Algebraic equations on L-spaces (Q1107056)

The author describes a certain class of topological spaces (L-spaces), which includes all connected locally compact Abelian groups and all locally connected spaces and starting with the ideas of his paper [Funkts. Anal. Prilozh. 16, No.4, 68-69 (1982; Zbl 0519.58012)], extends many of the results of that paper, to nonseparable polynomials with functional coefficients in L-spaces. The following are some of these results. Theorem (1). Let X be an L-space. (a) There exists a mapping \(\sigma_ n(A): P^ n_*(A)\cap P^ n_{loc}(A)\to H^ 1(X,S(n))\), such that \(\sigma_ n(A)^{-1}(O_ n)=P^ n_*(A)\cap P^ n_{gl}(A)\); (b) If the cohomological set \(H^ 1(X,S(n))\) consists of one element then \(P^ n_*(A)\cap P^ n_{loc}(A)\subset P^ n_{gl}(A).\) Theorem (2). Let p be an integral polynomial over the algebra B of conditionally periodic functions (or, equivalently, over C(T)). If the polynomial p is locally totally reducible over the algebra C(T), and if the set \(\{\) \(x\in T: d_ p(x)=0\}\) is locally refined in T, then the polynomial p is totally reducible over the algebra of uniform almost- periodic functions on the real line, and the Fourier spectrum of the roots of the polynomial p is contained in the set \(Q[\alpha_ 1,...,\alpha_ K]\). - The author states three corollaries to the first theorem and states and proves the proposition that ``a connected compact Abelian group X is an L-space'' and a corollay of it.