Electronic Edition Vol. 27, No. 1, 2001

Editors: H. Amann (Zürich), G. Auchmuty (Houston), D. Bao (Houston), H. Brezis (Paris), J. Damon (Chapel Hill), K. Davidson (Waterloo), C. Hagopian (Sacramento), R. M. Hardt (Rice), J. Hausen (Houston), J. A. Johnson (Houston), J. Nagata (Osaka), B. H. Neumann (Canberra), G. Pisier (College Station and Paris), S. W. Semmes (Rice)
Managing Editor: K. Kaiser (Houston)


G. Grätzer, Department of Mathematics, University of Manitoba, Winnipeg, MN R3T 2N2, Canada (gratzer@cc.umanitoba.ca) and E.T. Schmidt, Mathematical Institute of the Technical University of Budapest, Müegyetem RKP. 3, H-1521 Budapest, Hungary (schmidt@math.bme.hu).
Congruence-preserving extensions of finite lattices to semimodular lattices , pp. 1-9.
ABSTRACT. We prove that every finite lattice has a congruence-preserving extension to a finite semimodular lattice.

H. Pat Goeters, Department of Mathematics, Auburn University, Auburn, AL 36849-5310, (goetehp@mail.auburn.edu).
Locally Strongly Homogeneous Rings and Modules, pp. 11-33.
ABSTRACT. An important class of torsion-free abelian groups of finite rank is the class of strongly homogeneous groups. These groups are the torsion-free analogue of Kaplansky's transitive and fully transitive groups.
New developments and variations of the strong homogeneity condition are given.

Heath, Jo, Department of Mathematics, Auburn University, Auburn, Albama, 36844 (heathjw@mail.auburn.edu) and Van C. Nall, Department of Mathematics and Computer Science, University of Richmond, Richmond, Virginia, 23173 (vnall@richmond.edu).
Locally 1-to-1 Maps and 2-to-1 Retractions , pp. 35-44.
ABSTRACT. This paper considers the question of which continua are 2-to-1 retracts of continua.

M. Bonanzinga , Dipartimento di Matematica, Universita di Messina, Contrada Papardo, Salita Sperone, 98166 Messina, Italy ( milena@dipmat.unime.it ) and M. V. Matveev, Department of Mathematics, University of California, Davis, Davis, CA 95616, U.S.A. (misha@matveev.hotmail.com).
Products of star-Lindelöf and related spaces , pp. 45-57.
ABSTRACT. We survey and generalize known results on finite and infinite products of star-Lindelöf spaces and observe that non-trivial infinite box products are never linked-Lindelöf.

A.M. Mohamad, Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, New Zealand (mohamad@math.auckland.ac.nz).
Some Results on Quasi--sigma and theta Spaces , pp. 59-65.
ABSTRACT. In this paper we show that a quasi-G*δ-diagonal plays a central role in metrizability.

E. Santillan Zeron, Department of Mathematics, University of Toronto, M5S 3G3, Canada (eduardo@math.toronto.edu).
Cancellation Laws in Topological Products , pp. 67-74.
ABSTRACT. Given three spaces A, B and H such that AH is homeomorphic to BH, when are A and B homeomorphic? In this paper we answer positively this old question when A and B are subsets of the real line and H is connected.

Adam Bobrowski, Department of Mathematics, University of Houston, Houston TX 77204 (adambob@math.uh.edu).
Notes on the algebraic version of the Hille--Yosida--Feller--Phillips--Miyadera theorem, pp. 75-95.
ABSTRACT. While proving Kisynski's ``algebraic version'' of the generation theorem, we show that the Hille--Yosida--Feller--Phillips--Miyadera semigroup can be constructed as a quotient semigroup of the translation semigroup in a tensor product space. Then we exhibit some applications of the related characterization of the regularity space of the resolvent based on Cohen's factorization theorem. These include ``implemented semigroups'', eventually norm continuous semigroups, and non-continuous semigroups. We also introduce a new formula connecting a resolvent with the related integrated semigroup and the Hille--Yosida--Feller--Phillips--Miyadera semigroup. Finally, we prove an algebraic version of the generation theorem for semigroups with kernels

Jeffrey Fox, Mathematics Department, University of Colorado, Boulder, CO 80309 (jfox@euclid.colorado.edu), Cezary Gajdzinski, 10-437 Olsztyn, ul. Dworcowa 51/7, Poland (cezary@ro.com.pl) and Peter Haskell, Mathematics Dept. Virginia Tech Blacksburg, VA 24061-0123 (haskell@math.vt.edu) .
Homology Chern Characters of Perturbed Dirac Operators, pp. 97-121.
ABSTRACT. This paper studies the index theory of perturbed operators of Dirac type involving unbounded perturbations on complete Riemannian manifolds. The interplay between perturbation growth and manifold geometry determines what large-scale topology of a manifold is revealed by indices and higher indices of these operators.

Luigi Fontana, Dipartimento di Matematica, Via Saldini 50, Universita di Milano, 20133 Milano (Italy) (fontana@vmimat.mat.unimi.it), Steven G. Krantz , Department of Mathematics, Department of Mathematics, Washington University, St. Louis, MO 63130 (USA) (sk@math.wustl.edu) and Marco M. Peloso, Dipartimento di Matematica, Politecnico di Torino, 10129 Torino (Italy) (peloso@calvino.polito.it).
Estimates for the δ-Neumann problem in the Sobolev topology on Z(q) domains , pp. 123-175.
ABSTRACT. The authors formulate the Z(q) condition of Hormander for the inhomogeneous Cauchy-Riemann equations on a domain in multi-dimensional complex space, with the Hodge theory set up in a Sobolev topology. Suitable (and sharp) existence and regularity theorems are proved.

Bruce Barnes, Mathematics Department, University of Oregon, Eugene, OR 97403 (barnes@math.uoregon.edu).
Finite Rank Projections in Algebras Associated with a Bounded Linear Operator , pp. 177-188.
ABSTRACT. Assume that T is a bounded linear operator with commutant {T}'. When does {T}' contain a nonzero finite rank projection? What can we say about T if {T} contains a large family of such projections? When does {T}'' contain a nonzero finite rank operator? These questions and others are studied in this paper.

Douglas Szajda, University of Maryland Institute for Advanced Computer Studies, College Park, MD 20742 (szajda@umiacs.umd.edu),
Absolute Continuity of a Class of Unitary Operators , pp. 189-201.
ABSTRACT. Let F denote the Fourier transform on L2(R), and let T = DfMu, where Df is defined by Df = FMfF-1, f is the boundary value function of a bounded analytic function on the complex upper half plane and is inner, and |u|=1 almost everywhere. This paper gives a partial description of the spectral multiplicity theory of T. It is shown that T is absolutely continuous and is a bilateral shift of infinite multiplicity if f is not a finite Blaschke product. Similar results are obtained for the (isometric) restrictions of T to the invariant subspaces of the form L2(t, infinity). Specifically, these restrictions always have absolutely continuous unitary parts, and shift parts with multiplicity equal to the multiplicity of f.

Jari Taskinen, Department of Mathematics, P.O. Box 4, 00014 University of Helsinki, Finland (Jari.Taskinen@helsinki.fi).
Compact composition operators on general weighted spaces , pp. 203-218.
ABSTRACT. We study analytic composition operators on weighted Hinfty spaces defined on the open unit disc of C. In the unweighted case it is known that for every compact composition operator CΦ : Hinfty ->Hinfty, the symbol Φ maps the open unit disc D strictly inside D. We show that under surprisingly mild assumptions on the weight, there exist compact composition operators on the weighted spaces, the symbols of which do not have the above mentioned property.

L. De Carli, Dip. Mat. e Appl. Univ. Napoli “Federico II”, Napoli, 80126, ITALY (decarli@unina.it) and T. Okaji, Dept. Mathematics Kyoto Univ., Kyoto, 606-8502, Japan.
Unique continuation theorems for Schrödinger operators from a sphere, pp. 219-235.
ABSTRACT. The paper deals with topics mentioned in the title.