Factorization of polynomials over Banach algebras

Abstract

Introduction. A Banach algebra in this paper will be understood to mean a commutative, semi-simple Banach algebra with multiplicative unit e. By the carrier space DA of the Banach algebra A, we mean the space of multiplicative linear functionals on A to C, the complex numbers, and it is to be endowed with the usual weak* topology (cf. [6]). For a E A, a denotes the Gelfand transform of a defined on (DA and A will denote the collection of such functions. We set the following notation. x will be used to denote an indeterminate over A as well as over A and C. If x(x) = 7=Oa,ix' is a polynomial over A, let &(x) and ;,(x) denote, respectively, ,i oQX,(2) = 0}, ac(x) A[x], plays an important role in the present paper. Z(oc(x),A) is topologized with the relative product topology from (A x C. The mapping n is defined by r(h, 2) = h, (h, 2) e Z(oc(x), A). The multiplicity function M of a(x) is defined as follows: for (h, 2) E Z(x(x), A), M(h, A) is equal to the multiplicity of A as a root of cth(x) = 0. In ?1 we introduce the concept of M-neighborhood of a point in Z(a(x),A). We say that W c Z(cx(x), A) is a M-neighborhood of (ho, 2A) E Z(oc(x), A) if W is a neighborhood in Z(oc(x),A) of (ho,AO) and if, for each h e p(W), M(ho, 2O) is equal to the sum of the values of M at the points (h,2) in 7K-1(h) nl W. Proposition 1.1 states that M-neighborhoods exist and that they form a base for the neighborhood system at each point of Z(ox(x),A). The remainder of this section contains most of the topological lemmas needed for our work on factorization. Particular attention is paid to the case where Z(o(x),A) contains a compact open subset K (7t(K) = 'DA) on which M is constant. When this condition obtains, K and 4FD decompose topologically and this decomposition in turn forces a(x) to factor. The main factorization theorem (2.1) says that if a(x)eA[x] and if K (nr(K)-=(A) is a compact open subset of Z(a(x), A), then there exists a monic polynomial P(x) e A[x] such that ,B(x) is a factor of o(x), Z(fl(x), A) = K and Z(ax(x)/fl(x), A) = Z(a(x), A) K. A more detailed description of the factorization of monic polynomials follows. For example, it is shown that if A is indecomposable