K-theory provides powerful tools for classifying and understanding C-algebras. It captures information about projections and unitaries, offering insights into their structure. The K₀ and K₁ groups, along with additional invariants, form the basis for Elliott's classification program.

Applications of K-theory extend to extension theory, index theory, and noncommutative geometry. These areas connect operator algebras to topology and geometry, enabling the study of quantum spaces and singular structures through a noncommutative lens.

Classification and Invariants

Elliott's Classification Program and K-theoretic Invariants

• Elliott's classification program aims to categorize simple, separable, nuclear C-algebras using K-theoretic and tracial invariants
• K-theory provides powerful tools for distinguishing and classifying C-algebras
• K₀ group captures information about projections in a C-algebra
  • Consists of formal differences of equivalence classes of projections
  • Equipped with addition operation derived from direct sum of projections
• K₁ group encodes data about unitaries in a C-algebra
  • Composed of homotopy classes of unitaries in the algebra
  • Addition operation stems from multiplication of unitaries
• Ordered K₀ group incorporates additional order structure
  • Positive cone $K₀^+$ contains classes of non-zero projections
  • State space of $K₀$ relates to tracial states on the C-algebra
• Elliott invariant combines $K₀$, $K₁$, and tracial data
  • Includes ordered group $(K₀(A), K₀^+(A), [1_A])$
  • Incorporates abelian group $K₁(A)$
  • Encompasses tracial state space $T(A)$ and pairing map
• Classification results achieved for various classes of C-algebras (AF algebras, irrational rotation algebras)

Challenges and Refinements in Classification

• Counterexamples to Elliott's conjecture emerged in the early 2000s
  • Rørdam constructed simple, nuclear C-algebras with identical Elliott invariants but different $\mathcal{Z}$-stability properties
  • Toms discovered examples distinguishable by their Cuntz semigroup but not by Elliott invariant
• Refined classification program incorporates additional regularity properties
  • $\mathcal{Z}$-stability: tensorial absorption of Jiang-Su algebra $\mathcal{Z}$
  • Finite nuclear dimension: generalization of covering dimension to noncommutative setting
• Toms-Winter conjecture proposes equivalence of regularity properties for simple, separable, nuclear C-algebras
  • $\mathcal{Z}$-stability
  • Finite nuclear dimension
  • Strict comparison of positive elements
• Recent breakthroughs in classification theory (Gong, Lin, Niu, Winter, Elliott, Tikuisis)
  • Classified large classes of simple, separable, nuclear, $\mathcal{Z}$-stable C-algebras
  • Utilized techniques from K-theory, tracial approximation, and regularity properties

Extension and Index Theory

Extension Theory and Exact Sequences

• Extension theory studies short exact sequences of C-algebras
  • Sequence $0 \rightarrow I \rightarrow E \rightarrow A \rightarrow 0$ with $I$ ideal in $E$
  • Extensions classified by Ext group, measuring obstruction to splitting
• Busby invariant provides alternative description of extensions
  • Maps $A$ to Calkin algebra $Q(I) = M(I)/I$
  • Establishes bijection between extensions and $$$$-homomorphisms $A \rightarrow Q(I)$
• Six-term exact sequence in K-theory connects K-groups of $I$, $E$, and $A$
  • $K_0(I) \rightarrow K_0(E) \rightarrow K_0(A)$
  • $\uparrow \qquad \qquad \qquad \qquad \downarrow$
  • $K_1(A) \leftarrow K_1(E) \leftarrow K_1(I)$
• Boundary maps in six-term sequence encode important topological information
  • Index map $K_1(A) \rightarrow K_0(I)$ relates to Fredholm index
  • Exponential map $K_0(A) \rightarrow K_1(I)$ connects to winding numbers

Index Theory and Applications

• Index theory bridges operator theory and topology
  • Atiyah-Singer Index Theorem relates analytical and topological indices
  • Fredholm index of operators on Hilbert spaces: \text{index}(T) = \dim \ker T - \dim \ker T^
• K-theoretic formulation of index theory
  • Fredholm operators correspond to invertible elements in Calkin algebra
  • Index map in K-theory exact sequence computes Fredholm index
• Applications of index theory in noncommutative geometry
  • Spectral flow: measures spectral changes in families of self-adjoint operators
  • $\eta$-invariant: spectral asymmetry of elliptic operators on odd-dimensional manifolds
• Pimsner-Voiculescu (PV) exact sequence for crossed products
  • Relates K-groups of $C^(G,A,\alpha)$ to those of $A$ for $\mathbb{Z}$-actions
  • Six-term exact sequence involving K_*(A)$ and $K_*(C^(\mathbb{Z},A,\alpha))
  • Crucial tool for computing K-theory of crossed products and Cuntz-Pimsner algebras

Noncommutative Geometry

Applications of Noncommutative Geometry

• Noncommutative geometry extends classical geometric concepts to quantum spaces
  • Replaces commutative algebras of functions with noncommutative C-algebras
  • Provides framework for studying singular spaces and quantum phenomena
• Spectral triples $(A,H,D)$ generalize notion of Riemannian manifold
  • $A$: C-algebra representing noncommutative space
  • $H$: Hilbert space of "spinors"
  • $D$: Dirac operator encoding metric information
• Noncommutative torus: prototype of noncommutative manifold
  • Generated by unitaries $U$ and $V$ satisfying $VU = e^{2\pi i \theta} UV$
  • Irrational rotation algebras for irrational $\theta$
  • K-theory: $K_0(A_\theta) \cong \mathbb{Z}^2$, $K_1(A_\theta) \cong \mathbb{Z}^2$
• Cyclic cohomology as noncommutative analogue of de Rham cohomology
  • Pairs with K-theory via Chern character
  • Provides invariants for noncommutative spaces

Quantum Groups and Operator Algebras

• Quantum groups generalize classical Lie groups to noncommutative setting
  • Hopf algebra structure encodes group-like properties
  • Compact quantum groups admit Haar state and Peter-Weyl decomposition
• Woronowicz's compact quantum groups in C-algebraic framework
  • C-algebra $A$ with coproduct $\Delta: A \rightarrow A \otimes A$
  • Coassociativity and cancellation properties
• Examples of quantum groups and their K-theory
  • Quantum SU(2): deformation of classical SU(2)
  • $K_0(SU_q(2)) \cong \mathbb{Z}$, $K_1(SU_q(2)) \cong \mathbb{Z}$
• Applications to index theory and noncommutative index theorems
  • Baum-Connes conjecture relates K-theory of group C-algebras to equivariant K-homology
  • Connects representation theory, geometry, and operator algebras