Projection decomposition in multiplier algebras

0.21 MB·English

by  V. Kaftal

PROJECTION DECOMPOSITION IN MULTIPLIER ALGEBRAS
VICTORKAFTAL,P.W.NG,ANDSHUANGZHANG
2
1
0
Abstract. In this paper we present new structural information about the
2 multiplier algebra M(A) of a σ-unital purely infinite simple C∗-algebra A,
n by characterizing the positive elements A ∈ M(A) that are strict sums of
a projections belonging to A. If A 6∈ A and A itself is not a projection, then
J the necessary and sufficient condition for A to be a strict sum of projections
3 belongingtoAisthatkAk>1andthattheessentialnormkAkess≥1.
2 Based on a generalization of the Perera-Rordam weak divisibilityof sepa-
rable simple C∗-algebras of real rank zero to all σ-unital simple C∗-algebras
] ofrealrankzero,weshowthateverypositiveelementofAwithnormgreater
A than1canbeapproximatedbyfinitesumsofprojections. Basedonblocktri-
diagonal approximations, wedecompose anypositiveelementA∈M(A)with
O
kAk>1andkAkess≥1intoastrictlyconvergingsumofpositiveelementsin
. Awithnormgreater than1.
h
t
a
m
[ 1. Introduction and the main result

1 In [9] Fillmore raised the following question: Which positive bounded operators
v on a separable Hilbert space H can be written as (finite) sums of projections?
9
Fillmore obtained a characterization of the finite rank operators that are sums of
1
8 projections (see [9] Theorem1) and of the bounded operators that are the sums of
4 two projections (see [9] Theorem 2).
1. Forinfinitesumsofprojectionswithconvergenceinthestrongoperatortopology,
0 this question arose naturally from work on frame theory by Dykema, Freeman,
2 Kornelson, Larson, Ordower and Weber (see [5]). They proved that a sufficient
1 condition for a positive bounded operator A B(H) to be a (possibly infinite)
: ∈
v sum of projections converging in the strong operator topology is that its essential
i norm A is greaterthan1(see [5]Theorem2). This resultservedasabasisfor
X ess
k k
further work by Kornelson and Larson [13] and then by Antezana, Massey, Ruiz
r
a and Stojanoff [1] on decompositions of positive operators into strongly converging
sums of rank one positive operatorswith preassignednorms. In [12], the necessary
andsufficientconditionforapositiveboundedoperatortobeastronglyconverging
sum of projections was obtained by the three authors of this article for the B(H)
case and for the case of a countably decomposable type III von Neumann factor,
and for the “diagonalizable” case of type II von Neumann factors.
In this paper, we extend the characterization of the positive operators that are
sums of projections to the case of bounded module maps (with adjoints defined)
on Hilbert C∗-modules, namely, B(H) is replaced by the multiplier algebra M(A)
of A. Dealing with multiplier algebras, we replace the strong operator topology
by the strict topology. We point out that when A is reduced to the algebra of
complex numbers C, then B(H) =M(K), the multiplier algebra of the C∗-algebra
K of compact operators on a separable Hilbert space, and the *-strong operator
topology on B(H) is precisely the strict topology of M(K).
1
2 VICTORKAFTAL,P.W.NG,ANDSHUANGZHANG
In this article we generalize the main result of [5] to certain multiplier algebras,
stated as follws.
Theorem 1.1. Let A be a σ-unital simple purely infinite C∗-algebra and A be
a positive element of M(A). Then A is a strictly converging sum of projections
belonging to A if and only if one of the following mutually exclusive conditions
hold:
(i) A >1.
ess
k k
(ii) A =1 and A >1.
ess
(iii)kAk M(A) A iks akprojection.
(iv) A ∈is the su\m of finitely many projections belonging to A.
When A is unital and hence M(A)=A, if a positive element A A is a strictly
converging sum of (nonzero) projections belonging to A, then th∈e sum must be
finite (Proposition 3.1), stated as the case (iv).
The non-trivial case is thus when A M(A) A where A is σ-unital but non-
∈ \
unital. Notice that such a C∗-algebra (σ-unital but non-unital simple purely infi-
nite) is necessarily stable (see [32] and [22]) and has real rank zero ([29, 1.2]).
Thenecessityoftheconditions(i)–(iii)isgivenbyCorollary3.3. Thesufficiency
of (iii) being trivial, the main focus of this paper is to prove the sufficiency of (i)
and (ii).
The proof is arranged in the following way.
In section 2 we prove that all non-elementary, σ-unital, simple C∗-algebras of
real rank zero are weakly divisible in the sense of Perera-Rordam in [19], thus
generalizing the previous result of [19] from the separable category to the σ-unital
category. This weak divisibility property and Fillmore’s characterization of the
finite rank operators in B(H) enable us to approximate a positive element with a
norm greater than 1 by finite sums of projections (Lemma 2.5.)
In section 3 we prove that a positive element A of M(A) with essential norm
A >1 can be written as a strict sum of projections in A.
ess
k k
Section 4 deals with the crucialcase when A =1 and A >1. We employ
ess
k k k k
a block tri-diagonal approximation and operator theory techniques to construct
a strictly converging sequence of projections f A for which f Af > 1 for
k k k
∈ k k
all k (Lemma 4.4). From that, we decompose A into a strict sum of projections
(Proposition 4.6) and conclude the proof.
Aside from the works on the B(H) and von Neumann factors cases that have
been mentioned above, this paper employs some previous results and ideas on the
structures of multiplier algebras of simple purely infinite C∗-algebras scattered in
the severalpapers such as [14], [16], [17], [19], and [21] – [33].
ThefirstandsecondnamedauthorsparticipatedintheNSFsupportedWorkshop
in Analysis and Probability, Texas A & M University, Summer 2006, where they
firstheardfrom DavidLarsonaboutthe results in [5] and [13] that stimulated this
project.
The first and third author were partially supported by grants from the Charles
Phelps Taft Research Center.
2. Weak divisibility of σ-unital C∗-algebras of real rank zero
In this section we show that in a σ-unital simple purely infinite C∗-algebra A,
everypositive element with norm greaterthan 1 canbe approximatedfrom under-
neath by finite sums of projections. To do so we first extend to all non-elementary
PROJECTION DECOMPOSITION 3
σ-unitalC∗-algebrasofrealrankzerothe propertyofweakdivisibilityobtainedfor
separable non-elementary simple C∗-algebras of real rank zero by Perera and Ror-
dam in [19, 5.3]. Recall that a C∗-algebra is called non-elementary if it is neither
K nor M for any n. A C∗-algebra is called to be σ-unital, if it has a strictly pos-
n
itive element b, namely, (bA)− = (Ab)− = A. A C∗-algebra B is weakly divisible
([19, 5.1,5.2]) if and only if for any nonzero projection p of B there exists a unital
*-homomorphismfrom M M to pBp.
2 3
⊕
Proposition2.1. IfA is a non-elementaryσ-unital simple C∗-algebra of realrank
zero, then A is weakly divisible.
Proof. By [19, Lemma 5.2]it suffices to show that for eachnonzero projectionp of
A there exists a unital *-homomorphism from M M into pAp. To prove this,
2 3
⊕
we use the result of divisibility of all projections in any simple C∗-algebra of real
rank zero in [26, 1.1]: For each pair of projections (q,r) in A and each natural
numbern the projectionq canbe rewrittenasa directsumofmutually orthogonal
subprojections
q =p1 p2 p2n r0
⊕ ⊕···⊕ ⊕
such that p is equivalent to p for all pairs (i,j) in the sense of Murray-von Neu-
i j
mann and r is equivalent to a subprojection of r.
0
Applying this result to the case q =r =p and n=1, one has
p=p p r
1 2 0
⊕ ⊕
where p is equivalent to p and r is equivalent to a subprojection of p , say r .
1 2 0 1 1
Choose a partial isometry v such that p = vv∗, p = v∗v, and set r = v∗r v.
1 2 2 1
Then
p=(p r ) (p r ) r r r .
1 1 2 2 0 1 2
− ⊕ − ⊕ ⊕ ⊕
Thenp r andp r areequivalent,andsoarer ,r andr . Thisdecomposition
1 1 2 2 0 1 2
of p into−these five−projections leads to a unital *-homomorphism from M M
2 3
into pAp. ⊕ (cid:3)
The same idea above also proves the following lemma that will be used as one
of the technical ingredients in this article.
Lemma 2.2. Let A be a non-elementary σ-unital simple C∗-algebra of real rank
zero. Then for every integer n 1 and for every nonzero projection p of A there
exists a unital -embedding of M≥2n M2n+1 into pAp.
∗ ⊕
Proof. Applying [26, 1.1] to the case q = r = p and arbitrary natural number n,
one has
p=p p p r
1 2 2n 0
⊕ ⊕···⊕ ⊕
wherep is equivalentto p forallpair(i,j)andr isequivalenttoasubprojection
i j 0
of p , say r . For each k choose a partial isometry v pAp such that p = v v∗
1 1 k ∈ 1 k k
and p =v∗v . Let r =v∗r v . Then
k k k k k 1 k
p=(p r ) (p r ) (p r ) r r r r .
1 1 2 2 2n 2n 0 1 2 2n
− ⊕ − ⊕···⊕ − ⊕ ⊕ ⊕ ⊕···⊕
Then p r and p r are equivalent for all pairs (i,j), and so are r , r , ,
i i j j 0 1
r . This−decomposit−ionofpleadstoaunital*-homomorphismfromM M···
2n 2n 2n+1
into pAp. ⊕ (cid:3)
We need the following approximation property for positive elements in a C∗-
algebra of real rank zero.
4 VICTORKAFTAL,P.W.NG,ANDSHUANGZHANG
Lemma 2.3. Let C be a C∗-algebra of real rank zero and c be any positive element
in C. For ǫ > 0 there exist pairwise orthogonal projections p ,p ,...,p in C and
1 2 n
positive real numbers α ,α ,...,α such that
1 2 n
(i) α p +α p +...+α p c <ǫ
1 1 2 2 n n
k − k
(ii) α p +α p +...+α p c.
1 1 2 2 n n
≤
Proof. Without loss of generality, assume that ǫ < 2 c . Let g be the piecewise
k k
linear function
0 0 x ǫ/2
g(x)=: ≤ ≤
(x ǫ/2 x>ǫ/2.
−
The hereditary subalgebra of C generated by g(c)Cg(c) still has real rank zero
([3]). Thus one can find a positive element d g(c)Cg(c) with finite spectrum, say
∈
d=α p +α p +...+α p , such that g(c) d <ǫ/2. It follows that
1 1 2 2 n n
k − k
c α p +α p +...+α p c g(c) + g(c) d <ǫ/2+ǫ/2=ǫ.
1 1 2 2 n n
k − k≤k − k k − k
The key point is to prove that d c. Assume without loss of generality that C act
faithfully and non-degenerately o≤n a Hilbert space H. Let q =: χ (c). For
[ǫ/2,∞)
ξ qH, one has that
∈
<(c d)ξ,ξ >=<(c g(c))ξ,ξ >+<(g(c) d)ξ,ξ >
− − −
=ǫ<ξ,ξ >+<(g(c) d)ξ,ξ >
−
(ǫ g(c) d )<ξ,ξ >
≥ −k − k
0.
≥
If ξ q⊥H, then dξ = 0 because d g(c)Cg(c), and hence also <(c d)ξ,ξ >=<
cξ,ξ∈> 0. Therefore, c d, as wan∈ted. − (cid:3)
≥ ≥
We will use the following result due to Fillmore [9, Thm. 1] (see also [5, Prop.
6] and [12, 2.5, 2.6].)
Proposition 2.4. Let tr be the natural (non-normalized) trace on the algebra M
n
of n by n complex matrices and A M be a positive matrix. Then A is a sum of
n
projections in M if and only if tr(∈A) is an integer and tr(A) rank(A).
n
≥
Recall that all simple purely infinite C∗-algebras have real rank zero ([29, 1.2]).
The following lemma is one of the two central technical ingredients of this article.
Lemma 2.5. Let A be a σ-unital purely infinite simple C∗-algebra and A A be
∈
a positive element with A >1. Then for every ǫ>0 there exist positive elements
A ,A A such that k k
1 2
∈
(i) A=A +A ,
1 2
(ii) A is the sum of finitely many projections belonging to A, and
1
(iii) A <ǫ.
2
k k
Proof. ByLemma 2.3wecanassume withoutlossofgeneralitythatAis apositive
element with finite spectrum and with norm strictly greater than one. Then there
are nonzero pairwise orthogonalprojections
e ,e ,...,e ,f ,f ,...,f A
1 2 m 1 2 n
∈
PROJECTION DECOMPOSITION 5
and strictly positive real numbers λ ,λ ,...,λ ,µ ,µ ,...,µ such that
1 2 m 1 2 n
m n
A= λ e + µ f ,
i i j j
i=1 j=1
X X
where 1 < λ and 0 < µ 1 for 1 i m and 1 j n. Note that A > 1
i j
≤ ≤ ≤ ≤ ≤ k k
implies m 1; but n=0 is possible.
≥
Choose N large enough in the form 2k such that there are positive integers
k ,k′,l ,l′ for all 1 i m and 1 j n satisfying the following inequalities:
i i j j ≤ ≤ ≤ ≤
1<k /N <λ and 1<k′/(N +1)<λ ,
i i i i
l /N <µ and l′/(N +1)<µ ,
j j j j
ǫ ǫ
0<λ k /N < and 0<λ k′/(N +1)< ,
i− i 2 i− i 2
ǫ ǫ
0<µ l /N < and 0<µ l′/(N +1)< .
j − j 2 j − j 2
By Lemma 2.2 there exists for 1 i m a unital *-homomorphismfrom M
N
M onto a C∗-subalgebra B of ≤the≤corner e Ae , and for 1 j n there is⊕a
N+1 i i i
unital*-homomorphismfromM M ontoaC∗-subalgebraC≤of≤f Af . Notice
N N+1 j j j
thatforagiveni,the projections⊕inB thatcorrespondto the minimalprojections
i
of M are all mutually equivalent, but in general they are not comparable to the
N
minimal projections in Cj or in Bi′ for i = i′ or to those in Bi that correspond to
the minimal projections ofM . The id6entity of B is e and the identity of C is
N+1 i i j
f andthis way,eachsummand λ e is identified with a direct sumof two diagonal
j i i
matrices, say B = B B in B , where B is a matrix of size N N, B is a
i i1 i2 i i1 i2
⊕ ×
matrix of size (N +1) (N +1), and both have all diagonal entries λ . Similarly,
i
×
each summand µ f is identified with a direct sum of two diagonal matrices, say
j j
C =C C in C , where C is of size N N, C is of size (N +1) (N+1),
j j1 j2 j j1 j2
⊕ × ×
and both have all diagonal entries µ .
j
Modify B =B B to B′ =B′ B′ where B′ has the same matrix units
i i1⊕ i2 i i1⊕ i2 i1
as B but has all diagonalentries k /N insteadof λ and B′ has the same matrix
i1 i i i2
units as of B but has all diagonal entries k′/(N +1) instead of λ . Similarly,
i2 i i
modify C = C C to C′ = C′ C′ by replacing the diagonal entries µ of
j j1 ⊕ j2 j j1 ⊕ j2 j
C with l /N and the diagonal entries µ of C with l′/(N +1). Let
j1 j j j2 j
m n
A′ = B′+ C′.
i j
i=1 j=1
X X
Notice that all the matrices B′ and C′ have rank N and all the matrices B′
i1 j1 i2
and C′ have rank N +1. The conditions defining k ,k′,l ,l′ imply:
j2 j j j j
ǫ
0 A′ A and A A′ < .
≤ ≤ k − k 2
tr(B′ )=N(k /N)=k and tr(B′ )=(N +1)k′/(N +1)=k′.
i1 i i i2 i i
tr(C′ )=N(l /N)=l and tr(C′ )=(N +1)l′/(N +1)=l′.
j1 j j j2 j j
Since
tr(B′ )=k >N =rank(B′ ) and tr(B′ )=k′ >N +1=rank(B′ ),
i1 i i1 i2 i i2
6 VICTORKAFTAL,P.W.NG,ANDSHUANGZHANG
by Proposition 2.4 each B′ and B′ is a sum of projections. If n = 0, A′ =
i1 i2
m B′ B′ is a sum of projections and then setting A =A′ and A =A A′
i=1 i1⊕ i2 1 2 −
will satisfy the thesis.
PFrom now on assume n 1. Since m B′ B′ is a sum of projections, it is
enoughtoprovethatB′+≥n C′ isalsio=a2suim1⊕ofpi2rojections. Lete (resp.,e )
1 j=1 j P 11 12
betheidentityofthecopyPofMN (resp.,MN+1)inB1. ThenB1′ = kN1e11+Nk+1′1e12.
Since 1< k1,there exists foreach1 j n anintegermultiple ofN, sayL , such
N ≤ ≤ j
that
k
1
L 1) N l .
j j
N − ≥ −
For every 1 j n, let f be(cid:0)a minimal projection of C′ . Denote by N f
≤ ≤ j1 j1 · j1
the identity of the copy of M in C . Then C′ = lj(N f ). Since the corner
N j j1 N · j1
e Ae of A is still simple and purely infinite, one can recursively find n L
11 11 j=1 j
mutually orthogonal projections in e Ae , where for each 1 j n, L of these
11 11 j
≤ ≤ P
projections are equivalent to f and we denote their sum by L f . Then
j1 j j1
·
n n n
k k l
D := 1(L f )+ C′ = 1(L f )+ j (N f )
N j · j1 j1 N j · j1 N · j1
Xj=1 Xj=1 Xj=1(cid:16) (cid:17)
and for each j, k1(L f ) + lj(N f ) is a matrix of size L + N and trace
N j · j1 N · j1 j
Ljk1 +l L +N and hence is the sum of projections by Proposition2.4.
N j ≥ j
Similarly, there exists for each 1 j n an integer multiple of N +1, say L′,
≤ ≤ j
such that
k
L′ 1 1 N +1 l′.
j N +1 − ≥ − j
For every 1 j n, let f (cid:0) be a mini(cid:1)mal projection of C′ , (N +1) f the
≤ ≤ j2 j2 · j2
identity ofthe copyofM in C L′ f the sumof orthogonalsubprojectionsof
N+1 j j· j2
e equivalent to f so that L′ f are also mutually orthogonal. Then
12 j2 j · j2
n k n n k l′
D′ := 1 (L′ f )+ C′ = 1 (L′ f )+ j ((N+1) f )
N +1 j· j2 j2 N +1 j· j1 N +1 · j1
Xj=1 Xj=1 Xj=1(cid:16) (cid:17)
is the sum of projections by the same argument as for D.
Finally, let e′ =e n (L f ) n (L′ f ). Then
1− j=1 j · j1 − j=1 j · j
P n P k
B′ + C′ =D+D′+ 1e′.
1 j N
j=1
X
If e′ = 0, by the same argument as for the case n = 0 one can find a sum of
projectio6 ns D′′ for which D′′ k1e′ < ǫ. Then setting
k − N k 2
m
A := B′ B′ +D+D′+D′′
1 i1⊕ i2
i=2
X
and A :=A A satisfies the thesis. (cid:3)
2 1
−
PROJECTION DECOMPOSITION 7
3. The cases A A and A >1
ess
∈ k k
We first discuss when a positive operator A in a σ-unital simple purely infinite
C∗-algebra A is a strict sum of projections in A.
Proposition 3.1. Let A be a σ-unital C∗-algebra with an approximate identity of
projections and A be a positive element in A. If A is the strict sum of projections
belonging to A, then A must be the sum of finitely many projections belonging to
A.
Proof. Wewillreasonbycontradiction. Assumethat p ∞ isaninfinitesequence
of nonzero projections in A such that A = ∞ p {, wk}hke=re1the sum converges in
k=1 k
the strict topology in M(A).
Let e ∞ be an approximate unit for APconsisting of an increasing sequence
{ n}n=1
of projections. Note that such an increasing approximate identity of projections
indeed exists in A ([28]). Choose an integer N 1 such that for all n N,
≥ ≥
A e A <1/2. As a consequence,
n
k − k
(1 e )A(1 e ) = (A e A)(1 e ) A e A <1/2
N N N N N
k − − k k − − k≤k − k
Recallaclassicalresult(forexample,see[4,LemmaIII.3.1])thatforevery0<ǫ<1
there exists a δ > 0 such that p A with dist(p,(1 e )A(1 e )) < δ implies
N N
the existence of a projection q (∈1 e )A(1 e ) s−atisfying −p q <ǫ. Such a
N N
projection q is equivalent to p i∈n A.−For ǫ=1/−2 there exists δ >k 0−. Skince ∞ p
k=1 k
converges in the strict topology on M(A), let K 1 be such that p e < δ/3
k N
≥ k Pk
for all k K. Hence, for all k K,
≥ ≥
(1 e )p (1 e ) p p e e p +e p e 3 p e <δ
N k N k k N N k N k N k N
k − − − k≤k− − k≤ k k
Thus dist(p ,(1 e )A(1 e )) < δ. It follows from the classical result stated
K N N
− −
above that there is a projection
q (1 e )A(1 e ) with p q <1/2.
N N K
∈ − − k − k
Now let B = p +q. Then
k6=K k
P B A = q pK <1/2,
k − k k − k
and hence,
(1 e )B(1 e ) (1 e )A(1 e ) <1/2.
N N N N
k − − − − − k
Applying the triangle inequality, one has
(1 e )B(1 e ) <1/2+ (1 e )A(1 e ) <1.
N N N N
k − − k k − − k
On the other hand, (1 e )B(1 e ) q implies (1 e )B(1 e ) 1,
N N N N
− − ≥ k − − k ≥
a contradiction. Therefore, A, as the strict sum of projections, must be a finite
sum. (cid:3)
We now turn to handle the sufficient condition A > 1. Let us first review
ess
k k
some elementary facts about the essential norm, which are formulated only for the
special cases that we will work with. Let A be a non-unital C∗-algebra, let π be
the canonical homomorphism from M(A) onto the corona algebra M(A)/A, and
for every A M(A), let A := π(A) denote the essential norm.
ess
∈ k k k k
Lemma 3.2. Let A be a non-unital C∗-algebra.
(i) For every positive A M(A),
∈
A =inf A(I a) a A+, a 1 .
ess
k k {k − k| ∈ k k≤ }
8 VICTORKAFTAL,P.W.NG,ANDSHUANGZHANG
(ii) Let A M(A) A be a positive element, and let a be a monotone increasing
n
sequence o∈f positive\ elements of A converging to A in the strict topology. Then
A =inf A a .
ess n
k k n k − k
Proof.
(i) SinceAa Aforeverya A,itfollowsthat A = A(I a) A(I a)
ess ess
∈ ∈ k k k − k ≤k − k
and hence
A inf A(I a) a A+, a 1 .
ess
k k ≤ {k − k| ∈ k k≤ }
If A = A , then the reverse inequality holds by choosing a = 0, so assume
ess
k k k k
that A > A . Let 0 < ǫ < A A , let h be the positive continuous
ess ess
k k k k k k−k k
function on the interval [0, A ] defined as
k k
0 t [0, A ]
ess
∈ k k
h(t):= linear t [ A , A +ǫ],
 ess ess
∈ k k k k
1 t [ A ess+ǫ, A ]
∈ k k k k
and let a := h(A). Clearly, a 0 and a = 1. Via the Gelfand’s transforma-
≥ k k
tion, identify C∗(π(A)) with the algebra of complex-valued continuous functions
C(σ (A)) defined on the essentialspectrum σ (A) of A. Since h vanishes on σ (A)
e e e
and h π =π h, it follows that π(h(A))=0 and hence h(A) A. Moreover,
◦ ◦ ∈
A(I a) = t(1 h(t)) A +ǫ,
∞ ess
k − k k − k ≤k k
whence
inf A(I a) a A+, a 1 A .
ess
{k − k| ∈ k k≤ }≤k k
Thus equality holds, proving (i).
(ii) Since for every n
A = A a A a ,
ess n ess n
k k k − k ≤k − k
it follows that
A inf A a .
ess n
k k ≤ n k − k
For every positive contraction a A and every n
∈
A a 1/2 = (A a )1/2 (A a )1/2a + (A a )1/2(I a) .
n n n n
k − k k − k≤k − k k − − k
Since 0 A a A,
n
≤ − ≤
(A a )1/2(I a) 2 = (I a)(A a )(I a) (I a)A(I a) = A1/2(I a) 2.
n n
k − − k k − − − k≤k − − k k − k
But then
A1/2(I a) (A a )1/2(I a) A a 1/2 (A a )1/2a .
n n n
k − k≥k − − k≥k − k −k − k
Since A a 0 in the strict topology it follows that (A a )1/2a 0. Since
n n
− → k − k→
a ismonotoneincreasing,itfollowsthat A a 1/2 inf A a 1/2 andhence
n n n
k − k → k − k
A1/2(I a) inf A a 1/2.
n
k − k≥ n k − k
Thus
inf A1/2(I a) a A+, a 1 inf A a 1/2
n
{k − k| ∈ k k≤ }≥ n k − k
and by (i),
A1/2 inf A a 1/2.
ess n
k k ≥ n k − k
PROJECTION DECOMPOSITION 9
Since A = A1/2 2 , it follows that
k kess k kess
A inf A a ,
ess n
k k ≥ n k − k
which concludes the proof. (cid:3)
Corollary 3.3. Let A be a non-unital C∗-algebra and let A = ∞ a where
j=i j
a A+, a 1 for all j and the series converges in the strict topology of M(A).
j ∈ k jk≥ P
Then A 1.
ess
k k ≥
Everyσ-unitalC∗-algebraAhasastrictlypositiveelementb A,i.e.,apositive
element for which (bA)− = (Ab)− = A. As usual, one can ass∈ume that b = 1.
Define a seminorm on M(A), say . , by k k
b
kk
m := mb + bm for all m M(A).
b
k k k k k k ∈
Clearly, . generates the strict topology on M(A). Note that m 2 m for
b b
all m Mk(kA). k k ≤ k k
∈
Proposition 3.4. Let A be a σ-unital non-unital purely infinite simple C∗-algebra
and let A M(A) be a positive element with A > 1. Then A is a strict sum
ess
∈ k k
of projections.
Proof. Everyσ-unital, non-unital C∗-algebraof realrank zero has an approximate
identity of projections; such an approximate identity can always be chosen to be
countable and increasing, say e ([28]).
j
Let q =e e setting e{ =}0. Then ∞ q =I, where the convergence is
j j − j−1 o j=1 j
in the strict topology. Furthermore,
P
∞
A= A1/2q A1/2
j
j=1
X
where the convergence is also in the strict topology. By Lemma 3.2 (ii),
∞
A1/2q A1/2 A
j ess
k k≥k k
j=n
X
for every n. Thus the condition A > 1 allows us to find a strictly increasing
ess
k k
sequence of integers n starting with n =1 such that
k 0
nk−1
A1/2q A1/2 >1
k
k k
j=Xnk−1
for every k. Let
nk−1
a := A1/2q A1/2.
k k
j=Xnk−1
Then a is a positive element in A+ with a > 1 for every k and A = ∞ a
k k kk k=1 k
in the strict topology. Thus
∞ P
a 0.
k b
k k →
k=n
X
Apply Lemma 2.5 to a to obtain a finite sum of projections d A, d a with
1 1 1 1
∈ ≤
∞
1
d a < a .
1 1 k b
k − | 2k k
k=2
X
10 VICTORKAFTAL,P.W.NG,ANDSHUANGZHANG
Letb :=a d A+andhence b ∞ a . ThenA d =b + ∞ a ,
1 1− 1 ∈ k 1kb ≤k k=2 kkb − 1 1 k=2 k
and hence,
∞P ∞ P
A d b + a 2 a .
1 b 1 b k b k b
k − k ≤k k k k ≤ k k
k=2 k=2
X X
Next, since b +a A+ and b +a a > 1, we can apply Lemma 2.5 to
1 2 1 2 2
∈ k k ≥ k k
b +a to obtain a finite sum of projections d b +a with
1 2 2 1 2
≤
∞
1
b +a d a .
1 2 2 k b
k − k≤ 2k k
k=3
X
Thus, iterating, we can find for each k a finite sum d of projections in A so that
k
n ∞
A d 2 a 0.
k k b
k − k≤ k k →
k=1 k=n+1
X X
This provesthatthe sum ∞ d convergesto Ainthe stricttopology,andhence
k=1 k
that A is a strict sum of projections, as claimed. (cid:3)
P
Remark 3.5. In the course of the above proof we have proven that if A is a σ-
unital non-unital purely infinite simple C∗-algebra, and A= ∞ a in the strict
k=1 k
topology, where a A+ and a > 1 for all k, then A is a strict sum of pro-
k k
∈ k k P
jections. The condition ”purely infinite and simple” is the key assumption for the
conclusion to hold in the eyes of key Lemma 2.5.
4. The case A =1 and A >1
ess
k k k k
The objective of this section is to prove that A = 1 and A > 1 suffice
ess
to have A written as a strictly converging sum ofkprkojections in Ak. Wke start with
some technical preparations.
Lemma4.1. Let A beanyC∗-algebra of real rankzeroand Abea positive element
in M(A) such that A = 1 and A > 1. Then there exist a positive element
ess
A′ M(A), a real nkumkber λ>1, ankd aknonzero projection p A such that
∈ ∈
(i) A′ =1,
ess
k k
(ii) A′p=pA′ =0,
(iii) A′+λp A.
≤
Proof. Let δ = A 1. Define two positive continuous functions h (t) and h (t)
1 2
k k−
on [0, A ] as follows:
k k
0 t [0,1+ δ] t t [0,1+ δ]
∈ 2 ∈ 4
h (t):= linear t [1+ δ,1+ 3δ] and h (t):= linear t [1+ δ,1+ δ]
1  ∈ 2 4 2  ∈ 4 2
t t [1+ 3δ, A ] 0 t [1+ δ, A ]
∈ 4 k k ∈ 2 k k
andClhea(rAly),hh1((At))+=h02.(Lt)e≤t πtbanedthhe1(qtu)oht2i(etn)t=m0apfofrroamll tM,h(eAn)cet,oht1h(eAc)o+rohn2a(Aal)ge≤brAa
1 2
M(A)/A. Reasoning as in Lemma 3.2, h (A) A and h (A) = A = 1.
1 2 ess ess
∈ k k k k
Applying Lemma 2.3, approximate h (A) by a positive element of finite spectrum
1
satisfying
α p +α p + +α p h (A)
1 1 2 2 m m 1
··· ≤
wherep arepairwiseorthogonalnonzeroprojectionsin A. Fora sufficientapprox-
i
imation, α > 1 holds for at least one i . Set λ := α , p := p , and A′ = h (A).
i 0 i0 i0 2