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 operator-algebra chapters below have direct programme reading links. They are not included in this selected download; the complete programme is required to follow those links offline. 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.
Consumed proof sections: Nonempty spectrum · Spectral radius · Characters · 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.
Consumed proof sections: Normal-element norm · Real self-adjoint spectrum · Commutative representation · Spectral permanence · Continuous functional calculus · Contractive homomorphisms · Positive cone · Order and Hilbert-space positivity
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.html- RT-LIE-04
Theorem3.3 (and general-field4.1): finite complete reducibility.
Editable proof source. Companion-course prerequisite; not included in this selected compact-group download.
- RT-LIE-07
Theorems2.1/4.1/5.1 and Section6: root spaces, strings, Killing pairing, positive metric and classical roots.
theorem 2 1 · theorem 4 1 · theorem 5 1 · section 6
Editable proof source. Companion-course prerequisite; not included in this selected compact-group download.
- RT-LIE-14
Theorems2.1/3.1/4.1, Lemma4.3, Proposition5.1 and Section6.4: Verma characters, integral highest weights, multiplicity symmetry, exact weight set and G2 models.
theorem 2 1 · theorem 3 1 · theorem 4 1 · lemma 4 3 · proposition 5 1 · section 6
Editable proof source. Companion-course prerequisite; not included in this selected compact-group download.
- RT-LIE-15
Corollary5.2 and Proposition6.1: all-complex central-character linkage and ordered Casimir scalar.
corollary 5 2 · proposition 6 1
Editable proof source. Companion-course prerequisite; not included in this selected compact-group download.
spatial-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