Proof foundations and further programme reading
Representations of compact groups
The selected prerequisite chapters refer both to required proof foundations and to further applications. The six full published operator chapters below now have direct reading links. Banach and C*-algebra functional calculus supply required foundations for the general spectral proof of Schur’s lemma. The other operator chapters have their roles stated individually. Additional parent-course routes without an included proof home remain listed by filename.
For Schur’s lemma, a complete compact-averaging proof supplies a route using only the included Haar and compact spectral proofs, independently of the general C*-algebra functional-calculus route.
- Commutative operator algebras: measure, order and duality
Companion chapter on abelian operator algebras; the needed commutative C*-calculus is proved in the linked C*-algebra chapter.
- Banach algebras, spectrum, holomorphic functional calculus and Gelfand theory
Required by the continuous-calculus foundation of the general spectral Schur proof: nonempty spectrum, spectral radius and Gelfand representation.
- Compact and trace-class operators, the predual of B(H), and the operator topologies
Companion chapter on trace ideals, preduals and operator topologies. The compact spectral proof used here is in the included Hilbert-space chapter.
- Order, local units and quotients of C*-algebras
Required by the general spectral Schur proof: Theorem 5.1 provides isometry and commutation on B(H), for arbitrary Hilbert spaces.
HA-LCA-01.htmlreduction-and-the-holonomy-theorem.html- Building representations from positive functionals
Companion C*-algebra GNS and representation theory; not an imported prerequisite to the Schur or bounded spectral proofs. Its component terms remain at the linked source.
RT-FIN-01.htmlRT-FIN-02.htmlRT-FIN-04.htmlRT-LIE-01.htmlRT-LIE-03.htmlRT-LIE-04.htmlRT-LIE-07.htmlRT-LIE-14.htmlRT-LIE-15.htmlspatial-tensor-products-of-von-neumann-algebras.html- The double commutant theorem
Companion operator-algebra closure theorem; not an input to the bounded spectral chapter’s Sections 1–4.
vector-valued-functions-tensor-products-with-lp-and-preduals.html