Dense Morphisms in Commutative Banach Algebras

Using a new notion of stability we compute exactly the stable rank of the polydisc algebra, extend Oka's extension theorem to «-tuples of functions without common zeros and give an estimation for a question raised by Swan concerning the stable rank of a dense subalgebra of a given Banach algebra. 0. Introduction. The stable rank of a ring A [2] is an algebraic invariant of A which turns out to be closely related to the topology of the spectrum of A when A is a complex commutative Banach algebra (see [4, 7, 8]). In this paper we study certain approximation and interpolation problems which are related to that notion. Our main result (Theorem 2.7), whose statement is too technical to be described here, has several applications. First, we find exactly the stable rank of the polydisc algebra An of C": sr(An) = [n/2] + 1. Moreover, for every «-generated algebra A, sr(A) < [n/2] + 1 and the equality holds if the joint spectrum of a system of « generators of A has non void interior in C". Next, we obtain a result in connection with an old problem of Cartan [3] who looked for conditions on an /--tuple of holomorphic functions on a domain Û in C* without common zero to be completed to an invertible matrix of such functions. This problem has been considered by Lin [16] and Sibony and Wermer [20]. We prove that if A is «-generated then every w-tuple a = (ax,..., am), such that âx,...,âm have no common zero (see Definitions below), can be completed to an invertible matrix with entries in A for m > [n/2] + 1. Another application is a /c-dimensional version of Oka's extension theorem. Finally, we give a partial answer to a question of Swan [23, Remark, p. 206] if there exists a morphism f: A -> B with dense image such that "/(a) invertible =* a invertible" then sr(A) 4: sr(B) ^ sr(A) + 2. Preliminaries. In this paper Banach algebra means a complex commutative Banach algebra with identity. The spectrum of a Banach algebra A is the space X(A) of all nonzero complex homomorphisms of A; the elements of X(A), called characters, are continuous and X(A) is a compact Hausdorff space with the induced weak* topology. The Gelfand transformation A: A -* C(X(A)) is defined by â(h) = h(a) (a e A, he X(A)). For any «-tuple a = (ax,..., an) e A" we write â = (âx,..., â„): X(A) -> C"; the image of â is the joint spectrum of a, which we denote a(a) and we recall that a(a) = [X = (X1,...,\n) e C": Y."=XA ■ (a¡ A,) *A). Received by the editors November 18, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 46J15, 18F25; Secondary 32E20, 32E25, 55Q55. ©1987 American Mathematical Society 0002-9947/87 $1.00 + $.25 per page 537 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 538 GUSTAVO CORACH AND F. D. SUÁREZ If a = (a,),e/ is a (possibly infinite) system of generators of A and a(a) = {(«(a,)),e/ e Cr: h e X(A)}, then a(a) is a polynomially convex compact subset of C ' (where a compact subset X of C ' is polynomially convex if it coincides with its hull X = (z e C': \p(z)\ < sup{|/?(x)|: x e X) for every polynomial p) [22]). Moreover à: X(A) -> a(a) is a homeomorphism. For a compact subset .Yof C7 we define P(X) as the closure, in C(X), of the polynomial functions. The spectrum of P(X) is identified to X. Observe that there exists a continuous homomorphism A -> P(o(a)) (where a = (a¡) is a system of generators), with dense image. Sometimes we identify X(A) to a(a) and write P(X(A)) instead of P(a(a)). We put Un(A) = (a e A": 0 & a(a)}; its elements are called unimodulars. A unimodular {ax,..., an_x, an) is reducible if there exist xx,..., xn_x in A such that (ax + xxan,..., an_x + xn_xan) is unimodular. The stable rank of A is the least « such that every a e Un+X(A) is reducible. We use the symbol A for the closed unit disc of the complex plane C. Given spaces X, Y, C(X, Y) denotes the set of all maps from X into Y. Definition 1.1. Let E, B, and X be Hausdorff topological spaces. Suppose that B is metrizable with a metric d and A' is a compact space. A map p: E -> B has property (H) with respect to X if for every commutative diagram / X -» E a ip (i=[o,i],i(x) = (o,x)) I X X -» B f and e > 0 there exists a map F: I X X -> E such that Fi = f and sup{di pFit, x), fit, x)): tel, x e X) < e. A map />: £ -» B is a Serre gwaszfibration if it has property (H) with respect to every cube Im (m > 0). When e = 0 we get the classical notion of Serre fibration (see [14, pp. 61-64]). Proposition 1.2. Let : A —> B be a dense morphism of Banach algebras ii.e. is a continuous homomorphism with dense image). Consider the induced group homomorphism 4>: GYniA) -> GYniB), with image L. (1) for b e GYn(B), bel. (the closure of L in GYn(B)) if and only if there exists b' e L which belongs to the connected component ofb in GYn(B); (2) : GL„(v4) -» GL„(.B) is a quasi-fibration. Proof. (1)(=>) Let b eL; then there exists a e GL„(,4) such that ||(a) ¿>|| < ||ö_1||_1 and b + ti4>ia) b) it e I) defines an arc in GL„(£) joining b and (a): in fact, b~\b + t(4»(a) -b)) = l + tb^i^a) b) and \\b-\*(a)-b)\\<\ it el). ( «= ) We prove first that §(GYn(A)0) is dense in GL„(.B)0 (where in general G0 is the connected component of the neutral element); if y e GYn(B)0 then there exist bx.bse Mn(B) with y = exp^) • • • exp(bs) [18, Chapter I]. Using the continuity of exp and the density of the image of , we can approach y by License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use DENSE MORPHEMS IN COMMUTATIVE BANACH ALGEBRAS 539 exp <¡>(ax) ■ ■ • exp (exp(ax) ■ ■ ■ exp(as)) e 4>(GL„(.4)0) for some ax,...,ase MniA); this proves our assertion. Now, if Gx is a connected component of GL„(2?) and G0 = GL„(5)o> for u e Gx the translation x -* xu defines a homomorphism G0 ~* Gx whose inverse map is y -» yu~l. Then, if b e Gx and b' e Gx n L we get b(b'yl e GQczL, but b' el which is a subgroup of GL„(i?), thus b = b(b')'lb' e L, too. (2) Firstly, we prove that <#>: GL„(^4) -» GYn(B) has property (H) with respect to /°={0}. " For this, we consider the commutative diagram GY„iA) U (/(0) = (0,0)) GL„iB) and we look for a map F: I X {0} -* GL„(,4) such that f(0,0) = /(0) and sup{|| 0 (where the norm is that of M„(B), which is induced by that of B). Observe that we can think of / and /(0,0) as elements of GL„(C(7, B)); moreover, they belong to the some connected component of GL„(C(/, B)), for 5 -* f(st,0) is an arc which begins at /(0,0) and ends at /. Observe also that ¡¡>: A -> B induces a dense morphism C(I, B) which induces, as before, a group homomorphism GL„(C(7, A)) -> GYn(C(I, B)), which we denote again (/(0)) belongs to 0. We shall prove that, for e' small enough, F = /(0)a(0)_1a is the map we look for (where we are identifying / with / X {0} and looking at /, a: I X {0} -» GL„(^)). For this, keep t e I fixed; then \\^fiO)aiO)-lait))-fit)\ 4 \\i,{fiO)aiO)-lait)) iait)) \\ + \\<¡,{a(t)) -/(/) || <|*(/(0)fl(0)-1)-l|ll*(fl(0)ll + ll9(fl)-/lk < \\{fi0)ai0yl) 1 ¡iUiait)) -fit) || + ||/(0 ||) + e' <|*(/(o)fl(o)-1)-i|(e' + ||/(01|) + e'