Rudin's orthogonality problem and the Nevanlinna counting function

Let be a holomorphic function taking the open unit disk into itself. We show that the set of nonnegative powers of is orthogonal in if and only if the Nevanlinna counting function of , , is essentially radial. As a corollary, we obtain that the orthogonality of for a univalent implies for some constant . We also show that if is orthogonal, then the closure of must be a disk.

     

On the von Neumann-Jordan constant for Banach spaces

Let be the von Neumann-Jordan constant for a Banach space . It is known that for any Banach space ; and is a Hilbert space if and only if . We show that: (i) If is uniformly convex, is less than two; and conversely the condition implies that admits an equivalent uniformly convex norm. Hence, denoting by the infimum of all von Neumann-Jordan constants for equivalent norms of , is super-reflexive if and only if . (ii) If , (the same value as that of -space), is of Rademacher type and cotype for any with , where ; the converse holds if is a Banach lattice and is finitely representable in or .

     

Invariance of spectrum for representations of -algebras on Banach spaces

Let be a Banach space, a unital -algebra, and an injective, unital homomorphism. Suppose that there exists a function such that, for all , and all ,

(a) ,

(b) ,

(c) .
Then for all , the spectrum of in equals the spectrum of as a bounded linear operator on . If satisfies an additional requirement and is a -algebra, then the Taylor spectrum of a commuting -tuple of elements of equals the Taylor spectrum of the -tuple in the algebra of bounded operators on . Special cases of these results are (i) if is a closed subspace of a unital -algebra which contains as a unital -subalgebra such that , and only if , then for each , the spectrum of in is the same as the spectrum of left multiplication by on ; (ii) if is a unital -algebra and is an essential closed left ideal in , then an element of is invertible if and only if left multiplication by on is bijective; and (iii) if is a -algebra, is a Hilbert -module, and is an adjointable module map on , then the spectrum of in the -algebra of adjointable operators on is the same as the spectrum of as a bounded operator on . If the algebra of adjointable operators on is a -algebra, then the Taylor spectrum of a commuting -tuple of adjointable operators on is the same relative to the algebra of adjointable operators and relative to the algebra of all bounded operators on .

     

Inner derivations on ultraprime normed algebras

We show that, for every ultraprime Banach algebra , there exists a positive number satisfying for all in , where denotes the centre of and denotes the inner derivation on induced by . Moreover, the number depends only on the ``constant of ultraprimeness' of .

     

Continuous functions on compact groups

We show that every scalar valued continuous function on a compact group may be written as for all , where are vectors in a separable Hilbert space , and is a strongly continuous unitary valued function on which is a product of unitary representations and antirepresentations of on . This product is countable, but always converges uniformly on . Moreover the supremum norm of is matched by . This may be viewed as a `Fourier product representation' for , and complements a result of Eymard for the Fourier algebra. For `Fourier polynomials' we show that the Hilbert space may be taken to be finite dimensional, and the product finite, which is more or less obvious except in that we are able to match the correct norm. The main ingredients of the proof are the Peter-Weyl theory, Tannaka's duality theorem, and a method developed with Paulsen using a characterization of operator algebras due to the author, Ruan and Sinclair. We also give the analogues of these formulae for compact quantum groups.

     

On invariants dual to the Bass numbers

Let be a commutative Noetherian ring, and let be an -module. In earlier papers by Bass (1963) and Roberts (1980) the Bass numbers were defined for all primes and all integers by use of the minimal injective resolution of . It is well known that . On the other hand, if is finitely generated, the Betti numbers are defined by the minimal free resolution of over the local ring . In an earlier paper of the second author (1995), using the flat covers of modules, the invariants were defined by the minimal flat resolution of over Gorenstein rings. The invariants were shown to be somehow dual to the Bass numbers. In this paper, we use homologies to compute these invariants and show that

for any cotorsion module . Comparing this with the computation of the Bass numbers, we see that is replaced by and the localization is replaced by (which was called the colocalization of at the prime ideal by Melkersson and Schenzel).

     

Hypersurfaces in a sphere with constant mean curvature

Let be a closed hypersurface of constant mean curvature immersed in the unit sphere . Denote by the square of the length of its second fundamental form. If , is a small hypersphere in . We also characterize all with .

     

A Characterization of the Leinert property

Let be a discrete group and denote by its left regular representation on . Denote further by the free group on generators and its left regular representation. In this paper we show that a subset of has the Leinert property if and only if for some real positive coefficients the identity

holds. Using the same method we obtain some metric estimates about abstract unitaries satisfying the similar identity

     

A note on paracompactness in generalized ordered spaces

In this paper, we show that for generalized ordered spaces, paracompactness is equivalent to Property D, where a space is said to have Property D if, given any collection of open sets in satisfying for each , there is a closed discrete subset of satisfying .

     

On an optimality property of Ramanujan sums

We evaluate , where the is taken over sequences satisfying . In particular we show that it is attained by taking for all , which reduces the summation over to a Ramanujan sum .

     

Topologies on the ideal space of a Banach algebra and spectral synthesis

Let the space of closed two-sided ideals of a Banach algebra carry the weak topology. We consider the following property called normality (of the family of finite subsets of : if the net in converges weakly to , then for all we have (e.g. -algebras, with compact . For a commutative Banach algebra normality is implied by spectral synthesis of all closed subsets of the Gelfand space , the converse does not always hold, but it does under the following additional geometrical assumption: .

     

Bilocal derivations of standard operator algebras

In this paper, we shall show the following two results: (1) Let be a standard operator algebra with , if is a linear mapping on which satisfies that maps into for all , then is of the form for some in . (2) Let be a Hilbert space, if is a norm-continuous linear mapping on which satisfies that maps into for all self-adjoint projection in , then is of the form for some in .

     

Semi-free actions of zero-dimensional compact groups on Menger compacta

Let be the -dimensional universal Menger compactum, a -set in and a metrizable zero-dimensional compact group with the unit. It is proved that there exists a semi-free -action on such that is the fixed point set of every . As a corollary, it follows that each compactum with can be embedded in as the fixed point set of some semi-free -action on .

     

Hypercomplex structures on four-dimensional Lie groups

The purpose of this paper is to classify invariant hypercomplex structures on a -dimensional real Lie group . It is shown that the -dimensional simply connected Lie groups which admit invariant hypercomplex structures are the additive group of the quaternions, the multiplicative group of nonzero quaternions, the solvable Lie groups acting simply transitively on the real and complex hyperbolic spaces, and , respectively, and the semidirect product . We show that the spaces and possess an of (inequivalent) invariant hypercomplex structures while the remaining groups have only one, up to equivalence. Finally, the corresponding hyperhermitian -manifolds are determined.

     

A note on GK dimension of skew polynomial extensions

Let be a finitely generated commutative domain over an algebraically closed field , an algebra endomorphism of , and a -derivation of . Then if and only if is locally algebraic in the sense that every finite dimensional subspace of is contained in a finite dimensional -stable subspace.

Similarly, if is a finitely generated field over , a -endomorphism of , and a -derivation of , then if and only if is an automorphism of finite order.

     

Fixed points of the mapping class group in the moduli spaces

Let be a compact oriented surface with or without boundary components. In this note we prove that if then there exist infinitely many integers such that there is a point in the moduli space of irreducible flat connections on which is fixed by any orientation preserving diffeomorphism of . Secondly we prove that for each orientation preserving diffeomorphism of and each there is some such that has a fixed point in the moduli space of irreducible flat connections on . Thirdly we prove that for all there exists an integer such that the 'th power of any diffeomorphism fixes a certain point in the moduli space of irreducible flat connections on .

     

Collapsing successors of singulars

Let be a singular cardinal in , and let be a model such that for some -cardinal with . We apply Shelah's pcf theory to study this situation, and prove the following results. 1) is not a -c.c generic extension of . 2) There is no ``good scale for ' in , so in particular weak forms of square must fail at . 3) If then and also . 4) If then .

     

Superrigid subgroups of solvable Lie groups

Let be a discrete subgroup of a simply connected, solvable Lie group , such that has the same Zariski closure as . If is any finite-dimensional representation of , we show that virtually extends to a continuous representation of . Furthermore, the image of is contained in the Zariski closure of the image of . When is not discrete, the same conclusions are true if we make the additional assumption that the closure of is a finite-index subgroup of (and is closed and is continuous).

     

Constructing free subgroups of integral group ring units

Let be an arbitrary group. It is proved that if contains a bicyclic unit , then is a nonabelian free subgroup of invertible elements.

     

Intersection of sets with -connected unions

We show that if sets in a topological space are given so that all the sets are closed or all are open, and for each every of the sets have a -connected union, then the sets have a point in common. As a consequence, we obtain the following starshaped version of Helly's theorem: If every or fewer members of a finite family of closed sets in have a starshaped union, then all the members of the family have a point in common. The proof relies on a topological KKM-type intersection theorem.