Abstract Weiss The commutator structure of operator ideals (e.g., the class of single commutators, AB-BA, with A, B from two two-sided operator ideals, respectively, along with commutator ideals (their linear span)) and related trace extension problems will be investigated and exploited to determine the Hochschild and cyclic homology for two-sided ideals in B(H), the class of bounded linear operators acting on a separable, infinite-dimensional, complex Hilbert space. A close connection between certain algebraic K-groups has been found for a wide class of two-sided ideals in B(H). Recent important advances have been made in characterizing those ideals which have zero homology (i.e., those commutator ideals which are the ideals themselves and hence have no trace) and in generalizing N. J. Kalton's characterization of the commutator ideal of the Hilbert-Schmidt class, and also the commutator ideal of the trace class (with B(H)), to arbitrary ideals. A natural generalization is a characterization of those ideals whose commutator ideal (with B(H)) contains T for a given compact operator T. A question arising in this context is: Which normed ideals admit a continuous or positive trace? The answer to this question for symmetrically normed ideals with a dense finitely generated subideal is already settled. Techniques under development involve exploiting recently found obstructions preventing certain ideals from having a trace extension beyond certain smaller ideals, and are related to the associated commutator classes and commutator ideals. The study of operator algebras and operator ideals (i.e., special classes of operators each operator of which may be viewed as an infinite matrix (a square, infinite array of numbers)) involves the investigation of operations between their operators, especially addition and multiplication. Operator algebras with these operations play an indispensable roll throughout mathematics, science, business, computer science and many of their applications to other f ields. The well-known commutative law of multiplication is that AB = BA for all numbers A and B, i.e., one always obtains the same result no matter which order one multiplies numbers. Algebras whose elements have this property are called commutative algebras. But most algebras are not commutative and studying the structure of their non-zero commutators, AB-BA, becomes central in understanding the algebras and their relation to the more easily understood commutative algebras, inside which AB-BA is always zero. The famous Heisenberg Uncertainly Principle in physics provides an early example of the relevance of commutators. This proposal is to expand our knowledge of the commutator structure of a number of important operator algebras and to use the new information towards developing techniques to settle some old and new open problems in commutator theory and in operator algebras and ideals.
