The Second Dual Algebra of the Measure Algebra of a Compact Group

It is shown that for every compact group G, L¹(G)^ is uniqueand minimal among all the closed subsets I of M(G)** such thatI is a proper (0, M(G)**) algebraic ideal, and such that I issolid with respect to absolute continuity; that is, n L¹(G)^whenever n M(G)** and n << µ L¹(G)^. 1991 MathematicsSubject Classification 43A20, 43A22.     

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).     

Homological Properties of Modules Over Group Algebras

Let G be a locally compact group, and let L¹ (G) be the Banachalgebra which is the group algebra of G. We consider a varietyof Banach left L¹ (G)-modules over L¹ (G), and seek to determineconditions on G that determine when these modules are eitherprojective or injective or flat in the category. The answerstypically involve G being compact or discrete or amenable. Forexample, in the case where G is discrete and 1 < p < ,we find that the module ^p (G) is injective whenever G is amenable,and that, if it is amenable, then G is ‘pseudo-amenable’,a property very close to that of amenability. 2000 MathematicsSubject Classification 46H25, 43A20.     

Rank of Elliptic Curves over almost Separably Closed Fields

Let E be an elliptic curve over a finitely generated infinitefield K. Let K^s denote a separable closure of K, an elementof the Galois group G_K=Gal(K^s/K), and K^s() the invariant subfieldof K^s. If the characteristic of K is not 2 and belongs to asuitable open subgroup of G_K, then E(K^s()) has infinite rank.2000 Mathematics Subject Classification 11G05.     

A Liouville Theorem for Matrix-Valued Harmonic Functions on Nilpotent Groups

Let be a non-degenerate positive M_n-valued measure on a locallycompact group G with |||| = 1. An M_n-valued Borel function fon G is called -harmonic if for all x G. Given such a function f which is bounded and leftuniformly continuous on G, it is shown that every central elementin G is a period of f. Further, it is shown that f is constantif G is nilpotent or central. 2000 Mathematics Subject Classification31C05, 43A05, 45E10, 46G10.     

Ramsey and Freeness Properties of Polish Planes

Suppose that X is a Polish space which is not -compact. We provethat for every Borel colouring of X² by countably many colours,there exists a monochromatic rectangle with both sides closedand not -compact. Moreover, every Borel colouring of [X]² byfinitely many colours has a homogeneous set which is closedand not -compact. We also show that every Borel measurable functionf:X² X has a free set which is closed and not -compact. Ascorollaries of the proofs we obtain two results: firstly, theproduct forcing of two copies of superperfect tree forcing doesnot add a Cohen real, and, secondly, it is consistent with ZFCto have a closed subset of the Baire space which is not -compactand has the property that, for any three of its elements, noneof them is constructible from the other two. A similar proofshows that it is consistent to have a Laver tree such that noneof its branches is constructible from any other branch. Thelast four results answer questions of Goldstern and Brendle.2000 Mathematics Subject Classification: 03E15, 26B99, 54H05.     

Continuity of Multiplication in the Largest Compactification of a Locally Compact Group

G^LUC is the largest semigroup compactification of the locallycompact group G. When G is not compact, given q G* = G^LUC \G, there is p G* such that x qx is discontinuous at p (Theorem2). If G is -compact, there is one p which will serve for allq (Theorem 1).     

There is a Group of Every Strong Symmetric Genus

Let G be a finite group. The strong symmetric genus ⁰(G) isthe minimum genus of any Riemann surface on which G acts, preservingorientation. For any non-negative integer g, there is at leastone group of strong symmetric genus g. For g2, one such grouphas the form Z_k x D_n for some k and n. 2000 Mathematics SubjectClassification 57M60 (primary), 20H10, 30F99 (secondary).     

Finite CI-Groups are Soluble

For a finite group G and a subset S of G with 1 S and S = S^–1,the Cayley graph Cay(G, S) is the graph with vertex set G suchthat {x, y} is an edge if and only if yx^–1 S. The groupG is called a CI-group if, for all subsets S and T of G\{1},Cay(G, S) Cay(G, T) if and only if S = T for some Aut(G).In this paper, for each prime p 1 (mod 4), a symmetric graph(p) is constructed from PSL(2, p) such that Aut (p) = Z₂ x PSL(2,p); it is then shown that A₅ is not a CI-group, and that allfinite CI-groups are soluble. 1991 Mathematics Subject Classification05C25.     

On v-Distal Flows on 3-Manifolds

In [2], H. Furstenberg studied a distal action of a locallycompact group G on a compact metric space X, and establisheda structure theorem. As a consequence, he showed that if G isabelian, then a simply connected space X does not admit a minimaldistal G-action. In this paper we concern ourselves with a nonsingular flow = {^t} on a closed 3-manifold M. Recall that is called distalif for any distinct two points x, y M, the distance d(^tx, ^ty)is bounded away from 0. The distality depends strongly uponthe time parametrization. For example, there exists a time parametrizationof a linear irrational flow on T² which yields a nondistal flow[4, 6]. 1991 Mathematics Subject Classification 58F25, 57R30.     

Sub-Laplacians of Holomorphic Lp-Type on Exponential Solvable Groups

Let L denote a right-invariant sub-Laplacian on an exponential,hence solvable Lie group G, endowed with a left-invariant Haarmeasure. Depending on the structure of G, and possibly alsothat of L, L may admit differentiable L^p-functional calculi,or may be of holomorphic L^p-type for a given p 2. ‘HolomorphicL^p-type’ means that every L^p-spectral multiplier for Lis necessarily holomorphic in a complex neighbourhood of somenon-isolated point of the L²-spectrum of L. This can in factonly arise if the group algebra L¹(G) is non-symmetric. Assume that p 2. For a point in the dual g^* of the Lie algebrag of G, denote by ()=Ad^*(G) the corresponding coadjoint orbit.It is proved that every sub-Laplacian on G is of holomorphicL^p-type, provided that there exists a point g^* satisfying Boidol'scondition (which is equivalent to the non-symmetry of L¹(G)),such that the restriction of () to the nilradical of g is closed.This work improves on results in previous work by Christ andMüller and Ludwig and Müller in twofold ways: on theone hand, no restriction is imposed on the structure of theexponential group G, and on the other hand, for the case p>1,the conditions need to hold for a single coadjoint orbit only,and not for an open set of orbits. It seems likely that the condition that the restriction of ()to the nilradical of g is closed could be replaced by the weakercondition that the orbit () itself is closed. This would thenprove one implication of a conjecture by Ludwig and Müller,according to which there exists a sub-Laplacian of holomorphicL¹ (or, more generally, L^p) type on G if and only if there existsa point g^* whose orbit is closed and which satisfies Boidol'scondition.     

A Characterization of Fredholm Pseudo-Differential Operators

We give a necessary and sufficient condition on an ellipticsymbol of order m to ensure that the unique closed extensionin L^p(Rⁿ) for 1 < p < , of the pseudo-differential operatorT, initially defined on the Schwartz space, is a Fredholm operatorfrom L^p(Rⁿ) into L^p(Rⁿ) with domain H^{m, p}, where H^{m, p} is theL^p Sobolev space of order m.     

Intertwining Maps from Certain Group Algebras

In [17, 18, 19], we began to investigate the continuity propertiesof homomorphisms from (non-abelian) group algebras. Alreadyin [19], we worked with general intertwining maps [3, 12]. Thesemaps not only provide a unified approach to both homomorphismsand derivations, but also have some significance in their ownright in connection with the cohomology comparison problem [4]. The present paper is a continuation of [17, 18, 19]; this timewe focus on groups which are connected or factorizable in thesense of [26]. In [26], G. A. Willis showed that if G is a connectedor factorizable, locally compact group, then every derivationfrom L¹(G) into a Banach L¹(G)-module is automatically continuous.For general intertwining maps from L¹(G), this conclusion isfalse: if G is connected and, for some nN, has an infinite numberof inequivalent, n-dimensional, irreducible unitary representations,then there is a discontinuous homomorphism from L¹(G into aBanach algebra by [18, Theorem 2.2] (provided that the continuumhypothesis is assumed). Hence, for an arbitrary intertwiningmap from L¹(G), the best we can reasonably hope for is a resultasserting the continuity of on a ‘large’, preferablydense subspace of L¹(G). Even if the target space of is a Banachmodule (which implies that the continuity ideal I() of is closed),it is not a priori evident that is automatically continuous:the proofs of the automatic continuity theorems in [26] relyon the fact that we can always confine ourselves to restrictionsto L¹(G) of derivations from M(G) [25, Lemmas 3.1 and 3.4].It is not clear if this strategy still works for an arbitraryintertwining map from L¹(G) into a Banach L¹(G)-module.     

Rank Distributions on Coarse Spaces and Ideal Structure of Roe Algebras

The notions of controlled truncations for operators in the Roealgebras C^* (X) of a coarse space (X, ) with uniformly locallyfinite coarse structure, and rank distributions on (X, ) areintroduced. It is shown that the controlled propagation operatorsin an ideal I of C^* (X) are exactly the controlled truncationsof elements in I. It follows that the lattice of the idealsof C^* (X) in which controlled propagation operators are denseis isomorphic to the lattice of all rank distributions on (X,). If X is a discrete metric space with Yu's property A, thenthe ideal structure of the Roe algebra C^* (X is completely determinedby the rank distributions on X. 2000 Mathematics Subject Classification46L80, 46L89.     

On a p-Local Geometry Associated with a Complex of Non-Trivial p-Subgroups

Let D be a complex of non-trivial p-subgroups of a finite groupG, closed under G-conjugation. In this paper, a p-local geometry^p(G; D) is introduced for G associated with the complex D. Thehomotopy equivalence between ^p(G; D) and D is also studied.2000 Mathematics Subject Classification 20E15, 51E24.     

A Note on Ideal Spaces of Banach Algebras

In a previous paper, the second author introduced a compacttopology _r on the space of closed ideals of a unital Banachalgebra A. If A is separable, then _r is either metrizable orelse neither Hausdorff nor first countable. Here it is shownthat _r is Hausdorff if A is C¹[0, 1], but that if A is a uniformalgebra, then _r is Hausdorff if and only if A has spectral synthesis.An example is given of a strongly regular, uniform algebra forwhich every maximal ideal has a bounded approximate identity,but which does not have spectral synthesis. 1991 MathematicsSubject Classification 46H10.     

Global smooth fibrations of R3 by oriented lines

A smooth fibration of ³ by oriented lines is given by a smoothunit vector field V on ³ all of whose integral curves are straightlines. Such a fibration is said to be nondegenerate if dV vanishesonly in the direction of V. Let be the space of oriented linesof ³ endowed with its canonical pseudo-Riemannian neutral metric.We characterize the nondegenerate smooth fibrations of ³ byoriented lines as the closed (in the relative topology) definiteconnected surfaces in . In particular, local conditions on imply the existence of a global fibration. Besides, for anysuch fibration the base space is diffeomorphic to the open discand the directions of the fibers form an open convex set ofthe two-sphere. We characterize as well, in a similar way, thesmooth (possibly degenerate) fibrations.     

A remark on the representation of Zolotarev polynomials

Zolotarev polynomials are the polynomials that have minimaldeviation from zero on [–1, 1] with respect to the norm||xⁿ – x^n–1 + a_n–2 x^n–2 + ... + a₁x+ a_n|| for given and for all a_k . This note complements the paper of F. Pehersforfer [J. LondonMath. Soc. (1) 74 (2006) 143–153] with exact (not asymptotic)construction of the Zolotarev polynomials with respect to thenorm L₁ for || < 1 and with respect to the norm L₂ for || 1 in the form of Bernstein–Szegö orthogonal polynomials.For all in L₁ and L₂ norms, the Zolotarev polynomials satisfyexactly (not asymptotically) the triple recurrence relationof the Chebyshev polynomials.     

Spectral Synthesis and Operator Synthesis for Compact Groups

Let G be a compact group and C(G) be the C^*-algebra of continuouscomplex-valued functions on G. The paper constructs an imbeddingof the Fourier algebra A(G) of G into the algebra V(G) = C(G)^hC(G)(Haagerup tensor product) and deduces results about parallelspectral synthesis, generalizing a result of Varopoulos. Itthen characterizes which diagonal sets in G x G are sets ofoperator synthesis with respect to the Haar measure, using thedefinition of operator synthesis due to Arveson. This resultis applied to obtain an analogue of a result of Froelich: atensor formula for the algebras associated with the pre-ordersdefined by closed unital subsemigroups of G.