Noncommutative manifolds from graph and $k$-graph $C^*$-algebras

0.36 MB·English

by  David Pask

NONCOMMUTATIVE MANIFOLDS FROM GRAPH AND k-GRAPH
C∗-ALGEBRAS
7
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0
2 DAVIDPASK,ADAMRENNIE,AIDANSIMS

n
a
J School of Mathematical and Physical Sciences

9 University of Newcastle, Callaghan, NSW Australia, 2308
1 [email protected],

[email protected]
]
A Institute for Mathematical Sciences, University of Copenhagen
O Universitetsparken 5, DK-2100, Copenhagen
. [email protected]
h
t
a Abstract
m
In [PRen] we constructed smooth (1, )-summable semfinite spectral triples for graph algebras with
[ ∞
a faithful trace, and in [PRS] we constructed (k, )-summable semifinite spectral triples for k-graph

1 ∞
algebras. In this paper we identify classes of graphs and k-graphs which satisfy a version of Connes’
v
7 conditions for noncommutative manifolds.
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5
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0 Contents
/
h
t
a 1. Introduction 2
m
: 2. Background on Graph C∗-Algebras and Spectral Triples 2
v
Xi 2.1. The C∗-algebras of Graphs 2
r
a 2.2. Semifinite Spectral Triples 4
2.3. Summability 7
2.4. The Gauge Spectral Triple for a Graph C∗-Algebra 8
3. Conditions for Locally Compact Semifinite Manifolds 10
3.1. The Analytic Conditions 11
3.2. The Orientation and Closedness Conditions 12
3.3. The Bimodule Conditions 16
4. k-Graph Manifolds 22
References 29
1
2 DAVIDPASK,ADAMRENNIE,AIDANSIMS
1. Introduction
Theobject of this paper is to address the general definition of noncommutative manifolds. The phrase
‘noncommutative manifold’ is one which is still open to some degree of interpretation. Broadly speak-
ing, a noncommutative manifold is a spectral triple ( , , ) satisfying some additional conditions,
A H D
such as those originally proposed by Connes, [C1]. However, the conditions presented in [C1] only
make sense when is unital (that is, a compact noncommutative space). Moreover, the proof of
A
Connes’ spin manifold theorem in [RV] uses a modification of Connes’ conditions even in the compact
case.
We consider the set of conditions presented in [RV], and show that there is a natural generalisation
of each to noncommutative, nonunital and semifinite spectral triples. In the process, we show that
for certain graphs and k-graphs, the spectral triples constructed in [PRen, PRS] satisfy these condi-
tions, making them reasonable candidates for the title of noncommutative manifolds. Conditions for
noncompact noncommutative manifolds have previously been considered in [R1, R2, GGISV].
We have made an effort to generalise the conditions from [RV] in as minimal and stringent a way as
possible. Nevertheless, our conditions must be regarded as provisional. Additional examples and ap-
plications arerequired todeterminethe‘correct’ conditions characterising noncommutative manifolds.
The vast majority of examples of noncommutative manifolds in this paper come from nonunital al-
gebras (see [PRen, PRS]), so our conditions must address aspects of ‘noncompact noncommutative
manifolds.’ Moreover, most of our examples are semifinite, in that the trace employed is not the
operator trace on Hilbert space; it is a faithful normal semifinite trace on a different von Neumann
algebra. This is not to say that the C∗-algebras arising in our examples do not admit type I spectral
triples. By considering traces which reflect the geometry of the underlying graph (or k-graph), we are
naturally led to semifinite spectral triples.
For simplicity we discuss only graph algebras (i.e. algebras of 1-graphs) in detail; in a final section we
summarise the k-graph situation, since it is largely similar.
The conditions introduced in Section 3 do not employ Poincar´e Duality in K-theory, but rather
the Morita equivalence condition characterising spinc structures, [P]. The equivalence of these two
conditions in the compact commutative case (in the presence of the other conditions) was proved in
[RV]. In addition, we do not consider the metric condition, since this has recently been shown to be
redundant [RV2].
Acknowledgements We thank J. Varilly for useful discussions. This work was supported by the
ARC and the Danish Research Council.
2. Background on Graph C∗-Algebras and Spectral Triples
2.1. The C∗-algebras of Graphs. For a more detailed introduction to graph C∗-algebras we refer
the reader to [BPRS, KPR, R] and the references therein. A directed graph E = (E0,E1,r,s) consists
of countable sets E0 of vertices and E1 of edges, and maps r,s : E1 E0 identifying the range and
→
sourceofeach edge. Thegraphisrow-finite ifeach vertex emitsatmostfinitelymanyedgesandlocally
finite if it is row-finite and each vertex receives at most finitely many edges. We write En for the set
of paths µ = µ µ µ of length µ := n; that is, sequences of edges µ such that r(µ )= s(µ ) for
1 2 n i i i+1
··· | |
NONCOMMUTATIVE MANIFOLDS FROM GRAPH AND k-GRAPH C∗-ALGEBRAS 3
1 i < n. The maps r,s extend to E∗ := En in an obvious way. A loop in E is a path L E∗
≤ n≥0 ∈
with s(L) = r(L), we say that a loop L haSs an exit if there is v = s(Li) for some i which emits more
than one edge. If V E0 then we write V w if there is a path µ E∗ with s(µ) V and r(µ)= w.
⊆ ≥ ∈ ∈
A sink is a vertex v E0 with s−1(v) = , a source is a vertex w E0 with r−1(w) = .
∈ ∅ ∈ ∅
ACuntz-Krieger E-family inaC∗-algebra B consists ofmutually orthogonal projections p :v E0
v
{ ∈ }
and partial isometries S :e E1 satisfying the Cuntz-Krieger relations
e
{ ∈ }
S∗S = p for e E1 and p = S S∗. whenever v is not a sink.
e e r(e) ∈ v e e
X
{e:s(e)=v}
Theorem1.2of[KPR]showsthatthereisauniversalC∗-algebraC∗(E)generatedbyauniversalCuntz-
Krieger E-family S ,p . A product S := S S ...S is non-zero precisely when µ =µ µ µ
{ e v} µ µ1 µ2 µn 1 2··· n
is a path in En. Since the Cuntz-Krieger relations imply that the projections S S∗ are also mutually
e e
orthogonal, we have S∗S = 0 unless e = f, and words in S ,S∗ collapse to products of the form
e f { e f}
S S∗ for µ,ν E∗ satisfying r(µ)= r(ν) (cf. [KPR, Lemma 1.1]). Indeed, because the family S S∗
µ ν ∈ { µ ν}
is closed under multiplication and involution, we have
(1) C∗(E) = span S S∗ : µ,ν E∗ and r(µ)= r(ν) .
{ µ ν ∈ }
The algebraic relations and the density of span S S∗ in C∗(E) play a critical role throughout the
{ µ ν}
paper. We adopt the conventions that vertices are paths of length 0, that S := p for v E0, and
v v
∈
that all paths µ,ν appearing in (1) are non-empty; we recover S , for example, by taking ν = r(µ),
µ
so that S S∗ = S p = S .
µ ν µ r(µ) µ
If z S1, then the family zS ,p is another Cuntz-Krieger E-family which generates C∗(E), and
e v
∈ { }
the universal property gives a homomorphism γ : C∗(E) C∗(E) such that γ (S ) = zS and
z z e e
→
γ (p ) = p . The homomorphism γ is an inverse for γ , so γ AutC∗(E), and a routine ǫ/3
z v v z z z
∈
argument using (1) shows that γ is a strongly continuous action of S1 on C∗(E). It is called the gauge
action. Because S1 is compact, averaging over γ with respect to normalised Haar measure gives an
expectation Φ of C∗(E) onto the fixed-point algebra C∗(E)γ:
1
Φ(a) := γ (a)dθ for a C∗(E), z = eiθ.
z
2π Z ∈
S1
The map Φ is positive, has norm 1, and is faithful in the sense that Φ(a∗a) = 0 implies a = 0.
From Equation (1), it is easy to see that a graph C∗-algebra is unital if and only if the underlying
graph is finite. When we consider infinite graphs, formulas which involve sums of projections may
contain infinite sums. To interpret these, we use strict convergence in the multiplier algebra of C∗(E).
The following is proved in [PR].
Lemma 2.1. Let E be a row-finite graph, let A be a C∗-algebra generated by a Cuntz-Krieger E-family
T ,q , and let p be a sequence of projections in A. If p T T∗ converges for every µ,ν E∗, then
{ e v} { n} n µ ν ∈
p converges strictly to a projection p M(A).
n
{ } ∈
Another graph theoretic concept useful for the graphs we will be dealing with is the following.
Definition 2.2. Let E be a row-finite directed graph. An end will mean a sink, a loop without exit or
an infinite path with no exits.
4 DAVIDPASK,ADAMRENNIE,AIDANSIMS
Ends play two roles in this paper. One is connected to the existence of faithful gauge invariant traces.
The most basic statement about such traces is the following, [PRen].
Lemma 2.3. Let E be a row-finite directed graph.
(i) If C∗(E) has a faithful, semifinite trace then no loop can have an exit.
(ii) If C∗(E) has a faithful, semifinite, lower semicontinuous, gauge-invariant trace τ then τ Φ = τ
◦
and
τ(S S∗) = δ τ(p ).
µ ν µ,ν r(µ)
This result was sharpened using the notion of a graph trace, which we also require in this paper.
Definition 2.4 ([T]). If E is a row-finite directed graph, then a graph trace on E is a function
g :E0 R+ such that for any v E0 we have
→ ∈
(2) g(v) = g(r(e)).
X
s(e)=v
We showed that if E admits a faithful graph trace then no vertex v E0 connects to any other vertex
∈
orany endviainfinitelymany distinctpaths. Inparticular, E cancontain noloopswithexits. Wealso
shows that if E contains finitely many ends and every infinite path x E∞ is eventually contained in
∈
an end, then E admits a faithful graph trace.
The following proposition from [PRen] shows how to use the existence of graph traces on E to deduce
the existence of faithful, semifinite, lower semicontinuous, gauge invariant traces on C∗(E).
Proposition 2.5. Let E be a row-finite directed graph. Then there is a one-to-one correspondence
between faithful graph traces on E and faithful, semifinite, lower semicontinuous, gauge invariant
traces on C∗(E).
The other role of ends for graphs such that no loop has an exit is topological, and will be used when
we discuss the conditions. The following is proved in [RSz, Corollary 5.3].
Lemma 2.6. Let A= C∗(E) be a graph C∗-algebra such that no loop in the locally finite graph E has
an exit. Then,
K (C∗(E)) = Z#ends, K (C∗(E)) = Z#loops.
0 1
In particular, K (C∗(E)) is finitely generated if there are finitely many ends in E.
∗
2.2. Semifinite Spectral Triples. We begin with some semifinite versions of standard definitions
andresults. Letτ beafixedfaithful,normal, semifinitetrace onthevon Neumannalgebra . Let
N
N K
be the τ-compact operators in (that is the norm closed ideal generated by the projections E
N ∈ N
with τ(E) < ).
∞
Definition 2.7. A semifinite spectral triple ( , , ) consists of a Hilbert space , a -algebra
A H D H ∗ A ⊂ N
where is a semifinite von Neumann algebra acting on , and a densely defined unbounded self-
N H
adjoint operator affiliated to such that
D N
1) [ ,a] is densely defined and extends to a bounded operator in for all a , and
D N ∈ A
2) a(λ )−1 for all λ R and all a .
N
−D ∈ K 6∈ ∈ A
NONCOMMUTATIVE MANIFOLDS FROM GRAPH AND k-GRAPH C∗-ALGEBRAS 5
The triple is said to be even if there is Γ such that Γ∗ = Γ, Γ2 = 1, aΓ = Γa for all a and
∈ N ∈ A
Γ+Γ = 0. Otherwise it is odd.
D D
Definition 2.8. A semifinite spectral triple ( , , ) is QCk for k 1 (Q for quantum) if for all
A H D ≥
a the operators a and [ ,a] are in the domain of δk, where δ(T) = [ ,T] is the partial derivation
∈ A D |D|
on defined by . We say that ( , , ) is QC∞ if it is QCk for all k 1.
N |D| A H D ≥
Note. Thenotationismeanttobeanalogoustotheclassicalcase,butweintroducetheQsothatthere
is no confusion between quantum differentiability of a and classical differentiability of functions.
∈ A
Remarks concerning derivations and commutators. By partial derivation we mean that δ is
defined on some subalgebra of which need not be (weakly) dense in . More precisely, dom δ =
N N
T :δ(T) is bounded . We also note that if T , one can show that [ ,T] is bounded if and
{ ∈ N } ∈ N |D|
only if [(1+ 2)1/2,T] is bounded, by using the functional calculus to show that (1+ 2)1/2
D |D|− D
extends to a bounded operator in . In fact, writing = (1+ 2)1/2 and δ (T)= [ ,T] we have
1 1 1
N |D| D |D|
dom δn = dom δn for all n.
1
We also observe that if T and [ ,T] is bounded, then [ ,T] . Similar comments apply to
∈ N D D ∈ N
[ ,T], [(1+ 2)1/2,T]. The proofs of these statements can be found in [CPRS2].
|D| D
Definition 2.9. A -algebra is smooth if it is Fr´echet and -isomorphic to a proper dense subalgebra
∗ A ∗
i( ) of a C∗-algebra A which is stable under the holomorphic functional calculus.
A
Thus saying that is smooth means that is Fr´echet and a pre-C∗-algebra. Asking for i( ) to be
A A A
a proper dense subalgebra of A immediately implies that the Fr´echet topology of is finer than the
A
C∗-topology of A (since Fr´echet means locally convex, metrizable and complete.) We will sometimes
speak of = A, particularly when is represented on Hilbert space and the norm closure is
A A A
unambiguous. At other times we regard i : ֒ A as an embedding of in a C∗-algebra. We will
A → A
use both points of view.
It has been shown that if is smooth in A then M ( ) is smooth in M (A), [GVF, S]. This ensures
n n
A A
thattheK-theoriesofthetwoalgebrasareisomorphic,theisomorphismbeinginducedbytheinclusion
map i. This definition ensures that a smooth algebra is a ‘good’ algebra, [GVF], so these algebras
have a sensible spectral theory which agrees with that defined using the C∗-closure, and the group of
invertibles is open.
Stability under the holomorphic functional calculus extends to nonunital algebras, since the spectrum
of an element in a nonunital algebra is defined to be the spectrum of this element in the ‘one-point’
unitization, though we mustof course restrict to functionssatisfying f(0) = 0. Likewise, thedefinition
of a Fr´echet algebra does not require a unit. The point of contact between smooth algebras and QC∞
spectral triples is the following Lemma, proved in [R1].
Lemma 2.10. If ( , , ) is a QC∞ spectral triple, then ( , , ) is also a QC∞ spectral triple,
δ
A H D A H D
where is the completion of in the locally convex topology determined by the seminorms
δ
A A
q (a) = δndi(a) , n 0, i = 0,1,
n,i
k k ≥
where d(a) = [ ,a]. Moreover, is a smooth algebra.
δ
D A
We call the topology on determined by the seminorms q of Lemma 2.10 the δ-topology.
ni
A
6 DAVIDPASK,ADAMRENNIE,AIDANSIMS
Whilst smoothness does not depend on whether is unital or not, many analytical problems arise
A
because of the lack of a unit. As in [R1, R2, GGISV], we make two definitions to address these issues.
Definition 2.11. An algebra has local units if for every finite subset of elements a n , there
A { i}i=1 ⊂ A
exists φ such that for each i
∈ A
φa = a φ = a .
i i i
Definition 2.12. Let be a Fr´echet algebra and be a dense subalgebra with local units. Then
c
A A ⊆ A
we call a quasi-local algebra (when is understood.) If is a dense ideal with local units, we call
c c
A A A
local.
c
A ⊂ A
Quasi-local algebras have an approximate unit φ such that φ φ = φ , [R1].
n n≥1 c n+1 n n
{ } ⊂ A
Example For a graph C∗-algebra A= C∗(E), Equation (1) shows that
A = span S S∗ : µ,ν E∗ and r(µ)= r(ν)
c { µ ν ∈ }
is a dense subalgebra. It has local units because
S S∗ v = s(µ)
p S S∗ = µ ν .
v µ ν (cid:26) 0 otherwise
Similarcomments applytorightmultiplication byp . Bysummingthesourceandrangeprojections
s(ν)
(without repetitions) of all S S∗ appearing in a finite sum
µi νi
a= c S S∗
µi,νi µi νi
Xi
we obtain a local unit for a A . By repeating this process for any finite collection of such a A we
c c
∈ ∈
see that A has local units.
c
We also require that when we have a spectral triple the operator is compatible with the quasi-local
D
structure of the algebra, in the following sense.
Definition 2.13. If ( , , ) is a spectral triple, then we define ( ) to be the algebra generated by
D
A H D C A
and [ , ].
A D A
Definition 2.14. A local spectral triple ( , , ) is a spectral triple with quasi-local such that there
A H D A
exists an approximate unit φ for satisfying
n c
{ } ⊂ A A
( ) = ( ) ,
D c D n
C A C A
[n
where
( ) = ω ( ): φ ω = ωφ = ω .
D n D n n
C A { ∈ C A }
Remark A local spectral triple has a local approximate unit φ such that φ φ =
n n≥1 c n+1 n
{ } ⊂ A
φ φ = φ and φ [ ,φ ] = [ ,φ ]φ = [ ,φ ], [R2]. This is the crucial property we require
n n+1 n n+1 n n n+1 n
D D D
to prove most of our summability results, to which we now turn.
NONCOMMUTATIVE MANIFOLDS FROM GRAPH AND k-GRAPH C∗-ALGEBRAS 7
2.3. Summability. Inthe following, let bea semifinite von Neumannalgebra with faithfulnormal
N
trace τ. Recall from [FK] that if S , the t-th generalized singular value of S for each real t > 0 is
∈ N
given by
µ (S)= inf SE : E is a projection in with τ(1 E) t .
t
{|| || N − ≤ }
The ideal 1( ) consists of those operators T such that T := τ(T ) < where T = √T∗T.
1
L N ∈ N k k | | ∞ | |
In the Type I setting this is the usual trace class ideal. We will simply write 1 for this ideal in order
L
to simplify the notation, and denote the norm on 1 by . An alternative definition in terms of
1
singular values is that T 1 if T := ∞µ (T)dLt < k. · k
∈ L k k1 0 t ∞
R
Note that in the case where = ( ), 1 is not complete in this norm but it is complete in the norm
N 6 B H L
. + . . (where . is the uniform norm). Another important ideal for us is the domain of the
1 ∞ ∞
|||| |||| ||||
Dixmier trace:
1 t
(1,∞)( ) = T : T := sup µ (T)ds < .
L N (cid:26) ∈N k kL(1,∞) log(1+t)Z s ∞(cid:27)
t>0 0
We will suppress the ( ) in our notation for these ideals, as will always be clear from context.
N N
The reader should note that (1,∞) is often taken to mean an ideal in the algebra of τ-measurable
L N
operators affiliated to . Our notation is however consistent with that of [C] in the special case
N e
= ( ). With this convention the ideal of τ-compact operators, ( ), consists of those T
N B H K N ∈ N
(as opposed to ) such that
N
µ (T) := lim µ (T) = 0.
∞ t
e t→∞
Definition 2.15. A semifinite local spectral triple is (1, )-summable if
∞
a( λ)−1 (1,∞) for all a , λ C R.
c
D− ∈ L ∈ A ∈ \
RemarkIf is unital, ker is τ-finitedimensional. Note thatthesummability requirements areonly
A D
for a . We do not assume that elements of the algebra are all integrable in the nonunital case.
c
∈ A A
Strictly speaking, this definition describes local (1, )-summability, however we use the terminology
∞
(1, )-summable to be consistent with the unital case.
∞
We need to briefly discuss the Dixmier trace, but fortunately we will usually be applying it in reason-
ablysimplesituations. For moreinformation onsemifiniteDixmiertraces, see[CPS2]. For T (1,∞),
∈ L
T 0, the function
≥ 1 t
F :t µ (T)ds
T s
→ log(1+t)Z
0
is bounded. For certain generalised limits ω L∞(R+)∗, we obtain a positive functional on (1,∞) by
∈ ∗ L
setting
τ (T)= ω(F ).
ω T
This is the Dixmier trace associated to the semifinite normal trace τ, denoted τ , and we extend it to
ω
all of (1,∞) by linearity, where of course it is a trace. The Dixmier trace τ is defined on the ideal
ω
L
(1,∞), and vanishes on the ideal of trace class operators. Whenever the function F has a limit at
T
L
infinity, all Dixmier traces return the value of the limit. We denote the common value of all Dixmier
traces on measurable operators by . So if T (1,∞) is measurable, for any allowed functional
− ∈ L
ω L∞(R+)∗ we have R
∈ ∗
τ (T) = ω(F ) = T.
ω T
Z−
8 DAVIDPASK,ADAMRENNIE,AIDANSIMS
Example Let = 1 d act on L2(S1). Then it is well known that the spectrum of consists of
D i dθ D
eigenvalues n Z , each with multiplicity one. So, using the standard operator trace, Trace, the
{ ∈ }
function F is
(1+D2)−1/2
N
1
(1+n2)−1/2
log2N
nX=−N
and this is bounded. Hence (1+ 2)−1/2 (1,∞) and
D ∈ L
(3) Trace ((1+ 2)−1/2)= (1+ 2)−1/2 = 2.
ω
D Z− D
In [R1, R2] we proved numerous properties of local algebras. The introduction of quasi-local algebras
in [GGISV] led us to review the validity of many of these results for quasi-local algebras. Most of the
summability results of [R2] are valid in the quasi-local setting. In addition, the summability results of
[R2] are also valid for general semifinite spectral triples since they rely only on properties of the ideals
(p,∞), p 1, [C, CPS2], and the trace property. We quote the version of the summability results
L ≥
from [R2] that we require below.
Proposition 2.16 ([R2]). Let ( , , ) be a QC∞, local (1, )-summable semifinite spectral triple
A H D ∞
relative to ( ,τ). Let T satisfy Tφ= φT = T for some φ . Then
c
N ∈ N ∈ A
T(1+ 2)−1/2 (1,∞).
D ∈ L
For Re(s)> 1, T(1+ 2)−s/2 is trace class. If the limit
D
(4) lim (s 1/2)τ(T(1+ 2)−s)
s→1/2+ − D
exists, then it is equal to
1
T(1+ 2)−1/2.
2 Z− D
In addition, for any Dixmier trace τ , the function
ω
a τ (a(1+ 2)−1/2)
ω
7→ D
defines a trace on .
c
A ⊂A
2.4. The Gauge Spectral Triple for a Graph C∗-Algebra. In this section we summarise the
construction of a Kasparov module and a semifinite spectral triple for locally finite directed graphs
with no sources. This material is based on [PRen]. We begin by constructing a Kasparov module.
For E a row finite directed graph, we set A= C∗(E), F = C∗(E)γ, the fixed point algebra for the S1
gauge action. The algebras A ,F are defined as the finite linear span of the generators.
c c
Right multiplication makes A into a right F-module, and similarly A is a right module over F . We
c c
define an F-valued inner product ( ) on both these modules by
R
·|·
(ab) := Φ(a∗b),
R
|
where Φ is the canonical expectation A F.
→
Definition 2.17. Let X be the right C∗-F-module obtained by completing A (or A ) in the norm
c
x 2 := (xx) = Φ(x∗x) .
k kX k | RkF k kF
NONCOMMUTATIVE MANIFOLDS FROM GRAPH AND k-GRAPH C∗-ALGEBRAS 9
The algebra A acting by multiplication on the left of X provides a representation of A as adjointable
operators on X. We let X be the copy of A X.
c c
⊂
For each k Z, define an operator Φ on X by
k
∈
1
Φ (x) = z−kγ (x)dθ, z = eiθ, x X.
k z
2π Z ∈
S1
Observe that on generators we have
S S∗ α β = k
(5) Φ (S S∗)= α β | |−| | .
k α β (cid:26) 0 α β = k
| |−| | 6
Lemma 2.18. The operators Φ are adjointable endomorphisms of the F-module X such that Φ∗ =
k k
Φ = Φ2 and Φ Φ = δ Φ . If K Z then the sum Φ converges strictly to a projection in
k k k l k,l k ⊂ k∈K k
the endomorphism algebra. The sum k∈ZΦk convergePs to the identity operator on X.
P
Corollary 2.19. Let x X. Then with x = Φ x the sum x converges in X to x.
∈ k k k∈Z k
Proposition 2.20 ([PRen]). Let X be the right C∗-F-moduPle of Definition 2.17. Let
X = x X : k2(x x )< ,
D k k
{ ∈ | ∞}
kX∈Z
and define : X X by
D
D →
x = kx .
k k
D
kX∈Z kX∈Z
Then is closed, self-adjoint and regular.
D
We refer to Lance’s book, [L, Chapters 9,10], for information on unboundedoperators on C∗-modules.
Lemma 2.21. Assume that the directed graph E is locally finite and has no sources. For all a A
∈
and k Z, aΦ End0(X), the compact right endomorphisms of X. If a A then aΦ is finite
∈ k ∈ F ∈ c k
rank.
Theorem 2.22. Suppose that the graph E is locally finite and has no sources, and let X be the right
F module of Definition 2.17. Let V = (1+ 2)−1/2. Then (X,V) is an odd Kasparov module for
D D
A-F and so defines an element of KK1(A,F).
Given the hypotheses of the Theorem, we may describe as
D
= kΦ .
k
D
kX∈Z
To construct a semifinite spectral triple, we suppose that our graph C∗-algebra also has a faithful
gauge invariant trace τ : A C. Using the trace τ, we define a C-valued inner product , on X
c
→ h· ·i
by
x,y := τ((xy) ) = τ(Φ(x∗y)) = τ(x∗y),
R
h i |
the last equality following from the gauge invariance of τ. Denote the Hilbert space completion of X
c
by = L2(X,τ).
H
Proposition 2.23. The actions of A by left and right multiplication on X extend to give commuting
bounded faithful nondegenerate representations of A and Aop, where Aop is the opposite algebra of A.
Furthermore, any endomorphism of the right F module X which preserves X extends uniquely to a
c
bounded operator on .
H
10 DAVIDPASK,ADAMRENNIE,AIDANSIMS
The operator : dom X X also extends to a self-adjoint operator on , [PRen, Lemma 5.5].
D D ⊂ → H
It is shown in [PRen] that for all a A the commutator [ ,a] extends to a bounded operator on .
c
∈ D H
Lemma 2.24. The algebra A and the linear space [ ,A ] are contained in the smooth domain of the
c c
D
derivation δ where for T ( ), δ(T) = [ ,T]. So the completion of A in the δ-topology, which we
c
∈ B H |D|
denote by , is a Fr´echet pre-C∗-algebra. Moreover is a quasi-local algebra with dense subalgebra
A A
A .
c
Since [ ,A ] A , it is not hard to see that if ( , , ) is a spectral triple, then it is local.
c c
D ⊂ A H D
The only remaining piece of information we require to obtain a spectral triple is the von Neumann
algebra and trace which give us a semifinite spectral triple. Let End00(X ) be the finite rank endo-
F c
morphisms of the pre-C∗-module X .
c
Proposition 2.25. Let = (End00(X ))′′. Then there exists a faithful, normal, semifinite trace
N F c
τ˜: C such that for all rank one endomorphisms Θ of X we have
x,y c
N →
τ˜(Θ ) = τ((y x) ), x,y X .
x,y R c
| ∈
Moreover, for all a and λ C R the operator a(λ )−1 lies in .
N
∈ A ∈ \ −D K
Hence we obtain a semifinite spectral triple. However, more is true.
Theorem 2.26. Let E be a locally finite graph with no sources, and let τ be a faithful, semifinite, norm
lower-semicontinuous, gauge invariant trace on C∗(E). Then ( , , ) is a QC∞ (1, )-summable
A H D ∞
odd local semifinite spectral triple (relative to ( ,τ˜)). For all a the operator a(1+ 2)−1/2 is not
N ∈A D
trace class. For any v E0 which does not connect to a sink we have
∈
τ˜ (p (1+ 2)−1/2)= 2τ(p ),
ω v v
D
where τ˜ is any Dixmier trace associated to τ˜.
ω
The main observation of the proof is that for v E0 such that v does not connect to a sink, and for
∈
k Z we have
∈
τ˜(p Φ )= τ(p ).
v k v
This is the spectral triple we will be working with for the rest of the paper, and we refer to it as
the gauge spectral triple of the directed graph E (or algebra C∗(E)). We remind the reader that the
existence of this spectral triple depends only on the graph E being locally finite with no sources, and
the existence of a faithful, semifinite, gauge invariant, norm lower-semicontinuous trace τ : A C.
→
The latter is a nontrivial condition.
3. Conditions for Locally Compact Semifinite Manifolds
We now review in turn the conditions for noncommutative manifolds as presented in [RV]. We will
considernaturalgeneralisations oftheseconditionstothesemifiniteandnonunitalsettingandconsider
when the gauge spectral triple satisfies these generalisations.
We will present each condition as stated for the type I and unital case, where ( ,τ) = ( ( ),Trace)
N B H
and (1+ 2)−1/2 ( ), and then present the necessary modification of the condition, if it requires
D ∈ K H
modification.