ON THE LINEAR SPAN OF THE PROJECTIONS IN CERTAIN SIMPLE C -ALGEBRAS

ON THE LINEAR SPAN OF THE PROJECTIONS IN CERTAIN SIMPLE C -ALGEBRAS L.W. MARCOUX 1 Abstract. In this paper we show that if a C -algebra A admits a certain 3 3 matrix decomposition, then every commutator in A can be written as a linear combination of at most 84 projections in A. In certain C -algebras, this is sufficient to allow us to show that every element is a linear combination of a fixed finite number of projections from the algebra. 1. Introduction 1.1. Let H be an infinite dimensional, separable, complex Hilbert space. By B(H) we denote the C -algebra of bounded linear operators acting on H. From the point of view of single operator theory, a particularly useful fact about B(H) is that for each n 1, it is -isomorphic to M n (B(H)), the set of n n matrices with entries in B(H). The usefulness of this result stems from the fact that in some practical sense, there is more room in the n n matrices over an algebra than in the algebra itself. For example, constructions such as direct sums of elements do not take us out of B(H). This is one reason why arguments which work in infinite dimensions often cannot be extended to the finite dimensional case. Most C -algebras do not enjoy this kind of stability under tensoring with M n, and so much of the constructive geometric machinery available in B(H) does not carry over to these algebras either. This paper deals with a particular example of this phenomenon. While the Spectral Theorem for normal operators easily implies that the linear span of the projections is dense in B(H), it is not at all clear a priori that it coincides with B(H). Various proofs of this fact now exist, and it was P. Fillmore [16] who first showed that every operator T B(H) can be written as a linear combination of at most 257 projections. This estimate was substantially improved by C. Pearcy and D. Topping [24], who showed that 16 projections (or 8 idempotents) will suffice. Their proof uses a deep result of A. Brown and C. Pearcy [8] characterizing those Hilbert space operators which can be expressed as single commutators - i.e. T = [A, B] := AB BA for some A, B B(H), as well as the aforementioned isomorphism between B(H) and M 2 (B(H)). In finite dimensions, an interesting result of Y. Nakamura [23] shows that every operator in B(C n ) is a linear combination of at most 8 projections, independent of n. 1.2. For arbitrary C -algebras, we cannot expect such a result to exist. One inescapable reason for this is the existence of C -algebras A without projections - for eg., the space C 0 (0, 1) of complex valued, continuous functions f on (0, 1) such that lim x 0 f(x) = lim x 1 f(x) = 0. Far more interestingly, B. Blackadar [4] has provided an example of a simple, unital C -algebra with no nontrivial projections. 1 Research supported in part by NSERC (Canada). Nov. 14, 2001. 1

2 On the linear span of the projections Even if many projections exist, this might not be good enough; the linear span of the projections is easily seen to be dense in A = CI K(H), the unitisation of the algebra of compact operators on H. On the other hand, every projection in A either has finite or co-finite rank, and hence the same is true of any finite linear combination of projections in A. There are other C -algebras, however, for which the situation is less clear. Consider, for instance, the UHF C -algebra A = lim(m kn, ϕ n ). For each a A and ε > 0, we can find n 1 and b M kn so that a b < ε. Since b is a linear combination of 8 projections by Nakamura s result, the set 8 CP(A) of linear combinations of 8 projections in A is clearly dense. In [20], G.J. Murphy and the author showed that the linear span of the projections in A coincides with A by first showing that it is a Lie ideal and then characterising the Lie ideals of A. This technique also works for other classes of simple, unital C -algebras, including those which are infinite. (This addresses a remark made by A.G. Robertson [26].) 1.3. A question remains: is there a number κ > 0 so that every element of A can be written as a linear combination of at most κ projections in A? This is precisely the question we study in this paper, not only for UHF C -algebras, but more general inductive limit algebras as well. We first show that if a unital C -algebra A admits a certain 3 3 matrix decomposition, then every commutator in A can be written as a linear combination of at most 84 (and in some cases, 52) projections in A. We then identify a reasonably large class of C -algebras for which such a decomposition is possible, including simple, unital AF C -algebras, Bunce-Deddens algebras and irrational rotation algebras. Combining this with results of K. Thomsen and of T. Fack relating trace zero elements in A to finite sums of commutators, we obtain a positive answer to the above question for many of these classes of algebras, and we are able to explicitly compute an upper bound for κ. Along the way, we obtain auxiliary results for sums of nilpotents of order two and sums of idempotents in these algebras. We would like to thank D. Hadwin for some interesting conversations regarding this paper. 2. Preliminaries and notations 2.1. We now set some of the notation which we shall use below, as well as review some relevant definitions. Given a unital C -algebra A, we consider the set c(a) = {[x, y] := xy yx : x, y A} of commutators, the set N (2) (A) = {a A : a 2 = 0} of nilpotents of order two, the set E(A) = {e A : e 2 = e} of idempotents, and the set P(A) = {p A : p = p 2 = p } of projections in A. If S A is any subset, we let CS = {λs : λ C, s S} and for each n 1, we let n S = {s 1 s 2... s n : s i S, 1 i n}. A C -algebra A is said to be of real rank zero [10] if the set of selfadjoint elements with finite spectrum is norm dense in the set A sa of all selfadjoint elements of A. If p, q P(A), we say that p is (Murray-von Neumann) equivalent to q and write p q provided that we can find a partial isometry w A such that p = w w and q = w w. We write p q (resp. p q) if we can find a projection r q (resp. r < q) so that p r. We denote by I(A) the set of tracial states of A, i.e. the positive norm one linear functionals τ which satisfy τ(ab) = τ(ba) for all a, b A. 2.2. Let A be a finite factor von Neumann algebra and τ denote the standard trace on A. It is well-known that two projections p and q are equivalent in A if and only if τ(p) = τ(q),

in certain simple C -algebras 3 i.e. for all traces on A. How much such a result or a more general theory of equivalence of selfadjoint elements can be carried over to simple, unital C -algebras is an object of much study [2, 3, 6, 11, 12, 21]. In [6], B. Blackadar introduces a number of different notions of comparability of projections in simple, unital C -algebras in terms of trace conditions, and refers to the study of these notions as the Fundamental Comparability Question [FCQ]. One version of this question defined in that paper and which will be used extensively below is the following: 2.3. Definition. We shall say that a (simple, unital) C -algebra A satisfies [FCQ2] if for all p, q P(A) such that τ(p) < τ(q) for all τ I(A), we have p q. 2.4. One use of equivalent projections in C -algebras is that if p, q P(A) are orthogonal projections and p q, then (p q)a(p q) M 2 (pap) (see [28], Prop. 5.3.1). This allows us to embed a 2 2 matrix algebra in A, a fact we shall exploit more than once below. If A is a simple C -algebra which is not one dimensional, then B. Blackadar and D. Handelman [2] have shown that there always exist hereditary C -subalgebras B C A such that M 2 (B) is isomorphic to C. As pointed out in [6] (see Sec. 4.1), the algebra B cannot always be chosen to be a corner algebra - i.e. an algebra of the form pap for some p P(A). Since our techniques require B to be of this form, the [FCQ2] property of A is crucial to our arguments. 2.5. Given p P(A), A unital, we can express each a A uniquely as a sum a = pap pa(1 p) (1 p)ap (1 p)a(1 p). By writing this as a matrix, [ ] pap pa(1 p) a =, (1 p)ap (1 p)a(1 p) we establish a -isomorphism between A and the 2 2 matrix algebra [ ] pap pa(1 p) (1 p)ap (1 p)a(1 p) equipped with the usual matrix operations and the norm inherited from A. Let p 1, p 2, p 3 be mutually orthogonal projections in A with 1 = p 1 p 2 p 3. In a similar manner we can identify A with a 3 3 matrix algebra [p i Ap j ], and we refer to this matrix algebra as a decomposition of A relative to p 1, p 2, p 3. For operators on B(H), this is an absolutely standard operation. For C -algebras, we refer the reader to Section 5.3 of [28] for more details. In particular, if p 1 (1 p 1 ), say p 1 q 1 (1 p 1 ), then A contains an isomorphic copy of M 2 (p 1 Ap 1 ), as noted above. This simple observation is the basis of most of the computations of the next section. 2.6. The subsequent sections of the paper are devoted to showing that many of the standard C -algebras, including Bunce-Deddens algebras, irrational rotation algebras, simple, unital AF algebras and Cuntz algebras admit such a decomposition. We refer the reader to [13] as an excellent source for the definitions and basic properties of these and other C -algebras. 3. Commutators and projections in 3 3 matrix algebras. 3.1. In this section we shall show how the existence of 3 3 matrix decompositions in a C -algebra allow one to express every commutator in the algebra as a sum of a bounded number of nilpotents of order two, and then express every nilpotent in this sum as a linear combination of a bounded number of idempotents, and in turn, projections.

4 On the linear span of the projections 3.2. Lemma. Let A be a unital C -algebra and suppose that p 1 and p 2 are mutually orthogonal projections in A. (i) If a = p 1 ap 2 and b = p 2 bp 1, then ab ba 3 N(2) (A). (ii) If p 1 (1 p 1 ) and a, b p 1 Ap 1, then ab ba 4 N(2) (A). (i) Algebraically, we write ab ba = (ab aba b ba) ( aba) (b), and verify that each of the above terms in brackets is nilpotent of order 2. In the (hopefully more descriptive) matrix representation, we have ab 0 0 0 ba 0 p 1 p 2 1 (p 1 p 2 ) = ab aba 0 b ba 0 0 aba 0 b 0 0 If p 1 p 2 = 1, simply delete the third row and the third column from the matrix forms. (ii) Choose q 1 1 p 1 so that p 1 q 1. Let q 2 = 1 p 1 q 1. (Again, we will describe the case where q 2 0.) Then (p 1 q 1 )A(p 1 q 1 ) M 2 (p 1 Ap 1 ) by [28], and so ab ba 0 0 p 1 q 1 q 2 = ab aba 0 b ba 0 ba ba 0 ba ba 0 0 aba ba 0 b ba 0 0 is a decomposition of ab ba as a sum of four nilpotents of order 2 in A. [ ] z11 z 3.3. Lemma. Let B be a unital algebra and A = M 2 (B). Suppose z = 12 A. z 21 z 22 If z 11 z 22 2 c(b), then z 5 N(2) (A). This proof is due to C. Pearcy and D. Topping [24]. Let z 11 z 22 = (rs sr) (xy yx) for r, s, x, y B. Let w = rs xy z 11 ; u = z 12 w r x; v = z 21 srs yxy w. Then [ ] [ ] rs r xy x z = srs sr yxy yx [ ] [ 0 u 0 0 0 0 v 0 [ w w w w ], ] and so z 5 N(2) (A). The context for the Pearcy-Topping result is slightly different than ours. They assumed that A is a properly infinite von Neumann algebra (i.e. one which has no finite central projection other than zero), and they were able to deduce that every operator Z A admits such a matrix form, and hence is a sum of five nilpotents of order 2 in A.

in certain simple C -algebras 5 3.4. Lemma. Let A be a unital C -algebra and suppose p 1, p 2, and p 3 are mutually orthogonal projections in A satisfying p 1 p 2 p 3 = 1. Consider x = [x ij ] and y = [y ij ] A. (i) There exists w 9 N(2) (A) such that [x, y] = [x 11, y 11 ] 0 0 0 [x 22, y 22 ] 0 w. 0 0 [x 33, y 33 ] (ii) Alternatively, there exists v 8 N(2) (A) such that [x, y] = [x 11, y 11 ] t 12 0 t 21 [x 22, y 22 ] 0 v 0 0 [x 33, y 33 ] for some t 12 p 1 Ap 2 and t 21 p 2 Ap 1. (i) Let r = xy yx and write r = [r ij ]. Then r = r 11 0 0 0 r 22 0 0 0 r 33 r 31 r 32 0 0 r 12 r 13 r 21 0 r 23 Next, consider Equation ( ): r 11 0 0 0 r 22 0 [x 11, y 11 ] 0 0 0 [x 22, y 22 ] 0 0 0 r 33 0 0 [x 33, y 33 ] = x 12y 21 x 12 y 21 x 12 0 y 21 y 21 x 12 0 x 13y 31 0 x 13 y 31 x 13 y 31 0 y 31 x 13 0 x 23 y 32 x 23 y 32 x 23 0 y 32 y 32 x 23 0 s 12 s 13. y 12x 21 y 12 x 21 y 12 0 x 21 x 21 y 12 0 y 13x 31 0 y 13 x 31 y 13 x 31 0 x 31 y 13 s 31 s 32 0 0 y 23 x 32 y 23 x 32 y 23 0 x 32 x 32 y 23 s 21 0 s 23,

6 On the linear span of the projections where s 12 = x 12 y 21 x 12 y 12 x 21 y 12 s 13 = x 13 y 31 x 13 y 13 x 31 y 13 s 21 = y 21 x 21 s 23 = x 23 y 32 x 23 y 23 x 32 y 23 s 31 = y 31 x 31 s 32 = y 32 x 32. Denote by m 0 the sum of the first six matrices on the right-hand side of Equation ( ). Note that each term in m 0 is nilpotent of order two, i.e. m 0 6 N(2) (A). If t ij = r ij s ij when i j, then m 1 = 0 t 12 t 13, m 2 = t 31 t 32 0, m 3 = t 21 0 t 23 are also nilpotent of order two. Since r = xy yx = [x 11, x 22 ] 0 0 0 [x 22, y 22 ] 0 m 0 m 1 m 2 m 3, 0 0 [x 33, y 33 ] our claim is proven. (ii) Repeating the above calculations, we can also write r = [x 11, y 11 ] t 12 0 t 21 [x 22, y 22 ] 0 m 0 0 0 t 13 0 0 t 23 0 0 [x 33, y 33 ] t 31 t 32 0 We finish by letting v denote the sum of the last three terms on the right-hand side of this equation.. 3.5. Theorem. Let A be a unital C -algebra and suppose p 1, p 2 and p 3 are mutually orthogonal projections in A such that p 1 p 2 p 3 = 1. Suppose furthermore that p i (1 p i ), 1 i 3. Then (i) c(a) 21 N(2) (A). (ii) If p 1 p 2, then c(a) 13 N(2) (A). (i) By Lemma 3.4(i), given x = [x ij ], y = [y ij ] with respect to the decomposition induced by p 1, p 2, p 3, we can write r = [x, y] = [x 11, y 11 ] 0 0 [x 22, y 22 ] 0 0 w 0 0 [x 33, y 33 ] for some w 9 N(2) (A). By Lemma 3.2(ii), each of the first three terms on the right-hand side of this equation lies in 4 N(2) (A). From this it clearly follows that [x, y] 21 N(2) (A).

(ii) Using Lemma 3.4(ii), we write r = [x, y] = in certain simple C -algebras 7 [x 11, y 11 ] t 12 t 21 [x 22, y 22 ] 0 v with v 8 N(2) (A). Since p 1 p 2, we identify (p 1 p 2 )A(p 1 p 2 ) with M 2 (p 1 Ap 1 ); hence Lemma 3.3 applies. Since [x 11, y 11 ] [x 22, y 22 ] 2 c(p 1Ap 1 ), we find that [x 11, y 11 ] t 12 t 21 [x 22, y 22 ] N (2) (A), 0 5 and so [x, y] 58=13 N(2) (A). The following result is due to K.R. Davidson. We thank him for allowing us to reproduce it here. 3.6. Theorem. [K.R. Davidson] Let A be a unital C*-algebra. Then every idempotent in A is a linear combination of at most 5 projections in A. Suppose that a = pa(1 p) belongs to A and a 1. Notice that is unitary and u a = (1 aa a a) 1/2 a a p a = u apu a = 1 aa a a a(1 a a) 1/2 (1 a a) 1/2 a is a projection. Indeed, our matrix decomposition with respect to p and (1 p) yields the matrix form [ (1 aa u a = ) 1/2 ] a a (1 a a) 1/2 and [ 1 aa p a = a(1 a a) 1/2 ] (1 a a) 1/2 a 1 a. a Let a = u a be the polar decomposition of a in A. A routine calculation shows that 1 3 ī k p 4 i k a = a(1 a a) 1/2 = uf( a ), k=0 where f(x) = x(1 x 2 ) 1/2 1. Since f maps [0, 2 ] bijectively onto [0, 1 2 ], it has a continuous inverse, namely g(x) = ( 1 2 ( 1 4 x2 ) 1/2) 1/2. Thus if b = pb(1 p) = u b and b 1 2, let a = ug( b ). Note that a belongs to A because g(0) = 0. Then since b = uf( a ), the argument above shows that it is a linear combination of four projections in A. If q is an idempotent if A, then the projection p with the same range lies in A and q = p pq(1 p) (see, for example, [5], Prop. 4.6.2). Thus q lies in the span of at most five projections in A. Observe that Davidson s argument also shows that if p A is a projection and b = pb(1 p) A, then b 4 CP(A).

8 On the linear span of the projections 3.7. Corollary. Let A be a unital C -algebra and p 1, p 2 be mutually orthogonal projections in A. Let z A. Then z 2 CE(A) 4 CP(A) in each of the following cases: (i) z = p 1 z(1 p 1 ); (ii) z = (1 p 1 )zp 1 ; (iii) p 3 = 1 (p 1 p 2 ), p 1 p 2, and z = c c 0 c c 0 we are identifying (p 1 p 2 )A(p 1 p 2 ) with M 2 (p 1 Ap 1 ).) ab aba 0 (iv) p 3 = 1 (p 1 p 2 ) and z is of the form b ba 0 b = p 2 bp 1. p 1 p 2 p 3, for some c p 1 Ap 1. (Here p 1 p 2 p 3 for some a = p 1 ap 2, (i) Let e = p 1 z; then e, p 1 E(A) and z = e p 1. By the remark preceding this Corollary, z 4 CP(A). (ii) This follows immediately from (i) by interchanging p 1 and 1 p 1. (iii) Let u = 1 2 1 2 1 2 1 2 1 p 1 p 2 p 3 uzu = = u = u 1 A. Then 0 2c 0 By (i), uzu 2 CE(A) 4 CP(A). Since both sets on the right hand side of this equation are invariant under unitary conjugation, the assertion follows. (iv) If we let x = 1 a, then x 1 = 1 a and xzx 1 = b = p 2 bp 1 = p 2 b(1 p 2 ). Thus z = x 1 bx 0. = (x 1 p 2 x)(x 1 bx)(x 1 (1 p 2 )x) = e 2 z(1 e 2 ), where e 2 = x 1 p 2 x. Let q be the range projection of e 2, so that q A. Then qe 2 = e 2 and e 2 q = q. Thus z = e 2 z = (qe 2 )z = qz, and zq = z(e 2 q) = 0, so that z = z(1 q). That is to say, z = qz(1 q) where q A is a projection. From (i) above, we deduce that z 2 CE(A) 4 CP(A). 3.8. Theorem. Let A be a unital C -algebra with mutually orthogonal projections p 1, p 2 and p 3 such that 1 = p 1 p 2 p 3 and p i (1 p i ), 1 i 3. Then (i) c(a) 42 CE(A) 84 CP(A); (ii) if p 1 p 2, then c(a) 26 CE(A) 52 CP(A). (i) By Theorem 3.5 (i), c(a) 21 N(2) (A). Moreover, the construction of the nilpotents of order two from Lemma 3.2 yields nilpotents precisely of the form

in certain simple C -algebras 9 described in Corollary 3.7. As such, each one lies in 2 CE(A) 4 CP(A). Thus A 42 CE(A). (ii) If p 1 p 2, then by Theorem 3.5 (ii), c(a) 13 N(2) (A). The rest of the proof follows as in (i). 4. Finite, simple C -algebras 4.1. In the previous section we saw that if a unital C -algebra admits a certain 3 3 matrix algebra decomposition, then every commutator can be written as a linear combination of at most 84 projections. We now turn our attention to showing that the existence of such a matrix algebra decomposition is not as uncommon as one might initially suspect. Indeed, as previously mentioned, decompositions exist for irrational rotation algebras, Bunce-Deddens algebras and others. We first isolate the features common to each of our examples - features which allow us to construct the projections of Theorem 3.8. 4.2. Theorem. Suppose that A is a simple, unital C -algebra that satisfies [FCQ2]. If there exists a projection p 1 P(A) such that 1 4 < τ(p 1) < 1 2 for all τ I(A), then c(a) 13 N (2) (A) 52 CP(A). By assumption, τ(p 1 ) < τ(1 p 1 ) for all τ I(A). [FCQ2] is precisely the property that allows us to conclude that there exists a projection p 2 < 1 p 1 such that p 1 p 2. Let p 3 = 1 (p 1 p 2 ). Now τ(p 1 ) < τ(1 p 1 ) for all τ I(A) as just noted. Also, τ(p 2 ) = τ(p 1 ) < τ(1 p 1 ) = τ(1 p 2 ) for all τ I(A). Finally, 0 < τ(p 3 ) = 1 2τ(p 1 ) < 1 2, and so τ(p 3) < τ(1 p 3 ) for all τ I(A) as well. Again, [FCQ2] implies that p i (1 p i ), 1 i 3, and so Theorem 3.8 (ii) applies, from which we may draw the above conclusion. 4.3. As pointed out by the referee, the conclusion of Theorem 4.2 holds with a variety of different assumptions. For example, if one supposes that A is a unital C -algebra containing a finite dimensional unital C -algebra B such that B N i=1 M n i and n i 3 for all 1 i N, then it is possible to find projections p 1, p 2, p 3 satisfying the conditions of Theorem 3.8, and so the result follows. Furthermore, if A is a C -algebra of the form described in Theorem 4.4 below, the existence of such a finite dimensional unital subalgebra B of A is guaranteed by the argument in Remark 1.9 of [27]. From this we have: 4.4. Theorem. Let A be a simple, unital, infinite dimensional inductive limit of a sequence of C -algebras of the form A n = rn j=1 C(Ω n,j) M [n,j], where each Ω n,j is a compact, connected Hausdorff space, [n, j] is a positive integer all n, j, and {ϕ m,n } is a system of connecting unital homomorphisms ϕ m,n : A m A n, then c(a) 13 N(2) (A) 52 CP(A). We remark that our original version of this result required that the algebra have but one trace and slow dimension growth in the sense of [1].

10 On the linear span of the projections 4.5. It was shown by J. Cuntz and G.K. Pedersen in [12] that if A is a unital C -algebra with at least one tracial state and we define sl(a) = {ker τ : τ I(A)}, then sl(a) = [A, A], where [A, A] denotes the linear span of c(a). If I(A) = φ, then [A, A] = A. For certain algebras, including those of interest to us, there is a stronger result. We refer the reader to [22] for the definition of the covering dimension dim X of a compact, Hausdorff space X. For our purposes, it will suffice to note that if X R n is compact, then dim X n, and that the covering dimension of a compact n-dimensional (real) manifold is n. In particular, therefore, if T = {z C : z = 1}, then dim T = 1, and dim{x} = 0 for each point x R n. 4.6. Theorem. [K. Thomsen] [27] Let A = lim (A n, ϕ m,n ) be a unital, simple, infinite dimensional C -inductive limit, where each A n = rn j=1 C(Ω n,j) M [n,j], Ω n,j is a compact, connected, Hausdorff space, and [n, j] N. Suppose there exists d > 0 so that dim(ω n,j ) d for all n, j. If a A sa sl(a), then there exist x 1, x 2,..., x d7 A so that a = d7 i=1 [x i, x i]. This result is a generalisation of a theorem of T. Fack [15], who considered the case where A is a simple, unital AF C -algebra - i.e. the case where each Ω n,j is a point. The case where Ω n,j = T for all n, j is referred to as a limit circle algebra [13]. We now have assembled all of the ingredients to prove: 4.7. Theorem. Suppose that A is a simple, unital, infinite dimensional limit circle algebra with a unique trace. Then (i) c(a) 13 N(2) (A) 52 CP(A); (ii) A = 833 CP(A). (i) This follows immediately from Theorem 4.4. (ii) Since I(A) = {τ} is a singleton, sl(a) = ker τ has codimension 1 in A. By Thomsen s Theorem above, given a = a sl(a), we can find x 1, x 2,..., x 8 A so that a = 8 i=1 [x i, x i] 8 c(a). Now let b A be arbitrary, and let a = b τ(b)1 sl(a). Since the real and imaginary parts of a lie in 8 c(a) from the previous paragraph, part (i) implies that b C1 16 c(a) 833 CP(A). Since b was arbitary, we are done. 4.8. Of course, if we replace T with a compact, connected, Haudsdorff space X with dim X = d N and leave all other conditions untouched, then exactly the same proof can be used to show that c(a) 52 CP(A); sl(a) d7 c(a) 13(d7) N(2) (A) sl(a), and hence the containments can be replaced by equalities; and A = 26(d7) CE(A) = 152(2d14) CP(A). Note: The last containment of the second observation follows from the fact that if x N (2) (A), then x is quasinilpotent, and so τ(x) = 0 for all tracial states τ on A.

4.9. Corollary. in certain simple C -algebras 11 (i) Let A be any simple, unital AF C -algebra with a unique trace (this includes the case where A is a UHF algebra). Then A = 729 CP(A). (ii) Let A be any Bunce-Deddens algebra. Then A = 833 CP(A). (iii) Let A be any irrational rotational algebra. Then A = 833 CP(A). (i) In this case, we replace T by a single point {x}, where dim{x} = 0. Thus A 152(14)=729 CP(A). (ii) In [17], it is shown that all Bunce-Deddens algebras are limit circle algebras. (That they are simple and unital can be found in [13].) (iii) In [14], it is shown that all irrational rotation algebras are limit circle algebras. (Again, they are simple and unital as shown in [13].) 4.10. In Corollary 5.7, we improve the estimate on the number κ of projections required so that A = κ CP(A) where A is a UHF C -algebra whose supernatural number contains a factor of 2. 4.11. Theorem. Let A be a simple, unital, infinite dimensional AF C -algebra. Then (i) c(a) 13 N(2) (A) 52 CP(A). (ii) If, in addition, A has exactly m extremal tracial states (where 1 m < ), then A = 728m CP(A). (i) Again, this follows from Theorem 4.4, where Ω n,j is a single point for all n, j. (ii) Because A has m extremal tracial states, it follows that sl(a) has codimension m in A. By Fack s result [15], sl(a) = 14 c(a). By [20], A is the linear span of its projections. Let π : A A/sl(A) denote the (Banach space) quotient map, and choose x 1, x 2,..., x m A so that {π(x i )} m i=1 is a basis for A/sl(A). Using [20], choose projections {q j } s j=1 A so that {x i} m i=1 span{q j} s j=1. Then {π(q j)} s j=1 spans A/sl(A), and so we can find m linearly independent vectors in this set which, after reindexing if necessary, we may assume are {π(q j )} m j=1. If a A, then we can find {λ j } m j=1 C and b sl(a) so that a = m j=1 λ jq j b. Since b 14 c(a) 728 CP(A) by (i), it follows that a 728m CP(A). 5. Simple, infinite C -algebras and beyond 5.1. In this section we turn our attention to the case of simple, unital, infinite algebras. Recall that a projection p in a C -algebra A is said to be infinite if p q for some q < p. Then A is said to be infinite if it possesses an infinite projection. 5.2. Lemma. Suppose A is a simple, unital, infinite C -algebra. Then A possesses mutually orthogonal projections p 1, p 2, p 3 such that 1 = p 1 p 2 p 3, p i (1 p i ), 1 i 3 and p 1 p 2. By Theorem V.5.1 of [13], we can find an infinite projection q P(A) and partial isometries w 1, w 2, w 3 such that w 1 w 1 = w 2 w 2 = w 3 w 3 = q > 3 i=1 w i w i. Let p 1 = w 1 w 1

12 On the linear span of the projections and p 2 = w 2 w 2, so that p 1 q p 2. Let p 3 = 1 (p 1 p 2 ). Then p 1 p 2 < (1 p 1 ) and p 2 p 1 < (1 p 2 ). By Lemma V.5.4 of [13], since A is simple and q is infinite, any projection in A is equivalent to a subprojection of q. In particular, p 3 q. Since q p 1 < (1 p 3 ), it follows that p 3 (1 p 3 ), completing the proof. 5.3. Theorem. Let A be a unital, simple, infinite C -algebra. Then (i) c(a) 13 N(2 )(A) 26 CE(A) 52 CP(A); and (ii) A = 520 CP(A). (i) This follows by applying Lemma 5.2 and Theorem 3.8(ii). (ii) By Corollary 2.2 of T. Fack [15], A 10 c(a), from which the result clearly follows. 5.4. Corollary. Let O n be the Cuntz algebra, 2 n. Then O n = 520 CP(O n). Again, we refer the reader to [13] for the proof that O n is a simple, unital, infinite C -algebra. 5.5. Algebras admitting a 2 2 decomposition. It is perhaps an understatement to say that it is rather unlikely that the estimates on the number κ of projections required so that A = κ CP(A) obtained in Theorem 4.7 and Theorem 5.3 are close to optimal. In fact, under certain more stringent conditions, we can easily prove that they are not. The key lies in noticing that all of the calculations done in the first few lemmas of Section Three were based upon 2 2 matrix decompositions with projections p such that p (1 p). Thus, with minor modifications to those proofs, we obtain: 5.6. Theorem. Let B be a unital C -algebra and A = M 2 (B). Then c(a) 10 N (2) (A) 20 CE(A) 40 CP(A). Given x = [x ij ] and y = [y ij ] A, [ [x11, y [x, y] = 11 ] [x 22, y 22 ] ] [ ] x12 y 21 y 12 x 21 0. 0 x 21 y 12 y 21 x 12 By Lemma 3.3, each term on the right-hand side of this equation lies in 5 N(2) (A). The other estimates now follow from an easy modification of Corollary 3.7.

in certain simple C -algebras 13 5.7. Corollary. Let A be UHF C -algebra and suppose that the supernatural number of A contains a factor of 2. Then A = 281 CP(A). Let 2 s 0 p s 1 1 ps 2 2 denote the supernatural number for A, and let B be a UHF C - algebra whose supernatural number is 2 s0 1 p s 1 1 ps 2 2. Since M 2(B) and A are two UHF algebras with the same supernatural number, they are isomorphic [18]. Thus A satisfies the conditions of Theorem 5.6. By Fack s Theorem [15], sl(a) = 7 c(a). Using the previous Theorem combined with the fact that A has a unique trace, A = C1 sl(a) CP(A). 17(40)=281 5.8. Non-simple algebras. We can also make a few observations concerning non-simple C -algebras. Suppose 0 K j A π B 0 is a short exact sequence of C -algebras. Suppose also that B sa m CP(B) and K sa CP(K). If projections in B lift to projections in A, then n A sa mn CP(A), and hence A = 2m2n CP(A). Indeed, if a = a A, then π(a) = π(a) B sa m CP(B). Write π(a) = m i=1 λ ib i, where b i P(B), 1 i m. Thus π(a) = 1 m 2 i=1 (λ i λ i )b i. Choose projections p i A so that π(p i ) = b i, 1 i m. Then ( a = 1 m ) (λ i λ i )p i k 2 i=1 for some k = k K. Thus k n CP(K) n CP(A). Thus a mn CP(A). To make this useful, one needs conditions on B which guarantee that all projections in B lift to projections in A. Suppose that K and B are simple, unital AF C -algebras with a unique trace. A result of L.G. Brown [9] shows that A is then also an AF algebra, and all projections in B lift to projections in A. Since K = 729 CP(K) and B = 729 CP(B) by Corollary 4.9 above, we may conclude from the above analysis that A = 2896 CP(A). In [10] and [29] it is shown that if K and B have real rank zero, then projections in B lift to projections in A if and only if A also has real rank zero. Thus if K and B are simple, unital limit circle algebras with a unique trace, and if A is an extension with real rank zero, it follows as above that A = 4(833)=3332 CP(A). While this might not be an optimal bound, it does have the one saving grace of being finite! (We also draw the reader s attention to some related results of H. Lin and M. Rørdam [19] regarding extensions of limit circle algebras having real rank zero.)

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