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.     

Corrigendum: Elliptic Curves and Quadratic Recurrence Sequences

The statement of the numerical values and z₀ on page 167 of[1, Section 3] contained an error. The values that were actuallyused were (to nine decimal places): thesebeing shifted, by the periods 2₁ and 2₃ respectively, comparedwith the values given in [1] (with ₁ = 1.496729323 and ₃ = 1.225694691i).With ₀ = ₁ and (z) denoting the sigma function (z; g2, g3) withinvariants g2 = 4, g3 = –1 associated with the ellipticcurve given by equation (3.2), these values of and z₀ yield and the latter three values all agreewith those stated in the paper (apart from rounding down thelast digit in the imaginary part of A). 2000 Mathematics SubjectClassification 11B37 (primary), 33E05, 37J35 (secondary).     

SMALL C*-ALGEBRAS THAT FAIL TO HAVE SEPARABLE REPRESENTATIONS

It is shown that any -finite, wild AW*-algebra of type II₁ failsto have non-trivial separable representations. Here a wild AW*-algebrameans an AW*-algebra with no direct summand, which is *-isomorphicto a von Neumann algebra. Let be the -finite, wild type II₁AW*-algebra formed by taking the monotone complete tensor productof the Dixmier algebra and a factor of type II₁ acting ona separable Hilbert space. As both and act on separable Hilbertspaces, it seems natural to describe as small and to ask thefollowing question. Does have a non-trivial separable representation?The above described result answers negatively to this question.That is, fails to have non-trivial separable representations.     

Correction: on the Intersection Matrices of Curves lying on Irregular Algebraic Surfaces

Professor W. F. Hammond has kindly drawn my attention to a blunderin 4 of the above paper. He referred to the ( – 2r) xß submatrix D of the skew-symmetric matrix displayednear the top of page 181, of which it is asserted that it issquare and non-singular, and pointed out that, from the factthat the matrix of which D forms part is regular, it may onlybe deduced that the columns of D are linearly independent; thatis, it only follows that – 2r ß. The validity of the equation – 2r = ß is essentialto the succeeding argument and, fortunately, may be establishedby alternative means. Using the nomenclature of the paper, wehave on F the set ₁^*, ..., _2r^*, ₁^*, ..., _ß^* of independent3-cycles (independent because they cut independent 1-cycleson the curve C), which may be completed, to form a basis forsuch cycles on F, by a further set ₁', ..., _2q–2r–pof independent 3-cycles, each of which meets C in a cycle homologousto zero on C. The cycles ₁^*, ..., ^* are invariant cycles andare independent on F so that, if > 2r + ß, thereis a non-trivial linear combination ^* of these having zero intersectionon C with each of the cycles ₁^*, ..., _2r^*, ₁^*, ..., _ß^*.Thus we have. (^* ._k^*)c = 0 = (^* ._i^*)c i.e. (^* ._k^*) = 0 = (^* ._i^* on F (1 k 2r; 1 i ß). Furthermore, (_j . C) 0 on C and we have (^* ._j .C)_C = 0 i.e. (^* ._j) = 0 on F (1 j 2q – 2r – ß). It now follows that ^* 0 on F (for it has zero intersectionwith every member of a basic set of 3-cycles on F). But thiscondradicts the assumption that ^* is a non-trivial linear combinationof the independent cycles ₁^*, ...,^*; and hence < 2r + ß.     

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).     

Gyration Numbers, Fixed Points and K2

We give a formula for the Boyle-Krieger gyration numbers ofan involution in the symmetry group Aut (_A) of an even subshiftof finite type _A in terms of the number of fixed points of in the periodic orbits of _A. A relationship between gyrationnumbers, the algebraic K-theory group K₂, and the tame symbolis also discussed.     

Discontinuous Homomorphisms from Non-Commutative Banach Algebras

In the 1970s, a question of Kaplansky about discontinuous homomorphismsfrom certain commutative Banach algebras was resolved. Let Abe the commutative C*-algebra C(), where is an infinite compactspace. Then, if the continuum hypothesis (CH) be assumed, thereis a discontinuous homomorphism from C() into a Banach algebra[2, 7]. In fact, let A be a commutative Banach algebra. Then(with (CH)) there is a discontinuous homomorphism from A intoa Banach algebra whenever the character space _A of A is infinite[3, Theorem 3] and also whenever there is a non-maximal, primeideal P in A such that |A/P|=2^₀ [4, 8]. (It is an open questionwhether or not every infinite-dimensional, commutative Banachalgebra A satisfies this latter condition.) 1991 MathematicsSubject Classification 46H40.     

Beurling and Lipschitz Algebras

It is well known that there exist infinite closed subsets Eof T such that A(E) = C(E) (see, for example, [3]). Such setsare called Helson sets. Let E be a closed subset of T, let 0< < 1, and let A(E) be the restriction of the Beurlingalgebra A(T). Then A(E) lipE. We shall show that A(E) = lipEif and only if E is finite. This answers a question raised byPedersen [5], where partial results were obtained. 1991 MathematicsSubject Classification 46J10.     

Finite groups in Stone-Cech compactifications

Let be an infinite cardinal and let G = ₂. Now let β Gbe the Stone–ech compactification of G as a discrete semigroup,and let =_<c_{β G} {xG\{0}:minsupp (x)}. We show that thesemigroup contains no nontrivial finite group.     

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.     

Prime Non-Commutative JB*-Algebras

We prove that if A is a prime non-commutative JB*-algebra whichis neither quadratic nor commutative, then there exist a primeC*-algebra B and a real number with < 1 such that A =B as involutive Banach spaces, and the product of A is relatedto that of B (denoted by , say) by means of the equality xy= xy+(1 – )yx. 2000 Mathematics Subject Classification46K70, 46L70.     

Asymptotically Almost Periodic Solutions of Inhomogeneous Cauchy Problems on The Half-Line

Let u be a bounded, uniformly continuous, mild solution of aninhomogeneous Cauchy problem on R₊: u'(t)+Au(t)+(t)(t0). Supposethat u has uniformly convergent means, (A)iR is countable, and is asymptotically almost periodic. Then u is asymptoticallyalmost periodic. Related results have been obtained by Ruessand V, and by Basit, using different methods. A direct proofis given of a Tauberian theorem of Batty, van Neerven and Räbiger,and applications to Volterra equations are discussed. 1991 MathematicsSubject Classification 34C28, 44A10, 47D03.     

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).     

Asymptotic Cones of Finitely Generated Groups

Answering a question of Gromov [7], we shall present an exampleof a finitely generated group and two non-principal ultrafiltersA, B such that the asymptotic cones Con_A and Con_B are nothomeomorphic. 1991 Mathematics Subject Classification 20F06,20F32.     

Hopf C*-Algebras

In this paper we define and study Hopf C^*-algebras. Roughlyspeaking, a Hopf C^*-algebra is a C^*-algebra A with a comultiplication: A M(A A) such that the maps a b (a)(1 b) and a (a 1)(b)have their range in A A and are injective after being extendedto a larger natural domain, the Haagerup tensor product A _hA. In a purely algebraic setting, these conditions on are closelyrelated to the existence of a counit and antipode. In this topologicalcontext, things turn out to be much more subtle, but neverthelessone can show the existence of a suitable counit and antipodeunder these conditions. The basic example is the C^*-algebra C₀(G) of continuous complexfunctions tending to zero at infinity on a locally compact groupwhere the comultiplication is obtained by dualizing the groupmultiplication. But also the reduced group C^*-algebra of a locally compact group with thewell-known comultiplication falls in this category. In factall locally compact quantum groups in the sense of Kustermansand the first author (such as the compact and discrete ones)as well as most of the known examples are included. This theory differs from other similar approaches in that thereis no Haar measure assumed. 2000 Mathematics Subject Classification: 46L65, 46L07, 46L89.     

On Extensions of Myers' Theorem

Let M be a compact Riemannian manifold, and let h be a smoothfunction on M. Let p^h(x) = inf||–₁(Ric_x(,)–2Hess(h_x(,)).Here Ric_x denotes the Ricci curvature at x and Hess(h) is theHessian of h. Then M has finite fundamental group if ^h–p^h<0. Here ^h =:+2L_h is the Bismut-Witten Laplacian. This leadsto a quick proof of recent results on extension of Myers' theoremto manifolds with mostly positive curvature. There is also asimilar result for noncompact manifolds.     

The Series of Norms in a Soluble p-Group

The norm of a group G is the subgroup of elements of G whichnormalise every subgroup of G. We shall denote it (G). An ascendingseries of subgroups _i(G) in G may be defined recursively by:₀(G) = 1 and, for i 0, _i+1(G)/_i(G) = (G/_i(G)). For each i,the section _i+1(G)/i(G) clearly contains the centre of the groupG/_i(G). A result of Schenkman [8] gives a very close connectionbetween this norm series and the upper central series: _i(G) _i(G) _2i(G). 1991 Mathematics Subject Classification 20E15.     

Full Crossed Products by Coactions, C0(X)-Algebras and C*-Bundles

Let be a coaction of a locally compact group G on a C*-algebraA. We show that if I is a -invariant ideal in A, then for full crossed products, as Landstadet al. have done for spatial crossed products by coactions.We prove that for suitable coactions, the crossed products ofC₀(X)-algebras are again C₀(X)-algebras, and the crossed productsof continuous C*-bundles by a locally compact group are againcontinuous C*-bundles. 1991 Mathematics Subject Classification46L55.     

Calculs De Facteurs Epsilon De Paires Pour GLn Sur Un Corps Local, I

Soient F un corps commutatif localement compact non archimédienet un caractère additif non trivial de F. Soient unereprésentation du groupe de Weil–Deligne de F,et sa contragrédiente. Nous calculons le facteur (, , ). De manière analogue, nous calculons le facteur (x, , ) pour toute représentationadmissible irréductible de GL_n(F). En conséquence,si F est de caractéristique nulle et si et se correspondentpar la correspondance de Langlands construite par M. Harris,ou celle construite par les auteurs, alors les facteurs (, , s) et (x, , s) sont égaux pour tout nombre complexe s. Let F be a non-Archimedean local field and a non-trivial additivecharacter of F. Let be a representation of the Weil–Delignegroup of F and its contragredient representation. We compute (, , ). Analogously, we compute (x, , ) for all irreducible admissible representations of GL_n(F).Consequently, if F has characteristic zero, and , correspondvia the Langlands correspondence established by M. Harris orthe correspondence constructed by the authors, then we have(, , s) = (x, , s) for all sC. 1991 Mathematics Subject Classification22E50.