🪡K-Theory Unit 9 – KK–Theory and its Applications

What's KK-Theory?

• KK-theory is a powerful tool in operator algebras and noncommutative geometry that generalizes K-theory
• Provides a unified framework for studying the K-theory of C*-algebras and their bimodules
• Introduces the concept of KK-groups, which are abelian groups associated with pairs of C*-algebras
• KK-groups capture essential information about the structure and relationships between C*-algebras
• Allows for the construction of a category whose objects are C*-algebras and morphisms are elements of KK-groups
• Enables the formulation of a bivariant K-theory that encompasses both K-theory and K-homology
• Facilitates the study of extensions and deformations of C*-algebras through the use of KK-theory invariants

Key Concepts and Definitions

• C*-algebras are complex Banach algebras equipped with an involution satisfying the C*-identity $\|a^*a\| = \|a\|^2$
• Hilbert modules are generalizations of Hilbert spaces that allow for a C*-algebra-valued inner product
• Kasparov modules are triples $(E, \phi, F)$ consisting of a Hilbert module $E$, a *-homomorphism $\phi$, and an operator $F$
  • Kasparov modules are the building blocks of KK-theory and represent abstract elliptic operators
• KK-groups, denoted $KK(A, B)$, are abelian groups associated with pairs of C*-algebras $A$ and $B$
  • Elements of KK-groups are equivalence classes of Kasparov modules
• The Kasparov product is a bilinear map $KK(A, B) \times KK(B, C) \to KK(A, C)$ that generalizes the composition of morphisms
• Bott periodicity in KK-theory states that $KK(A, B) \cong KK(A \otimes \mathcal{C}_0(\mathbb{R}^2), B)$, where $\mathcal{C}_0(\mathbb{R}^2)$ is the C*-algebra of continuous functions vanishing at infinity on $\mathbb{R}^2$
• The universal coefficient theorem relates the KK-groups to the K-theory and K-homology of C*-algebras

Historical Development

• KK-theory was introduced by Gennadi Kasparov in the early 1980s as a generalization of K-theory for C*-algebras
• Kasparov's work built upon earlier developments in operator algebras and K-theory, such as:
  • The Atiyah-Singer index theorem, which relates the analytical index of elliptic operators to topological invariants
  • The Brown-Douglas-Fillmore theory of extensions of C*-algebras and their K-homology
• Kasparov's original motivation was to prove the Novikov conjecture for hyperbolic groups using operator algebraic methods
• The development of KK-theory led to significant advances in the study of C*-algebras and their classification
• KK-theory has since found applications in various areas of mathematics and mathematical physics, including:
  • Index theory and the geometry of elliptic operators
  • Noncommutative geometry and quantum field theory
  • Representation theory and harmonic analysis on locally compact groups

Mathematical Foundations

• KK-theory is built upon the theory of C*-algebras and their representations on Hilbert spaces
• The construction of KK-groups relies on the notion of Kasparov modules, which generalize elliptic operators
• Kasparov modules are triples $(E, \phi, F)$, where:
  • $E$ is a countably generated Hilbert module over a C*-algebra $B$
  • $\phi: A \to \mathcal{L}(E)$ is a -homomorphism from a C-algebra $A$ to the C*-algebra of adjointable operators on $E$
  • $F \in \mathcal{L}(E)$ is an operator satisfying certain conditions related to $\phi$ and the compact operators on $E$
• The KK-groups $KK(A, B)$ are defined as the set of homotopy equivalence classes of Kasparov modules
• The Kasparov product provides a composition operation on KK-groups, making them into a category
• Bott periodicity and the universal coefficient theorem are fundamental results in KK-theory that relate KK-groups to K-theory and K-homology
• The KK-theory framework allows for the formulation of generalized index theorems and the study of extensions of C*-algebras

KK-Theory in Action

• KK-theory provides a powerful toolbox for studying the structure and classification of C*-algebras
• The KK-groups serve as invariants that capture essential information about the relationships between C*-algebras
• KK-theory allows for the construction of maps between K-theory groups of different C*-algebras using the Kasparov product
• The Kasparov product is used to define the notion of KK-equivalence, which provides a way to compare C*-algebras
  • C*-algebras are KK-equivalent if there exist elements in the KK-groups that are invertible under the Kasparov product
• KK-theory is used to formulate and prove index theorems for elliptic operators on noncommutative spaces
  • The noncommutative Atiyah-Singer index theorem relates the index of abstract elliptic operators to K-theory invariants
• KK-theory is a key tool in the classification of C*-algebras, particularly in the study of nuclear C*-algebras
  • The Kirchberg-Phillips classification theorem uses KK-theory to classify certain classes of simple, nuclear C*-algebras
• KK-theory has applications in the study of group actions on C*-algebras and the Baum-Connes conjecture
  • The Baum-Connes conjecture relates the K-theory of group C*-algebras to the equivariant K-homology of classifying spaces

Applications Across Fields

• KK-theory has found significant applications in various areas of mathematics and mathematical physics, beyond operator algebras
• In noncommutative geometry, KK-theory is used to study the geometry of noncommutative spaces
  • Alain Connes' noncommutative geometry program relies heavily on KK-theory to define and study geometric invariants
• KK-theory has connections to index theory and the geometry of elliptic operators on manifolds
  • The Atiyah-Singer index theorem can be formulated and generalized using KK-theory
• In mathematical physics, KK-theory is used to study quantum field theories and their symmetries
  • KK-theory provides a framework for understanding the K-theory of operator algebras arising from quantum field theories
• KK-theory has applications in representation theory and harmonic analysis on locally compact groups
  • The Baum-Connes conjecture, formulated using KK-theory, relates the representation theory of groups to their geometric properties
• KK-theory has been used to study the geometry of group actions on C*-algebras and the structure of crossed product algebras
• In topology, KK-theory has been applied to the study of the Novikov conjecture and the classification of manifolds
  • KK-theory provides tools for proving the Novikov conjecture for certain classes of groups, such as hyperbolic groups

Challenges and Limitations

• Despite its power and wide range of applications, KK-theory also has some challenges and limitations
• The theory can be technically demanding, requiring a deep understanding of operator algebras and functional analysis
• Computations in KK-theory can be difficult, often involving intricate constructions and homotopy arguments
  • Explicit computations of KK-groups and the Kasparov product can be challenging, especially for more complex C*-algebras
• The interpretation of KK-theory results in terms of the original C*-algebras can be nontrivial
  • Translating KK-theoretic statements back to the language of C*-algebras and their structure requires care and expertise
• Some C*-algebras may not be amenable to study using KK-theory, particularly those with poor structural properties
  • C*-algebras that are not nuclear or lack certain regularity properties can be more difficult to analyze with KK-theory
• The theory relies heavily on the existence of suitable Kasparov modules, which may not always be easy to construct
• KK-theory does not provide a complete classification of C*-algebras, and there are still many open problems in this area
  • The classification of non-simple C*-algebras and those without good regularity properties remains a challenge

Future Directions and Open Problems

• KK-theory continues to be an active area of research, with many open problems and potential future directions
• One major open problem is the Baum-Connes conjecture, which seeks to relate the K-theory of group C*-algebras to the equivariant K-homology of classifying spaces
  • The conjecture has been proven for certain classes of groups but remains open in general
• The classification of C*-algebras using KK-theory is an ongoing area of research
  • Extending the Kirchberg-Phillips classification theorem to broader classes of C*-algebras is an important goal
• Developing computational tools and techniques for KK-theory is an active area of research
  • Finding efficient methods for computing KK-groups and the Kasparov product could make the theory more accessible and applicable
• Exploring the connections between KK-theory and other areas of mathematics, such as homotopy theory and category theory, is a promising direction
  • Categorical and homotopical approaches to KK-theory have been developed and continue to be investigated
• Applying KK-theory to new areas of mathematics and mathematical physics is an ongoing effort
  • Potential applications include the study of quantum groups, noncommutative geometry, and topological phases of matter
• Generalizing KK-theory to other settings, such as locally convex algebras or algebras over other fields, is an area of active research
  • Extending the theory to encompass a wider range of algebraic structures could lead to new insights and applications