We formulate a conjecture analogous to Gross' refinement of the Stark conjectures on special values of abelian L-series at s = 0. Some evidence for the conjecture can be obtained, thanks to the fundamental ideas of F. Thaine.

Let T be an invertible measure-preserving transformation on a σ-finite measure space (X, μ) and let 1 < p < ∞. This paper uses an abstract method developed by José Luis Rubio de Francia which allows us to give a unified approach to the problems of characterizing the positive measurable functions v such that the limit of the ergodic averages or the ergodic Hilbert transform exist for all f ∈ Lp(νdμ). As a corollary, we obtain that both problems are equivalent, extending to this setting some results of R. Jajte, I. Berkson, J. Bourgain and A. Gillespie. We do not assume the boundedness of the operator Tf(x) = f(Tx) on Lp(νdμ). However, the method of Rubio de Francia shows that the problems of convergence are equivalent to the existence of some measurable positive function u such that the ergodic maximal operator and the ergodic Hilbert transform are bounded from LP(νdμ) into LP(udμ). We also study and solve the dual problem.

Let U(KλG) be the group of units of the infinite twisted group algebra KλG over a field K. We describe the FC-centre ΔU of U(KλG) and give a characterization of the groups G and fields K for which U(KλG) = ΔU. In the case of group algebras we obtain the Cliff-Sehgal-Zassenhaus theorem.

Let an interval I ⊂ ℝ and subsets D0, D1 ⊂ I with D0 ∪ D1 = I and D0 ∩ D1 = Ø be given, as well as functions r0: D0 → I, r1: D1 → I. We investigate the system (S) of two functional equations for an unknown function f: I → [0, 1]: We derive conditions for the existence, continuity and monotonicity of a solution. It turns out that the binary expansion of a solution can be computed in a simple recursive way. This recursion is algebraic for, e.g., inverse trigonometric functions, but also for the elliptic integral of the first kind. Moreover, we use (S) to construct two kinds of peculiar functions: surjective functions whose intervals of constancy are residual in I, and strictly increasing functions whose derivative is 0 almost everywhere.

If X is a separable non-reflexive Banach space, then the set NA of all norm-attaining elements of X* is not a w*-Gδ subset of X*. However if the norm of X is locally uniformly rotund, then the set of norm attaining elements of norm one is w*-Gδ. There exist separable spaces such that NA is a norm-Borel set of arbitrarily high class. If X is separable and non-reflexive, there exists an equivalent Gâteaux-smooth norm on X such that the set of all Gâteaux-derivatives is not norm-Borel.

We examine the problem of factoring a positive invertible operator in an AF C*-algebra as T*T for some invertible operator T with both T and T-1 in a triangular AF subalgebra. A factorization theorem for a certain class of positive invertible operators in AF algebras is proven. However, we explicitly construct a positive invertible operator in the CAR algebra which cannot be factored with respect to the 2∞ refinement algebra. Our main result generalizes this example, showing that in any AF algebra, there exist positive invertible operators which fail to factor with respect to a given triangular AF subalgebra. We also show that in the context of AF algebras, the notions of having a factorization and having a weak factorization are the same.

The intersection IW(G) of the normalizers of the infinite subnormal subgroups of a group G is a characteristic subgroup containing the Wielandt subgroup W(G) which we call the generalized Wielandt subgroup. In this paper we show that if G is infinite, then the structure of IW(G)/ W(G) is quite restricted, being controlled by a certain characteristic subgroup S(G). If S(G) is finite, then so is IW(G)/ W(G), whereas if S(G) is an infinite Prüfer-by-finite group, then IW(G)/W(G) is metabelian. In all other cases, IW(G) = W(G).

We will generalize Ringrose's notion of a simple chain of closed invariant subspaces of a compact operator acting on a Banach space, to that of an elementary chain of invariant subspaces of a subalgebra of compact operators. With this we expand the notion of diagonal coefficients to that of diagonal representations and subsequently generalize Ringrose's theorem equating the spectrum of an operator to the collection of diagonal coefficients. This in turn, in conjunction with some results from the theory of Polynomial Identity algebras, allows us to generalize Murphy's theorem which states that a closed subalgebra of compact operators is simultaneously triangularizable if and only if / rad () is commutative. Let be an algebra of compact operators acting on a Banach space with a norm ‖ · ‖ which dominates the operator norm, and under which 𝒜 is complete. Then has an elementary chain of invariant subspaces of bound n if and only if / rad () satisfies the standard polynomial .

From Koornwinder's interpretation of big q-Legendre polynomials as spherical elements on the quantum SU(2) group an addition formula is derived for the big g-Legendre polynomial. The formula involves Al-Salam-Carlitz polynomials, little q-Jacobi polynomials and dual q-Krawtchouk polynomials. For the little q-ultraspherical polynomials a product formula in terms of a big q-Legendre polynomial follows by q-integration. The addition and product formula for the Legendre polynomials are obtained when q tends to 1.

We determine the structure of representations induced from discrete series of parabolic subgroups of quasi-split p-adic groups G with G/G° a cyclic group of prime order. We attach to each such representation an R-group which extends the definition of the Knapp-Stein R-group. We show that this R-group has the properties predicted by Arthur. We apply our results to the case of Orthogonal groups.

We obtain a new version of the Sergeev–Pragacz formula for supersymmetric functions of standard shape–one applicable to arbitrary skew shape. The result involves an antisymmetrized sum of determinants that are themselves flagged supersymmetric functions. The proof is combinatorial, and follows by means of lattice path transformations.

Let p be a prime number, and let F be a free pro-p group of rank two. Consider an automorphism α of F of finite order m, and let FixF(α) = {x ∈ F | α(x) = x} be the subgroup of F consisting of the elements fixed by α. It is known that if m is prime to p and α = idF, then the rank of FixF(α) is infinite. In this paper we show that if m is a finite power pr of p, the rank of FixF(α) is at most 2. We conjecture that if the rank of F is n and the order of a is a power of α, then rank (FixF(α)) ≤ n.

In the context of a locally compact abelian group, we establish maximal theorem counterparts for weak type (1,1) multipliers of the classical de Leeuw theorems for individual strong multipliers. Special methods are developed to handle the weak type (1,1) estimates involved since standard linearization methods such as Lorentz space duality do not apply to this case. In particular, our central result is a maximal theorem for convolutions with weak type (1,1) multipliers which opens avenues of approximation. These results complete a recent series of papers by the authors which extend the de Leeuw theorems to a full range of strong type and weak type maximal multiplier estimates in the abstract setting.

In this paper, we study the boundary version of the classical Cartan theorem. We show that for some weakly pseudoconvex domains, when a holomorphic self-mapping has a sufficiently high order of contact (which depends only on the geometric properties of the domains) with the identical map at some boundary point, then it must coincide with the identity.