ((a)) dt
t
Tk
Tk
hence (a) 6= 0 and is faithful on C () as required.
Corollary 3.5.
(i) Let ( d) be a k-graph and let G be its associated groupoid. Then there is an
isomorphism C () = C (G ) such that s 7! 1Z (s()) for 2 . Moreover,
the canonical map C (G ) ! Cr (G ) is an isomorphism.
(ii) Let fM1 : : : Mk g be a collection of matrices satisfying (H0){(H3) of RS2]
and W the k-graph dened in 1.7(iv). Then C (W ) = A, via the map s 7!
ss() for 2 W .
Alex Kumjian and David Pask
12
(iii) If is a k-graph and f : N` ! Nk is injective, then the -homomorphism f :
C (f ()) ! C () (see 1.11) is injective. In particular the C {algebras of
the coordinate graphs i for 1 i k form a generating family of subalgebras
of C (). Moreover, if f is surjective then C (f ()) = C () C (T`;k ).
(iv) Let (i di ) be ki -graphs for i = 1 2, then C (1 2 ) = C (1 ) C (2 )
via the map s(1 2 ) 7! s1 s2 for (1 2 ) 2 1 2 .
Proof. For (i) we note that s 7! 1Z(s()) for 2 is a -representation of hence there is a -homomorphism : C () ! C (G ) such that (s ) = 1Z (s())
for 2 (see 1.6(i)). Let denote the Tk -action on C (G ) induced by the Zk valued 1{cocycle dened on G by (x k y) 7! k (see R, II.5.1]) one checks that
t = t for all t 2 Tk . Clearly for v 2 0 we have 1Z (vv) 6= 0, since 1 (v) 6= and is injective. Surjectivity follows from the fact that (s s ) = 1Z () together
with the observation that C (G ) = spanf1Z () g. The same argument shows that
Cr (G ) = C () and so Cr (G ) = C (G )1 .
For (ii) we note that there is a surjective -homomorphism : C (W ) ! A
such that (s ) = ss() for 2 W (see 1.7(iv)) which is clearly equivariant for
the respective Tk {actions. Moreover by RS2, Lemma 2.9] we have svv 6= 0 for all
v 2 W0 = A and so the result follows.
For (iii) note that the injection f : N` ! Nk extends naturally to a homomorphism f : Z` ! Zk which in turn induces a map f^ : Tk ! T` characterised by
f^(t)p = tf (p) for p 2 N`. Let B be the xed point algebra of the gauge action of Tk
on C () restricted to the kernel of f^. The gauge action restricted to B descends
to an action of T` = Tk =Ker f^ on B which we denote . Observe that for t 2 Tk
and ( n) 2 f () we have
t (f (s(n) )) = tf (n) s = f^(t)n s hence Im f B (if t 2 Ker f^, then f^(t)n = 1). By the same formula we see that
f = f and the result now follows by 3.4. The last assertion follows from
part (i) together with the fact that Gf () = G Z`;k (see 2.10).
For (iv), dene a map : C (1 2 ) ! C (1 ) C (2 ) given by s(1 2 ) 7!
s1 s2 this is surjective as these elements generate C (1 ) C (2 ). We note
that C (1 ) C (2 ) carries a Tk1 +k2 action dened for (t1 t2 ) 2 Tk1 +k2 and
(0 1 ) 2 1 2 by (t1 t2 ) (s1 s2 ) = t1 s1 t2 s2 . Injectivity then follows
by 3.4, since is equivariant and for (v w) 2 (1 2 )0 we have pv pw 6= 0.
Henceforth we shall tacitly identify C () with C (G ).
Remark 3.6. Let be a k-graph and suppose that f : N` ! Nk is an injective
morphism for which H , the image of f , is conal. Then f induces an isomorphism
of C (f ()) with its range, the xed point algebra of the restriction of the gauge
action to H ? .
Aperiodicity and its consequences
4.
The aperiodicity condition we study in this section is an analog of condition (L)
used in KPR]. We rst dene what it means for an innite path to be periodic or
aperiodic.
1
This can be also deduced from the amenability of G (see 5.5).
Higher Rank Graph C -Algebras
13
Denitions 4.1. For x 2 1 and p 2 Zk we say that p is a period of x if for
every (m n) 2 with m + p 0 we have x(m + p n + p) = x(m n). We say that
x is periodic if it has a nonzero period. We say that x is eventually periodic if
n x is periodic for some n 2 Nk , otherwise x is said to be aperiodic.
Remarks 4.2. For x 2 1 and p 2 Zk , p is a period of x if and only if mx = n x
for all m n 2 Nk such that p = m ; n. Similarly x is eventually periodic, with
eventual period p =
6 0 if and only if m x = n x for some m n 2 Nk such that
p = m ; n.
Denition 4.3. The k-graph is said to satisfy the aperiodicity condition (A)
if for every v 2 0 there is an aperiodic path x 2 1 (v).
Remark 4.4. Let E be a directed graph which is row nite and has no sinks.
Then the associated 1-graph E satises the aperiodicity condition if and only if
every loop in E has an exit (i.e., satises condition (L) of KPR]). However, if we
consider the 2-graph f (E ) where f : N2 ! N is given by f (m1 m2 ) = m1 + m2
then p = (1 ;1) is a period for every point in f (E )1 (even if E has no loops).
Proposition 4.5. The groupoid G is essentially free (i.e., the points with trivial
isotropy are dense in G0 ) if and only if satises the aperiodicity condition.
Proof. Observe that if x 2 1 is aperiodic then mx = n x implies that m = n
and hence x 2 1 = G0 has trivial isotropy, and conversely. Hence G is essentially
free if and only if aperiodic points are dense in 1 . If aperiodic points are dense
in 1 then clearly satises the aperiodicity condition, for Z (v) = 1 (v) must
then contain aperiodic points for every v 2 0 . Conversely, suppose that satises
the aperiodicity condition, then for every 2 there is x 2 1 (s()) which is
aperiodic. Then x 2 Z () is aperiodic. Hence the aperiodic points are dense in
1 .
The isotropy group of an element x 2 1 is equal to the subgroup of its eventual
periods (including 0).
Theorem 4.6. Let : C () ! B be a {homomorphism and suppose that satises the aperiodicity condition. Then is faithful if and only if (pv ) 6= 0 for
all v 2 0 .
Proof. If (pv ) = 0 for some v 2 0 then clearly is not faithful. Conversely,
suppose (pv ) 6= 0 for all v 2 0 then by 3.5(i) we have C () = Cr (G ) and hence
from KPR, Corollary 3.6] it suces to show that is faithful on C0 (G0 ). If the
kernel of the restriction of to C0 (G0 ) is nonzero, it must contain the characteristic
function 1Z () for some 2 . It follows that (s s ) = 0 and hence (s ) = 0 in
which case (ps() ) = (s s ) = 0, a contradiction.
Denition 4.7. We say that is conal if for every x 2 1 and v 2 0 there is
2 and n 2 Nk such that s() = x(n) and r() = v.
Proposition 4.8. Suppose satises the aperiodicity condition, then C () is
simple if and only if is conal.
Proof. By 3.5(i) C () = Cr (G ) since G is essentially free, C () is simple if
and only if G is minimal. Suppose that is conal and x x 2 1 and 2 14
Alex Kumjian and David Pask
then by conality there is a 2 and n 2 Nk so that s(
) = x(n) and r(
) = s().
Then y = n x 2 Z () and y is in the same orbit as x hence all orbits are dense
and G is minimal.
Conversely, suppose that G is minimal and that x 2 1 and v 2 0 . Then
there is y 2 Z (v) such that x y are in the same orbit. Hence there exist m n 2 Nk
such that n x = m y then it is easy to check that = y(0 m) and n have the
desired properties.
Notice that second hypothesis used in the following corollary is the analog of the
condition that every vertex connects to a loop and it is equivalent to requiring that
for every v 2 0 , there is an eventually periodic x 2 1 (v) with positive eventual
period (i.e., the eventual period lies in Nk nf0g). The proof follows the same lines
as KPR, Theorem 3.9].
Proposition 4.9. Let satisfy the aperiodicity condition. Suppose that for every
v 2 0 there are , 2 with d(
) 6= 0 such that r() = v and s() = r(
) = s(
).
Then C () is purely innite in the sense that every hereditary subalgebra contains
an innite projection.
Proof. Arguing as in KPR, Lemma 3.8] one shows that G is locally contracting.
The aperiodicity condition guarantees that G is essentially free, hence by A-D,
Proposition 2.4] (see also LS]) we have C () = Cr (G ) is purely innite.
5.
Skew products and group actions
Let G be a discrete group, a k-graph and c : ! G a functor. We introduce
an analog of the skew product graph considered in KP, x2] (see also GT]) the
resulting object, which we denote G c , is also a k-graph. As in KP] if G is
abelian the associated C {algebra is isomorphic to a crossed product of C () by
the natural action of Gb induced by c (more generally it is a crossed product by
a coaction | see Ma, KQR]). As a corollary we show that the crossed product
of C () by the gauge action, C () o Tk , is isomorphic to C (Zk d ), the
C {algebra of the skew-product k-graph arising from the degree map. It will then
follow that C () o Tk is AF and that G is amenable.
Denition 5.1. Let G be a discrete group, ( d) a k-graph. Given c : ! G a
functor then dene the skew product G c as follows: the objects are identied
with G0 and the morphisms are identied with G with the following structure
maps
s(g ) = (gc() s()) and r(g ) = (g r()):
If s() = r(
) then (g ) and (gc() ) are composable in G c and
(g )(gc() ) = (g ):
The degree map is given by d(g ) = d().
One must check that G c is a k-graph. If k = 1 then any function c : E 1 ! G
extends to a unique functor c : E ! G (as in KP, x2]). The skew product graph
E (c) of KP] is related to our skew product in a simple way: G c E = E (c) . A
key example of this construction arises by regarding the degree map d as a functor
with values in Zk .
Higher Rank Graph C -Algebras
15
The functor c induces a cocycle c~ : G ! G as follows: given (x ` ; m y) 2 G
so that ` x = m y then set
c~(x ` ; m y) = c(x(0 `))c(y(0 m));1 :
As in KP] one checks that this is well-dened and that c~ is a (continuous) cocycle
regarding the degree map d as a functor with values in Zk , we have d~(x n y) = n for
(x n y) 2 G . In the following we show that the skew product groupoid obtained
from c~ (as dened in R]) is the same as the path groupoid of the skew product
(cf. KP, Theorem 2.4]).
Theorem 5.2. Let G be a discrete group, a k-graph and c : ! G a functor.
Then GGc = G (~c) where c~ : G ! G is dened as above.
Proof. We rst identify G 1 with (G c )1 as follows: for (g x) 2 G 1
dene (g x) : ! G c by
(g x)(m n) = (gc(x(0 m)) x(m n))
it is straightforward to check that this denes a degree{preserving functor and thus
an element of (G c )1 . Under this identication n (g x) = (gc(x(0 n)) n x) for
all n 2 Nk , (g x) 2 (G c )1 . As in the proof of KP, Theorem 2.4] dene a map
: G (~c) ! GGc as follows: for x y 2 1 with ` x = m y set (x ` ; m y] g) =
(x0 ` ; m y0) where x0 = (g x) and y0 = (gc~(x ` ; m y) y). Note that
m y0 = m (gc~(x ` ; m y) y) = m (gc(x(0 `))c(y(0 m));1 y)
= (gc(x(0 `)) m y) = (gc(x(0 `)) ` x) = ` (g x)
= ` x0 and hence (x0 `;m y0) 2 GGc . The rest of the proof proceeds as in KP, Theorem
2.4] mutatis mutandis.
Corollary 5.3. Let G be a discrete abelian group, a k-graph and c : ! G a
functor. There is an action c : Gb ! Aut C () such that for 2 Gb and 2 c (s ) = h c()is :
Moreover C () oc Gb = C (G c ). In particular the gauge action is of the form,
= d , and so C () o Tk = C (Zk d ).
Proof. Since C () is dened to be the universal C {algebra generated by the
s 's subject to the relations (1.5) and c preserves these relations it is clear that it
denes an action of Gb on C (). The rest of the proof follows in the same manner
as that of KP, Corollary 2.5] (see R, II.5.7]).
In order to show that C () o Tk is AF, we need the following lemma.
Lemma 5.4. Let be a k-graph and suppose there is a map b : 0 ! Zk such that
d() = b(s()) ; b(r()) for all 2 , then C () is AF.
Proof. For every n 2 Zk let An be the closed linear span of elements of the form
s s with b(s()) = n. Fix , 2 with b(s()) = b(s(
)) = n. We claim that
s s = 0 if 6= . If s s 6= 0 then by 3.1 there are , 2 with s() = r() and
s(
) = r( ) such that = but then we have
d() + n = d() + b(s()) = b(s()) = b(s(
)) = d( ) + b(s(
)) = d( ) + n:
16
Alex Kumjian and David Pask
Thus d() = d( ) and hence by the factorisation property = . Consequently
= by cancellation and the claim is established. It follows that for each v with
b(v) = n the elements s s with s() = s(
) = v form a system of matrix units
and two systems associated to distinct v's are orthogonal (see 3.2). Hence we have
M ; 2 ;1 An K ` (s ( v ) :
=
b(v)=n
By an argument similar to that in the proof of Lemma 3.2, if n m then An Am
(see equation (9)) our conclusion now follows.
Note that An in the above proof is the C {algebra of a subgroupoid of G which
is isomorphic to the disjoint union
G
Rv 1 (v)
b(v)=n
where Rv is the transitive principal groupoid on s;1 (v). Since G is the increasing
union of these elementary groupoids, it is an AF-groupoid and hence amenable (see
R, III.1.1]). The existence of such a function b : 0 ! Zk is not necessary for
C () to be AF since there are 1{graphs with no loops which do not have this
property (see KPR, Theorem 2.4]).
Theorem 5.5. Let be a k-graph, then C () o Tk is AF and the groupoid G is
amenable. Moreover, C () falls in the bootstrap class N of RSc] and is therefore
nuclear. Hence, if C () is simple and purely innite (see Proposition 4.9), then
it may be classied by its K -theory.
Proof. Observe that the map b : (Zk d )0 ! Zk given by b(n v) = n satises
b(s(n )) ; b(r(n )) = b(n + d() ) ; b(n r()) = n + d() ; n = d(n )
The rst part of the result then follows from 5.4 and 5.3. To show that G is
amenable we rst observe that G (d~) = GZk d is amenable. Since Zk is amenable,
we may apply R, Proposition II.3.8] to deduce that G is amenable. Since C ()
is strongly Morita equivalent to the crossed product of an AF algebra by a Zk {
action, it falls in the bootstrap class N of RSc]. The nal assertion follows from
the Kirchberg-Phillips classication theorem (see K, P]).
We now consider free actions of groups on k-graphs (cf. KP, x3]). Let be a kgraph and G a countable group, then G acts on if there is a group homomorphism
G ! Aut (automorphisms are compatible with all structure maps, including the
degree): write (g ) 7! g. The action of G on is said to be free if it is free on
0 . By the universality of C () an action of G on induces an action on C ()
such that g s = sg .
Given a free action of a group G on a k-graph one forms the quotient =G
by the equivalence relation if = g for some g 2 G. One checks that all
structure maps are compatible with and so =G is also a k-graph.
Remark 5.6. Let G be a countable group and c : ! G a functor, then G acts
freely on G c by g(h ) = (gh ) furthermore (G c )=G = .
Suppose now that G acts freely on with quotient =G we claim that is
isomorphic, in an equivariant way, to a skew product of =G for some suitably
chosen c (see GT, Theorem 2.2.2]). Let q denote the quotient map. For every
Higher Rank Graph C -Algebras
17
v 2 (=G)0 choose v0 2 0 with q(v0 ) = v and for every 2 =G let 0 denote the
unique element in such that q(0 ) = and r(0 ) = r()0 . Now let c : =G ! G
be dened by the formula
s(0 ) = c()s()0 :
We claim that c(
) = c()c(
) for all , 2 with s() = r(
). Note that
r(c()
0 ) = c()r(
0 ) = c()r(
)0 = c()s()0 = s(0 )
hence, we have (
)0 = 0 (c()
0 ) (since the image of both sides agree under q and
r). Thus
c(
)s(
)0 = c(
)s(
)0 = s(
)0 ] = s(c()
0 ) = c()s(
0 ) = c()c(
)s(
)0
which establishes the desired identity (since G acts freely on ). The map (g ) 7!
g0 denes an equivariant isomorphism between G c (=G) and as required.
The following is a generalization of KPR, 3.9, 3.10] and is proved similarly.
Theorem 5.7. Let be a k-graph and suppose that the countable group G acts
freely on , then
; C () o
G = C (=G) K `2 (G) :
Equivalently, if c : 0 ! G is a functor, then
; C (G c 0 ) o
G = C (0 ) K `2 (G)
where , the action of G on C (G c 0 ), is induced by the natural action on G c 0 .
If G is abelian this action is dual to c under the identication of 5.3.
Proof. The rst statement follows from the second with 0 = =G indeed, by 5.6
there is a functor c : =G ! G such that = G c (=G) in an equivariant way.
The second statement follows from applying KP, Proposition 3.7] to the natural
G-action on GGc 0 = G0 (~c). The nal statement follows from the identications
C () oc Gb = C (G c ) = C (G (~c))
and R, II.2.7].
6.
2-graphs
Given a k-graph one obtains for each n 2 Nk a matrix
Mn(u v) = #f 2 n : r() = u s() = vg:
By our standing assumption the entries are all nite and there are no zero rows.
Note that for any m n 2 Nk we have Mm+n = Mm Mn (by the factorisation
property) consequently, the matrices Mm and Mn commute for all m n 2 Nk . If
W is the k-graph associated to the commuting matrices fM1 : : : Mk g satisfying
conditions (H0){(H3) of RS2] which was considered in Example 1.7(iv), then one
checks that MWei = Mit . Further, if = E is a 1-graph derived from the directed
graph E , then M1 is the vertex matrix of E .
Now suppose that A and B are 1-graphs with A0 = B 0 = V such the associated
vertex matrices commute. Set A1 B 1 = f( ) 2 A1 B 1 : s() = r( )g and
B 1 A1 = f( ) 2 B 1 A1 : s( ) = r()g since the associated vertex matrices
commute there is a bijection : ( ) 7! ( 0 0 ) from A1 B 1 to B 1 A1 such that
r() = r( 0 ) and s( ) = s(0 ). We construct a 2-graph from A, B and . This
18
Alex Kumjian and David Pask
construction is very much in the spirit of RS2] roughly speaking an element in of degree (m n) 2 N2 will consist of a rectangular grid of size (m n) with edges
of A horizontally, edges of B vertically and nodes in V arranged compatibly. First
identify 0 = V . For (m n) 2 N2 set W (m n) = f(i j ) 2 N2 : (i j ) (m n)g.
An element in (mn) is given by v(i j ) 2 V for (i j ) 2 W (m n), (i j ) 2 A1 for
(i j ) 2 W (m ; 1 n) and (i j ) 2 B 1 for (i j ) 2 W (m n ; 1) (set W (m n) = if
m or n is negative) satisfying the following compatibility conditions wherever they
make sense:
i. r((i j )) = v(i j ) and r( (i j )) = v(i j )
ii. s((i j )) = v(i + 1 j ) and s( (i j )) = v(i j + 1)
iii. ((i j ) (i + 1 j )) = ( (i j ) (i j + 1))
for brevity and with a slight abuse of notation we regard this element as a triple
(v ) (note that disappears if m = 0 and disappears if n = 0 and v is
determined by and/or if mn 6= 0). Set
(mn)
=
(mn)
and dene s(v ) = v(m n) and r(v ) = v(0 0).
Note that if 2 Am and 2 B n with m n > 0 such that s() = r(
) there is
a unique element (v ) 2 (mn) such that = (0 0)(1 0) (m ; 1 0) and
= (m 0) (m 1) (m n ; 1) denote this element . Further if 2 Am and
2 B n with m n > 0 such that r() = s(
) there is a unique element (v ) in
(mn) such that = (0 n)(1 n) (m ; 1 n) and = (0 0) (0 1) (0 n ;
1) denote this element . Using these two facts it is0 not
dicult to verify that
given elements (v ) 2 (mn) and (v0 0 0 )0 2 (0m n0 ) with v(m n) = v0 (0 0)
there is a unique element (v00 00 00 ) 2 (m+m n+n ) such that v00 (i j ) = v(i j ),
00 (i j ) = (i j ), 00 (i j ) = (i j ), v00 (m + i n + j ) = v0 (i j ), 00 (m + i n +
j ) = 0 (i j ) and 00 (m + i n + j ) = 0 (i j ) wherever these formulas make sense.
Write (v00 00 00 ) = (v )(v0 0 0 ). This denes composition in note that
associativity and the factorisation property are built into the construction (as in
RS2]). Finally, we write = A B . It is straightforward to verify that up to
isomorphism any 2-graph may be obtained from its constituent 1-graphs in this
way.
If A = B , then we may take = the identity map. In that case one has
A A = f (A) where f : N2 ! N is given by f (m n) = m + n. Hence, by
Corollary 3.5(iii) we have C (A A) = C (A) C (T).
To further emphasise the dependence of the product A B on the bijection
: A1 B 1 ! B 1 A1 consider the following example.
Example 6.1. Let A = B be the 1-graph derived from the directed graph which
consists of one vertex and two edges, say A1 = fe f g (note C (A) = O2 ). Then
A1 A1 = f(e e) (e f ) (f e) (f f )g, and we dene the bijection to be the &ip.
It is easy to show that A A = A A hence,
C (A A) = O2 O2 = O2
where the rst isomorphism follows from Corollary 3.5(iv) and the second from the
Kirchberg-Phillips classication theorem (see K, P]). But
C (A A) = O2 C (T)
Higher Rank Graph C -Algebras
19
hence, A A 6 A A.
=
References
A-D]
A-DR]
ALNR]
BPRS]
B]
CK]
D]
GT]
H]
aHR]
KQR]
K]
KPRR]
KPR]
KP]
LS]
MacL]
Ma]
Mu]
P]
R]
RS1]
RS2]
RS3]
RSc]
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20
Alex Kumjian and David Pask
Department of Mathematics (084), University of Nevada, Reno NV 89557-0045, USA.
alex@unr.edu http://equinox.comnet.unr.edu/homepage/alex/
Department of Mathematics, University of Newcastle, NSW 2308, Australia
davidp@maths.newcastle.edu.au http://maths.newcastle.edu.au/~davidp/
This paper is available via http://nyjm.albany.edu:8000/j/2000/6-1.html.