Very few Banach spaces E are known for which the lattice of closed ideals in the Banach algebra B(E) of all (bounded, linear) operators on E is fully understood. Indeed, up to now the only such Banach spaces are, up to isomorphism, Hilbert spaces and the sequence spaces c0 and `p for 1 6 p < ∞. We add a new member to this family by showing that there are exactly four closed ideals in B(E) for t...

We investigate a connection between two important classes of Euclidean lattices: well-rounded and ideal lattices. A lattice of full rank in a Euclidean space is called well-rounded if its set of minimal vectors spans the whole space. We consider lattices coming from full rings of integers in number fields, proving that only cyclotomic fields give rise to well-rounded lattices. We further study ...

This paper is part of an eight paper project [14]-[20] studying the arithmetic mean operator ideals in B(H) introduced by Dykema, Figiel, Weiss and Wodzicki in [10]. Every ideal I is generated by diagonal operators with positive decreasing sequences, and its arithmetic mean ideal Ia is generated by diagonal operators with the arithmetic means of those sequences. In this paper we focus on lattic...

The main result in [24] on the structure of commutators showed that arithmetic means play an important role in the study of operator ideals. In this survey we present the notions of arithmetic mean ideals and arithmetic mean at infinity ideals. Then we explore their connections with commutator spaces, traces, elementary operators, lattice and sublattice structure of ideals, arithmetic mean idea...

Lattice games are real-valued functions de...ned on a ...nite lattice L. The basic players are the nonzero join-irreducible elements of the lattice and the coalitions are its elements. If L is the Boolean algebra 2 then we obtain a n-person game. Gilboa and Lehrer introduced the global games, which are lattice games where L = ¦n, the lattice of all partitions of N ordered by re...nement. Faigle...

Inspired by engineering of high-speed switching with quality of service, this paper introduces a new approach to classify finite lattices by the concept of cut-through coding. An n-ary cut-through code of a finite lattice encodes all lattice elements by distinct n-ary strings of a uniform length such that for all j, the initial j encoding symbols of any two elements x and y determine the initia...

We study the topology of the lcm-lattice of edge ideals and derive upper bounds on the Castelnuovo-Mumford regularity of the ideals. In this context it is natural to restrict to the family of graphs with no induced 4-cycle in their complement. Using the above method we obtain sharp upper bounds on the regularity when the complement is a chordal graph, or a cycle, or when the primal graph is cla...

Introduction 5 0.1. What is Commutative Algebra? 5 0.2. Why study Commutative Algebra? 5 0.3. Acknowledgments 7 1. Commutative rings 7 1.1. Fixing terminology 7 1.2. Adjoining elements 10 1.3. Ideals and quotient rings 11 1.4. The monoid of ideals of R 14 1.5. Pushing and pulling ideals 15 1.6. Maximal and prime ideals 16 1.7. Products of rings 17 1.8. A cheatsheet 19 2. Galois Connections 20 2...

Every ring R with identity satisfies the following property: the factor ideals of R (i.e., those ideals I such that I+ J= R and In J= (0) for some ideal J) form a Boolean sublattice of the lattice of all ideals of R. The universal algebraic abstraction of this property is known as Boolean factor congruences (BFC) or as the strict refinement property; more examples of algebras having BFC are lat...