# Additional and planned readings

## Planned bivariant prerequisites

The bivariant conclusions that invoke the following programme lessons remain conditional on their stated proof prerequisites. This collection does not supply those planned proofs.

- Kasparov products: planned proof prerequisite.
- Thom isomorphisms and K-orientations in KK: planned proof prerequisite.
- Asymptotic morphisms and E-theory: planned proof prerequisite.
- Wrong-way maps for K-oriented maps: planned proof prerequisite.

## Additional readings

The following lessons occur in later examples, comparison notes or further reading of the supporting texts. They are outside the exact proof sections imported by KT-CP and are not included in this supporting collection.

- Cauchy data from jumps and residues (AN-03).
- Causal kernels, initial data, and short-time geometry (AN-03).
- Changing an interior frame to extend an invertible matrix (AN-03).
- Solving an elliptic system from compatible boundary measurements (AN-03).
- Inductive limits and the K-theory of AF and AT algebras (KT-OPK).
- Hilbert modules and fields on the leaf space (NCG-FOLIATIONS).
- K-theory of the leaf space (NCG-FOLIATIONS).
- Transverse measures of foliations (NCG-FOLIATIONS).
- Tensor positivity and nuclearity (OA-APPROX).
- Atomic representations and measurable lifts (OA-FOUND-REMAINDER).
- Monotone approximation and semicontinuous operators (OA-FOUND-REMAINDER).
- Multipliers and essential extensions (OA-FOUND-REMAINDER).
- Analytic elements and strip arguments (analytic-elements-strips-and-kms).
- Completely positive maps (foundations-of-von-neumann-algebras).
- Integral representations of states (foundations-of-von-neumann-algebras).
- Projections and types of von Neumann algebras (foundations-of-von-neumann-algebras).
- The spectral theorem for bounded self-adjoint operators (foundations-of-von-neumann-algebras).
- The universal enveloping von Neumann algebra of a \(C^*\)-algebra, and \(W^*\)-algebras (foundations-of-von-neumann-algebras).
- Weak topologies: Tychonoff, Banach–Alaoglu, Mazur, bipolars, Krein–Milman and Eberlein–Šmulian (foundations-of-von-neumann-algebras).
- Square-integrable representations and random operators (noncommutative-integration).
- Traces on von Neumann algebras (traces-and-noncommutative-integration).

- Lebesgue’s covering theorem and the dimension of cubes (index-theory-of-elliptic-operators): further reading on the reverse dimension inequality.