Cohomological detection and regular elements in group cohomology

Suppose that is a compact Lie group or a discrete group of finite virtual cohomological dimension and that is a field of characteristic . Suppose that is a set of elementary abelian -subgroups such that the cohomology is detected on the centralizers of the elements of . Assume also that is closed under conjugation and that is in whenever some subgroup of is in . Then there exists a regular element in the cohomology ring such that the restriction of to an elementary abelian -subgroup is not nilpotent if and only if is in . The converse of the result is a theorem of Lannes, Schwartz and the second author. The results have several implications for the depth and associated primes of the cohomology rings.

     

The range of a ring homomorphism from a commutative -algebra

We prove that if a commutative semi-simple Banach algebra is the range of a ring homomorphism from a commutative -algebra, then is -equivalent, i.e. there are a commutative -algebra and a bicontinuous algebra isomorphism between and . In particular, it is shown that the group algebras , and the disc algebra are not ring homomorphic images of -algebras.

     

A topological characterization of linearity for quasi-traces

Let be a -algebra, and let be a (local) quasi-trace on . Then is linear if, and only if, the restriction of to the closed unit ball of is uniformly weakly continuous.

     

On contravariant finiteness of subcategories of modules of projective dimension

Let be an artin algebra. This paper presents a sufficient condition for the subcategory of to be contravariantly finite in , where is the subcategory of consisting of --modules of projective dimension less than or equal to . As an application of this condition it is shown that is contravariantly finite in for each when is stably equivalent to a hereditary algebra.

     

Isometries of certain operator algebras

To a given basis on an -dimensional Hilbert space , we associate the algebra of all linear operators on having every as an eigenvector. So, is commutative, semisimple, and -dimensional. Given two algebras of this type, and , there is a natural algebraic isomorphism of and . We study the question: When does preserve the operator norm?

     

-analyticity

Let be a finite-dimensional commutative algebra over and let , and be the ring of -differentiable functions of class , the ring of real analytic mappings with values in and the ring of -analytic functions, respectively, defined on an open subset of . We prove two basic results concerning -differentiability and -analyticity: ) , ) if and only if is defined over .

     

Comparative probability on von Neumann algebras

We continue here the study begun in earlier papers on implementation of comparative probability by states. Let be a von Neumann algebra on a Hilbert space and let denote the projections of . A comparative probability (CP) on (or more correctly on is a preorder on satisfying:

with for some .

If , then either or .

If , and are all in and , , then .

A state on is said to implement a on if for , . In this paper, we examine the conditions for implementability of a CP on a general von Neumann algebra (as opposed to only type I factors). A crucial tool used here, as well as in earlier results, is the interval topology generated on by . A will be termed continuous in a given topology on if the interval topology generated by is weaker than the topology induced on by the given topology. We show that uniform continuity of a comparative probability is necessary and sufficient if the von Neumann algebra has no finite direct summand. For implementation by normal states, weak continuity is sufficient and necessary if the von Neumann algebra has no finite direct summand of type I. We arrive at these results by constructing an appropriate additive measure from the CP.

     

Minimal upper bounds of commuting operators

Let be a finite collection of commuting self-adjoint elements of a von Neumann algebra . Then within the (abelian) C*-algebra they generate, these elements have a least upper bound . We show that within , is a minimal upper bound in the sense that if is any self-adjoint element such that for all , then . The corresponding assertion for infinite collections is shown to be false in general, although it does hold in any finite von Neumann algebra. We use this sort of result to show that if are von Neumann algebras, is a faithful conditional expectation, and is positive, then converges in the strong operator topology to the ``spectral order majorant' of in .

     

Compact operators and the geometric structure of -algebras

Given a -algebra and an element , we give necessary and sufficient geometric conditions equivalent to the existence of a representation of so that is a compact or a finite-rank operator. The implications of these criteria on the geometric structure of -algebras are also discussed.

     

Completely distributive CSL algebras with no complements in

Anoussis and Katsoulis have obtained a criterion for the space to have a closed complement in , where is a completely distributive commutative subspace lattice. They show that, for a given , the set of for which this complement exists forms an interval whose endpoints are harmonic conjugates. Also, they establish the existence of a lattice for which has no complement for any . However, they give no specific example. In this note an elementary demonstration of a simple example of this phenomenon is given. From this it follows that for a wide range of lattices , fails to have a complement for any .

     

On the pluricanonical map of threefolds of general type

Let be a smooth minimal threefold of general type and let be an integer . Assume that the image of the pluricanonical map of is a curve. Then a simple computation shows that is necessarily or . When with a numerical condition or when , we obtain two inequalities and , where is the irregularity of and is the Euler characteristic of .

     

Paracompact subspaces in the box product topology

In 1975 E. K. van Douwen showed that if is a family of Hausdorff spaces such that all finite subproducts are paracompact, then for each element of the box product the -product is paracompact. He asked whether this result remains true if one considers uncountable families of spaces. In this paper we prove in particular the following result: Let be an infinite cardinal number, and let be a family of compact Hausdorff spaces. Let be a fixed point. Given a family of open subsets of which covers , there exists an open locally finite in refinement of which covers .

We also prove a slightly weaker version of this theorem for Hausdorff spaces with ``all finite subproducts are paracompact" property. As a corollary we get an affirmative answer to van Douwen's question.

     

Radial images by holomorphic mappings

Let be a nonexceptional Riemann surface, other than the punctured disk. We prove that if is a holomorphic mapping from the unit disk of the complex plane into , then the set of radial images that remain bounded in the Poincaré metric of has Hausdorff dimension at least , the exponent of convergence of . The result is best possible. This is a hyperbolic analog of the result of N. G. Makarov that Bloch functions are bounded on a set of radii of dimension one.

     

Almost linearity of -bi-Lipschitz maps between real Banach spaces

Let and be real Banach spaces. A map between and is called an -bi-Lipschitz map if for all . In this note we show that if is an -bi-Lipschitz map with from onto , then is almost linear. We also show that if is a surjective -bi-Lipschitz map with , then there exists a linear isomorphism such that

where as and .

     

Extending finite group actions from surfaces to handlebodies

We show that every action of a finite dihedral group on a closed orientable surface extends to a 3-dimensional handlebody , with . In the case of a finite abelian group , we give necessary and sufficient conditions for a -action on a surface to extend to a compact -manifold, or, equivalently in this case, to a 3-dimensional handlebody; in particular all (fixed-point) free actions of finite abelian groups extend to handlebodies. This is no longer true for free actions of arbitrary finite groups: we give a procedure which allows us to construct free actions of finite groups on surfaces which do not extend to a handlebody. We also show that the unique Hurwitz action of order of on a surface of genus does not extend to any compact 3-manifold with , thus resolving the only case of Hurwitz actions of type of low order which remained open in an earlier paper (Math. Proc. Cambridge Philos. Soc. 117 (1995), 137--151).

     

Rational nodal curves with no smooth Weierstrass points

Let denote the rational curve with nodes obtained from the Riemann sphere by identifying 0 with and with for , where is a primitive th root of unity. We show that if is even, then has no smooth Weierstrass points, while if is odd, then has smooth Weierstrass points.

     

Operator versions of the Kantorovich inequality

The Operator Kantorovich Inequality

holds for a wide class of operators on a Hilbert space and all operators for which is a partial isometry, being the range projection of

     

Representations of the Gupta-Sidki group

If is an odd prime, then the Gupta-Sidki group is an infinite -generated -group. It is defined in a recursive manner as a particular subgroup of the automorphism group of a regular tree of degree . In this note, we make two observations concerning the irreducible representations of the group algebra with an algebraically closed field. First, when , we obtain a lower bound for the number of irreducible representations of any finite degree . Second, when , we show that if has one nonprincipal irreducible representation, then it has infinitely many. The proofs of these two results use similar techniques and eventually depend on the fact that the commutator subgroup of has a normal subgroup of finite index isomorphic to the direct product of copies of .