A note on ideal spaces of Banach Algebras

In a previous paper the second author introduced a compact topology on the space of closed ideals of a unital Banach algebra A. If A is separable then this topology is either metrizable or else neither Hausdorff nor first countable. Here it is shown that this topology is Hausdorff if A is the algebra of once continuously differentiable functions on an interval, but that if A is a uniform algebra then this topology is Hausdorff if and only if A has spectral synthesis. An example is given of a strongly regular, uniform algebra for which every maximal ideal has a bounded approximate identity, but which does not have spectral synthesis.Comment: 9 pages plain te

Cosmologies with Two-Dimensional Inhomogeneity

We present a new generating algorithm to construct exact non static solutions of the Einstein field equations with two-dimensional inhomogeneity. Infinite dimensional families of $G_1$ inhomogeneous solutions with a self interacting scalar field, or alternatively with perfect fluid, can be constructed using this algorithm. Some families of solutions and the applications of the algorithm are discussed.Comment: 9 pages, one postscript figur

On the Entropy and the Density Matrix of Cosmological Perturbations

We look at the transition to the semiclassical behaviour and the decoherence process for the inhomogeneous perturbations in the inflationary universe. Two different decoherence mechanisms appear: one dynamical, accompanied with a negligible, if at all, entropy gain, and the other, effectively irreversible dephasing, due to a rapid variation in time of the off-diagonal density matrix elements in the post-inflationary epoch. We thus settle the discrepancies in the entropy content of perturbations evaluated by different authors.Comment: LaTeX2e with the epsf packag

Penrose Limits, the Colliding Plane Wave Problem and the Classical String Backgrounds

We show how the Szekeres form of the line element is naturally adapted to study Penrose limits in classical string backgrounds. Relating the "old" colliding wave problem to the Penrose limiting procedure as employed in string theory we discuss how two orthogonal Penrose limits uniquely determine the underlying target space when certain symmetry is imposed. We construct a conformally deformed background with two distinct, yet exactly solvable in terms of the string theory on R-R backgrounds, Penrose limits. Exploiting further the similarities between the two problems we find that the Penrose limit of the gauged WZW Nappi-Witten universe is itself a gauged WZW plane wave solution of Sfetsos and Tseytlin. Finally, we discuss some issues related to singularity, show the existence of a large class of non-Hausdorff solutions with Killing Cauchy Horizons and indicate a possible resolution of the problem of the definition of quantum vacuum in string theory on these time-dependent backgrounds.Comment: Some misprints corrected. Matches the version in print. To appear in Classical & Quantum Gravit

Initial Conditions and the Structure of the Singularity in Pre-Big-Bang Cosmology

We propose a picture, within the pre-big-bang approach, in which the universe emerges from a bath of plane gravitational and dilatonic waves. The waves interact gravitationally breaking the exact plane symmetry and lead generically to gravitational collapse resulting in a singularity with the Kasner-like structure. The analytic relations between the Kasner exponents and the initial data are explicitly evaluated and it is shown that pre-big-bang inflation may occur within a dense set of initial data. Finally, we argue that plane waves carry zero gravitational entropy and thus are, from a thermodynamical point of view, good candidates for the universe to emerge from.Comment: 18 pages, LaTeX, epsfig. 3 figures included. Minor changes; paragraph added in the introduction, references added and typos corrected. Final version published in Classical and Quantum Gravit

Two-flux Colliding Plane Waves in String Theory

We construct the two-flux colliding plane wave solutions in higher dimensional gravity theory with dilaton, and two complementary fluxes. Two kinds of solutions has been obtained: Bell-Szekeres(BS) type and homogeneous type. After imposing the junction condition, we find that only Bell-Szekeres type solution is physically well-defined. Furthermore, we show that the future curvature singularity is always developed for our solutions.Comment: 16 pages, Latex; typoes corrected; references added, minor modification