For any holomorphic map in the unit disk, the set of radial limits at a Borel set on the unit circle is a Suslin-analytic set. Here it is proved that, for a conformal map, this set is, in fact, Borel. As a consequence, the sets of accessible boundary points, of cut points and of transition points are Borel. In addition, it is shown that the set of end points is a $G_{\delta}$-set.

The automorphism group of a finitely generated free group is the normal closure of a single element of order 2. If $m<n$, then a homomorphism ${\rm Aut}(F_n)\to {\rm Aut}(F_m)$ can have image of cardinality at most 2. More generally, this is true of homomorphisms from ${\rm Aut}(F_n)$ to any group that does not contain an isomorphic image of the symmetric group $S_{n+1}$. Strong restrictions are also obtained on maps to groups that do not contain a copy of $W_n=({\mathbb Z}/2)^n\rtimes S_{n}$, or of ${\mathbb Z}^{n-1}$. These results place constraints on how ${\rm Aut}(F_n)$ can act. For example, if $n\ge 3$, any action of ${\rm Aut}(F_n)$ on the circle (by homeomorphisms) factors through ${\rm det} : {\rm Aut}(F_n)\to{\mathbb Z}_2$.The research of the first author was supported by an EPSRC Advanced Fellowship. The research of the second author is supported in part by NSF grant DMS-9307313.

This paper studies the spectrum that results when all height one polynomials are evaluated at a Pisot number. This continues the research theme initiated by Erdős, Joó and Komornik in 1990. Of particular interest is the minimal non-zero value of this spectrum. Formally, this value is denoted as $l^1(q)$ , and this definition is extended to all height $m$ polynomials as \[ l^m(q):= \inf(\vert y\vert: y = \epsilon_0 + \epsilon_1 q^1 + \cdots + \epsilon_n q^n,\, \epsilon_i \in {\bb Z},\, \vert\epsilon_i\vert \leqslant m,\, y \ne 0). \] A recent result in 2000, of Komornik, Loreti and Pedicini gives a complete description of $l^m(q)$ when $q$ is the Golden ratio. This paper extends this result to include all unit quadratic Pisot numbers. A main theorem is as follows.

THEOREM. Let $q$ be a quadratic Pisot number that satisfies a polynomial of the form $p(x) = x^2 - ax \pm 1$ , with conjugate $r$ . Let $q$ have convergents $\{C_k/D_k\}$ and let $k$ be the maximal integer such that \[ \vert D_k r - C_k\vert \leqslant m \frac{1}{1-\vert r\vert}; \] then \[ l^m (q) = \vert D_k q - C_k \vert. \]

A value related to $l(q)$ is $a(q)$ , the minimal non-zero value when all ${\pm}1$ polynomials are evaluated at $q$ . Formally, this is \[ a(q) := \inf(\vert y\vert: y = \epsilon_0 + \epsilon_1 q^2 + \cdots + \epsilon_n q^n,\, \epsilon_i = {\pm} 1,\, y \ne 0). \] An open question concerning how often $a(q) = l(q)$ is also answered in this paper.

In this paper, Conway's group, $\Co_3$, is characterized among simple groups of even type with $e(G)=3$, by a restriction on the $p$-local structure for some odd prime $p$ for which $m_{2,p}(G) = 3$.

Given a coaction $\alpha$ of a Hopf ${\rm C}^*$-algebra $A$ on a ${\rm C}^*$-algebra $B$ with an $\alpha$-invariant ${\rm C}^*$-subalgebra $C$, and a conditional expectation $E:B \rightarrow C$ commuting with $\alpha$, it is shown that if $(\pi, u)$ is a covariant representation of the system ($C, A, \alpha\mid_C$), then there is an associated covariant representation ($\tilde{\pi}, \tilde{u}$) of the system ($B, A, \alpha$), where $\tilde{\pi}$ is the representation induced from $\pi$ up to $B$ via $E$, and $\tilde{u}$ is a unitary corepresentation of $A$ naturally 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 multiplicative unitary operators via infinite tensor product covariant representations.

Let $\sigma$ be a non-degenerate positive $M_n$-valued measure on a locally compact group $G$ with $\|\sigma\|\,{=}\,1$. An $M_n$-valued Borel function $f$ on $G$ is called $\sigma$-harmonic if $f(x) = \int_G f(xy^{-1})\,d\sigma(y)$ for all $x\,{\in}\,G$. Given such a function $f$ which is bounded and left uniformly continuous on $G$, it is shown that every central element in $G$ is a period of $f$. Further, it is shown that $f$ is constant if $G$ is nilpotent or central.

In this paper, some new results concerning the existence and uniqueness of nontrivial solutions to nonlinear Volterra integral equations with convolution kernels are presented.

A relatively long-standing problem concerning the best constants of approximation by entire functions of exponential type on the class $W^rH_{\omega}$ is solved. The central tool is a limit relation between the best constants of approximation by trigonometric polynomials and entire functions of exponential type on $W^rH_{\omega}$ and its periodic analogue.

Co-Hopfian finitely generated torsion-free nilpotent groups are characterized in this paper in terms of their Lie algebra automorphisms. Many examples of such groups are also constructed.Partially supported by NSF Grant # DMS-0203979.

The authors study the algebra of uniformly continuous holomorphic symmetric functions on the ball of ${\cal L}_p$ , investigating in particular the spectrum of such algebras. To do so, they examine the algebra of symmetric polynomials on ${\cal L}_p$ -spaces, as well as finitely generated symmetric algebras of holomorphic functions. Such symmetric polynomials determine the points in ${\cal L}_p$ up to a permutation.

In a beautiful result, Herrero (D. A. Herrero, ‘Normal limits of nilpotent operators’, Indiana Univ. Math. J. 23 (1973/74) 1097–1108) showed that a normal operator on $l^2$ lies in the closure of the set of nilpotent operators if and only if its spectrum is connected and contains zero. In the quest for an automatic continuity result for algebra homomorphisms between ${\rm C}^*$-algebras, Dales showed that, if a discontinuous algebra homomorphism $\theta :{\cal A} \rightarrow {\cal U}$ exists between ${\rm C}^*$-algebras ${\cal A}$ and ${\cal U}$, and if $\theta({\cal A})$ is dense in ${\cal U}$, then there is a ${\rm C}^*$-algebra ${\cal U}_2$ with a dense subalgebra ${\cal N} \subset {\cal U}_2$ such that every $x \in {\cal N}$ is quasinilpotent (see p. 685 of H. G. Dales, Banach algebras and automatic continuity, London Mathematical Society Monographs 24, Oxford University Press, 2001). (A discontinuous homomorphism $\theta_2 :{\cal A}_2 \rightarrow {\cal U}_2$ can be defined with the same basic properties as $\theta$, but the revised target space ${\cal U}_2$ has a dense subalgebra consisting of quasinilpotent elements.) As remarked by Dales, no such ${\rm C}^*$-algebra was then known; but here we present one. Indeed, using the full power of Herrero's result, one may arrange that every $x \in {\cal N}$ is nilpotent. The ${\rm C}^*$-algebra is constructed in a ‘neat’ way; it is most naturally constructed as a non-separable, concrete ${\rm C}^*$-algebra of operators on a separable Hilbert space ${\cal K}$; but one can arrange that the algebra ${\cal U}$ itself be separable if desired.

An element u of a norm-unital Banach algebra A is said to be unitary if u is invertible in A and satisfies $\Vert u\Vert =\Vert u^{-1}\Vert =1$. The norm-unital Banach algebra A is called unitary if the convex hull of the set of its unitary elements is norm-dense in the closed unit ball of A. If X is a complex Hilbert space, then the algebra $\BL(X)$ of all bounded linear operators on X is unitary by the Russo–Dye theorem. The question of whether this property characterizes complex Hilbert spaces among complex Banach spaces seems to be open. Some partial affirmative answers to this question are proved here. In particular, a complex Banach space X is a Hilbert space if (and only if) $\BL(X)$ is unitary and, for Y equal to $X,$ $X^*$ or $X^{**},$ there exists a biholomorphic automorphism of the open unit ball of Y that cannot be extended to a surjective linear isometry on Y.

It is proved that if $P_G(s)$ has an Euler product expansion with all factors of the form $1-c_i/q_i^s$ where each $q_i$ is a prime power, then $G$ is soluble.

Higher algebraic K-theoretic analogues of Hasse's norm theorem in class field theory are proved. The result gives a partial affirmative answer to a question posed by Bak and Rehmann.

Suppose that $R$ goes to infinity through a second-order lacunary set. Let $S_R$ denote the $R$th spherical partial inverse Fourier integral on ${\rm I\!R}^d$. Then $S_R f$ converges almost everywhere to $f$, provided that $f$ satisfies \[ \int \widehat{f}(\xi)\log\log(8+|\xi|)^2\,d\xi < \infty. \]

If there are infinitely many p-Mersenne prime numbers, there is no bad field of positive characteristic p.

It is shown that a function whose critical locus is an isolated complete intersection singularity of arbitrary dimension, and that has finite codimension (in the sense of R. Pellikaan, Proc. London Math. Soc. (3) 57 (1998) 357–382) with respect to the ideal defining the isolated complete intersection singularity, can be approximated by a function whose critical locus is a finite number of Morse points together with the Milnor fibre of the isolated complete intersection singularity, having there well-known types of singularities.

Let $M$ be a compact complex Kähler manifold of dimension $n\geq 3$, and suppose that $N\hookrightarrow M$ is a closed complex submanifold of dimension $n-1$. Denote by $\mathcal{KC}(1)$ the space of classes of Kähler forms $[\omega]\in H^{1,1}(M,\mathbb R)$ that define Kähler metrics of volume $1$ on $M$, and define $\mathbf f:\mathcal{KC}(1)\longrightarrow\mathbb R$ by $\mathbf f([\omega])=({1}/{(n-1)!})\int _N\omega^{n-1}$, equal to the volume of $N$ in the metric induced by $\omega$. In this paper, the critical points of $\mathbf f$ are studied. The Riemann–Hodge bilinear relations of $M$ and $N$ are shown to imply that any critical point of $\mathbf f$ is the strict global minimum; also, the hard Lefschetz theorem determines the critical point $[\omega]$. A positive multiple of $[\omega]\in H^2(M,\mathbb R)$ is the Poincaré dual of the homology class of $N$. If this is applied to the Theta divisor of the Jacobian $J(C)$ of a compact Riemann surface $C$, the Theta metric minimizes the volume of the Theta divisor within all Kähler metrics of volume one on $J(C)$.

Akemann and Anderson made a conjecture about ‘paving’ projections in finite-dimensional matrix algebras which, if true, would settle the well-known Kadison–Singer problem. Their conjecture is falsified in this paper by an explicit sequence of counterexamples.

Let $B$ be a unital commutative Banach algebra, and let $A$ be an arbitrary subalgebra of $B$. Suppose that an ideal $I\subset A$ consists of elements that are non-invertible in $B$. Then there exists an ideal $J$ in $A$ that: (i) contains $I$, (ii) also consists of elements non-invertible in $B$, and (iii) is of codimension one in $A$. This result is used in the study of a class of subspectra on $A$.