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 .

     

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 .

     

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.

     

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 .

     

On a theorem of Privalov and normal functions

A well known result of Privalov asserts that if is a function which is analytic in the unit disc , then has a continuous extension to the closed unit disc and its boundary function is absolutely continuous if and only if belongs to the Hardy space . In this paper we prove that this result is sharp in a very strong sense. Indeed, if, as usual, we prove that for any positive continuous function defined in with , as , there exists a function analytic in which is not a normal function and with the property that , for all sufficiently close to .

     

A commutativity theorem for semibounded operators in hilbert space

Let and be semibounded (bounded from below) operators in a Hilbert space and a dense linear manifold contained in the domains of , , , and , and such that for all in . It is shown that if the restriction of to is essentially self-adjoint, then and are essentially self-adjoint and and commute, i.e. their spectral projections permute.

     

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.

     

Invariants of Skew Derivations

If is an automorphism and is a -derivation of a ring , then the subring of invariants is the set The main result of this paper is Theorem. Let be a -derivation of an algebra over a commutative ring such that

for all , where and .

(i)

If , then .

(ii)

If is a -stable left ideal of such that , then .

This theorem generalizes results on the invariants of automorphisms and derivations.

     

Rigidity of compact manifolds with boundary and nonnegative Ricci curvature

Let be an ()-dimensional compact Riemannian manifold with nonnegative Ricci curvature and nonempty boundary . Assume that the principal curvatures of are bounded from below by a positive constant . In this paper, we prove that the first nonzero eigenvalue of the Laplacian of acting on functions on satisfies with equality holding if and only if is isometric to an -dimensional Euclidean ball of radius . Some related rigidity theorems for are also proved.

     

Weighted weak-type inequalities for the maximal function of nonnegative integral transforms over approach regions

The relation between approach regions and singularities of nonnegative kernels is studied, where , , , and is a homogeneous space. For , a sufficient condition on approach regions () is given so that the maximal function

is weak-type with respect to a pair of measures and . It is shown that this condition is also necessary for operators of potential type in the sense of Sawyer and Wheedon (Amer. J. Math. 114 (1992), 813-874).

     

The structure of hypersurfaces with some curvature conditions

Let be a hypersurface in , and let denote the mean curvature and the scalar curvature of respectively. We show that if is compact and , then is diffeomorphic to . Also we prove that if is complete, is constant and , then is or or .

     

Finite linear groups and theorems of Minkowski and Schur

Let be a finite group with a faithful rational valued character of degree . A theorem of I. Schur gives a bound for the order of in terms of , generalizing an earlier result of H. Minkowski who showed that the same bound holds if . This note contains strengthened versions of these results which in particular show that a -subgroup of of maximum possible order contains a reflection.

     

A formula with nonnegative terms for the degree of the dual variety of a homogeneous space

Let be a reductive group and a parabolic subgroup. For every -regular dominant weight let denote the variety embedded in the projective space by the embedding corresponding to the ample line bundle . Writing , we prove that the degree of the dual variety to is a polynomial with nonnegative coefficients in . In the case of homogeneous spaces we find an expression for the constant term of this polynomial.

     

On Bernstein-Sato polynomials

We show that for fixed and the set of Bernstein-Sato polynomials of all the polynomials in at most variables of degrees at most is finite. As a corollary, we show that there exists an integer depending only on and such that generates as a module over the ring of the -linear differential operators of , where is an arbitrary field of characteristic 0, is the ring of polynomials in variables over and is an arbitrary non-zero polynomial of degree at most .

     

There is no degree invariant half-jump

We prove that there is no degree invariant solution to Post's problem that always gives an intermediate degree. In fact, assuming definable determinacy, if is any definable operator on degrees such that on a cone then is low or high on a cone of degrees, i.e., there is a degree such that for every or for every .

     

Less Saturated Ideals

We prove the following:

(1)

If is weakly inaccessible then is not -saturated.

(2)

If is weakly inaccessible and is regular then is not -saturated.

(3)

If is singular then is not -saturated.

Combining this with previous results of Shelah, one obtains the following:

(A)

If then is not -saturated.

(B)

If then is not -saturated.

     

Exact topological analogs to orthoposets

An arbitrary orthoposet is shown to be isomorphic to , being a subbasis of a Hausdorff topological space satisfying 1) , 2) , and 3) every covering of by elements of possesses an at most 2-element subcovering. The couple turns out to be unique.

     

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

     

Extreme points of unit balls in Lipschitz function spaces

We give a new characterization of the set of all extreme points of the unit ball in the Banach space of all Lipschitz functions on a metric space This result is applied to get a total variation characterization of in the particular case when is a convex subset of a Banach space.