Volume 111, Issue 1
February 2001, pages 1-137
pp 1-31
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
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
Limits of Commutative Triangular Systems on Locally Compact Groups
On a locally compact group 𝐺, if $𝜈_{n}^{k_{n}} →𝜇$, (𝑘_{𝑛} →∞), for some probability measures 𝜈_{𝑛} and 𝜇 on 𝐺, then a sufficient condition is obtained for the set 𝐴 = {$𝜈_n^m|$ 𝑚 ≤ 𝑘_{𝑛}} 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 if 𝐺 is a discrete linear group over $\mathsf{R}$ 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
Topological ^{∗}-algebras with 𝐶*-enveloping Algebras II
Universal 𝐶*-algebras 𝐶*(𝐴) exist for certain topological ^{∗}-algebras called algebras with a 𝐶*-enveloping algebra. A Frechet ^{∗}-algebra 𝐴 has a 𝐶*-enveloping algebra if and only if every operator representation of 𝐴 maps 𝐴 into bounded operators. This is proved by showing that every unbounded operator representation 𝜋, continuous in the uniform topology, of a topological ^{∗}-algebra 𝐴, which is an inverse limit of Banach ^{∗}-algebras, is a direct sum of bounded operator representations, thereby factoring through the enveloping pro-𝐶* algebra 𝐸(𝐴) of 𝐴. Given a 𝐶*-dynamical system (𝐺, 𝐴, 𝛼), any topological ^{∗}-algebra 𝐵 containing 𝐶_{𝑐}(𝐺, 𝐴) as a dense ^{∗}-subalgebra and contained in the crossed product 𝐶*-algebra 𝐶*(𝐺, 𝐴, 𝛼) satisfies 𝐸(𝐵)=𝐶*(𝐺, 𝐴, 𝛼). If $G = \mathbb{R}$, if 𝐵 is an 𝛼-invariant dense Frechet ^{∗}-subalgebra of 𝐴 such that 𝐸(𝐵) = 𝐴, and if the action 𝛼 on 𝐵 is 𝑚-tempered, smooth and by continuous ^{∗}-automorphisms: then the smooth Schwartz crossed product $S(\mathbb{R}, B, 𝛼)$ satisfies $E(S(\mathbb{R}, B, 𝛼)) = C^*(\mathbb{R}, A, 𝛼)$. When 𝐺 is a Lie group, the 𝐶^{∞}-elements 𝐶^{∞}(𝐴), the analytic elements 𝐶^{𝜔}(𝐴) as well as the entire analytic elements 𝐶^{𝑒𝜔}(𝐴) carry natural topologies making them algebras with a 𝐶*-enveloping algebra. Given a non-unital 𝐶*-algebra 𝐴, an inductive system of ideals 𝐼_{𝛼} is constructed satisfying $A = C^*-\mathrm{ind} \lim I_𝛼$; and the locally convex inductive limit $\mathrm{ind}\lim I_𝛼$ is an 𝑚-convex algebra with the 𝐶*-enveloping algebra 𝐴 and containing the Pedersen ideal 𝐾_{𝐴} of 𝐴. Given generators 𝐺 with weakly Banach admissible relations 𝑅, we construct universal topological ^{∗}-algebra 𝐴(𝐺, 𝑅) and show that it has a 𝐶*-enveloping algebra if and only if (𝐺, 𝑅) is 𝐶*-admissible.
pp 95-106
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
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 field 𝑓(𝑡, 𝑥, 𝑦) is Caratheodory and in some instances the continuity condition on 𝑥 or 𝑦 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
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 Errata
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