On the Primitive Ideals of Nest Algebras
We show that Ringrose's diagonal ideals are primitive ideals, subject to the Continuum Hypothesis, and provide examples of a range of types of primitive ideals. Separately we provide a standard presentation for left ideals in a nest algebras, and explore their properties.
Proc. Edinburgh Math. Soc., 63 (3), 2020, 737-760
A New Class of Maximal Triangular Algebras
We describe a new class of maximal tringular algebras on B(H) which have uniform infinite multiplicity nests.
Proc. Edinburgh Math. Soc., 61 (4), 2018, 909-931
Course Builder Skill Maps
In this paper, we present a new set of features introduced in Course Builder that allow instructors to add skill maps to their courses. We show how skill maps can be used to provide up-to-date and actionable information on students' learning behavior and performance.
Proceedings of the Third (2016) ACM Conference on Learning @ Scale, 2016, 89-92
The Maximal Two-Sided Ideals of Nest Algebras
We give a necessary and sufficient criterion for an operator in a nest algebra to belong to a proper two-sided ideal of that algebra. Using this result, we describe the strong radical of a nest algebra, and give a general description of the maximal two-sided ideals.
J. Operator Theory, 73, (2), 2015, 407-416
The Stable Ideals of a Continuous Nest Algebra, II
The paper presents a unified description of stable ideals of a continuous nest agebra as the kernels of limits of certain diagonal compressions. This description leads to natural formulas for the quotient norm, and criteria for when two limits give rise to the same ideal. Detailed information about sums of ideals is also obtained.
J. Operator Theory, 57 (1), 2007, 67 - 94
Randomized Interval Analysis Checks for the Equivalence of Mathematical Expressions
The paper presents two algorithms that use Interval Analysis to make quick and effective comparisons of mathematical expressions. The algorithms are based on random sampling and admit only one-sided error, in the sense that equivalent expressions are never judged unequal.
Preprint, June 2000. Manuscript 12 pages.
Stable Ideals of Continuous Nest Algebras
The paper characterizes those closed ideals of a continuous nest algebra which are fixed by the automorphism group. This provides a framework in which to organize previously known ideals, and introduce new examples.
J. Operator Theory, 45, (2), 2001, 377-412
eGrade Student Learning Guide
Manual on effective study skills for students using my eGrade on-line assessment system.
(ISBN: 0471395579), John Wiley & Sons, New York, 2000
On-line Assessment Using the Wiley Web-Tests
Describes our development of an on-line homework management system at the University of Nebraska.
Focus on Calculus 16, 1999, 7-8
On-line Gateway Exams in Calculus
A description of the on-line Gateway Exam system which we developed at the University of Nebraska. This was an on-line mastery-learning system to ensure core computational competency in students in Calculus I and II.
Focus on Calculus 14, 1998, 6-7
Principal Bimodules of Nest Algebras
We study weakly closed bimodules of nest algebras, and completely characterize those which are singly generated.
J. Functional Analysis, 157, 1998, 488-533
Factorization of Triangular Operators and Ideals Through the Diagonal
Let D be a fixed diagonal operator. We give necessary and sufficient conditions for an upper triangular operator X to factor through X as ADB, where A and B are upper triangular operators. This leads to a description of the ideal generated by a diagonal operator in the algebra of upper triangular operators.
Proc. Edinburgh Math. Soc., 40, 1997, 227-241
Connectedness of the Invertibles in Certain Nest Algebras
In an earlier paper (The Invertibles are Connected in Infinite Multiplicity Nest Algebras) it was shown that the group of inveribles in a nest algebra is connected, provided the algebra has infinite multiplicity. This paper extends that result to essentially all nest algebras that do not contain a copy of the algebra of infinite upper triangular matrices. It is still unknown whether the invertible group is connected in such an algebra.
Canadian Math. Bulletin. 38, (4), 1995, 412-420
Shuffling of Linear Orders
Canadian Math. Bulletin, 38, (2), 1995, 223-229
Some Representations of TAF Algebras
Pacific J. Math., 167, (1), 1995, 129-161
The Invertibles are Connected in Infinite Multiplicity Nest Algebras
This paper shows that the group of invertibles is connected in nest algebras with infinite multiplicity. (Infinite multiplicity in this context means that the nest has no finite rank atoms.)
Bull. London Math. Soc., 27, 1995, 155-161
Triangular Algebras and Ideals of Nest Algebras
This work is motivated by a construction proposed by Kadison and Singer for building new maximal triangular algebras out of nest algebras and their ideals. This leads to the study of diagonal-disjoint ideals of nest algebras. We show that Larson's ideal is the largest diagonal-disjoint ideal in any nest algebra. From this we can construct the first concrete examples of maximal triangular algebras in B(H) which are not nest algebras. These examples allow us to answer longstanding questions on the structure of maximal triangular algebras. We also introduce and classify new families of maximal triangular algebras.
Memoirs of the Amer. Math. Soc., 562 (117), 1995.
Epimorphisms of Nest Algebras
We attempt a classification of the epimorphisms that can map between two nest algebras. In nearly all cases this classification is achieved. In all cases the epimorphisms are automatically continuous.
International J. Math., 6, (5), 1995, 657-687
The Jacobson Radical of a CSL Algebra
We develop a general framework to characterize the Jacobson radical of a completely distributive CSL algebra, which reduces the problem to a combinatoric problem. We solve this combinatoric problem in two dimensions, hence characterizing the Jacobson radical of width-two CSLs (both the completely distributive case, and the non CD ).
Transactions of the Amer. Math. Soc., 334, (2), 1994, 925-947
The Maximal Ideals of a Nest Algebra
This paper gives a concrete description of the maximal ideals of a continuous nest algebra. The concrete form enables us to describe all ideals in the lattice of ideals generated by the maximal ideals. It is shown that this lattice contains all closed ideals which contain the ideal which is the meet of the maximal ideals. This lattice is shown to be closed under sums and products, which coincide with joins and meets of ideals.
J. Functional Analysis, 124, (1), 1994, 119-134
An Estimate on the Norm of the Product of Infinite Block Operator Matrices
Using methods of infinite Ramsey theory, a lower bound is found for the quantity sup( ||XDY|| ) where X and Y are infinite block operator matrices and the supremum is taken as D ranges over all contractive block diagonal operators.
J. Combinatorial Theory (Series A), 63, (2), 1993, 195-209
Representation and Refinement for Reflexive Operator Algebras with Completely Distributive Commutative Subspace Lattice
Indiana U. Math. J., 40, (2), 1991, 617-638
On the closure of triangular algebras
An example is presented of a maximal triangular subalgebra of B(H) which is not norm-closed. Variants of this example show that transitive maximal triangular subalgebras of B(H) and maximal triangular subalgebras of the II1 factor can also fail to be norm-closed.
Amer. J. Math., 112, 1990, 481-497
Triangular algebras and ideals of nest algebras
This announcement reports on the results of Triangular Algebras and Ideals of Nest Algebras and The Maximal Ideals of a Nest Algebra
Bull. Amer. Math. Soc., 23, (2), 1990, 461-467
On generators of the radical of a nest algebra
Ths paper characterizes those nest algebras for which the Jacobson radical is singly generated. It is shown that the radical is singly generated if and only if it is countably generated.
J. London Math. Soc., 40, (2), 1989, 547-562
A note on quasicentral approximate units in B(H)
Proc. Amer. Math. Soc., 105, (1), 1989, 149-150
Diagonal-disjoint ideals of nest algebras
Ph.D. Thesis, University of London, 1989