Volume 111, Issue 1
February 2001, pages 1-137
pp 1-31 February 2001
This is an expository article on the theory of algebraic stacks. After introducing the general theory, we concentrate in the example of the moduli stack of vector bundles, giving a detailed comparison with the moduli scheme obtained via geometric invariant theory.
pp 33-47 February 2001
Variational formulae for Fuchsian groups over families of algebraic curves
We study the problem of understanding the uniformizing Fuchsian groups for a family of plane algebraic curves by determining explicit first variational formulae for the generators.
pp 49-63 February 2001
Limits of commutative triangular systems on locally compact groups
On a locally compact group G, if$$v_n^{k_n } \to \mu ,(k_n \to \infty )$$, for some probability measuresv_{n} and μ onG, then a sufficient condition is obtained for the set$$A = \{ v_n^m \left| {m \leqslant k_n } \right.\} $$ to be relatively compact; this in turn implies the embeddability of a shift of μ. The condition turns out to be also necessary when G is totally disconnected. In particular, it is shown that ifG is a discrete linear group over R then a shift of the limit μ is embeddable. It is also shown that any infinitesimally divisible measure on a connected nilpotent real algebraic group is embeddable.
pp 65-94 February 2001
Topological *-algebras withC*-enveloping algebras II
UniversalC*-algebrasC*(A) exist for certain topological *-algebras called algebras with aC*-enveloping algebra. A Frechet *-algebraA has aC*-enveloping algebra if and only if every operator representation ofA mapsA into bounded operators. This is proved by showing that every unbounded operator representation π, continuous in the uniform topology, of a topological *-algebraA, which is an inverse limit of Banach *-algebras, is a direct sum of bounded operator representations, thereby factoring through the enveloping pro-C*-algebraE(A) ofA. Given aC*-dynamical system (G,A,α), any topological *-algebraB containingC_{c}(G,A) as a dense *-subalgebra and contained in the crossed productC*-algebraC*(G,A,α) satisfiesE(B) =C*(G,A,α). IfG = ℝ, ifB is an α-invariant dense Frechet *-subalgebra ofA such thatE(B) =A, and if the action α onB ism-tempered, smooth and by continuous *-automorphisms: then the smooth Schwartz crossed productS(ℝ,B,α) satisfiesE(S(ℝ,B,α)) =C*(ℝ,A,α). WhenG is a Lie group, theC^{∞}-elementsC^{∞}(A), the analytic elementsC^{ω}(A) as well as the entire analytic elementsC^{є}(A) carry natural topologies making them algebras with aC*-enveloping algebra. Given a non-unitalC*-algebraA, an inductive system of idealsI_{α} is constructed satisfyingA =C*-ind limI_{α}; and the locally convex inductive limit ind limI_{α} is anm-convex algebra with theC*-enveloping algebraA and containing the Pedersen idealK_{a} ofA. Given generatorsG with weakly Banach admissible relationsR, we construct universal topological *-algebraA(G, R) and show that it has aC*-enveloping algebra if and only if (G, R) isC*-admissible.
pp 95-106 February 2001
On the equisummability of Hermite and Fourier expansions
We prove an equisummability result for the Fourier expansions and Hermite expansions as well as special Hermite expansions. We also prove the uniform boundedness of the Bochner-Riesz means associated to the Hermite expansions for polyradial functions.
pp 107-125 February 2001
Periodic and boundary value problems for second order differential equations
Nikolaos S Papageorgiou Francesca Papalini
In this paper we study second order scalar differential equations with Sturm-Liouville and periodic boundary conditions. The vector fieldf(t,x,y) is Caratheodory and in some instances the continuity condition onx ory is replaced by a monotonicity type hypothesis. Using the method of upper and lower solutions as well as truncation and penalization techniques, we show the existence of solutions and extremal solutions in the order interval determined by the upper and lower solutions. Also we establish some properties of the solutions and of the set they form.
pp 127-135 February 2001
Boundary controllability of integrodifferential systems in Banach spaces
Sufficient conditions for boundary controllability of integrodifferential systems in Banach spaces are established. The results are obtained by using the strongly continuous semigroup theory and the Banach contraction principle. Examples are provided to illustrate the theory.
pp 137-137 February 2001 Erratum
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