Rosenberg–Schochet

Rosenberg–Schochet
Name	Rosenberg–Schochet
Field	Algebraic topology; Operator algebras
Introduced	1987
Authors	Jonathan Rosenberg; Claude Schochet
Related concepts	K-theory; C*-algebras; Baum–Connes conjecture; Atiyah–Singer index theorem

Rosenberg–Schochet

Rosenberg–Schochet is a theorem in the interaction of K-theory and C*-algebras that gives an algebraic tool for computing the K-theory of certain crossed products and extensions. It was introduced by Jonathan Rosenberg and Claude Schochet and situates at the crossroads of techniques from Kasparov, Baum–Connes conjecture, and the Atiyah–Singer index theorem. The result is influential in work on classification of AF algebras, analysis of the Cuntz algebras, and computations related to the Pimsner–Voiculescu exact sequence.

Introduction

The theorem addresses how the K-theory groups of a separable, nuclear C*-algebra relate through a spectral sequence to the sheaf cohomology of the primitive ideal space and to continuous fields studied by Dixmier. It builds on earlier machinery of Brown–Douglas–Fillmore, Rieffel, and Thomsen and is frequently applied where methods of E-theory and KK-theory by Gennadi Kasparov are relevant. The original statement uses the language of continuous fields over a compact Hausdorff space and employs a filtration by ideals similar to techniques used by Jordan in algebraic decomposition problems. The theorem has been cited in subsequent research by Higson, Rørdam, Kirchberg, and Voiculescu.

Statement of the Theorem

Rosenberg–Schochet provides, for a separable, nuclear C*-algebra A with a finite filtration of ideals associated to a finite CW-complex or a finite-dimensional Hausdorff primitive ideal space, a convergent homological spectral sequence

- E^2_{p,q} ≅ H_p(X; K_q(F))

that converges to K_{p+q}(A), where X is the base space of the continuous field and F denotes the fiber algebras. This aligns with earlier spectral sequences of Atiyah–Hirzebruch type and is compatible with the six-term exact sequences from Brown and Pimsner–Voiculescu. The theorem requires hypotheses comparable to nuclearity and UCT-type conditions used in the Universal Coefficient Theorem (UCT) for KK-theory by Rosenberg and Schochet themselves. It organizes the computation of K-theory via the cellular or skeletal filtration of X, making use of machinery from homological algebra in the guise developed by Cartan–Eilenberg.

Proofs and Techniques

Proofs employ a mixture of functional-analytic and homological techniques: the construction of the spectral sequence uses exact couples arising from filtrations by ideals, analogous to constructions in Grothendieck spectral sequence theory and influenced by the use of derived functors in Weibel's exposition. Central to the argument is separability and nuclearity to invoke lifting theorems akin to those used by Choi–Effros and positive approximation techniques pioneered by Kirchberg. The identification of E^2-terms with sheaf or cellular homology utilizes comparison with the Atiyah–Hirzebruch spectral sequence and with computations in Bott periodicity contexts familiar from Bott and Milnor.

Key technical inputs include the six-term exact sequence for K-theory associated to extensions (from Brown–Douglas–Fillmore theory), Kasparov's KK-theory product to manage functoriality, and mapping cone constructions similar to those in Connes's noncommutative geometry program. Excision properties invoked are closely related to the hypotheses in the Universal Coefficient Theorem by Rosenberg and Schochet that links Ext and Hom groups in the computation of KK-groups.

Applications and Examples

The theorem has been used to compute K-theory for various classes of C*-algebras arising from dynamical systems studied by Anantharaman-Delaroche and Giordano–Putnam–Skau. Notable applications include calculations for irrational rotation algebras associated to Rieffel and for crossed products by actions of Z and Z^n where the Pimsner–Voiculescu exact sequence interacts with the Rosenberg–Schochet spectral sequence. It plays a role in classification results for simple, nuclear, separable C*-algebras in the program led by Elliott and in obstructions considered in work by Dadarlat and Loring.

Explicit examples treated with the theorem include continuous fields over spheres and tori where fibers are AF algebras (relating to Bratteli diagrams) and computations for graph algebras studied by Cuntz–Krieger and Raeburn. In noncommutative topology, it assists in relating topological invariants of the primitive spectrum to analytic invariants of the algebra, echoing themes from Connes and Voiculescu.

Variations and Generalizations

Subsequent work generalized the spectral sequence to broader settings: extensions to noncompact or infinite-dimensional base spaces were explored by Dadarlat and Echterhoff, while adaptations to equivariant K-theory involve inputs from Green–Julg and the Baum–Connes conjecture framework advocated by Baum and Connes. Connections with the Universal Coefficient Theorem and with classifications by the Kirchberg–Phillips theorem have been clarified, and variants replacing K-theory by cyclic homology or by L-theory have been investigated in directions suggested by Chern character considerations of Connes–Chern.

Further generalizations incorporate advances in triangulated category methods, derived functor approaches from Neeman, and enhancements via bivariant theories like E-theory by Connes–Higson. These broaden the applicability to groupoid C*-algebras studied by Renault and to deformation quantization contexts influenced by Fedosov and Rieffel.

Category:K-theory