Mathematics / research
Operator algebras and noncommutative structure
Study algebras of observables through positivity, representations, states, spectral calculus, and von Neumann structure.
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Module 01 research
Operator algebras and noncommutative structure
Study algebras of observables through positivity, representations, states, spectral calculus, and von Neumann structure.
- 01
Banach algebras and spectrum
Study invertibility in complex normed algebras, derive the Neumann resolvent expansion, and distinguish spectral radius from norm.
31 min - 02
C*-algebras and the C*-identity
Connect the adjoint, multiplication, and norm through the C*-identity and test why seemingly reasonable matrix norms can fail it.
31 min - 03
Positive elements and continuous spectral calculus
Apply continuous functions to normal elements, construct positive square roots, and distinguish scalar monotonicity from operator-order monotonicity.
33 min - 04
States and the Gelfand–Naimark–Segal construction
Turn a positive normalized functional into a Hilbert-space representation and distinguish a faithful state from a faithful representation.
34 min - 05
Commutative C*-algebras and Gelfand duality
Recover a compact space from characters of a commutative unital algebra and track the reversed direction of pullback maps.
32 min - 06
Representations and irreducibility
Identify invariant subspaces through commutants and distinguish faithfulness, cyclicity, purity, and irreducibility in concrete examples.
33 min - 07
Von Neumann algebras and bicommutants
Compare norm, strong, and weak closure, apply the bicommutant theorem under its unital *-algebra hypotheses, and introduce normality.
34 min - 08
Projections, traces, and factor types
Compare projections by partial isometries and use minimality, finiteness, and normal traces to distinguish factor types.
35 min - 09
Tensor products and completely positive maps
Require positivity under matrix amplification, construct a unital Stinespring dilation, and diagnose the transpose map with an entangled positive input.
36 min - 10
Modular theory and KMS states
Construct modular flow from a faithful normal state and verify its analytic equilibrium boundary identity with an explicit sign convention.
38 min