Isometries of some Banach spaces of analytic functions

We characterize the surjective isometries of a class of analytic functions on the disk which include the Analytic Besov spaceB _p and the Dirichlet spaceD ^p . In the case ofB _p we are able to determine the form of all linear isometries on this space. The isometries for these spaces are finite rank perturbations of integral operators. This is in contrast with the classical results for the Hardy and Bergman spaces where the isometries are represented as weighted compositions induced by inner functions or automorphisms of the disk.     

Zur Idealtheorie in Gruppenalgebren von [FIA] B − -Gruppen

In this paper we consider closedB-invariant ideals in the group algebraL ¹(G), whereG is a locally compact group with a relatively compact groupB of topological automorphisms, which contains the set of all inner automorphisms. We study conditions when closedB-invariant ideals are completely determined by their hull. Also questions concerning the existence of approximate units in these ideals will be answered. Above all, we shall study these properties with regard to the relations between ideals inL ¹(G),L ¹ (G/N) andL ¹(N), whereN is a closedB-invariant subgroup ofG.     

Closed ideals of homogeneous algebras

LetG denote a compact group andB a homogeneous Banach algebra of pseudomeasures onG (B is left translation invariant with continuous shifts; multiplication is convolution). It is shown that the closed ideal structure ofB is precisely that ofL ¹ or of one of the closed translation invariant subspaces,L _g ¹ , ofL ¹. The closed ideal structure ofL _g ¹ is presented.     

On the bounded part of a Hilbert module over a locally C *-algebra

We will show that the bounded part of the locally C^*-algebra of all adjointable operators on the Hilbert A-module E is isomorphic to the C^*-algebra L _b(A)(b(E)) of all adjointable operators on the Hilbert b(A)-module b(E). This revised version was published online in June 2006 with corrections to the Cover Date.     

Ellipticity and invertibility in the cone algebra onL p -Sobolev spaces

Given a manifoldB with conical singularities, we consider the cone algebra with discrete asymptotics, introduced by Schulze, on a suitable scale ofL _p -Sobolev spaces. Ellipticity is proven to be equivalent to the Fredholm property in these spaces; it turns out to be independent of the choice ofp. We then show that the cone algebra is closed under inversion: whenever an operator is invertible between the associated Sobolev spaces, its inverse belongs to the calculus. We use these results to analyze the behaviour of these operators onL _p (B).     

PrincipalG-bundles on nodal curves

LetG be a connected semisimple affine algebraic group defined over C. We study the relation between stable, semistable G-bundles on a nodal curveY and representations of the fundamental group ofY. This study is done by extending the notion of (generalized) parabolic vector bundles to principal G-bundles on the desingularizationC ofY and using the correspondence between them and principal G-bundles onY. We give an isomorphism of the stack of generalized parabolic bundles onC with a quotient stack associated to loop groups. We show that if G is simple and simply connected then the Picard group of the stack of principal G-bundles onY is isomorphic to ⊕_m Z,m being the number of components ofY.     

Remarks on Gelfand Pairs Associated to Non-Type-I Solvable Lie Groups

LetSbe a connected and simply connected unimodular solvable Lie group andKa connected compact Lie group acting onSas automorphisms. We call the pair (K S) a Gelfand pair if the Banach ∗-algebraL¹_K(S) of allK-invariant integrable functions onSis a commutative algebra. In this paper we give a necessary and sufficient condition for the pair (K; S) to be a Gelfand pair using the representation theory of non-type-I solvable Lie groups. For a Gelfand pair (K; S) we realize all irreducibleK-spherical representations ofK?Sfrom irreducible unitary representations ofS.     

Structure of spaces ofC ∞-functions on nuclear spaces

LetE be a real nuclear locally convex space; we prove that the space ℰ_ub(E), of allC ^∞-functions of uniform bounded type onE, coincides with the inductive limit of the spaces ℰ_Nbc(E _v) (introduced by Nachbin-Dineen), whenV ranges over a basis of convex balanced 0-neighbourhoods inE. LetE be a real nuclear bornological vector space; we prove that the space ℰ(E) of allC ^∞-functions onE coincides with the projective limit of the spaces ℰ_Nbc(E _B), whenB is a closed convex balanced bounded subset ofE. As a consequence we obtain some density results and a version of the Paley-Wiener-Schwartz theorem. Research done during the stay of this author at the University of Bordeaux (France) in the academic year 1980–1981.     

Fredholm composition operators on algebras of analytic functions on Banach spaces

We prove that Fredholm composition operators acting on the uniform algebra H^∞(BE) of bounded analytic functions on the open unit ball of a complex Banach space E with the approximation property are invertible and arise from analytic automorphisms of the ball.     

Connected components in the space of composition operators on<Emphasis Type="Italic">H∞</Emphasis> functions of many variables

LetE be a complex Banach space with open unit ballB _e. The structure of the space of composition operators on the Banach algebra H∞, of bounded analytic functions onB _e with the uniform topology, is studied. We prove that the composition operators arising from mappings whose range lies strictly insideB _e form a path connected component. WhenE is a Hilbert space or aC _o(X)- space, the path connected components are shown to be the open balls of radius 2. The research of this author was supported by grant number SAB1999-0214 from the Ministerio de Educación, Cultura y Deporte during his stay at the Universidad de Valencia. The research of this author was partially supported DGES(Spain) pr. 96-0758. The research of this author was partially supported by Magnus Ehrnrooths stiftelse.     

A note on bounded automorphisms of infinite graphs

LetX be a connected locally finite graph with vertex-transitive automorphism group. IfX has polynomial growth then the set of all bounded automorphisms of finite order is a locally finite, periodic normal subgroup ofAUT(X) and the action ofAUT(X) onV(X) is imprimitive ifX is not finite. IfX has infinitely many ends, the group of bounded automorphisms itself is locally finite and periodic.     

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.     

Trace polynomials and invariant theory

LetA be a finite-dimensional simple (non-associative) algebra over an algebraically closed fieldF of characteristic 0. LetG be the group of its automorphisms which acts onkA, the direct sum ofk copies ofA. SupposeA is generated byk elements. In this paper, generators of the field of rational invariantF(kA) ^G are described in terms of operations of the algebraA.     

Integral extensions of noncommutative rings

In this paper we study integral extensions of noncommutative rings. To begin, we prove that finite subnormalizing extensions are integral. This is done by proving a generalization of the Paré-Schelter result that a matrix ring is integral over the coefficient ring. Our methods are similar to those of Lorenz and Passman, who showed that finite normalizing extensions are integral. As corollaries we note that the (twisted) smash product over the restricted enveloping algebra of a finite dimensional restricted Lie algebra is integral over the coefficient ring and then prove a Going Up theorem for prime ideals in these ring extensions. Next we study automorphisms of rings. In particular, we prove an integrality theorem for algebraic automorphisms. Combining group gradings and actions, we show that if a ringR is graded by a finite groupG, andH is a finite group of automorphisms ofR that permute the homogeneous components, with the order ofH invertible inR, thenR is integral overR ₁ ^H , the fixed ring of the identity component. This, in turn, is used to prove our final result: Suppose that ifH is a finite dimensional semisimple cocommutative Hopf algebra over an algebraically closed field of positive characteristic. IfR is anH-module algebra, thenR is integral overR ^H , its subring of invariants.     

Spectral invariance for algebras of pseudodifferential operators on besov-triebel-lizorkin spaces

The algebra of pseudodifferential operators with symbols inS _1,δ ⁰ , δ<1, is shown to be a spectrally invariant subalgebra of ℒ(b _p,q ^s ) and ℒ(F _p,q ^s ). The spectrum of each of these pseudodifferential operators acting onB _p,q ^s orF _p,q ^s is independent of the choice ofs, p, andq.     

Nilpotent subgroups of self equivalences of torsion spaces

LetX be a finitep-torsion based connected nilpotent CW-complex. We give a criterion of a subgroup of ε(X), the group of self equivalences ofX, to be a nilpotent group, in terms of its action onE _*(X), whereE is a CW-spectrum, satisfying some technical conditions.     

A conjecture on convolution operators,and a non-Dunford-Pettis operator onL 1

There exists a non-Dunford-Pettis operator fromL ¹ into a Banach latticeE that does not contain a copy ofc ₀ orL ¹. This problem is related to regularisation properties of convolution operators onL ¹. Work partially supported by an N.S.F. Grant.     

On the enright functor in the highest weight category

LetG be a complex semisimple Lie group,B its Borel subgroup andX a flag variety ofG. We define a functor on the category ofB-equivariantD _X-modules that corresponds, under the global section functor, to the Enright functor on the highest weight category. We use this to lift Enright functor to the mixed version of the highest weight category. As an application we obtain that the socle and the cosocle filtration of a primitive quotient of the enveloping algebra coincide.     

Über Bewertungen,welche unter einer reduktiven Gruppe invariant sind

Summary LetG be a reductive group defined over an algebraically closed fieldk and letX be aG-variety. In this paper we studyG-invariant valuationsv of the fieldK of rational functions onX. These objects are fundamental for the theory of equivariant completions ofX. LetB be a Borel subgroup andU the unipotent radical ofB. It is proved thatv is uniquely determined by its restriction toK ^U . Then we study the set of invariant valuations having some fixed restrictionv ₀, toK ^B . Ifv ₀ is geometric (i.e., induced by a prime divisor) then this set is a polyhedron in some vector space. In characteristic zero we prove that this polyhedron is a simplicial cone and in fact the fundamental domain of finite reflection groupW ^X . Thus, the classification of invariant valuations is almost reduced to the classification of valuations ofK ^B .

Unterstützt durch den Schweizerischen Nationalfonds zur Förderung der wissenschaftlichen Forschung.