Maximality of analytic operator algebras

LetM be a σ-finite von Neumann algebra andα be an action ofR onM. LetH ^∞(α) be the associated analytic subalgebra; i.e.H ^∞(α)={X ∈M: sp_∞(X) [0, ∞]}. We prove that every σ-weakly closed subalgebra ofM that containsH ^∞(α) isH ^∞(γ) for some actionγ ofR onM. Also we show that (assumingZ(M)∩M ^α = Ci)H ^∞(α) is a maximal σ-weakly closed subalgebra ofM if and only if eitherH ^∞(α)={A ∈M: (I−F)xF=0} for some projectionF ∈M, or sp(α)=Γ(α).     

Quantum commutative algebras and their duals

If an algebraA is quantum commutative with respect to the action of a quasitriangular Hopf algebraH, then the monoidal structure on the category_H ^ℳ of modules overH induces a rnonoidal structure on the category_A#H ^ℳ of modules over the associated smash productA # H. The condition under which the braiding structure of_H ^ℳ induces a braiding structure on_A#H ^ℳ is further investigated. Dually, the notion of quantum cocommutativity is introduced, and similar result in this dual situation is obtained.     

Finite dimensional representations and subgroup actions on homogeneous spaces

LetH be an ℝ-subgroup of a ℚ-algebraic groupG. We study the connection between the dynamics of the subgroup action ofH onG/G _ℤ and the representation-theoretic properties ofH being observable and epimorphic inG. We show that ifH is a ℚ-subgroup thenH is observable inG if and only if a certainH orbit is closed inG/G _ℤ; that ifH is epimorphic inG then the action ofH onG/G _ℤ is minimal, and that the converse holds whenH is a ℚ-subgroup ofG; and that ifH is a ℚ-subgroup ofG then the closure of the orbit underH of the identity coset image inG/G _ℤ is the orbit of the same point under the observable envelope ofH inG. Thus in subgroup actions on homogeneous spaces, closures of ‘rational orbits’ (orbits in which everything which can be defined over ℚ, is defined over ℚ) are always submanifolds.     

On tests of hypotheses about treatment effects and treatmentX places interactions,in two heteroscedastic experiments

Summary Suppose we have two independent experiments conducted with a set of ‘t’ treatments each, at different places. This paper deals with two interesting problems of testing of hypotheses associated with these experiments. The first problem deals with the test of the equality of the respective treatment effects in the two experiments. The second problem is concerned with the testing of the equality of treatment into places interactions. Though we assume normality, the variance σ ₁ ² in one experiment is assumed different from the variance σ ₂ ² in the other experiment. When no information is available aboutR=σ ₁ ² /(σ ₁ ² +σ ₂ ² ) except that 0≦R≦1, tests known as ‘bilateral tests’ are proposed in the literature, to test the hypotheses mentioned above. This paper studies some important small sample properties of these bilateral tests. More specifically we study the probability of the first and second kind of error of these bilateral tests as a function ofR. When the two experiments have the same number of observations on each treatment, the bilateral test is shown to control the first kind of error. Fort=1,2, the level of these tests is a strictly convex function ofR and hence these tests can be very conservative. Some power properties of these tests are also obtained. Two tests which are equivalent to the bilateral tests for large sample sizes, and which are superior to the bilateral tests for small sample sizes, are obtained.     

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.     

On subgroups of the general linear group that contain a maximal nonsplit torus

The paper deals with the structure of intermediate subgroups of the general linear group GL(n, k) of degree n over a field k of odd characteristic that contain a nonsplit maximal torus related to a radical extension of degree n of the ground field k. The structure of ideal nets over a ring that determine the structure of intermediate subgroups containinga transvection is given. Let K = k( ⁿ?{d} ) K = k\left( {\sqrt[n]{d}} \right) be a radical degree-n extension of a field k of odd characteristic, and let T =(d) be a nonsplit maximal torus, which is the image of the multiplicative group of the field K under the regular embedding in G =GL(n, k). In the paper, the structure of intermediate subgroups H, T ≤ H ≤ G, that contain a transvection is studied. The elements of the matrices in the torus T = T (d) generate a subring R(d) in the field k.Let R be an intermediate subring, R(d) ⊆ R ⊆ k, d ∈ R. Let σ_R denote the net in which the ideal dR stands on the principal diagonal and above it and all entries of which beneath the principal diagonal are equal to R. Let σ^R denote the net in which all positions on the principal diagonal and beneath it are occupied by R and all entries above the principal diagonal are equal to dR. Let E(σ_R) be the subgroup generated by all transvections from the net group G(σ_R). In the paper it is proved that the product TE(σ_R) is a group (and thus an intermediate subgroup). If the net σ associated with an intermediate subgroup H coincides with σ_R,then TE(σ_R) ≤ H ≤ N(σ_R),where N(σ_R) is the normalizer of the elementary net group E(σ_R) in G. For the normalizer N(σ_R),the formula N(σ_R)= TG(σ^R) holds. In particular, this result enables one to describe the maximal intermediate subgroups. Bibliography: 13 titles.     

Artinian property of constants of algebraicq-skew derivations

Let δ denote aq-skew σ-derivation of an algebraR andR ^(δ)={r εR│δ(r)=0} stand for the subalgebra of invariants. We prove thatR ^(δ) is left artinian iffR is left artinian providedR is semiprime and the action of δ onR is algebraic. This research was supported by the grant 190/R97/R98 in the frame of French-Polish joint projects and by Polish KBN grant no. 2 PO3A 039 14. We would like to thank all three universities for their hospitality.     

r-entropy,equipartition, and Ornstein’s isomorphism theorem inR n

A new approach is given to the entropy of a probability-preserving group action (in the context ofZ and ofR _n ), by defining an approximate “r-entropy”, 0<r<1, and lettingr → 0. If the usual entropy may be described as the growth rate of the number of essential names, then ther-entropy is the growth rate of the number of essential “groups of names” of width≦r, in an appropriate sense. The approach is especially useful for actions of continuous groups. We apply these techniques to state and prove a “second order” equipartition theorem forZ _m ×R _n and to give a “natural” proof of Ornstein’s isomorphism theorem for Bernoulli actions ofZ _m ×R _n , as well as a characterization of such actions which seems to be the appropriate generalization of “finitely determined”.     

Polar actions on Hilbert space

An isometricH-action on a Riemannian manifoldX is calledpolar if there exists a closed submanifoldS ofX that meets everyH-orbit and always meets orbits orthogonally (S is called a section). LetG be a compact Lie group equipped with a biinvariant metric,H a closed subgroup ofG ×G, and letH act onG isometrically by (h ₁,h ₂) ·x = h ₁ xh ₂ ⁻¹ · LetP(G, H) denote the group ofH ¹-pathsg: [0, 1] →G such that (g(0),g (1)) ∈H, and letP(G, H) act on the Hilbert spaceV = H ⁰([0, 1], g) isometrically byg * u = gug ⁻¹ −g′g ⁻¹. We prove that if the action ofH onG is polar with a flat section then the action ofP(G, H) onV is polar. Principal orbits of polar actions onV are isoparametric submanifolds ofV and are infinite-dimensional generalized real or complex flag manifolds. We also note that the adjoint actions of affine Kac-Moody groups and the isotropy action corresponding to an involution of an affine Kac-Moody group are special examples ofP(G, H)-actions for suitable choice ofH andG. Work supported partially by NSF Grant DMS 8903237 and by The Max-Planck-Institut für Mathematik in Bonn.     

The quotient field as a torsion-free covering module

R will denote a commutative integral domain with quotient fieldQ. A torsion-free cover of a moduleM is a torsion-free moduleF and anR-epimorphism σ:F→M such that given any torsion-free moduleG and λ∈Hom _R (G, M) there exists μ∈Hom _R (G,F) such that σμ=λ. It is known that ifM is a maximal ideal ofR, R→R/M is a torsion-free cover if and only ifR is a maximal valuation ring. LetE denote the injective hull ofR/M thenR→R/M extends to a homomorphismQ→E. We give necessary and sufficient conditions forQ→E to be a torsion-free cover.     

Higher dimensional extensions of substitutions and their dual maps

Given a substitution σ ond letters, we define itsk-dimensional extension,E _k (σ), for 0≤k≤d. Thek-dimensional extension acts on the set ofk-dimensional faces of unit cubes inR ^d with integer vertices. The extensions of a substitution satisfy a commutation relation with the natural boundary operator: the boundary of the image is the image of the boundary. We say that a substitution is unimodular (resp. hyperbolic) if the matrix associated to the substitution by abelianization is unimodular (resp. hyperbolic). In the case where the substitution is unimodular, we also define dual substitutions which satisfy a similar coboundary condition. We use these constructions to build self-similar sets on the expanding and contracting space for an hyperbolic substitution.     

Projective bases of division algebras and groups of central type

Letk be a field. For each finite groupG and two-cocylef inZ ² (G, k ^x ) (with trivial action), one can form the twisted group algebra wherex _σ x _τ =f(σ,τ)x _στ for all σ, τ∃G. Our main result is a short list ofp-groups containing all thep-groupsG for which there is a fieldk and a cocycle such that the resulting twisted group algebra is ak-central division algebra. We also complete the proof (presented in all but one case in a previous paper by Aljadeff and Haile) that everyk-central division algebra that is a twisted group algebra is isomorphic to a tensor product of cyclic algebras.     

Jacobson radical of skew polynomial rings and skew group rings

LetR be a ring and σ an automorphism ofR. We prove the following results: (i)J(R _σ[x])={Σ_ir_i x ⁱ:r₀∈I∩J(R]), r _i∈I for alliε 1} whereI↪ {r∈R:rx ∈J(R _Σ[x])|s= (ii)J(R _σ<x>)=(J(R _σ<x>)∩R)_σ<x>. As an application of the second result we prove that ifG is a solvable group such thatG andR, + have disjoint torsions thenJ(R)=0 impliesJ(R(G))=0.     

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.     

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.     

On quasi-similarity of subnormal operators

For a compact subset K in the complex plane, let Rat(K) denote the set of the rational functions with poles off K. Given a finite positive measure with support contained in K, let R2(K,v) denote the closure of Rat(K) in L2(v) and let Sv denote the operator of multiplication by the independent variable z on R2(K, v), that is, Svf = zf for every f∈R2(K, v). SupposeΩis a bounded open subset in the complex plane whose complement has finitely many components and suppose Rat(Ω) is dense in the Hardy space H2(Ω). Letσdenote a harmonic measure forΩ. In this work, we characterize all subnormal operators quasi-similar to Sσ, the operators of the multiplication by z on R2(Ω,σ). We show that for a given v supported onΩ, Sv is quasi-similar to Sσif and only if v/■Ω■σ and log(dv/dσ)∈L1(σ). Our result extends a well-known result of Clary on the unit disk.     

The structure of injective covers of special modules

In this paper, we prove that the injective cover of theR-moduleE(R/B)/R/B for a prime ideal B ofR is the direct sum of copies ofE(R/B) for prime ideals D ⊃ B, and if B is maximal, the injective cover is a finite sum of copies ofE(R/B). For a finitely generatedR-moduleM withn generators andG an injectiveR-module, we argue that the natural mapG ⁿ →G ⁿ/Hom _R (M, G) is an injective precover if Ext _R ¹ (M, R) = 0, and that the converse holds ifG is an injective cogenerator ofR. Consequently, for a maximal ideal R ofR, depth_R R ≧ 2 if and only if the natural mapE(R/R) →E(R/R)/R/R is an injective cover and depth_R R > 0.     

On the finiteness of a field-theoretic invariant for commutative rings

If T is a (commutative unital) ring extension of a ring R, then Λ(T /R) is defined to be the supremum of the lengths of chains of intermediate fields between R _P /P R _P and T _Q /QT _Q , where Q varies over Spec(T) and P:= Q ∩ R. The invariant σ(R):= sup Λ(T/R), where T varies over all the overrings of R. It is proved that if Λ(S/R)< ∞ for all rings S between R and T, then (R, T) is an INC-pair; and that if (R, T) is an INC-pair such that T is a finite-type R-algebra, then Λ(T/R)< ∞. Consequently, if R is a domain with σ(R) < ∞, then the integral closure of R is a Prüfer domain; and if R is a Noetherian G-domain, then σ(R) < ∞, with examples showing that σ(R) can be any given non-negative integer. Other examples include that of a onedimensional Noetherian locally pseudo-valuation domain R with σ(R)=∞.