Unimodular Matrices in Banach Algebra Theory

Let A be a ring with 1 and denote by L (resp. R) the set of left (resp. right) invertible elements of A. If A has an involution *, there is a natural bijection between L and R. In general, it seems that there is no such bijection; if A is a Banach algebra, L and R are open subsets of A, and they have the same cardinality. More generally, we prove that the spaces Uk(A") of n X i-left-invertible matrices and kU(A") of k X n-right-invertible matrices are homotopically equivalent. As a corollary, we answer negatively two questions of Rieffel [12]. 1. Let A be a ring (with an identity element 1) and k < n. A matrix x G A"xk is unimodular if there exists y g Akx" such that y ■ x =* 1 e Mk(A) = Akxk. We denote by Í4(A") the set of all unimodular matrices of A"xk, and Sk(A") = {(x, y) g A"xk X Akx": y • x = 1}. The first projection induces a mapping />: ,SA.(A") -» Uk(A"). 1.1. Proposition. Le/ A be a ring, x g ,4bX*, and define Tx: Mn(A) -» ^"x/c ¿y TA(a) = ax. TTzeTj x is unimodular if and only if Tx is a direct epimorphism; i.e., there is an A-linear map S: AnXk —> Mn(A) such that Tx° S = \A„xk. Proof. Obvious. From now on, A denotes a (real or complex) Banach algebra. Our purpose is to study the fibration properties of the mappings tx = rjGL,U):GLB(,l)-+t4(^), for* g Uk(A") and p: Sk(A") -* Uk(An). 1.2. Lemma, (i) Uk(A") is open in A"xk. (ii) tx is continuous, for every x G Uk(A"). (iii) p is continuous. Proof. It is clear that tx and p are continuous, because they are restrictions of linear mappings. If x0 g A"xk is unimodular and y ■ x0 = 1 g Mk(A), there is a neighborhood U of x0 in A"xk such that y ■ x g GL^(vl) for every x G U, for the set of invertible elements of a Banach algebra is open. But it is evident that U is contained in Uk(A"). Thus Uk(A") is open as claimed. In view of the theory of Banach manifolds, as developed in Lang [7], we can prove the next result. Recall that a differentiable map /: X -* y is a submersion at x g X if the derivative of / at x G X is a direct epimorphism from the tangent space of X at x onto the tangent space of Y at f(x). Received by the editors June 19, 1984 and, in revised form, March 1, 1985. 1980 Mathematics Subject Classification. Primary 46H05, 18F25; Secondary 55R10, 55P10, 16A25. ©1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page 473 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 474 GUSTAVO CORACH AND A. R. LAROTONDA 1.3. Proposition. Let A be a Banach algebra and x g A"x . Then the following hold. (1) Ifx g Uk(A"), then tx: GLn(A) -» Uk(A") is a submersion. (2) Sk(A") is a Banach manifold. Proof. (1) If x g Uk(A"), the derivative of tx at any a g GL„(^4) coincides with Tx, which is a direct epimorphism, by 1.1, so tx is a submersion at a. (2) Following Raeburn [11], we need only prove that the derivative of (x, y) = y ■ x, at any (x0, y0) G Sk(A") is a direct epimorphism for Sk(A") = _1(l). Now, the derivative of <> at (x0, y0) is the ^-linear map (w, z) -* z ■ xQ + y0 ■ w (w g A"xk, z g Zkx"), and the map a -» \/2(xQa, ay0) (a g Mk(A)) is a right inverse of D GL„(^4) ■ x. (ii) tx. GL„(^) -» Uk(A") is a Serrefibration. (iii) tx is surjective if and only if the induced mapping tr0(tx): 7r0(GL„(,4)) —> ir0( Uk(A")) is onto. In particular, tx is surjective if Uk(An) is connected. Proof. Let Gx be the stabilizer of x; i.e., Gx = {a g GL„(^1): a x = x). Then Gx is a Banach-Lie subgroup of GL„(,4), and, by 1.4, GL„(^) • x may be identified with the homogeneous space GLn(A)/Gx. So, assertion (i) follows from the general theory of Banach-Lie groups as Bourbaki [1] or Raeburn [11]. For the proof of (ii), it is well known that any locally trivial bundle over a paracompact base space is a Serre fibration (see Dold [3]). In our case, Uk(A") is metrizable, a fortiori paracompact. The proof of (iii) is now obvious. The problem of determining whether an element z of Uk(A") belongs to GLn(A") ■ x for a fixed x g Uk(A") is, in general, very difficult. Let us give an example. Let A be the algebra of real continuous functions on the unit sphere S"_1 of R", and let x¡, i = 1,...,«, be the coordinate functions. Clearly, x = (xx,...,xn) belongs to License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use UNIMODULAR MATRICES IN BANACH ALGEBRA THEORY 475 UX(A") for x -'x = T.',Lixf = 1. Let / be the map a —> a ■ ex (where ex is the first canonical vector of A") from GL„(,4) into UX(A"). We shall show that x belongs to GLn(A) ■ ex only if n = 2,4, or 8. In fact, x g GL„(vl) • ex means that there is a basis {vx, v2,■■.,"„} of A" with vx = x. Let S be the free supplement of A ■ x generated by {v2,...,v„} and S'= [v ^ A": (v,x) = 0}. Then S' is another supplement of A ■ x, so S and S' are isomorphic and S" must be free. But 5" identifies with the ^-module of continuous sections of the tangent bundle t(S"~1) of S"1 (see Swan [13]). Thus, since S' is free, t(S"_1) must be a trivial bundle, and this may happen only if n = 2,4, or 8, as claimed. Let A be a ring, and let e be the n X ¿c-matrix whose columns are, in order, the first k canonical vectors of A". Let us set t = te: GLn(A) -* Uk(A"). A is (n, k)-Hermite if t is onto. (This terminology is consistent with that of Lam [6], at least for commutative rings.) The following lemma shows that the choice of e is irrelevant. 1.7. Lemma. Let A be a ring. Then A is (n, k)-Hermite if and only if tx: GLn(A) -* Uk(A") is onto for every x G Uk(A"). Proof. Suppose that A is (n, rV)-Hermite, and take z and x in Uk(A"). Then there exist t and a in GLn(A) such that z = t • e and x = o ■ e; then tx(ra~l) = t • a-1 • x = z. The converse is trivial. From 1.6(iii) we get 1.8. Proposition. Let A be a Banach algebra. Then A is (n, k)-Hermite if and only if 7r0(r) is onto. In particular, A is (n, k)-Hermite if Uk(A") is connected. 1.9. Examples. (1) Let A be a complex commutative Banach algebra with spectrum X(A). It is well known that tT0(Uk(A")) coincides with the set of homotopy classes of maps from X(A) into Uk(C") (see Novodvorski [9], Taylor [14] and Raeburn [11]). Lin [8] proved that A is (n, &)-Hermite if and only if C(X(A)), the algebra of complex continuous functions on X(A), is; moreover, for x and z in Uk(A"), z belongs to GLn(A) ■ x if and only if the Gelfand transform z belongs to GLn(C(X(A))) ■ x. In particular, if X(A) is dominated by an w-dimensional space, with m ^ 2(w — k), then A is (n, &)-Hermite. This last fact has also been proved by Lin [8]. (2) Let 3V be a Hubert space and A the algebra of bounded linear operators on Of. Then GL„(A) is contractible for every n (Kuiper [5]). Now, A is isomorphic to A"xk (as ,4-modules and as Banach spaces), and it is easy to see that Uk(A") is homeomorphic to Ux(Al), which is disconnected, as we shall see 2.6(iii). Then A cannot be (n, A:)-Hermite for any k < n. (3) If the stable range of A is < n — k, A is (n, &)-Hermite (see Corach and Larotonda [21]). 2. In this section we study the fibration properties of the map p: Sk(A") -* Uk(A") when A is a Banach algebra. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 476 GUSTAVO CORACH AND A R. LAROTONDA 2.1. Theorem. Let A be a Banach algebra. Then the projection onto the first coordinate p: Sk(A") -» Uk(A") is a locally trivial bundle with contrac tibie fibre. Proof. For (x0,y0) g Sk(A") we define : AnXk -* GL„(A) ■ x0 C yl"** by <í>(z) = exp(x, j>0) • x0. The derivative of is a diffeomorphism from {/onto V. We define a section of tx over K, s: V -» GLn(^4), by i(x) = exp(«/>_1(^) " J'o)1 In ^act' '* ° ■*(■*) = s(x) • x0 = x for every x g V and s(x0) = \M (A). We now define a trivializing map g: p~l(V) -» F X A/ by g(x, }>) = (x, y • s(x) jp0), where A/ = [y g yl**": _v • x0 = 0}; the inverse of g is A: Vx N -^> p~l(V), h(x, z) = (x,(z + y0) ■ í(x)"1). The fibre p'x(x), for x g V, is the affine manifold {(x,(y + y0) • s(x)~x): yg A/}, obviously contractible. 2.2. Corollary. The projection p is a homotopy equivalence and it admits a global section. Proof. From the examination of the homotopy sequence of p, it follows that p is a weak homotopy equivalence, so, by a result of Palais [10, Theorem 15], since Sk(A") and Uk(A") are metrizable manifolds (by 1.2 and 1.3), p is a homotopy equivalence. The second assertion follows from a result of Godement (see Dold [3, p. 223]) which states that a fibre bundle with contractible fibre over a paracompact space has a global section. 2.3. Corollary. For every compact pair (Z,Y), every map f: (Z,Y)-> (Uk(A"), x0), and every y0 g Akx" such that y0 ■ x0 = 1 g Mk(A), there is a lifting/: (Z,Y)^(Sk(A"),(x0,y0)); i.e., pf = f. 2.4. Corollary. For every compact space X the natural inclusion Uk(A(X)n) -* C(X,Uk(A")) (where A(X) is the algebra of maps from X into A) is a homeomorphism. We have been studying left unimodular matrices. We can also consider right unimodular matrices: y G Akx" (k < n) is right unimodular if there exists x G A"xk such that y • x = 1 S Mk(A). Of course, the set kU(A") of right unimodular matrices is the image of the second projection p2: Sk(A") -* Akx". We may reproduce all the results obtained for Uk(A") in this new context. In particular, we may prove that p2: Sk(A") -* kU(A") is a homotopy equivalence. Thus, we have 2.5. Corollary. Uk(A") and kU(A") are homotopically equivalent. Remark. This result answers negatively question 4.8 of Rieffel [12] and generalizes his Proposition 10.2 to any Banach algebra A. We now consider the case n = k = 1, so Ux(Al) = L and xU(Al) = R. 2.6. Theorem. Let A be a Banach algebra. Then the following hold. (i) L and R are homotopically equivalent. (ii) The connected components of 1 in L, R and G (= group of units of A) coincide; in particular, G is connected if L or R are connected and, in this case, G = L = R. Thus, L and R are disconnected if G ¥= L or G # R. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use UNIMODULAR MATRICES IN BANACH ALGEBRA THEORY 477 The proof follows directly from 2.5 and the following obvious lemma. 2.7. Lemma. G is open and closed in L and R.