Picard Groups of Irrational Rotation C*-Algebras

Let be an irrational number in [0, 1] and A the correspondingirrational rotation C*-algebra. Let Aut (A) be the group ofall automorphisms of A and Int (A) the normal subgroup of Aut(A) of all inner automorphisms of A. Let Pic (A) be the Picardgroup of A. In the present note we shall show that if is notquadratic, then Pic (A)Aut (A)/Int (A) and that if is quadratic,then Pic (A) is isomorphic to a semidirect product of Aut (A)/Int(A) with Z. Furthermore, in the last section we shall discussPicard groups of certain Cuntz algebras.     

Extremal Subgroups in Chevalley Groups

Throughout this paper G(k) denotes a Chevalley group of rankn defined over the field k, where n3. Let be the root systemassociated with G(k) and let ={₁, ₂, ..., _n} be a set of fundamentalroots of , with ⁺ being the set of positive roots of with respectto . For and ⁺, let n() be the coefficient of in the expressionof as a sum of fundamental roots; so =n(). Also we recall thatht(), the height of , is given by ht()=n(). The highest rootin ⁺ will be denoted by . We additionally assume that the Dynkindiagram of G(k) is connected.     

Base Sizes and Regular Orbits for Coprime Affine Permutation Groups

Let G be a permutation group on a finite set . A sequence B=(₁,..., _b) of points in is called a base if its pointwise stabilizerin G is the identity. Bases are of fundamental importance incomputational algorithms for permutation groups. For both practicaland theoretical reasons, one is interested in the minimal basesize for (G, ), For a nonredundant base B, the elementary inequality2^|B||G|||^|B| holds; in particular, |B|log|G|/log||. In the casewhen G is primitive on , Pyber [8, p. 207] has conjectured thatthe minimal base size is less than Clog|G|/log|| for some (large)universal constant C. It appears that the hardest case of Pyber's conjecture is thatof primitive affine groups. Let H=GV be a primitive affine group;here the point stabilizer G acts faithfully and irreduciblyon the elementary abelian regular normal subgroup V of H, andwe may assume that =V. For positive integers m, let mV denotethe direct sum of m copies of V. If (v₁, ..., v_m)mV belongsto a regular G-orbit, then (0, v₁, ..., v_m) is a base for theprimitive affine group H. Conversely, a base (₁, ..., _b) forH which contains 0V= gives rise to a regular G-orbit on (b–1)V. Thus Pyber's conjecture for affine groups can be viewed asa regular orbit problem for G-modules, and it is therefore aspecial case of an important problem in group representationtheory. For a related result on regular orbits for quasisimplegroups, see [4, Theorem 6].     

Regularity Conditions and Bernoulli Properties of Equilibrium States and g-Measures

When T : X X is a one-sided topologically mixing subshift offinite type and : X R is a continuous function, one can definethe Ruelle operator L : C(X) C(X) on the space C(X) of real-valuedcontinuous functions on X. The dual operator always has a probability measure as an eigenvectorcorresponding to a positive eigenvalue ( = with > 0). Necessary and sufficient conditionson such an eigenmeasure are obtained for to belong to twoimportant spaces of functions, W(X, T) and Bow (X, T). For example, Bow(X, T) if and only if is a measure with a certain approximateproduct structure. This is used to apply results of Bradleyto show that the natural extension of the unique equilibriumstate µ of Bow(X, T) has the weak Bernoulli propertyand hence is measure-theoretically isomorphic to a Bernoullishift. It is also shown that the unique equilibrium state ofa two-sided Bowen function has the weak Bernoulli property.The characterizations mentioned above are used in the case ofg-measures to obtain results on the ‘reverse’ ofa g-measure.     

Stability and Asymptotic Stability of Functional-Differential Equations

We investigate asymptotic behaviour of solutions of the functional-differentialequation where f and g arelocally Lipschitz functions, C is a continuous matrix and thesmooth lag function obeys 0 (t) t for t 0. We transformthe equation into a delay equation with an infinity of delaysand use a theorem of Söderlind to derive sufficient conditionsfor stability and for asymptotic stability in the case lim_t (t) = . The situation is qualitatively different when lim_t (t) = * < and we outline stability conditions for thatcase by employing direct techniques.     

Character Quotients for Coprime Acting Groups

Let the finite group A be acting on a finite group G with (|A|,|G|)=1. Let be the semidirect product of A and G. Let be acharacter of irreducible after restriction to G. In a previouspaper by Brian Hartley and the author, we proved that the restrictionof to S belongs to the set C(S) obtained by running over all that arise in this manner, by assuming, in addition, that Gis a product of extraspecial groups. This was proved in general,assuming only some condition on the Green functions of groupsof Lie type that is not as yet fully verified. In the presentpaper, we define the map Q(): SC by Q()(s)=|C_G(s)|/(s). We provethat Q()C(S) under the same hypotheses. In particular, the characterquotient Q() is an ordinary character.     

Free Groups of Outer Commutator Varieties of Groups

If F is a free group, 1 < i j 2i and i k i + j + 1 thenF/[_j(F), _i(F), _k(F)] is residually nilpotent and torsion-free.This result is extended to 1 < i j 2i and i k 2i + 2j.It is proved that the analogous Lie rings, L/[L^j, Lⁱ, L^k] whereL is a free Lie ring, are torsion-free. Candidates are foundfor torsion in L/[L^j, Lⁱ, L^k] whenever k is the least of {i,j, k}, and the existence of torsion in L/[L^j, Lⁱ, L^k] is provedwhen i, j, k 5 and k is the least of {i, j, k}.     

On strongly asymptotic lp spaces and minimality

Let 1 p and let X be a Banach space with a semi-normalizedstrongly asymptotic _p basis (e_i). If X is minimal and 1 p <2, then X is isomorphic to a subspace of _p. If X is minimaland 2 p < , or if X is complementably minimal and 1 p , then (e_i) is equivalent to the unit vector basis of _p (orc₀ if p = ).     

Support Varieties of Non-Restricted Modules over Lie Algebras of Reductive Groups

Let G be a connected semisimple group over an algebraicallyclosed field K of characteristic p>0, and g=Lie (G). Fixa linear function g* and let Z_g() denote the stabilizer of in g. Set N_p(g)={xg|x^[p]=0}. Let C(g) denote the category offinite-dimensional g-modules with p-character . In [7], Friedlanderand Parshall attached to each MOb(C(g)) a Zariski closed, conicalsubset V_g(M)N_p(g) called the support variety of M. Suppose thatG is simply connected and p is not special for G, that is, p2if G has a component of type B_n, C_n or F₄, and p3 if G has acomponent of type G₂. It is proved in this paper that, for anynonzero MOb(C(g)), the support variety V_g(M) is contained inN_p(g)Z_g(). This allows one to simplify the proof of the Kac–Weisfeilerconjecture given in [18].     

Contractions et Hyperdistributions a Spectre de Carleson

Let =(_n)_n1 be a log concave sequence such that lim inf_n+n/n^c>0for some c>0 and ((log _n)/n)_n1 is nonincreasing for some<1/2. We show that, if T is a contraction on the Hilbertspace with spectrum a Carleson set, and if ||T^–n||=O(_n)as n tends to + with _n11/(n log _n)=+, then T is unitary. Onthe other hand, if _n11/(n log _n)<+, then there exists a (non-unitary)contraction T on the Hilbert space such that the spectrum ofT is a Carleson set, ||T^–n||=O(_n) as n tends to +, andlim sup_n+||T^–n||=+.     

Theta-characteristics on singular curves

On a smooth curve a theta-characteristic is a line bundle L,the square of which is the canonical line bundle . The equivalentcondition om(L, ) L generalizes well to singular curves, asapplications show. More precisely, a theta-characteristic isa torsion-free sheaf of rank 1 with om(, ) . If the curvehas non-ADE singularities, then there are infinitely many theta-characteristics.Therefore, theta-characteristics are distinguished by theirlocal type. The main purpose of this article is to compute thenumber of even and odd theta-characteristics (that is withh⁰(C, ) 0 and h⁰(C, ) 1 modulo 2, respectively) in terms ofthe geometric genus of the curve and certain discrete invariantsof a fixed local type.     

The Isolated Points of G{rho} and the w*-Strongly Exposed Points of P{rho}(G)0

Let G be a separable locally compact group and let be its dualspace with Fell's topology. It is well known that the set P(G)of continuous positive-definite functions on G can be identifiedwith the set of positive linear functionals on the group C^*-algebraC^*(G). We show that if is discrete in , then there exists anonzero positive-definite function associated with such that is a w^*-strongly exposed point of P(G)₀, where P(G)₀={f P(G):f(e)1. Conversely, if some nonzero positive-definite function associatedwith is a w^*-strongly exposed point of P(G)₀, then is isolatedin . Consequently, G is compact if and only if, for every ,there exists a nonzero positive-definite function associatedwith that is a w^*-strongly exposed point of P(G)₀. If, in addition,G is unimodular and , then is isolated in if and only if somenonzero positive-definite function associated with is a w^*-stronglyexposed point of P(G)₀, where is the left regular representationof G and is the reduced dual space of G. We prove that if B(G)has the Radon–Nikodym property, then the set of isolatedpoints of (so square-integrable if G is unimodular) is densein . It is also proved that if G is a separable SIN-group, thenG is amenable if and only if there exists a closed point in. In particular, for a countable discrete non-amenable groupG (for example the free group F₂ on two generators), there isno closed point in its reduced dual space .     

Movement and Separation of Subsets of Points Under Group Actions

Let G be a permutation group on a set , and let m and k be integerswhere 0<m<k. For a subset of , if the cardinalities ofthe sets ^g\, for gG, are finite and bounded, then is said tohave bounded movement, and the movement of is defined as move()=max_gG|^g\|. If there is a k-element subset such that move()m, it is shown that some G-orbit has length at most (k²–m)/(k–m).When combined with a result of P. M. Neumann, this result hasthe following consequence: if some infinite subset has boundedmovement at most m, then either is a G-invariant subset withat most m points added or removed, or nontrivially meets aG-orbit of length at most m²+m+1. Also, if move ()m for allk-element subsets and if G has no fixed points in , then either||k+m (and in this case all permutation groups on have thisproperty), or ||5m–2. These results generalise earlierresults about the separation of finite sets under group actionsby B. J. Birch, R. G. Burns, S. O. Macdonald and P. M. Neumann,and groups in which all subsets have bounded movement (by theauthor).     

Asymptotics for Fractional Nonlinear Heat Equations

The Cauchy problem is studied for the nonlinear equations withfractional power of the negative Laplacian where (0,2), with critical = /n and sub-critical (0,/n)powers of the nonlinearity. Let u₀ L^1,a L C, u₀(x) 0 in Rⁿ, = . The case of not small initial data is of interest. It is proved that the Cauchy problemhas a unique global solution u C([0,); L L^1,a C) and the largetime asymptotics are obtained.     

Projective Modules over the Non-Commutative Sphere

The positive cone of the K₀-group of the non-commutative sphereB is explicitly determined by means of the four basic unboundedtrace functionals discovered by Bratteli, Elliott, Evans andKishimoto. The C*-algebra B is the crossed product A x Z₂ ofthe irrational rotation algebra A by the flip automorphism defined on the canonical unitary generators U, V by (U) = U*,(V) = V*, where VU = e²ⁱ UV and is an irrational real number.This result combined with Rieffel's cancellation techniquesis used to show that cancellation holds for all finitely generatedprojective modules over B. Subsequently, these modules are determinedup to isomorphism as finite direct sums of basic modules. Italso follows that two projections p and q in a matrix algebraover B are unitarily equivalent if, and only if, their vectortraces are equal: [p] = [q]. These results will have the following ramifications. They areused (elsewhere) to show that the flip automorphism on A isan inductive limit automorphism with respect to the basic buildingblock construction of Elliott and Evans for the irrational rotationalgebra. This will, in turn, yield a two-tower proof of thefact that B is approximately finite dimensional, first provedby Bratteli and Kishimoto.     

Antisupercyclic Operators and Orbits of the Volterra Operator

We say that a bounded linear operator T acting on a Banach spaceB is antisupercyclic if for any x B either Tⁿx = 0 for somepositive integer n or the sequence {Tⁿx/||Tⁿx||} weakly convergesto zero in B. Antisupercyclicity of T means that the angle criterionof supercyclicity is not satisfied for T in the strongest possibleway. Normal antisupercyclic operators and antisupercyclic bilateralweighted shifts are characterized. As for the Volterra operator V, it is proved that if 1 p and any f L_p [0,1] then the limit lim_n (n!||Vⁿf||_p)^1/n doesexist and equals 1 – inf supp (f). Upon using this asymptoticformula it is proved that the operator V acting on the Banachspace L_p[0,1] is antisupercyclic for any p (1,). The same statementfor p = 1 or p = is false. The analogous results are provedfor operators when the real part of z C is positive.     

Higher Generation Subgroup Sets and the Virtual Cohomological Dimension of Graph Products of Finite Groups

We introduce panels of stabilizer schemes (K, G_*) associatedwith finite intersection-closed subgroup sets of a given groupG, generalizing in some sense Davis' notion of a panel structureon a triangulated manifold for Coxeter groups. Given (K, G_*),we construct a G-complex X with K as a strong fundamental domainand simplex stabilizers conjugate to subgroups in . It turnsout that higher generation properties of in the sense of Abels-Holzare reflected in connectivity properties of X. Given a finite simplicial graph and a non-trivial group G()for every vertex of , the graph product G() is the quotientof the free product of all vertex groups modulo the normal closureof all commutators [G(), G(w)] for which the vertices , w areadjacent. Our main result allows the computation of the virtualcohomological dimension of a graph product with finite vertexgroups in terms of connectivity properties of the underlyinggraph .     

Skew Derivations with Central Invariants

If is an automorphism and is a -derivation of a ring R, thenthe subring of invariants is the set R⁽⁾={rR|(r)=0}. The mainresult of this paper is ‘let R be a semiprime ring withan algebraic -derivation such that R⁽⁾ is central; then R iscommutative’. This theorem generalizes results on theinvariants of automorphisms and derivations and is proved byreducing down to the special cases of automorphisms and derivations.     

Barrelled Countable Enlargements and the Bounding Cardinal

Replacing c with the bounding cardinal improves two standardBCE results of Robertson, Tweddle and Yeomans, and is optimalfor the BCE codimension/inheritance result of Bonet and PérezCarreras. Indeed, is the smallest infinite-dimensionality formetrizable barrelled spaces, and is the largest cardinal suchthat every subspace of codimension less than in a metrizablebarrelled space is itself barrelled. We thus reconfirm as oneof two ‘optimal cardinals for metrizable barrelled spaces’.Esthetically pleasing, these properties of immediately solvethe normable BCE problem without extra-ZFC axiomatic assumptions,reduce the separable quotient problem to Banach spaces E with density character of E c, and are part of the solution toone version of the metrizable BCE problem found in our sequel,devoted to the other ‘optimal cardinal’.