Are All Uniform Algebras Amnm?

A Banach algebra a is AMNM if whenever a linear functional on a and a positive number satisfy |(ab)–(a)(b)|||a||·||b||for all a, b a, there is a multiplicative linear functional on a such that ||–||=o(1) as 0. K. Jarosz [1] asked whetherevery Banach algebra, or every uniform algebra, is AMNM. B.E. Johnson [3] studied the AMNM property and constructed a commutativesemisimple Banach algebra that is not AMNM. In this note weconstruct uniform algebras that are not AMNM. 1991 MathematicsSubject Classification 46J10.     

Mobius Instability of Sampling for Small Weighted Spaces of Analytic Functions

In this paper, the space A (D)is considered, consisting of thoseholomorphic functions f on the unit disk D such that || f ||= sup_{z D} | f(z)|(|z|) < +, with (1) = 0. The sampling problemis studied for weights satisfying ln (r)/ln(1 – r) 0.Möbius stability of sampling is shown to fail in this space.2000 Mathematics Subject Classification 30H05 (primary), 30D60(secondary).     

Sub- and Superadditive Properties of Fejer's sine Polynomial

Let be Fejér's sine polynomial. We prove the following statements.

1. The inequality holds for all x, y (0, ) with x + y < if and only if 0 and + rß 1.
2. The converse of the above inequality is valid for allx, y (0, ) with x + y < if and only if 0 and + rß 1.
3. For all n N and x, y [0, ] we have . Both bounds are best possible.

2000 Mathematics Subject Classification 42A05, 26D05 (primary),39B62 (secondary).     

Frame duality properties for projective unitary representations

Let be a projective unitary representation of a countable groupG on a separable Hilbert space H. If the set B of Bessel vectorsfor is dense in H, then for any vector x H the analysis operator_x makes sense as a densely defined operator from B to ²(G)-space.Two vectors x and y are called -orthogonal if the range spacesof _x and _y are orthogonal, and they are -weakly equivalent ifthe closures of the ranges of _x and _y are the same. These propertiesare characterized in terms of the commutant of the representation.It is proved that a natural geometric invariant (the orthogonalityindex) of the representation agrees with the cyclic multiplicityof the commutant of (G). These results are then applied to Gaborsystems. A sample result is an alternate proof of the knowntheorem that a Gabor sequence is complete in L²( ^d) ifand only if the corresponding adjoint Gabor sequence is ²-linearlyindependent. Some other applications are also discussed.     

A Variation on a Theorem of Galvin and Hajnal

Let be a singular cardinal of regular uncountable cofinality. Let {(): < } be a continuous increasing sequence withlimit , and let =₍₎₊₍₎, < be regular cardinals. Let I be a normal ideal on , and assume that the reduced product_</I admits a cofinal -scale of ordinal functions. Then ₊, where =||||_I is the I-norm of .     

The structure of the normalisers of the congruence subgroups of the Hecke group G5

Let = 2cos (/5) and let []. Denote the normaliser ofG₀() of the Hecke group G₅ in PSL₂() by N(G₀()). Then N(G₀())= G₀(/h), where h is the largest divisor of 4 such that h² divides. Further, N(G₀())/G₀() is either 1 (if h = 1), ₂ x ₂ (if h= 2) or ₄ x ₄ (if h = 4).     

Approximation of Ground State Eigenvalues and Eigenfunctions of Dirichlet Laplacians

Let be a bounded connected open set in R^N, N 2, and let –0be the Dirichlet Laplacian defined in L²(). Let > 0 be thesmallest eigenvalue of –, and let > 0 be its correspondingeigenfunction, normalized by ||||₂ = 1. For sufficiently small>0 we let R() be a connected open subset of satisfying Let – 0 be the Dirichlet Laplacian on R(), and let >0and >0 be its ground state eigenvalue and ground state eigenfunction,respectively, normalized by ||||₂=1. For functions f definedon , we let Sf denote the restriction of f to R(). For functionsg defined on R(), we let Tg be the extension of g to satisfying 1991 Mathematics SubjectClassification 47F05.     

Intersections of Odd Order Hall Subgroups

Given a set of odd primes and a -solvable finite group G, onecan choose three (not necessarily distinct) Hall -subgroupsof G whose intersection is O(G). This statement relies on aresult concerning long orbits in coprime actions of groups ofodd order. 2000 Mathematics Subject Classification 20D20.     

Packing, Tiling, Orthogonality and Completeness

Let R^d be an open set of measure 1. An open set DR^d is calleda ‘tight orthogonal packing region’ for if D–Ddoes not intersect the zeros of the Fourier transform of theindicator function of , and D has measure 1. Suppose that isa discrete subset of R^d. The main contribution of this paperis a new way of proving the following result: D tiles R^d whentranslated at the locations if and only if the set of exponentialsE = {exp 2i, x: } is an orthonormal basis for L²(). (This resulthas been proved by different methods by Lagarias, Reeds andWang [9] and, in the case of being the cube, by Iosevich andPedersen [3]. When is the unit cube in R^d, it is a tight orthogonalpacking region of itself.) In our approach, orthogonality ofE is viewed as a statement about ‘packing’ R^d withtranslates of a certain non-negative function and, additionally,we have completeness of E in L²() if and only if the above-mentionedpacking is in fact a tiling. We then formulate the tiling conditionin Fourier analytic language, and use this to prove our result.2000 Mathematics Subject Classification 52C22, 42B99, 11K70.     

Boundaries of reduced free group C*-algebras

We prove that the crossed product C*-algebra C*_r(, ) of a freegroup with its boundary sits naturally between the reducedgroup C*-algebra C*_r and its injective envelope I(C*_r). In otherwords, we have natural inclusion C*_r C*_r(, ) I(C*_r) of C*-algebras.     

A separation property of the zeros of Eisenstein series for SL(2, Z)

Let E_k(z) be the Eisenstein series with weight k for the modulargroup SL(2, ). We prove that the zeros of E_k(eⁱ) interlace withthe zeros of E_k+12(eⁱ) on /2 < < 2/3. That is, any zeroof E_k(eⁱ) lies between two consecutive zeros of E_k+12(eⁱ) on/2 < < 2/3.     

On Transitive Permutation Groups with Primitive Subconstituents

Let G be a transitive permutation group on a set such that,for , the stabiliser G induces on each of its orbits in \{}a primitive permutation group (possibly of degree 1). Let Nbe the normal closure of G in G. Then (Theorem 1) either N factorisesas N=GG for some , , or all unfaithful G-orbits, if any exist,are infinite. This result generalises a theorem of I. M. Isaacswhich deals with the case where there is a finite upper boundon the lengths of the G-orbits. Several further results areproved about the structure of G as a permutation group, focussingin particular on the nature of certain G-invariant partitionsof . 1991 Mathematics Subject Classification 20B07, 20B05.     

On the Behaviour of the First Eigenfunction of the p-Laplacian Near its Critical Points

In this paper, the behaviour of the positive eigenfunction of in u_| = 0, p > 1, isstudied near its critical points. Under some convexity and symmetryassumptions on , is seen to have a unique critical point atx = 0; also, the behaviour of both and is determined nearby.Positive solutions u to some general problems –_pu = f(u)in , u_| = 0, are also considered, with some convexity restrictionson u. 2000 Mathematics Subject Classification 35B05 (primary),35J65, 35J70 (secondary).     

The A-Polynomial has Ones in the Corners

1. Definition of the A-polynomial The A-polynomial was introduced in [3] (see also [5]), and wepresent an alternative definition here. Let M be a compact 3-manifoldwith boundary a torus T. Pick a basis , µ of ₁T, whichwe shall refer to as the longitude and meridian. Consider thesubset R_U of the affine algebraic variety R = Hom (₁M, SL₂C)having the property that () and (µ) are upper triangular.This is an algebraic subset of R, since one just adds equationsstating that the bottom-left entries in certain matrices arezero. There is a well-defined eigenvalue map given by taking the top-left entries of () and (µ).1991 Mathematics Subject Classification 57M25, 57M50.     

The Glauberman Correspondence and Subgroups of Operator Groups: a Counterexample

Let G and A be finite groups with coprime orders, and supposethat A acts on G by automorphisms. Let (G, A):Irr_A(G)Irr(C_G(A))be the Glauberman–Isaacs correspondence. Let B A andIrr_A(G). We exhibit a counterexample to the conjecture that(G, A) is an irreducible constituent of the restriction of (G,B) to C_G(A). 1991 Mathematics Subject Classification 20C15.     

Cardinal Invariants and Eventually Different Functions

In this paper we study several kinds of maximal almost disjointfamilies. In the main result of this paper we show that forsuccessor cardinals , there is an unexpected connection betweeninvariants a_e(), b() and a certain cardinal invariant m_d(⁺)on ⁺. As a corollary we get for example the following result.For a successor cardinal , even assuming that ^< = and 2= ⁺, the following is not provable in Zermelo–Fraenkelset theory. There is a ⁺-cc poset which does not collapse andwhich forces a() = ⁺ < a_e() = ⁺⁺ = 2. We also apply the ideasfrom the proofs of these results to study a = a() and non(M).2000 Mathematics Subject Classification 03E17 (primary), 03E05(secondary).     

On the Reflexivity of the Space {pi}p(E, F) of p-Absolutely Summing Operators, 1 <= p < + {infty}

We give a representation of the dual of the space _p(E,F) ofp-absolutely summing operators (1 p < + ) under certainconditions on E and F. One deduces that the space _p(E, F), 1 p < + , is reflexive if and only if E and F are reflexive.We improve results of Gordon, Lewis, Retherford and Saphar.     

Quasi-Affinity in certain Classes of Operators

The family of operators S + V (, C, Re > 0), where V isan injective S-Volterra operator (that is, [S, V[ = V²) and— A — V^–1 generates a uniformly bounded C₀-semigroup,is studied in the context of similarity and of the weaker quasi-affinityrelation. It is shown that S is similar to S + V for all , C,Re > 1, and is a quasi-affine transform of S + tV for allt 0 and 0 < < 1.     

Covariant Representations for Coactions of Hopf C*-Algebras

Given a coaction of a Hopf C*-algebra A on a C*-algebra B withan -invariant C*-subalgebra C, and a conditional expectationE: B C commuting with , it is shown that if (, u) is a covariantrepresentation of the system (C, A, |_C), then there is an associatedcovariant representation of the system (B, A, ), where is the representation induced from up to B via E, and is a unitary corepresentation of Anaturally associated with u. Some applications are also discussed,including a lifting of ergodic coactions to von Neumann algebras,and a characterization of the amenability of multiplicativeunitary operators via infinite tensor product covariant representations.2000 Mathematics Subject Classification 46L55, 46L60, 46L87,46L89, 81R15, 81R50, 81T05.     

Correction: Abelian Varieties defined over their Fields of Moduli, I

Bull London Math. Soc, 4 (1972), 370–372. The proof of the theorem contains an error. Before giving acorrect proof, we state two lemmas. LEMMA 1. Let K/k be a cyclic Galois extension of degree m, let generate Gal (K/k), and let (A, I, ) be defined over K. Supposethat there exists an isomorphism :(A,I,) (A, I, ) over K suchthat v^m–1 ... = 1, where v is the canonical isomorphism(A^m, I^m, ^m) (A, I, ). Then (A, I, ) has a model over k, whichbecomes isomorphic to (A, I, ) over K. Proof. This follows easily from [7], as is essentially explainedon p. 371. LEMMA 2. Let G be an abelian pro-finite group and let : G Q/Z be a continuous character of G whose image has order p.Then either: (a) there exist subgroups G' and H of G such that H is cyclicof order p^m for some m, (G') = 0, and G = G' x H, or (b) for any m > 0 there exists a continuous character m ofG such that p^m _m = . Proof. If (b) is false for a given m, then there exists an element G, of order p^r for some r m, such that () ¦ 0. (Considerthe sequence dual to 0 Ker (p^m) G ^pm G). There exists an opensubgroup G_o of G such that (G₀) = 0 and has order p^r in G/G₀.Choose H to be the subgroup of G generated by , and then aneasy application to G/G₀ of the theory of finite abelian groupsshows the existence of G' (note that () ¦ 0 implies that is not a p-th. power in G). We now prove the theorem. The proof is correct up to the statement(iv) (except that (i) should read: F' k₁ F'ab). To removea minor ambiguity in the proof of (iv), choose to be an elementof Gal (F'_ab/k₂) whose image $$\stackrel{\&macr;}{\sigma}$$ in Gal (k₁/k₂) generates this last group. The error occursin the statement that the canonical map v : A^P A acts on pointsby sending a^p a; it, of course, sends a a. The proof is correct, however, in the case that it is possibleto choose so that ^p = 1 (in Gal (F'/k₂)). By applying Lemma 2 to G = Gal (F'_ab/k₂) and the map G Gal(k₁/k₂) one sees that only the following two cases have to beconsidered. (a) It is possible to choose so that ^pm = 1, for some m, andG = G' x H where G' acts trivially on k₁ and H is generatedby . (b) For any m > 0 there exists a field K, F'ab K k₁ k₂is a cyclic Galois extension of degree p^m. In the first case, we let K F'_ab be the fixed field of G'.Then (A, I, ), regarded as being defined over K, has a modelover k₂. Indeed, if m = 1, then this was observed above, butwhen m > 1 the same argument applies. In the second case, let : (A, I, ) (A$$\stackrel{\&macr;}{\sigma}$$, I$$\stackrel{\&macr;}{\sigma }$$, $$\stackrel{\&macr;}{\sigma}$$) be an isomorphism defined over k₁ and let v ... ^p–1 = µ(R). If is replaced by for some Aut_k1((A, I, )) then is replacedby ^P. Thus, as µ(R) is finite, we may assume that ^pm–1= 1 for some m. Choose K, as in (b), to be of degree p^m overk₂. Let _m be a generator of Gal (K/k₂) whose restriction tok₁ is $$\stackrel{\&macr;}{\sigma }$$. Then : (A, I, ) (A^{$$\stackrel{\&macr;}{\sigma }$$}, I^{$$\stackrel{\&macr;}{\sigma}$$}, ^{$$\stackrel{\&macr;}{\sigma }$$} = (A^{$$\stackrel{\&macr;}{\sigma}$$m}, I^{$$\stackrel{\&macr;}{\sigma }$$m}, ^{$$\stackrel{\&macr;}{\sigma}$$m} is an isomorphism defined over K and v ^mpm–1, ... ^m =^pm–1 = 1, and so, by) Lemma 1, (A, I, ) has a model overk₂ which becomes isomorphic to (A, I, over K. The proof may now be completed as before. Addendum: Professor Shimura has pointed out to me that the claimon lines 25 and 26 of p. 371, viz that µ(R) is a puresubgroup of _R^*t, does not hold for all rings R. Thus this condition,which appears to be essential for the validity of the theorem,should be included in the hypotheses. It holds, for example,if µ(R) is a direct summand of µ(F).