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.     

Deformation retracts and the Hochschild homology of polynomial rings

We assume given a ringA with unit, and a subcomplex of the reduced bar complex ofA. We assume that this subcomplex is a deformation retract of the whole complex and thus has homology equal to the Hochschild homology ofA, but it will typically be smaller and easier to calculate with. We use these to construct (accordingly small) deformation retracts for the reduced bar complexes ofA[t] andA[t,t ⁻¹]. WhenA is a Banach algebra, we also do this construction forC ^∞(S¹;A). Partially supported by N.S.F. Grant No. DMS 92-03398.     

Sums of corestrictions of cyclic algebras

By a cyclic layer of a finite Galois extension,E/K, of fields one means a cyclic extension,L/F, of fields whereE⊇L⊇F⊇K. LetC(E/K) denote the subgroup of the relative Brauer group, Br(E/K), generated by the various subgroups cor(Br(L/F)) asL/F ranges over all cyclic layers ofE/K and where cor denotes the corestriction map into Br(E/K). We show that forK global, [Br(E/K) :C(E/K)]<∞ and we produce examples whereC(E/K)≠Br(E/K). In memory of S.A. Amitsur, our teacher, friend, collaborator, and inspiration. Supported in part by NSA Grant No. MDA904-95-H-1054. Supported in part by NSA Grant No. MDA904-95-H-1022.     

Prime ideals in restricted differential operator rings

In this paper restricted differential operator rings are studied. A restricted differential operator ring is an extension of ak-algebraR by the restricted enveloping algebra of a restricted Lie algebra g which acts onR. This is an example of a smash productR #H whereH=u (g). We actually deal with a more general twisted construction denoted byR * g where the restricted Lie algebra g is not necessarily embedded isomorphically inR * g. Assume that g is finite dimensional abelian. The principal result obtained is Incomparability, which states that prime idealsP ₁ ⊆P ₂ ⊂R * g have different intersections withR. We also study minimal prime ideals ofR * g whenR is g-prime, showing that the minimal primes are precisely those having trivial intersection withR, that these primes are finite in number, and their intersection is a nilpotent ideal. Prime and primitive ranks are considered as an application of the foregoing results.     

A sum-product estimate in algebraic division algebras

LetA be a finite subset of some normed division algebra over ℝ with cardinality ⋎A⋎. We prove that either the sum set or the product set ofA has cardinality ⋎A⋎^1+δ for some δ>0. Partially supported by NSA grant No. MDA 904-03-1-0045.     

Almost everywhere convergence of Riesz means on noncompact symmetric spaceSL(3,H)/Sp(3)

LetG/K be the noncompact Riemannian symmetric spaceSL(3,H)/Sp(3). We shall prove in this paper that forf∈L ^p(SL(3,H)/Sp(3)), 1≤p≤2, the Riesz means of orderz off with respect to the eigenfunctions expansion of Laplace operator almost everywhere converge tof for Rez >δ(n,p). The critical index δ(n,p) is the same as in the classical Stein's result for Euclidean space, and as in the noncompact symmetric spaces of rank one and of complex type. Partially supported by National Natural Science Foundation of China     

Almost everywhere convergence of the inverse spherical transform onSL(2,R)

We prove the almost everywhere convergence of the inverse spherical transform ofL _p bi-K-invariant functions on the groupSL(2,R), 4/3<p≤2. The result appears to be sharp. Partially supported by the MPI Partially supported by the CMA     

Compactly supported box-spline wavelets

A general procedure for constructing multivariate non-tensor-product wavelets that generate an orthogonal decomposition ofL ²(R)^s,s s≥1, is described and applied to yield explicit formulas for compactly supported spline-wavelets based on the multiresolution analysis ofL ²(R)^s 1≤s≤3, generated by any box spline whose direction set constitutes a unimodular matrix. In particular, when univariate cardinal B-splines are considered, the minimally supported cardinal spline-wavelets of Chui and Wang are recovered. A refined computational scheme for the orthogonalization of spaces with compactly supported wavelets is given. A recursive approximation scheme for “truncated” decomposition sequences is developed and a sharp error bound is included. A condition on the symmetry or anti-symmetry of the wavelets is applied to yield symmetric box-spline wavelets. Partially supported by ARO Grant DAAL 03-90-G-0091 Partially supported by NSF Grant DMS 89-0-01345 Partially supported by NATO Grant CRG 900158.     

The unitary representation theory of GL(n) of an infinite discrete field

IfK is an infinite field and ifG=GL(n, K) with the discrete topology, then all principal-series representations ofG are irreducible, and any two such with the same central character ψ are weakly equivalent to one another and to the ψ-regular representation. In addition, every irreducible unitary representation ofG which is not one-dimensional weakly contains a representation of the principal series. We deduce that every maximal ideal ofC*(G) is either of codimension 1 or else a kernel of a principal-series representation. In particular, except in the exceptional case whereK is an infinite algebraic extension of a finite field, the reducedC*-algebra of PGL(n, K) is simple, as was also shown by de la Harpe in many cases. Partially supported by NSF Grant DMS-85-06130. It is a pleasure also to acknowledge the hospitality of the Institute for Advanced Studies, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel, from January to August, 1988. Partially supported by NSF Grants DMS-84-00900 and DMS-87-00551. Much of this work was done while visiting at, and partially supported by, the Department of Mathematics and Computer Science, Bar-Ilan University, 52100 Ramat Gan, Israel.     

Linear liftings for non-complete probability spaces

We show that it is consistent with ZFC thatL ^∞(Y,B,ν) has no linear lifting for many non-complete probability spaces (Y,B,ν), in particular forY=[0,1]^A,B=Borel subsets ofY, ν=usual Radon measure onB. Research supported by UPEI Senate Grant no 602101, by the Research Institute for Mathematical Sciences at Bar-Ilan University, and by the Landau Center for Mathematical Research in Analysis (supported by the Minerva Foundation). The author would like to thank the organizers of the Winter Institute on the Set Theory of the Reals for their hospitality while part of this research was being carried out. Partially supported by the Foundation for Basic Research of the Israel Academy of Science. Publication number 437.     

New central polynomials for the matrix algebra

Forn≥3 we find a central polynomial of degree (n−1)²+4 for then×n matrix algebra over a field of characteristic 0. Forn=3,4 our polynomial coincides with the known central polynomials of minimal degree and forn>4 the result gives new central polynomials. Until now, forn>4 the minimal degree of the known central polynomials wasn ². Partially supported by the Alexander von Humboldt Foundation in Germany and by Grant MM2/91 of the Ministry of Education and Science in Bulgaria.     

Enveloping Σ-C *-algebra of a smooth Frechet algebra crossed product by ℝ,K-theory and differential structure inC *-algebras

Given anm-tempered strongly continuous action α of ℝ by continuous^*-automorphisms of a Frechet^*-algebraA, it is shown that the enveloping ↡-C ^*-algebraE(S(ℝ, A^∞, α)) of the smooth Schwartz crossed productS(ℝ,A ^∞, α) of the Frechet algebra A^∞ of C^∞-elements ofA is isomorphic to the Σ-C ^*-crossed productC ^*(ℝ,E(A), α) of the enveloping Σ-C ^*-algebraE(A) ofA by the induced action. WhenA is a hermitianQ-algebra, one getsK-theory isomorphismRK _*(S(ℝ, A^∞, α)) =K ^*(C ^*(ℝ,E(A), α) for the representableK-theory of Frechet algebras. An application to the differential structure of aC ^*-algebra defined by densely defined differential seminorms is given.     

Bin packing can be solved within 1 + ε in linear time

For any listL ofn numbers in (0, 1) letL* denote the minimum number of unit capacity bins needed to pack the elements ofL. We prove that, for every positive ε, there exists anO(n)-time algorithmS such that, ifS(L) denotes the number of bins used byS forL, thenS(L)/L*≦1+ε for anyL providedL* is sufficiently large. The work of this author was supported by NSF Grant MCS 70-04997.     

Cusp forms

LetG andH ⊂G be two real semisimple groups defined overQ. Assume thatH is the group of points fixed by an involution ofG. Letπ ⊂L ²(H\G) be an irreducible representation ofG and letf επ be aK-finite function. Let Γ be an arithmetic subgroup ofG. The Poincaré seriesP _f(g)=Σ_H∩ΓΓ f(γ{}itg) is an automorphic form on Γ\G. We show thatP _f is cuspidal in some cases, whenH ∩Γ\H is compact. Partially supported by NSF Grant # DMS 9103608.     

Spectra of smash products

LetT=R #H be a smash product whereH is a finite dimensional Hopf algebra. We show that ideals ofT invariant under the dualH* ofH are extended fromH-invariant ideals ofR. This allows us to transport the study of ideals inT to invariant ideals. When the Hopf algebra is pointed the relationship between an ideal and its invariant ideal is shown to be manageable. Restricting to prime ideals, this yields results on the prime spectra ofR andT. We obtain Krull relations forR ⊆T for someH, including Incomparability wheneverH is commutative (or more generally whenH* is pointed after base extension). The results generalize and unify a number of results known in the context of group and restricted Lie actions.     

X-inner automorphisms of crossed products and semi-invariants of Hopf algebras

LetR*G be a crossed product of the groupG over the prime ringR and assume thatR*G is also prime. In this paper we study unitsq in the Martindale ring of quotientsQ ₀(R*G) which normalize bothR and the group of trivial units ofR*G. We obtain quite detailed information on their structure. We then study the group ofX-inner automorphisms ofR*G induced by such elements. We show in fact that this group is fairly close to the group of automorphisms ofR*G induced by certain trivial units inQ ₀(R)*G. As an application we specialize to the case whereR=U(L) is the enveloping algebra of a Lie algebraL. Here we study the semi-invariants forL andG which are contained inQ ₀(R*G) and we obtain results which extend known properties ofU(L). Finally, every cocommutative Hopf algebraH over an algebraically closed field of characteristic 0 is of the formH=U(L)*G. Thus we also obtain information on the semi-invariants forH contained inQ ₀(H). Research supported in part by N.S.F. Grant Nos. MCS 83-01393 and MCS 82-19678.     

Amitsur cohomology of cubic extensions of algebraic integers

LetK be the rational fieldQ or a complex quadratic number field other than . LetL be a normal three-dimensional field extension onK. IfR andS are the rings of algebraic integers ofK andL respectively, then the Amitsur cohomology groupH ² (S/R, U) is trivial. Inflation and class numbers give information about cohomology arising from certain nonnormal cubic extensions. This work was supported in part by NSF Grant GP-28409.     

A fixed point theorem for a class of star-shaped sets inc 0

A subsetK ofc ₀ is coordinatewise star-shaped (c.s.s.) if there exists a center pointx ∈K such that fory ∈K andz ∈c ₀, ifz is coordinatewise betweenx andy thenz ∈K. We prove that a weakly compact c.s.s. subset ofc ₀ has the fixed point property for nonexpansive mappings and that a fixed point for such a mapping can be obtained in a constructive manner. Research of the first two authors was partially supported by NSF Grant MCS78-01344 and of the last author by MCS78-01501.