The closure of two-sided multiplications on C*-algebras and phantom line bundles

0.35 MB·

by  Ilja Gogić

THE CLOSURE OF TWO-SIDED MULTIPLICATIONS ON
C*-ALGEBRAS AND PHANTOM LINE BUNDLES
6
1
0 ILJAGOGIC´1,2 ANDRICHARDM.TIMONEY2
2

ar Abstract. For a C∗-algebra A we consider the problem of when the set
M TM0(A) of all two-sided multiplications x 7→ axb (a,b ∈ A) on A is norm
closed,asasubsetofB(A). WefirstshowthatTM0(A)isnormclosedforall
2 primeC∗-algebras A. Onthe other hand, ifA∼=Γ0(E) is ann-homogeneous
2 C∗-algebra, where E isthe canonical Mn-bundle over the primitivespectrum
XofA,weshowthatTM0(A)failstobenormclosedifandonlyifthereexists
] aσ-compactopensubsetU ofX andaphantomcomplexlinesubbundleLof
A E overU (i.e. Lisnotgloballytrivial,butistrivialonallcompactsubsetsof
U). This phenomenon occurs whenever n≥2 and X is a CW-complex (or a
O
topological manifold)ofdimension3≤d<∞.
.
h
t
a
m
1. Introduction
[
LetAbeaC∗-algebraandletIB(A)(resp. ICB(A))denotethesetofallbounded
2
(resp. completely bounded) maps φ : A A that preserve (closed two-sided)
v →
ideals of A (i.e. φ(I) I for all ideals I of A). The most prominent class of maps
8
⊆
4 φ ICB(A) IB(A) are elementary operators, i.e. those that can be expressed as
∈ ⊂
8 finite sumsoftwo-sided multiplications M :x axb,where aandbareelements
a,b
6 of the multiplier algebra M(A). 7→
0
Elementary operators play an important role in modern quantum information
.
1 andquantumcomputationtheory. Inparticular,mapsφ:M M (M aren n
n n n
60 matrices over C) of the form φ= ℓi=1Ma∗i,ai (ai ∈Mn such→that ℓi=1a∗iai =×1)
represent the (trace-duals of) quantum channels, which are mathematical models
1
P P
: ofthe evolutionofan‘open’ quantumsystem(see e.g. [21]). Elementaryoperators
v
also provide ways to study the structure of C∗-algebras (see [2]).
i
X Let ℓ(A), TM(A) and TM (A) denote, respectively, the sets of all elementary
0
E
r operatorson A, two-sidedmultiplications onA and two-sidedmultiplications onA
a
with coefficients in A (i.e. TM (A)= M :a,b A ).
0 a,b
{ ∈ }
TheelementaryoperatorsarealwaysdenseinIB(A)inthetopologyofpointwise
convergence (by [23, Corollary 2.3]). However, more subtle considerations enter
in when one asks if φ ICB(A) can be approximated pointwise by elementary
∈
operators of cb-norm at most φ ([24] shows that nuclearity of A suffices; see
cb
k k
also [26]).
Date:VersionMarch23,2016.
2010 Mathematics Subject Classification. Primary46L05,46M20;Secondary 46L07.
Keywordsandphrases. C∗-algebra,homogeneous,two-sidedmultiplication,elementaryoper-
ator,complexlinebundle.
The work of the first author was supported by the Irish Research Council under grant
GOIPD/2014/7.
TheworkofthesecondauthorwassupportedbytheScienceFoundation Irelandunder grant
11/RFP/MTH3187.
1
2 I.GOGIC´ ANDR.M.TIMONEY
It is an interesting problem to describe those operators φ IB(A) (resp. φ
∈ ∈
ICB(A))thatcanbeapproximatedinoperatornorm(resp. cb-norm)byelementary
operators. Earlier works, which we cite below, revealed that this is an intricate
question in general, and can involve many and varied properties of A and φ. In
this paper, we show that the apparently much simpler problems of describing the
normclosuresofTM(A)andTM (A)canhavecomplicatedanswersevenforrather
0
well-behaved C∗-algebras.
In some cases ℓ(A) = IB(A) (which implies ℓ(A) = ICB(A)); or ℓ(A) is
E E E
normdense in IB(A); or ℓ(A) ICB(A) is dense in cb-norm. The conditions just
E ⊂
mentioned are in fact all equivalent for separable C∗-algebras A. More precisely,
Magajna [25] shows that for separable C∗-algebras A, the property that ℓ(A) is
E
norm (resp. cb-norm) dense in IB(A) (resp. ICB(A)) characterizes finite direct
sums of homogeneous C∗-algebras with the finite type property. Moreover, in this
situation we already have the equality ICB(A)=IB(A)= ℓ(A).
E
Itcanhappenthat ℓ(A)isalreadynormclosed(orcb-normclosed). In[13,14],
E
the first author showed that for a unital separable C∗-algebra A, if ℓ(A) is norm
E
(or cb-norm) closed then A is necessarily subhomogeneous, the homogeneous sub-
quotientsofAmusthavethe finite typepropertyandestablishedfurthernecessary
conditions on A. In [14, 15] he gave some partial converse results.
There is a considerable literature on derivations and inner derivations of C∗-
algebras. Inner derivations d on a C∗-algebra A, (i.e. those of the form d (x) =
a a
ax xa with a M(A)) are important examples of elementary operators. In [35,
− ∈
Corollary 4.6] Somerset shows that if A is unital, d : a A is norm closed if
a
{ ∈ }
and only if Orc(A) < , where Orc(A) is a constant defined in terms of a certain
∞
graph structure on Prim(A) (the primitive spectrum of A). If Orc(A) = , the
∞
structure of outer derivations that are norm limits of inner derivations remains
undescribed. In addition, if A is unital and separable, then by [19, Theorem 5.3]
and[35,Corollary4.6]Orc(A)< ifandonlyiftheset Mu,u∗ :u A,u unitary
∞ { ∈ }
of inner automorphisms is norm closed.
In[12,15]the firstauthorconsideredtheproblemofwhichderivationsonunital
C∗-algebras A can be cb-norm approximated by elementary operators. By [15,
Theorem 1.5] every such a derivation is necessarily inner in a case when every
Glimm ideal of A is prime. When this fails, it is possible to produce examples
which have outer derivations that are simultaneously elementary operators ([12,
Example 6.1]).
Whileconsideringderivationsdthatareelementaryoperatorsand/ornormlimits
of inner derivations, we realized that they are sometimes expressible in the form
d=M M even though they are not inner. We have not been able to decide
a,b b,a
−
when all such d are of this form, but this led us to the seemingly simpler question
of considering the closures of TM(A) and TM (A). In this paper, we see that
0
nontrivialconsiderationsenterintothesequestionsabouttwo-sidedmultiplications.
Of course the left multiplications M : a M(A) are already norm closed, as
a,1
{ ∈ }
are the right multiplications. So TM(A) is a small subclass of ℓ(A), and seems to
E
be the basic case to study.
Thispaperisorganizedasfollows. Webeginin 2withsomegeneralitiesandan
§
explanation that the set of elementary operators of length at most ℓ has the same
completion in the operator and cb-norms (for each ℓ 1).
≥
TWO-SIDED MULTIPLICATIONS AND PHANTOM LINE BUNDLES 3
In 3, we show that for a prime C∗-algebra A, we always have TM(A) and
§
TM (A) both norm closed.
0
In 4, we recall the description of (n-)homogeneous C∗-algebras A as sections
§
Γ ( ) of M -bundles over X = Prim(A) and some general results about IB(A),
0 n
E E
ICB(A) and ℓ(A) for such A.
E
In 5, for homogeneous C∗-algebras A = Γ ( ), we consider subclasses IB (A)
0 1
§ E
and IB (A) of IB(A) that seem (respectively) to be the most obvious choices for
0,1
thenormclosureofTM(A)andofTM (A),extrapolatingfromfibrewiserestrictions
0
on φ TM(A). For each φ IB (A) we associate a complex line subbundle of
1 φ
∈ ∈ L
therestriction toanopensubsetU X =Prim(A), whereU isdeterminedby
U
E| ⊆
φ as the cozero set of φ (U identifies the fibres of on which φ acts by a nonzero
E
operator). For separable A, the main result of this section is Theorem 5.15, where
we characterize the condition φ TM(A) in terms of triviality of the bundle .
φ
∈ L
We close 5 with Remark 5.21 comparing our bundle considerations to slightly
§
similar results in the literature for innerness of C(X)-linear automorphisms when
X is compact, or for some more general unital A.
Our final 6 is the main section of this paper. For homogeneous C∗-algebras
§
A = Γ ( ), we characterize operators φ in the norm closure of TM (A) as those
0 0
E
operatorsin IB (A) for which the associatedcomplex line bundle is trivial on
0,1 φ
L
eachcompactsubsetofU,whereU isasabove(Theorem6.9). Asaconsequence,we
obtain that TM (A) fails to be norm closed if and only if there exists a σ-compact
0
open subset U of X and a phantom complex line subbundle of (i.e. is not
U
L E| L
globally trivial, but is trivial on each compact subset of U). Using this and some
algebraictopologicalideas,weshowthatTM(A)andTM (A)bothfailtobe norm
0
closed whenever A is n-homogeneous with n 2 and X contains an open subset
≥
homeomorphic to Rd for some d 3 (Theorem 6.18).
≥
2. Preliminaries
Throughout this paper A will denote a C∗-algebra. By an ideal of A we always
mean a closed two-sided ideal. As usual, by Z(A) we denote the centre of A, by
M(A)themultiplier algebra ofA,andbyPrim(A)theprimitive spectrum ofA(i.e.
thesetofkernelsofallirreduciblerepresentationsofAequippedwiththeJacobson
topology).
Every φ IB(A) is linear over Z(M(A)) and, for any ideal I of A, φ induces a
∈
map
(2.1) φ :A/I A/I, which sends a+I to φ(a)+I.
I
→
It is easy to see that the norm (resp. cb-norm) of an operator φ IB(A) (resp.
∈
φ ICB(A)) can be computed via the formulae
∈
φ =sup φ : P Prim(A) resp.
P
k k {k k ∈ }
(2.2) φ =sup φ : P Prim(A) .
cb P cb
k k {k k ∈ }
The length of a non-zero elementary operator φ ℓ(A) is the smallest positive
∈E
integer ℓ=ℓ(φ) such that φ= ℓ M for some a ,b M(A). We also define
i=1 ai,bi i i ∈
ℓ(0)=0. We write ℓ (A) for the elementary operators of length at most ℓ. Thus
ℓ
E P
ℓ (A)=TM(A).
1
E
4 I.GOGIC´ ANDR.M.TIMONEY
We will also consider the following subsets of TM(A):
TMcp(A) = Ma,a∗ :a M(A) ,
{ ∈ }
InnAutalg(A) = Ma,a−1 :a M(A), a invertible , and
{ ∈ }
(2.3) InnAut(A) = Mu,u∗ :u M(A), u unitary
{ ∈ }
(where cp and alg signify, respectively, completely positive and algebraic). Note
that InnAut(A)=TM (A) InnAut (A).
cp alg
∩
It is well known that elementary operators are completely bounded with the
following estimate for their cb-norm:
(2.4) M a b ,
(cid:13) ai,bi(cid:13) ≤(cid:13) i⊗ i(cid:13)
(cid:13)Xi (cid:13)cb (cid:13)Xi (cid:13)h
(cid:13) (cid:13) (cid:13) (cid:13)
where is the Haager(cid:13)up tensor n(cid:13)orm o(cid:13)n the algeb(cid:13)raic tensor product M(A)
k·kh (cid:13) (cid:13) (cid:13) (cid:13) ⊗
M(A), i.e.
1 1
2 2
u =inf a a∗ b∗b : u= a b .
k kh (cid:13) i i(cid:13) (cid:13) i i(cid:13) i⊗ i
(cid:13)Xi (cid:13) (cid:13)Xi (cid:13) Xi 
(cid:13) (cid:13) (cid:13) (cid:13)
By inequality (2.4) the(cid:13)mapping(cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13) 
(M(A) M(A), ) ( ℓ(A), ) given by a b M .
⊗ k·kh → E k·kcb i⊗ i 7→ ai,bi
i i
X X
definesawell-definedcontraction. ItscontinuousextensiontothecompletedHaagerup
tensorproductM(A) M(A) is knownas a canonical contraction fromM(A)
h h
⊗ ⊗
M(A) to ICB(A) and is denoted by Θ .
A
We have the following result (see [2, Proposition 5.4.11]):
Theorem 2.1 (Mathieu). Θ is isometric if and only if A is a prime C∗-algebra.
A
The next result is a combination of [36, Corollary 3.8], (2.2) and the facts that
for φ = ℓ M , we have φ = φ and φ = φ where for
i=1 ai,bi k πk k kerπk k πkcb k kerπkcb
irreducible representation π: A (H ), φ = ℓ M ℓ( (H )) (as
P →B π π i=1 π(ai),π(bi) ∈E B π
in [36, 4]).
§ P
Theorem 2.2 (Timoney). For A a C∗-algebra and arbitrary φ ℓ(A) of length
∈ E
ℓ we have
φ = φ(ℓ) √ℓ φ ,
cb
k k k k≤ k k
where φ(ℓ) denotes the ℓ-th amplification of φ on M (A), φ(ℓ) :[x ] [φ(x )].
ℓ i,j i,j
7→
In particular, on each ℓ (A) the metric induced by the cb-norm is equivalent to
ℓ
E
the metric induced by the operator norm.
3. Two-sided multiplications on prime C∗-algebras
IfAisaprimeC∗-algebra,weprovehere(Theorem3.4)thatTM(A)andTM (A)
0
must be closed in (A).
B
The crucial step is the following lemma.
Lemma 3.1. Let a,b,c and d be norm-one elements of an operator space V. If
a b c d <ε 1/3,
h
k ⊗ − ⊗ k ≤
TWO-SIDED MULTIPLICATIONS AND PHANTOM LINE BUNDLES 5
then there exists a complex number µ of modulus one such that
max a µc , b µd <6ε.
{k − k k − k}
Proof. First we dispose of the simpler cases where a and c are linearly dependent
orwherebanddarelinearlydependent. Ifaandcaredependentthena=µcwith
µ =1. So a b c d=c (µb d) and a b c d = µb d <ε<6ε.
h
| | ⊗ − ⊗ ⊗ − k ⊗ − ⊗ k k − k
Similarly if b and d are dependent, b=µ¯d with µ =1 and µ¯a c <ε.
| | k − k
Leaving aside these cases, a b c d is a tensor of rank 2. By [5, Lemma 2.3]
⊗ − ⊗
there is an invertible matrix S M such that
2
∈
b
a c S <ε and S−1 <ε.
k − k d
(cid:13) (cid:20) (cid:21)(cid:13)
(cid:2) (cid:3) (cid:13) (cid:13)
Write α for the i,j entry of S and β for(cid:13)the i,j e(cid:13)ntry of S−1.
i,j i,j (cid:13) (cid:13)
Since α β +α β = 1, at least one of the absolute values α , β ,
1,1 1,1 1,2 2,1 1,1 1,1
| | | |
α or β must be at least 1/√2. We treat the four cases separately, by very
1,2 2,1
| | | |
similar arguments.
α 1/√2: Now from
1,1
| |≥
a c S = α a α c α a α c
1,1 2,1 1,2 2,2
− − −
we have α(cid:2) a α(cid:3) c <(cid:2) ε, so (cid:3)
1,1 2,1
k − k
α ε
a 2,1c < √2ε.
− α α ≤
(cid:13) 1,1 (cid:13) | 1,1|
(cid:13) (cid:13)
Let λ=α /α . (cid:13)Then (cid:13)
2,1 1,1 (cid:13) (cid:13)
1 λ = a λ c a λc √2ε,
| −| || |k k−| |k k|≤k − k≤
and so λ [1 √2ε,1+√2ε]. Also
| |∈ −
λ 1 λ √2ε
λ = λ | −| || (1+√2ε) <(√2+2)ε
λ − | | λ ≤ 1 √2ε
(cid:12)| | (cid:12) | | −
(cid:12) (cid:12)
(sinc(cid:12)e ε 1/(cid:12)3). So for µ= λ we have µ =1 and
(cid:12) ≤ (cid:12) |λ| | |
a µc a λc + λ µ c <√2ε+(√2+2)ε<5ε.
k − k≤k − k | − |k k
Then
a b c d = a b (µc µ¯d)
⊗ − ⊗ ⊗ − ⊗
= (a b) (a µ¯d)+(a µ¯d) (µc µ¯d)
⊗ − ⊗ ⊗ − ⊗
= (a (b µ¯d))+((a µc) µ¯d)
⊗ − − ⊗
and thus
b µ¯d = a (b µ¯d)
h
k − k k ⊗ − k
a b c d + (a µc) µ¯d
h h
≤ k ⊗ − ⊗ k k − ⊗ k
< ε+5ε=6ε.
α 1/√2: Westartnowwith α a α c <εandproceedinthesame
1,2 1,2 2,2
| |≥ k − k
way (with λ=α /α ).
2,2 1,2
6 I.GOGIC´ ANDR.M.TIMONEY
β 1/√2: In this case we use
1,1
| |≥
b β b+β d
S−1 = 1,1 1,2
d β b+β d
2,1 2,2
(cid:20) (cid:21) (cid:20) (cid:21)
and β b+β d <ε,leadingtoasimilarargument(withλ= β /β
1,1 1,2 1,2 1,1
k k −
and b taking the role of a).
β 1/√2: In this case use β b+β d <ε. (cid:3)
2,1 2,1 2,2
| |≥ k k
Corollary 3.2. If V is an operator space, the set S = a b : a,b V of all
1
{ ⊗ ∈ }
elementary tensors forms a closed subset of V V.
h
⊗
Proof. Suppose a b u V V (for a ,b V). If u=0, certainly u S
n n h n n 1
⊗ → ∈ ⊗ ∈ ∈
and otherwise we may assume u =1 and also that
h
k k
a b =1= a = b (n 1).
n n h n n
k ⊗ k k k k k ≥
Passing to a subsequence, we may suppose
1
a b a b (n 1).
k n⊗ n− n+1⊗ n+1kh ≤ 6 2n ≥
·
By Lemma 3.1, we may multiply a and b by complex conjugate modulus one
n n
scalars chosen inductively to get a′ and b′ such that
n n
a b =a′ b′ , a′ a′ 1/2n and b′ b′ 1/2n (n 1).
n⊗ n n⊗ n k n− n+1k≤ k n− n+1k≤ ≥
Inthiswaywefinda=lim a′ andb=lim b′ inV withu=a b S . (cid:3)
n→∞ n n→∞ n ⊗ ∈ 1
Question 3.3. If V is an operator space and ℓ>1, is the set
ℓ
S = a b :a ,b V
ℓ i i i i
( ⊗ ∈ )
i=1
X
of all tensors of rank at most ℓ closed in V V? In particular, can we extend
h
⊗
Lemma 3.1 as follows.
Let V be an operator space and let a b and c d be two norm-one tensors
⊙ ⊙
of the same (finite) rank ℓ in V V, where a,c and b,d are, respectively, 1 ℓ
h
⊗ ×
and ℓ 1 matrices with entries in V. Suppose that a b c d <ε for some
h
× k ⊙ − ⊙ k
ε>0. Can we find absolute constants C and δ (which depend only on ℓ and ε) so
that δ 0 as ε 0 with the following property:
→ →
There exists an invertible matrix S M such that
ℓ
∈
S , S−1 C, aS−1 c <δ and Sb d <δ?
k k k k≤ k − k k − k
Theorem 3.4. If A is a prime C∗-algebra, then both TM(A) and TM (A) are
0
norm closed.
Proof. By Theorem 2.2 we may work with the cb-norm instead of the (operator)
norm. Since A is prime, by Mathieu’s theorem (Theorem 2.1) the canonical map
Θ : M(A) M(A) ICB(A), Θ : a b M , is isometric. By Corollary 3.2,
h a,b
⊗ → ⊗ 7→
the set S of all elementary tensors in M(A) M(A) is closed in the Haagerup
1 h
⊗
norm. Therefore, TM(A)=Θ(S) is closed in the cb-norm.
ForthecaseofTM (A),weusethesameargumentbutworkwiththerestriction
0
of Θ to A A. (cid:3)
h
⊗
Corollary 3.5. If A is a prime C∗-algebra, then the sets TM (A), InnAut (A),
cp alg
and InnAut(A) (see (2.3)) are all norm closed.
TWO-SIDED MULTIPLICATIONS AND PHANTOM LINE BUNDLES 7
Proof. Suppose that an operator φ in the norm closure of any of these sets. Then,
by Theorem 3.4 there are b,c M(A) such that φ=M . Let ε>0.
b,c
∈
SupposethatφisintheclosureofTM (A). ByTheorem2.2wemayworkwith
cp
thecb-norminsteadofthe(operator)norm. Wemayalsoassumethat φ =1=
cb
k k
b = c . Then there is a M(A) such that
k k k k ∈
Mb,c Ma,a∗ cb = b c a a∗ h <ε
k − k k ⊗ − ⊗ k
(Theorem 2.1). If ε 1/3, by Lemma 3.1 we can find a complex number µ of
≤
modulus one such that b µa < 6ε and c µa∗ < 6ε. Then b c∗ 12ε.
k − k k − k k − k ≤
Hence c=b∗, so φ=M TM (A).
b,c cp
∈
Suppose that φ is in the closure of InnAut (A). Then there is an invertible
alg
element a∈M(A) such that kMb,c−Ma,a−1k<ε. Since A is an essential ideal in
M(A), this implies bxc axa−1 < ε for all x M(A), x 1. Letting x = 1
k − k ∈ k k ≤
we obtain bc 1 <ε. Hence c=b−1, so φ=M InnAut (A).
b,c alg
k − k ∈
InnAut(A) is norm closed as an intersection TM (A) InnAut (A) of two
cp alg
closed sets. ∩ (cid:3)
4. On homogeneous C∗-algebras
We recall that a C∗-algebra A is called n-homogeneous (where n is finite) if
every irreducible representation of A acts on an n-dimensional Hilbert space. We
say that A is homogeneous if it is n-homogeneous for some n. We will use the
following definitions and facts about homogeneous C∗-algebras:
Remark 4.1. Let A be an n-homogeneous C∗-algebra. By [20, Theorem 4.2]
Prim(A) is a (locally compact) Hausdorff space. If there is no danger of confusion,
we simply write X for Prim(A).
(a) A well-known theorem of Fell [10, Theorem 3.2], and Tomiyama-Takesaki
[37, Theorem 5] asserts that for any n-homogeneous C∗-algebra, A, there
is a locally trivial bundle over X with fibre M and structure group
n
E
PU(n) = Aut(M ) such that A is isomorphic to the C∗-algebra Γ ( ) of
n 0
E
continuous sections of which vanish at infinity.
E
Moreover,any two such algebras A =Γ ( ) with primitive spectra X
i 0 i i
E
(i=1,2)are isomorphicif andonly ifthere is ahomeomorphismf :X
1
→
X such that = f∗( ) (the pullback bundle) as bundles over X (see
2 E1 ∼ E2 1
[37, Theorem 6]). Thus, we may identify A with Γ ( ).
0
E
(b) Fora Aandt X wedefineπ (a)=a(t). Then,afteridentifyingthefibre
t
∈ ∈
with M , π :a π (a) (for t X) gives all irreducible representations
t n t t
E 7→ ∈
of A (up to the equivalence).
For a closed subset S X we define
⊆
I = kerπ = a A : a(t)=0 for all t S .
S t
{ ∈ ∈ }
t∈S
\
By [11, VII 8.7.] any closed two-sidedideal of A is of the form I for some
S
closed subset S X. Further, by the the generalized Tietze Extension
⊂
Theorem we may identify A = A/I with Γ ( ) (see [11, II. 14.8. and
S S 0 S
E|
VII 8.6.]). If S = t we just write A .
t
{ }
(c) If φ IB(A) and S X closed, we write φ for the operator φ on A
∈ ⊂ S IS S
(see(2.1)). IfS = t wejustwriteφ . IfAistrivial(i.e. A=C (X,M )),
t 0 n
{ }
8 I.GOGIC´ ANDR.M.TIMONEY
we will consider φ as an operator : M M (after identifying A with
t n n t
→
M in the obvious way).
n
IfU X isopen,wecanregardB =Γ ( )astheidealI ofA(by
0 U X\U
⊂ E|
extending sections to be zero outside U) and for φ IB(A), we then have
∈
a restriction φ IB(B) of φ to this ideal (with (φ ) =φ for t U).
U U t t
| ∈ | ∈
(d) IB(A)=ICB(A). Indeed,forφ IB(A)andt X wehave φ n φ
t cb t
∈ ∈ k k ≤ k k
([29, p. 114]), so by (2.2) we have φ n φ . Hence φ ICB(A).
cb
k k ≤ k k ∈
(e) Since each a Z(A) has a(t) a multiple of the identity in the fibre for
t
∈ E
each t X, we can identify Z(A) with C (X). Observe that A is quasi-
0
∈
central (i.e. no primitive ideal of A contains Z(A)).
(f) By [25, Lemma 3.2] we can identify M(A) with Γ ( ) (the C∗-algebra of
b
E
bounded continuous sections of ). As usual, we will identify Z(M(A))
E
with C (X) (using the Dauns-Hofmann theorem [33, Theorem A.34]).
b
IfA=C (X,M ),itiswellknownthatM(A)=C (X,M )=C(βX,M )
0 n b n n
[1, Corollary 3.4], where βX denotes the Stone-Cˇech compactification.
(g) On each fibre we can introduce an inner product , as follows.
t 2
E h· ·i
Choose an open covering U of X such that each is isomorphic
{ α} E|Uα
to U M (as an M -bundle), say via isomorphism Φ . Let
α n n α
×
(4.1) ξ,η =tr(Φ (ξ)Φ (η)∗) (ξ,η ),
2 α α t
h i ∈E
where α is chosen so that t U and tr() is the standard trace on M .
α n
∈ ·
This is independent of the choice of α since all automorphisms of M are
n
innerandtr()isinvariantunderconjugationbyunitaries. Ifa,b M(A)=
· ∈
Γ ( ) then t a(t),b(t) is in C (X).
b 2 b
E 7→h i
The norm on associated with , satisfies
2 t 2
k·k E h· ·i
(4.2) ξ ξ √n ξ (ξ ).
2 t
k k≤k k ≤ k k ∈E
Intheterminologyof[8],( , , )isa(complexcontinuous)Hilbertbundle
2
E h· ·i
of rank n2 with fibre norms equivalent to the original C∗-norms (by (4.2)).
(h) A is said to have the finite type property if can be trivialized over some
E
finite open cover of X. By [25, Remark 3.3] M(A) is homogeneous if and
onlyifAhasthe finite typeproperty. Whenthisfails,itispossibletohave
Prim(M(A)) non-Hausdorff[4, Theorem 2.1]. On the other hand, M(A) is
always quasi-standard (see [3, Corollary 4.10]).
For completeness we include a proof of the following.
Proposition4.2. LetX bealocallycompact HausdorffspaceandA=C (X,M ).
0 n
(a) IB(A) can be identified with C (X, (M )) by a mapping which sends an
b n
B
operator φ IB(A) to the function (t φ ).
t
∈ 7→
(b) Any φ IB(A) can be written in the form
∈
n
(4.3) φ= M ,
ek,i,ak,i
k,i=1
X
where (e )n are standard matrix units of M (considered as constant
k,i k,i=1 n
functions in C (X,M )=M(A)) and a M(A) depend on φ.
b n k,i
∈
Thus, we have
IB(A)=ICB(A)=Cb(X, (Mn))= ℓ(A)= ℓn2(A).
B E E
TWO-SIDED MULTIPLICATIONS AND PHANTOM LINE BUNDLES 9
Proof. Let φ IB(A).
∈
(a) Suppose that the function t φ : X (M ) is discontinuous at some
t n
7→ → B
pointt X. Thenthereisanet(t )inX convergingtot suchthat φ φ
0 ∈ α 0 k tα− t0k≥
δ >0forallα. Sothereisu M ofnormatmost1with φ (u ) φ (u ) δ.
α ∈ n k tα α − t0 α k≥
Passing to a subnet we may suppose u u and then (since φ φ and
α → k tαk ≤ k k
φ φ ) we must have
k t0k≤k k
φ (u) φ (u) >δ/2
k tα − t0 k
for α large enough.
Nowchoosef C (X)equalto1onaneighbourhoodoft andputa(t)=f(t)u.
0 0
∈
We then have a A and
∈
π (φ(a))=f(t )φ (u)=φ (u)
tα α tα tα
for large α and this contradicts continuity of φ(a) at t .
0
So t φ must be continuous (and also bounded by φ ).
t
7→ k k
Conversely,assumethatthefunctiont φ iscontinuousanduniformlybounded
t
7→
by some M > 0. Then for a A, t φ (π (a)) is continuous, bounded and van-
t t
∈ 7→
ishes at infinity, hence in A. So there is an associatedmapping φ: A A which is
→
easily seen to be bounded and linear. Moreoverφ IB(A) since all ideals of A are
∈
of the form I for some closed S X.
S
⊂
(b) First assume that A is unital, so that X is compact. Then each x A is a
∈
linear combination over C(X) = Z(A) of the e and since φ is C(X)-linear, we
i,j
have
n n
x= x e φ(x)= x φ(e ).
i,j i,j i,j i,j
⇒
i,j=1 i,j=1
X X
We may write
n n n
φ(e )= φ e = e e φ e
i,j i,j,k,ℓ k,ℓ k,i i,j i,j,k,ℓ j,ℓ
!
k,ℓ=1 k=1 ℓ=1
X X X
where φ C(X). It follows that
i,j,k,ℓ
∈
n n n
φ(x)= e x φ e φ= M ,
k,i  i,j,k,ℓ j,ℓ ⇒ ek,i,ak,i
k,i=1 j,ℓ=1 k,i=1
X X X
 
where a = n φ e M(A).
k,i j,ℓ=1 i,j,k,ℓ j,ℓ ∈
Now suppose that A is non-unital (so that X is non-compact). By (a) we can
P
identify φ with the function t φ : X (M ), which can be then uniquely ex-
t n
7→ →B
tended to a continuous function βX (M ). This extension defines an operator
n
→B
in IB(C(βX,M )) = IB(M(A)), which we also denote by φ. By the first part of
n
the proof, φ can be represented as (4.3). (cid:3)
Remark 4.3. Infact,inthecaseofgeneralseparableC∗-algebrasA,Magajna[25]
establishes the equivalence of the following properties:
(a) IB(A)= ℓ(A).
E
(b) ℓ(A) is norm dense in IB(A).
E
(c) Ais afinite directsums ofhomogeneousC∗-algebraswiththe finite typeprop-
erty.
10 I.GOGIC´ ANDR.M.TIMONEY
Analyzing the arguments in [25], for the implication (c) (a) it is sufficient to
⇒
assume that X is paracompact.
Since any n-homogeneous C∗-algebra is locally of the form C(K,M ) for some
n
compactsubsetK ofX with K◦ = ,we havethe following consequence ofPropo-
6 ∅
sition 4.2:
Corollary 4.4. If A is a homogeneous C∗-algebra, then for any φ IB(A) the
∈
function t φ is continuous on X. Hence the cozero set coz(φ) = t X :
t
7→ k k { ∈
φ =0 is σ-compact and open in X.
t
6 }
5. Fibrewise length restrictions
HereweconsiderahomogeneousC∗-algebraA=Γ ( )andoperatorsφ IB(A)
0
E ∈
such that φ is a two-sided multiplication on each fibre A (with t X, and X =
t t
∈
Prim(A) as usual). We will write φ IB (A) for this hypothesis. For separable A,
1
∈
themainresultinthissection(Theorem5.15)characterizeswhenallsuchoperators
φ aretwo-sidedmultiplications, in terms of triviality ofcomplex line subbundles of
for U X open.
U
E| In addit⊂ion to IB (A), we introduce various subsets IBnv(A) and IB (A) (No-
1 1 0,1
tation 5.5) which are designed to facilitate the description of TM(A), TM (A)
0
and both of their norm closures in terms of complex line bundles. The sufficient
condition that ensures IBnv(A) TM(A) is that X is paracompactwith vanishing
secondintegralCˇechcoho1molog⊂ygroupHˇ2(X;Z)(Corollary5.11). ForX compact
of finite covering dimension d and A = C(X,M ) we show that TM(A) ( IB (A)
n 1
provided Hˇ2(X;Z) = 0 and n2 (d+1)/2 (Proposition 5.12). We get the same
6 ≥
conclusion TM(A) ( IB (A) for σ-unital n-homogeneous C∗-algebras A = Γ ( )
1 0
E
withn 2providedX hasanonemptyopensubsethomeomorphicto(anopenset
≥
in) Rd with d 3 (Corollary 5.16).
≥
Notation 5.1. Let A be an n-homogeneous C∗-algebra. For ℓ 1 we write
≥
IB (A)= φ IB(A):φ ℓ (A ) for all t X .
ℓ t ℓ t
{ ∈ ∈E ∈ }
Lemma 5.2. Let A=Γ ( ) be a homogeneous C∗-algebra and φ IB (A).
0 ℓ
E ∈
If t X is such that φ ℓ (A ) ℓ (A ) (that is, such that φ has
0 ∈ t0 ∈ E ℓ t0 \E ℓ−1 t0 t0
lengthexactlythemaximalℓ),thentherearea ,...,a ,b ,...,b Aandacompact
1 ℓ 1 ℓ
∈
neighbourhood N of t such that φ agrees with the elementary operator ℓ M
0 i=1 ai,bi
modulo the ideal I , that is
N
P
ℓ
φ(x) a xb I for all x A.
i i N
− ∈ ∈
i=1
X
Moreover, we can choose N so that φ ℓ (A ) ℓ (A ) for all t N (that is,
t ℓ t ℓ−1 t
∈E \E ∈
φ is of the maximal length ℓ for t in a neighbourhood of t ).
t 0
Proof. Choose a compact neighbourhood K of t such that A = C(K,M ) and
0 K ∼ n
let φ be the induced operator (Remark 4.1 (b), (c)).
K
Then, for x A we have φ (x) = n2 c xd for some c ,d A (by
∈ K K i=1 i i i i ∈ K
Proposition 4.2 (b)). Moreover we can assume that c1(t),...,cn2(t) are linearly
P { }
independent for each t K, and even independent of t. Since (φ ) = φ has
∈ K t0 t0