revised June 1999 to appear in Pacific J. Math.

K_{1} OF SEPARATIVE EXCHANGE RINGS AND

C*-ALGEBRAS WITH REAL RANK ZERO

P. Ara, K. R. Goodearl, K. C. O’Meara, and R. Raphael

Abstract. For any (unital) exchange ring R whose finitely generated projective modules satisfy the separative cancellation property (A ⊕ A ≅ A ⊕ B ≅ B ⊕ B ⟹ A ≅ B), it is shown that all invertible square matrices over R can be diagonalized by elementary row and column operations. Consequently, the natural homomorphism GL_{1}(R) → K_{1}(R) is surjective. In combination with a result of Huaxin Lin, it follows that for any separative, unital C*-algebra A with real rank zero, the topological K_{1}(A) is naturally isomorphic to the unitary group U(A) modulo the connected component of the identity. This verifies, in the separative case, a conjecture of Shuang Zhang.

Introduction

The extent to which matrices over a ring R can be diagonalized is a measure of the complexity of R, as well as a source of computational information about R and its free modules. Two natural properties offer themselves as “best possible”: (1) that an arbitrary matrix can be reduced to a diagonal matrix on left and right multiplication by suitable invertible matrices, or (2) that an arbitrary invertible matrix can be reduced to a diagonal one by suitable elementary row and column operations. The second property has an immediate K-theoretic benefit, in that it implies that the Whitehead group K_{1}(R) is a natural quotient of the group of units of R. Our main goal here is to prove property (2) for exchange rings (definition below) satisfying a cancellation condition which holds very widely (and conceivably for all exchange rings). This theorem, when applied to C*- algebras with real rank zero (also defined below), verifies a conjecture of Shuang Zhang in an extensive class of C*-algebras.

The class of exchange rings has recently taken on a unifying role for certain direct sum cancellation problems in ring theory and operator algebra. In particular, exchange rings encompass both (von Neumann) regular rings (this is an old and easy observation) on the one hand, and C*-algebras with real rank zero [3, Theorem 7.2] on the other. Within this class, a unifying theme for a number of open problems is the property of separative cancellation for finitely generated projective modules, namely the condition

A ⊕ A ≅ A ⊕ B ≅ B ⊕ B ⟹ A ≅ B

Typeset by AMS-TEX

The research of the first author was partially supported by grants from the DGICYT (Spain) and the Comissionat per Universitats i Recerca de la Generalitat de Catalunya, that of the second by a grant from the NSF (USA), and that of the fourth by a grant from the NSERC (Canada).

(see [2, 3]). For example, if R is a separative exchange ring, then the (K-theoretic) stable rank of R can only be 1, 2, or ∞ [3, Theorem 3.3], and every regular square matrix over R is equivalent (via multiplication by invertible matrices) to a diagonal matrix [2, Theorem 2.4]. We prove below that invertible matrices over separative exchange rings can be diagonalized via elementary row and column operations. Recently, Perera [25] has applied our methods to the problem of lifting units modulo an ideal I in a ring R, assuming that I satisfies non-unital versions of separativity and the exchange property. In this case, a unit u of R/I lifts to a unit of R if and only if the class of u in K_{1}(R/I) is in the kernel of the connecting homomorphism K_{1}(R/I) → K_{0}(I) [25, Theorem 3.1].

We defer discussion of the C*-algebraic aspects of our results to Section 3, except for the following remark. While earlier uses of the exchange property and separativity for C*- algebras can easily be written out in standard C*-theoretic terms – e.g., with direct sums and isomorphisms of finitely generated projective modules replaced by orthogonal sums and Murray-von Neumann equivalences of projections – our present methods do not lend themselves to such a translation. In particular, although our main C*-algebraic application may be stated as a diagonalization result for unitary matrices, all of the steps in our proofs involve manipulations with non-unitary matrices.

Throughout the paper, we consider only unital rings and C*-algebras. We reserve the term elementary operation for the row (respectively, column) operation in which a left (respectively, right) multiple of one row (respectively, column) of a matrix is added to a different row (respectively, column). Similarly, we reserve the name elementary matrix for a transvection I + re_{ij} where I is an identity matrix, e_{ij} is one of the usual matrix units for some i ≠ j, and r is an element of the base ring. Thus, as usual, an elementary row (respectively, column) operation on a matrix A corresponds to multiplying A on the left (respectively, right) by an elementary matrix.

Note that while odd permutation matrices usually cannot be expressed as products of elementary matrices, certain signed permutation matrices can be. For example,

[ \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ] = [ \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 0 \\ −1 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ] .

In particular, the operation of replacing rows R_{i} and R_{j} (respectively, columns C_{i} and C_{j}) with the rows R_{j} and −R_{i} (respectively, the columns C_{j} and −C_{i}) can be achieved as a sequence of three elementary operations. Therefore any entry of a matrix can be moved to any other position by a sequence of elementary row and column operations, at the possible expense of moving other entries and multiplying some by −1.

For any ring R, let E_{n}(R) denote the subgroup of GL_{n}(R) generated by the elementary matrices. If GL_{n}(R) is generated by E_{n}(R) together with the subgroup D_{n}(R) of invertible diagonal matrices, then R is said to be a GE_{n}-ring [10, p. 5]. Further, R is a GE-ring provided it is a GE_{n}-ring for all n. If R is a GE_{n}-ring then E_{n}(R) is a normal subgroup of GL_{n}(R), and so GL_{n}(R) = D_{n}(R)E_{n}(R) = E_{n}(R)D_{n}(R). Of course, this means that every invertible n × n matrix over a GE_{n}-ring can be diagonalized using only elementary row (respectively, column) operations.

K_{1} OF SEPARATIVE EXCHANGE RINGS 3

It is easy to check that all rings with stable rank 1 are GE-rings. Note that if R is a GE-ring, then the natural homomorphism from GL_{1}(R), the group of units of R, to K_{1}(R) is surjective. For comparison, we recall the well-known fact that if R has stable rank d, then the natural map GL_{d}(R) → K_{1}(R) is surjective (e.g., [12, Theorem 40.42]).

1. Exchange rings and separativity

Although our notions and results will be right-left symmetric, all modules considered in this paper will be right modules. A module M over a ring R has the finite exchange property [11] if for every R-module A and any decompositions

A = M′ ⊕ N = A_{1} ⊕ ⋯ ⊕ A_{n}

with M′ ≅ M, there exist submodules A′_{i} ⊆ A_{i} such that

A = M′ ⊕ A′_{1} ⊕ ⋯ ⊕ A′_{n}.

(It follows from the modular law that A′_{i} must be a direct summand of A_{i} for all i.) It should be emphasized that the direct sums in this definition are internal direct sums of submodules of A. One advantage of the resulting internal direct sum decompositions (as opposed to isomorphisms with external direct sums) rests on the fact that direct summands with common complements are isomorphic – e.g., N ≅ ⨁_{i=1}^{n} A′_{i} above since each of these summands of A has M′ as a complementary summand.

Following Warfield [29], we say that R is an exchange ring if R_{R} satisfies the finite exchange property. By [29, Corollary 2], this definition is left-right symmetric. If R is an exchange ring, then every finitely generated projective R-module has the finite exchange property (by [11, Lemma 3.10], the finite exchange property passes to finite direct sums and to direct summands), and so the endomorphism ring of any such module is an exchange ring. Further, idempotents lift modulo all ideals of an exchange ring [24, Theorem 2.1, Corollary 1.3].

The class of exchange rings is quite large. It includes all semiregular rings (i.e., rings which modulo the Jacobson radical are (von Neumann) regular and have idempotentlifting), all π-regular rings, and more; see [1, 28, 29]. Further, all unital C*-algebras with real rank zero are exchange rings [3, Theorem 7.2].

The following criterion for exchange rings was obtained independently by Nicholson and the second author.

Lemma 1.1. [17, p. 167; 24, Theorem 2.1] A ring R is an exchange ring if and only if for every element a ∈ R there exists an idempotent e ∈ R such that e ∈ aR and 1 − e ∈ (1 − a)R. □

In the above lemma, it is equivalent to ask that for any a_{1}, a_{2} ∈ R with a_{1}R + a_{2}R = R, there exists an idempotent e ∈ a_{1}R such that 1−e ∈ a_{2}R. We shall also need the analogous property corresponding to sums of more than two right ideals:

Lemma 1.2. [24, Theorem 2.1, Proposition 1.11] Let R be an exchange ring. If I_{1}, …, I_{n} are right ideals of R such that I_{1} + ⋯ + I_{n} = R, then there exist orthogonal idempotents e_{1}, …, e_{n} ∈ R such that e_{1} + ⋯ + e_{n} = 1 and e_{j} ∈ I_{j} for all j. □

We reiterate that a ring R is separative provided the following cancellation property holds for finitely generated projective right (equivalently, left) R-modules A and B:

A ⊕ A ≅ A ⊕ B ≅ B ⊕ B ⟹ A ≅ B.

See [3] for the origin of this terminology and for a number of equivalent conditions. We shall need the following one:

Lemma 1.3. [2, Proposition 1.2; 3, Lemma 2.1] A ring R is separative if and only if whenever A, B, C are finitely generated projective right R-modules such that A⊕C ≅ B⊕C and C is isomorphic to direct summands of both A^{n} and B^{n} for some n, then A ≅ B. □

Note, in particular, that if R is separative and A, B, C are finitely generated projective right R-modules, then we can certainly cancel C from A ⊕ C ≅ B ⊕ C whenever A and B are generators in Mod-R.

Separativity seems to hold quite widely within the class of exchange rings; for instance, it holds for all known classes of regular rings (cf. [3]). In fact, the existence of non-separative exchange rings is an open problem.

It is clear from either form of the condition that a ring R is separative in case the finitely generated projective R-modules enjoy cancellation with respect to direct sums, which in turn holds in case R has stable rank 1. In fact, for exchange rings, cancellation of finitely generated projective modules is equivalent to stable rank 1 [31, Theorem 9]. Separativity, however, is much weaker than stable rank 1. For example, any regular right self-injective ring is separative (e.g., [15, Theorem 10.34(b)]), but such rings can have infinite stable rank – e.g., the ring of all linear transformations on an infinite dimensional vector space.

2. K_{1} of separative exchange rings

We use the notation A ≲^{⊕} B to denote that a module A is isomorphic to a direct summand of a module B.

Lemma 2.1. Let R be an exchange ring and e_{1}, …, e_{n} ∈ R idempotents. Then there exists an idempotent e ∈ e_{1}R + ⋯ + e_{n}R such that e_{1}R ≤ eR and e_{i}R ≲^{⊕} eR for all i. In particular, ReR = Re_{1}R + ⋯ + Re_{n}R.

Proof. By induction, it suffices to do the case n = 2. Now

R = e_{1}R ⊕ (1 − e_{1})R = e_{2}R ⊕ (1 − e_{2})R

and e_{1}R has the finite exchange property, so there exist decompositions e_{2}R = A ⊕ B and (1 − e_{2})R = A′ ⊕ B′ such that R = e_{1}R ⊕ A ⊕ A′. Then we can choose an idempotent e ∈ R such that eR = e_{1}R ⊕ A. Obviously e_{1}R ≤ eR, and since

e_{1}R ≅ R/(A ⊕ A′) ≅ B ⊕ B′,

we have e_{2}R ≲^{⊕} A ⊕ e_{1}R = eR. □

Corollary 2.2. Let R be an exchange ring and a ∈ R such that RaR = R. Then there exist idempotents e ∈ aR and f ∈ Ra such that ReR = RfR = R.

Proof. Write R = ∑_{i=1}^{n} x_{i}aR for some x_{i}. By Lemma 1.2, there exist orthogonal idempotents g_{1}, …, g_{n} ∈ R such that g_{1} + ⋯ + g_{n} = 1 and g_{i} ∈ x_{i}aR for all i. Set g_{i} = x_{i}ay_{i} with y_{i} = y_{i}g_{i}. Then e_{i} := ay_{i}x_{i} is an idempotent in aR and e_{i}R ≅ g_{i}R. By Lemma 2.1, there exists an idempotent e ∈ ∑_{i=1}^{n} e_{i}R such that e_{i}R ≲^{⊕} eR for all i. Then e ∈ aR and g_{i}R ≲^{⊕} eR for all i, so all g_{i} ∈ ReR, and thus ReR = R.

The existence of f follows by symmetry. □

Lemma 2.3. Let R be any ring and A ∈ GL_{n}(R). If A has an idempotent entry, then A can be reduced by elementary row and column operations to the form

[ \begin{matrix}1 & 0 & ⋯ & 0 \\ 0 & ∗ & ⋯ & ∗ \\ ⋮ & ⋮ & & ⋮ \\ 0 & ∗ & ⋯ & ∗ \end{matrix} ] .

Proof. By elementary operations, we can move the idempotent entry, call it e, into the 1, 1 position. If n = 1, then e is invertible, so e = 1 and we are done. Now assume that n > 1, and let

[ e b_{2} b_{3} ⋯ b_{n}]

be the first row of A. By elementary column operations, we can subtract eb_{i} from the i-th entry for each i ≥ 2. Thus, we can assume that b_{2}, …, b_{n} ∈ (1 − e)R. Since A is invertible, eR + b_{2}R + ⋯ + b_{n}R = R, and so it follows that b_{2}R + ⋯ + b_{n}R = (1 − e)R. Hence, by elementary column operations we can add 1 − e to the first entry. Now we have a 1 in the 1, 1 position, and the rest is routine. □

Since we shall need to perform a number of operations on the top rows of invertible matrices, it is convenient to work with the rows alone. Recall that any row [ a_{1} a_{2} ⋯ a_{n}] of an invertible matrix over a ring R is right unimodular, that is, ∑_{i=1}^{n} a_{i}R = R. Elementary column operations apply to such a row just by viewing it as a 1 × n matrix. Such operations amount to multiplying the row on the right by an elementary matrix. Since our rings need not be commutative, elementary column operations can only introduce right-hand coefficients.

Lemma 2.4. Let R be an exchange ring and α = [ a_{1} a_{2} ⋯ a_{n}] a right unimodular row over R. Then α can be transformed by elementary column operations to a row [ b_{1} b_{2} ⋯ b_{n}] such that R = b_{1}R ⊕ ⋯ ⊕ b_{n}R and each b_{i} ∈ a_{i}Ra_{i}.

Proof. Since ∑_{i=1}^{n} a_{i}R = R, Lemma 1.2 gives us orthogonal idempotents e_{1}, …, e_{n} ∈ R such that e_{1} + ⋯ + e_{n} = 1 and e_{i} ∈ a_{i}R for all i, say e_{i} = a_{i}r_{i}. By elementary column operations, we can subtract e_{i}a_{1} = a_{i}r_{i}a_{1} from the first entry of α for each i ≥ 2. This transforms α to α′ = [ e_{1}a_{1} a_{2} a_{3} ⋯ a_{n}]. Note that e_{1} ∈ e_{1}a_{1}R. Thus, we can repeat the above process for each entry, and transform α′ to the row [ e_{1}a_{1} e_{2}a_{2} ⋯ e_{n}a_{n}], with entries e_{i}a_{i} ∈ a_{i}Ra_{i}. Moreover, e_{i}a_{i}R = e_{i}R, and therefore R = ⨁_{i=1}^{n} e_{i}a_{i}R. □

Corollary 2.5. Let R be an exchange ring and α = [ a_{1} a_{2} ⋯ a_{n}] a right unimodular row over R, with n ≥ 2. Then α can be transformed by elementary column operations to a row [ b_{1} b_{2} ⋯ b_{n}] such that Rb_{1}R = R and b_{i} ∈ a_{i}Ra_{i} for all i ≥ 2.

Proof. By Lemma 2.4, we may assume that R = ⨁_{i=1}^{n} a_{i}R. It follows that all a_{i} ∈ Rb_{1} where b_{1} = a_{1}+⋯+a_{n} (multiply b_{1} on the left by the orthogonal idempotents arising from the given decomposition of R_{R}). Thus Rb_{1}R = R. By elementary column operations, we can add a_{2}, …, a_{n} to the first entry of α, and thus transform it to [ b_{1} a_{2} ⋯ a_{n}]. □

Recall that an element x in a ring R is (von Neumann) regular provided there exists an element y ∈ R such that xyx = x, equivalently, provided xR is a direct summand of R_{R}.

If y can be chosen to be a unit in R, then x is said to be unit-regular. A regular element x ∈ R is unit-regular if and only if R/xR ≅ r.ann(x), where r.ann(x) denotes the right annihilator of x in R (cf. [15, proof of Theorem 4.1]).

Corollary 2.6. Let R be an exchange ring and α = [ a_{1} a_{2} ⋯ a_{n}] a right unimodular row over R, with n ≥ 2. Then α can be transformed by elementary column operations to a row [ c_{1} c_{2} ⋯ c_{n}] such that c_{2} is a regular element, c_{2} ∈ Ra_{2}, and c_{2}R = (1 − g)R for an idempotent g with RgR = R.

Proof. By Corollary 2.5, we may assume that Ra_{1}R = R. By Corollary 2.2, there exists an idempotent e ∈ a_{1}R such that ReR = R. By elementary column operations, we can subtract ea_{2} from the second entry of α, so there is no loss of generality in assuming that a_{2} ∈ (1 − e)R. (At this stage, our current a_{2} is only a \underline{left} multiple of the original a_{2}. This is why the conclusions of the lemma state c_{2} ∈ Ra_{2} rather than c_{2} ∈ a_{2}Ra_{2}.) Now using Lemma 2.4, we can transform α to a row [ c_{1} c_{2} ⋯ c_{n}] such that R = ⨁_{i=1}^{n} c_{i}R and c_{2} ∈ a_{2}Ra_{2}. Then c_{2}R = (1 − g)R for some idempotent g, and c_{2} is regular. Moreover, (1 − g)R = c_{2}R ⊆ a_{2}R ⊆ (1 − e)R and so Re ⊆ Rg. Therefore RgR = R. □

Lemma 2.7. Let R be an exchange ring and A ∈ GL_{n}(R), with n ≥ 2. Then A can be transformed by elementary row and column operations to a matrix whose 1, 1 entry d is regular, with dR = (1 − p)R and Rd = R(1 − q) for some idempotents p, q such that RpR = RqR = R.

Proof. By Corollary 2.6, we can assume that the 1, 2 entry of A is a regular element c such that cR = (1 − g)R for some idempotent g with RgR = R. With elementary operations, we can move c to the 2, 1 position.

Now apply the transpose of Corollary 2.6 to the first column of A. Thus, A can be transformed by elementary row operations to a matrix whose 2, 1 entry is a regular element d such that d ∈ cR and Rd = R(1 − q) for some idempotent q with RqR = R. Since d is regular, dR = (1 − p)R for some idempotent p. Then (1 − p)R ⊆ (1 − g)R, whence Rg ⊆ Rp and so RpR = R.

Finally, use elementary operations to move d to the 1, 1 position. □

Theorem 2.8. If R is a separative exchange ring, then R is a GE-ring, and so the natural homomorphism GL_{1}(R) → K_{1}(R) is surjective.

Proof. We need to show that R is a GE_{n}-ring for all n. This is trivial for n = 1, hence we assume, by induction, that n ≥ 2 and that R is a GE_{n−1}-ring. Let A be an arbitrary invertible n × n matrix over R.

By Lemma 2.7, we may assume that the 1, 1 entry d of A is regular, with dR = (1 − p)R and Rd = R(1 − q) for some idempotents p, q such that RpR = RqR = R. We claim that d is unit-regular. Note that because RpR = RqR = R, the projective modules pR and qR are generators.

Now R = r.ann(d)⊕B = dR⊕C for some B, C, and we have to prove that r.ann(d) ≅ C. Since B ≅ dB = dR, we have r.ann(d) ⊕ B ≅ C ⊕ B. From Rd = R(1 − q), we get r.ann(d) = qR and so r.ann(d) is a generator. Since C ≅ R/dR ≅ pR, we see that C is a generator too. By Lemma 1.3, r.ann(d) ≅ C as desired.

The unit-regularity of d gives d = ue for some unit u and idempotent e. Set

U = [ \begin{matrix}u & 0 & ⋯ & 0 \\ 0 & 1 & ⋯ & 0 \\ . & . & . \\ 0 & 0 & ⋯ & 1 \end{matrix} ];

then the matrix U^{−1}A has an idempotent entry. By Lemma 2.3, there exist E, F ∈ E_{n}(R) such that

EU^{−1}AF = [ \begin{matrix}1 & 0 \\ 0 & A′ \end{matrix} ]

where A′ ∈ GL_{n−1}(R). By our induction hypothesis, A′ ∈ E_{n−1}(R)D_{n−1}(R). It follows that

A ∈ D_{n}(R)E_{n}(R)D_{n}(R)E_{n}(R),

and therefore we have shown that R is a GE_{n}-ring. This establishes the induction step and completes the proof. □

Remarks 2.9. (a) Observe that the proof of Theorem 2.8 did not use the full force of separativity, only the cancellation property (A ⊕ C ≅ B ⊕ C ⟹ A ≅ B) for finitely generated projective R-modules A, B, C with A and B generators.

(b) Theorem 2.8 includes, in particular, the result of Menal and Moncasi that every factor ring of a right self-injective ring is a GE-ring [22, Theorem 2.2]. To make the connection explicit, recall that right self-injective rings are semiregular (e.g., [13, Theorem 2.16, Lemma 2.18]) and hence exchange; thus, all their factor rings are exchange rings. Further, any right self-injective ring is separative (e.g., [14, Theorem 3]). It follows that factor rings of right self-injective rings are separative [3, Theorem 4.2].

(c) As a special case of Theorem 2.8, we obtain that any separative regular ring is a GE-ring, which gives a partial affirmative answer to a question of Moncasi [23, Questió 5].

(d) In the situation of Theorem 2.8, one naturally asks for a description of the kernel of the epimorphism GL_{1}(R) → K_{1}(R). This has been answered for unit-regular rings and regular right-self-injective rings by Menal and Moncasi [22, Theorems 1.6, 2.6], and for exchange rings with primitive factors artinian by Chen and Li [9, Theorem 3]. In all the above cases, K_{1}(R) ≅ GL_{1}(R)^{ab} provided \frac{1}{2} ∈ R [22, Theorems 1.7, 2.6; 9, Corollary 7]. Further, K_{1}(R) ≅ GL_{1}(R)^{ab} when R is either a C*-algebra with unitary 1-stable range or an AW*-algebra [22, Theorem 1.3, Corollary 2.11] (here the algebraic K_{1} is meant). The unit-regular and AW* results correct and extend earlier work of Handelman [18, Theorem 2.4; 19, Theorem 7].

Theorem 2.10. If R is a separative exchange ring and A is a (von Neumann) regular n×n matrix over R, then A can be diagonalized using elementary row and column operations.

Proof. By [2, Theorem 2.4], there exist P, Q ∈ GL_{n}(R) such that P AQ is diagonal. By Theorem 2.8, P = U_{1}V_{1} and Q = V_{2}U_{2}, where U_{1}, U_{2} ∈ D_{n}(R) and V_{1}, V_{2} ∈ E_{n}(R). So V_{1}AV_{2} is a diagonal matrix obtained from A by elementary row and column operations. □

Remark 2.11. When applying Theorem 2.10, note the distinction between invertible matrices and general matrices. An invertible matrix over a separative exchange ring can be diagonalized from either side (by Theorem 2.8), whereas the diagonalization of a general regular matrix sometimes requires elementary operations on both the rows and the columns. For example, the 2 × 2 matrix [ \begin{matrix}1 & 1 \\ 0 & 0 \end{matrix} ] over a field cannot be diagonalized using only elementary row operations.

Example 2.12. Non-regular matrices over separative exchange rings need not be diagonalizable by elementary operations, even over finite dimensional algebras. For example, choose a field F and let

R = F [x_{1}, x_{2}, x_{3}, x_{4}]/⟨x_{1}, x_{2}, x_{3}, x_{4}⟩^{2}.

Then R has a basis 1, a_{1}, a_{2}, a_{3}, a_{4} such that a_{i}a_{j} = 0 for all i, j. Since R is clearly semiregular, it is an exchange ring; separativity is an easy exercise. In fact, since R is artinian, it has stable rank 1. Recall that this also implies that R is a GE-ring. Observe that every element of R is a sum of a scalar plus a nilpotent element, and that the product of any two nilpotent elements of R is zero.

Now consider the matrix A = [ \begin{matrix}a_{1} & a_{2} \\ a_{3} & a_{4} \end{matrix} ], whose entries are linearly independent nilpotent elements of R. We claim that any sequence of elementary row or column operations on A can only produce a matrix whose entries are linearly independent nilpotent elements. For instance, consider a product

[ \begin{matrix}1 & b \\ 0 & 1 \end{matrix} ] [ \begin{matrix}c_{11} & c_{12} \\ c_{21} & c_{22} \end{matrix} ] = [ \begin{matrix}c_{11} + bc_{21} & c_{12} + bc_{22} \\ c_{21} & c_{22} \end{matrix} ]

where c_{11}, c_{12}, c_{21}, c_{22} are linearly independent nilpotent elements. Then b = β+n for some β ∈ F and some nilpotent element n, whence bc_{21} = βc_{21} and bc_{22} = βc_{22}, and so the entries in the matrix product above are linearly independent (they are clearly nilpotent). The same thing happens with other elementary operations, establishing the claim.

Therefore no sequence of elementary operations on A can produce a matrix with a zero entry. In particular, A cannot be diagonalized by elementary operations. Since R is a GE-ring, it follows that A cannot be diagonalized by invertible matrices either, i.e., there do not exist X, Y ∈ GL_{2}(R) such that XAY is diagonal. Thus the first of the natural properties discussed in the introduction is not implied by the second. □

3. K_{1} of separative C*-algebras with real rank zero

In connection with his work on the structure of multiplier algebras (e.g., [32, 33, 34]), Shuang Zhang has conjectured [unpublished] that if A is any unital C*-algebra with real rank zero, the topological K_{1}(A) is isomorphic to the unitary group U(A) modulo the connected component of the identity, U(A)^{∘}. We confirm this conjecture in case A is separative, which at the same time provides a unified approach to all known cases of the conjecture. The main interest of Zhang’s conjecture is in the case when the stable rank of A is greater than 1, since it has long been known that K_{1}^{top}(A) ≅ U(A)/U(A)^{∘} for all unital C*-algebras A with stable rank 1 (e.g., this is equivalent to [26, Theorem 2.10]).

We consider only unital, complex C*-algebras in this section, and we refer the reader to [4, 30] for background and notation for C*-algebras. In particular, we use ∼ and ≲ to denote Murray-von Neumann equivalence and subequivalence of projections, and we write M_{∞}(A) for the (non-unital) algebra of ω × ω matrices with only finitely many nonzero entries from an algebra A. We write U(A) for the unitary group of a unital C*-algebra A, and U(A)^{∘} for the connected component of the identity in U(A).

In the theory of operator algebras, it is customary to write K_{1}(A) for the topological K_{1}-group of A (e.g., [4, Definition 8.1.1; 30, Definition 7.1.1]), and we shall follow that practice here. Thus, K_{1}(A) = GL_{∞}(A)/GL_{∞}(A)^{∘}. We then use the notation K_{1}^{alg}(A) to denote the algebraic K_{1}-group of A. Since K_{1}^{alg}(A) is the abelianization of GL_{∞}(A) (e.g., [27, Proposition 2.1.4, Definition 2.1.5]) and K_{1}(A) is abelian (e.g., [4, Proposition 8.1.3; 30, Proposition 7.1.2]), there is a natural surjective homomorphism K_{1}^{alg}(A) → K_{1}(A). Finally, following Brown [5, p. 116], we say that A has K_{1}-surjectivity (respectively, K_{1}-injectivity) provided the natural homomorphism U(A)/U(A)^{∘} → K_{1}(A) is surjective (respectively, injective).

The concept of real rank zero for a C*-algebra A has a number of equivalent characterizations (see [6]). One is the requirement that each self-adjoint element of A can be approximated arbitrarily closely by real linear combinations of orthogonal projections. (This is usually phrased as saying that the set of self-adjoint elements of A with finite spectrum is dense in the set of all self-adjoint elements.) It was proved in [3, Theorem 7.2] that A has real rank zero if and only if it is an exchange ring. Hence, the C*-algebras with real rank zero are exactly the C*-algebras to which our results above can be applied.

Given a C*-algebra A, all idempotents in matrix algebras M_{n}(A) are equivalent to projections (e.g., [4, Proposition 4.6.2; 27, Proposition 6.3.12]). Hence, A is separative if and only if

p ⊕ p ∼ p ⊕ q ∼ q ⊕ q  ⟹  p ∼ q

for projections p, q ∈ M_{∞}(A). An equivalent condition (analogous to Lemma 1.3) is that p ⊕ r ∼ q ⊕ r ⟹ p ∼ q whenever r ≲ n.p and r ≲ n.q for some n. Separativity in A is thus equivalent to the requirement that all matrix algebras M_{n}(A) satisfy the weak cancellation introduced by Brown and Pedersen [5, p. 116; 7, p. 114]. They have shown that every extremally rich C*-algebra (see [7, p. 125]) with real rank zero is separative ([8], announced in [5, p. 116]). We would like to emphasize the question of whether non-separative exchange rings exist by focusing on the C* case:

Problem. Is every C*-algebra with real rank zero separative?

By combining Theorem 2.8 with a result of Lin, we obtain the following theorem.

Theorem 3.1. If A is a separative, unital C*-algebra with real rank zero, then the natural map U(A)/U(A)^{∘} → K_{1}(A) is an isomorphism.

Proof. Lin proved K_{1}-injectivity for C*-algebras with real rank zero in [20, Lemma 2.2]. Hence, it only remains to show K_{1}-surjectivity. It is a standard fact that U(A) and GL_{1}(A) have the same image in K_{1}(A) (e.g., [4, pp. 66, 67] or [30, proof of Proposition 4.2.6]). Now the natural map GL_{1}(A) → K_{1}(A) factors as the composition of natural maps GL_{1}(A) → K_{1}^{alg}(A) → K_{1}(A), the second of which is surjective. Since A has real rank zero, it is an exchange ring, and so the map GL_{1}(A) → K_{1}^{alg}(A) is surjective by Theorem 2.8. Therefore the image of U(A) in K_{1}(A) is all of K_{1}(A), as desired. □

Brown and Pedersen have proved that every separative, extremally rich C*-algebra has K_{1}-surjectivity ([8], announced in [5, p. 116; 7, p. 114]). Since there are C*-algebras with real rank zero that are not extremally rich [5, Example, p. 117], Theorem 3.1 can be viewed as a partial extension of the Brown-Pedersen result within the class of C*-algebras with real rank zero.

We thank the referee for the following remark.

Remark 3.2. While Theorem 3.1 is neither unexpected nor new in the case of stable rank 1 (cf. the result of Rieffel cited above), it is perhaps surprising that there are many C*-algebras of real rank zero and stable rank 2 to which the theorem applies. To see this, consider C*-algebra extensions

0 → I → A → B → 0

in which I and B have real rank zero and A is unital. By theorems of Zhang ([35, Lemma 2.4]; cf. [6, Theorem 3.14]) and Lin and Rørdam [21, Proposition 4], A has real rank zero if and only if projections lift from B to A, if and only if the connecting map K_{0}(B) → K_{1}(I) in topological K-theory vanishes. In this case, by [3, Theorem 7.5], A will be separative provided I and B are both separative, and in particular if I and B have stable rank 1. However, by [21, Proposition 4], if I and B have stable rank 1, then A will have stable rank 2 provided the connecting map K_{1}(B) → K_{0}(I) does not vanish. It is easy to find specific extensions satisfying the above conditions, such as the examples analyzed in [21, End of Section 1] or [16].

We conclude with an application of Theorem 3.1 that extends an argument of Brown [5, Theorem 1], relating homotopy and unitary equivalence of projections, to a wider context within real rank zero. Projections p and q in a C*-algebra A are unitarily equivalent provided there exists a unitary element u ∈ A such that upu^{∗} = q; they are homotopic provided there is a continuous path f : [0, 1] → {projections in A} such that f(0) = p and f(1) = q. It is a standard fact that homotopic projections are unitarily equivalent (e.g., [4, Propositions 4.3.3, 4.6.5; 30, Proposition 5.2.10]).

Theorem 3.3. Let A be a separative, unital C*-algebra with real rank zero, let p, q ∈ A be projections, and let B = \overline{ApA} + ℂ · 1. Then p and q are homotopic in A if and only if q ∈ B and p, q are unitarily equivalent in B.

Proof. If p and q are homotopic in A, they are connected by a path of projections within A. Each projection along this path is homotopic to p and hence is unitarily equivalent to p. Thus, these projections all lie in ApA. In particular, q ∈ B, and p and q are homotopic in B. Consequently, p and q must be unitarily equivalent in B.

Conversely, assume that q ∈ B and p, q are unitarily equivalent in B. By [6, Corollary 2.8, Theorem 2.5], the closed ideal I = \overline{ApA} has real rank zero (as a non-unital C*-algebra), and so the unital C*-algebras B and pIp have real rank zero. We do not need separativity for B, just K_{1}-injectivity (by Lin’s result). Since I is an ideal of A, any projections in M_{∞}(I) which are (Murray-von Neumann) equivalent in M_{∞}(A) are also equivalent in M_{∞}(I) (any implementing partial isometry necessarily lies in M_{∞}(I)). Hence, the separativity of A implies that I is separative, and so pIp is separative. Therefore, by Theorem 3.1, pIp has K_{1}-surjectivity.

With the above information in hand, Brown’s proof [5, Theorem 1] carries through in the present setting. We sketch the details for the reader’s convenience. By hypothesis, q = upu^{∗} for some unitary u ∈ U(B); let α denote the image of u in K_{1}(B). Now K_{1}(B) = K_{1}(I^{∼}) = K_{1}(I), and because pIp is a full hereditary sub-C*-algebra of I, the natural map K_{1}(pIp) → K_{1}(I) is an isomorphism [5, Remark, p. 117]. Thus α is the image of some β ∈ K_{1}(pIp). Since pIp has K_{1}-surjectivity, β is the image of some unitary v_{1} ∈ U(pIp). Let v = v_{1} + 1 − p and w = uv^{∗}. Then w is a unitary in B such that q = wpw^{∗}, and the image of w in K_{1}(B) is zero. Since B has K_{1}-injectivity, w ∈ U(B)^{∘}, from which it follows that p and q are homotopic. □

Acknowledgements

We thank Michael Barr for a stimulating conversation and Larry Levy for helpful comments.

References

1. P. Ara, Extensions of exchange rings, J. Algebra 197 (1997), 409-423.

2. P. Ara, K. R. Goodearl, K. C. O’Meara, and E. Pardo, Diagonalization of matrices over regular rings, Linear Algebra Applic. 265 (1997), 147-163.

3. , Separative cancellation for projective modules over exchange rings, Israel J. Math. 105 (1998), 105-137.

4. B. Blackadar, K-Theory for Operator Algebras, MSRI Publ. 5, Springer-Verlag, New York, 1986.

5. L. G. Brown, Homotopy of projections in C*-algebras of stable rank one, in Recent Advances in Operator Algebras (Orléans 1992), Astérisque 232 (1995), 115-120.

6. L. G. Brown and G. K Pedersen, C*-algebras of real rank zero, J. Func. Anal. 99 (1991), 131-149.

7. , On the geometry of the unit ball of a C*-algebra, J. reine angew. Math. 469 (1995), 113-147.

8. , Non-stable K-theory and extremally rich C*-algebras (to appear).

9. H. Chen and F.-u. Li, Whitehead groups of exchange rings with primitive factors artinian, Preprint, 1998.

10. P. M. Cohn, On the structure of the GL_{2} of a ring, Publ. Math. I.H.E.S. 30 (1966), 365-413 (5-53).

11. P. Crawley and B. Jónsson, Refinements for infinite direct decompositions of algebraic systems, Pacific J. Math. 14 (1964), 797-855.

12. C. W. Curtis and I. Reiner, Methods of Representation Theory, Vol. II, Wiley-Interscience, New York, 1987.

13. K. R. Goodearl, Ring Theory: Nonsingular Rings and Modules, Dekker, New York, 1976.

14. , Direct sum properties of quasi-injective modules, Bull. Amer. Math. Soc. 82 (1976), 108-110.

15. , Von Neumann Regular Rings, Pitman, London, 1979; Second Ed., Krieger, Malabar, Fl., 1991.

16. , C*-algebras of real rank zero whose K_{0}’s are not Riesz groups, Canad. Math. Bull. 39 (1996), 429-437.

17. K. R. Goodearl and R. B. Warfield, Jr., Algebras over zero-dimensional rings, Math. Ann. 223 (1976), 157-168.

18. D. Handelman, K_{1} of noncommutative von Neumann regular rings, J. Pure Appl. Algebra 8 (1976), 105-118.

19. , Stable range in AW*-algebras, Proc. Amer. Math. Soc. 76 (1979), 241-249.

20. H. Lin, Approximation by normal elements with finite spectra in C*-algebras of real rank zero, Pacific J. Math. 173 (1996), 443-489.

21. H. Lin and M. Rørdam, Extensions of inductive limits of circle algebras, J. London Math. Soc. (2) 51 (1995), 603-613.

22. P. Menal and J. Moncasi, K_{1} of von Neumann regular rings, J. Pure Appl. Algebra 33 (1984), 295-312.

23. J. Moncasi, Rang estable en anells regulars, Ph.D. Thesis, Universitat Autònoma de Barcelona, 1984.

24. W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977), 269-278.

25. F. Perera, Lifting units modulo exchange ideals and C*-algebras with real rank zero, Univ. of Copenhagen Math. Inst. Preprint Series 1999, No. 5.

26. M. A. Rieffel, The homotopy groups of the unitary groups of non-commutative tori, J. Operator Theory 17 (1987), 237-254.

27. J. Rosenberg, Algebraic K-Theory and Its Applications, Grad. Texts in Math. 147, Springer-Verlag, New York, 1994.

28. J. Stock, On rings whose projective modules have the exchange property, J. Algebra 103 (1986), 437-453.

29. R. B. Warfield, Jr., Exchange rings and decompositions of modules, Math. Ann. 199 (1972), 31-36.

30. N. E. Wegge-Olsen, K-Theory and C*-Algebras, Oxford Univ. Press, Oxford, 1993.

31. H.-P. Yu, Stable range one for exchange rings, J. Pure Appl. Algebra 98 (1995), 105-109.

32. S. Zhang, A Riesz decomposition property and ideal structure of multiplier algebras, J. Operator Theory 24 (1990), 204-225.

33. , Diagonalizing projections in multiplier algebras and in matrices over a C*-algebra, Pacific J. Math. 145 (1990), 181-200.

34. , K_{1}-groups, quasidiagonality and interpolation by multiplier projections, Trans. Amer. Math. Soc. 325 (1991), 793-818.

35. , Certain C*-algebras with real rank zero and their corona and multiplier algebras. Part I, Pacific J. Math. 155 (1992), 169-197.

P. Ara: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain

E-mail address: para@mat.uab.es

K. R. Goodearl: Department of Mathematics, University of California, Santa Barbara, California 93106, USA

E-mail address: goodearl@math.ucsb.edu

K. C. O’Meara: Department of Mathematics, University of Canterbury, Christchurch, New Zealand

E-mail address: komeara@math.canterbury.ac.nz

R. Raphael: Department of Mathematics and Statistics, Concordia University, Montréal, Québec H4B 1R6, Canada

E-mail address: raphael@alcor.concordia.ca