9.3 Applications to operator algebras and noncommutative geometry
9 min read•july 30, 2024
bridges the gap between operator algebras and geometry, offering powerful tools for analyzing noncommutative spaces. It extends classical concepts like to C*-algebras, enabling us to study the structure of these abstract objects through a geometric lens.
This framework has far-reaching applications in mathematics and physics. From classifying C*-algebras to probing quantum spaces, KK-theory provides a unified approach to tackling complex problems in noncommutative realms, revolutionizing our understanding of abstract algebraic structures.
KK-Theory for C*-Algebras
Extensions and K-Theory Computations
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KK-theory is a powerful tool for classifying of C*-algebras, which are short exact sequences of the form 0→I→A→A/I→0, where I is an ideal in A
Extensions capture important structural information about C*-algebras and their ideals
Examples of extensions include the Toeplitz extension (0→K→T→C(T)→0) and the extension of the compact operators by the Calkin algebra (0→K→B(H)→B(H)/K→0)
The KK(A,B) is a from the category of C*-algebras to the category of abelian groups, capturing essential information about the structure of extensions
KK(A,B) generalizes the notion of morphisms between C*-algebras, allowing for the study of more general "morphisms" that are not necessarily *-homomorphisms
The bifunctoriality of KK-theory allows for the composition of extensions and the study of their structural properties
The in KK-theory relates the K-theory of the C*-algebras in an extension, enabling the computation of the K-theory of the middle algebra A in terms of the K-theory of I and A/I
The six-term exact sequence is a powerful computational tool, reducing the computation of K-theory for extensions to the computation of K-theory for simpler C*-algebras
Example: The six-term exact sequence for the Toeplitz extension allows for the computation of the K-theory of the Toeplitz algebra T in terms of the K-theory of K and C(T)
The Kasparov product in KK-theory allows for the composition of extensions, crucial for understanding the structure of more complex extensions
The Kasparov product is a bilinear pairing KK(A,B)×KK(B,C)→KK(A,C), generalizing the composition of morphisms in the category of C*-algebras
The associativity of the Kasparov product enables the study of higher-order extensions and their structural properties
The in KK-theory provides a powerful tool for computing the KK-groups in terms of the K-theory and of the C*-algebras involved
The Universal Coefficient Theorem states that there is a short exact sequence 0→Ext(K∗(A),K∗(B))→KK(A,B)→Hom(K∗(A),K∗(B))→0, relating the KK-group to the K-theory and K-homology of A and B
This theorem simplifies the computation of KK-groups, reducing it to the computation of K-theory, K-homology, and the Ext and Hom groups
Kasparov Product and Universal Coefficient Theorem
The Kasparov product in KK-theory is a bilinear pairing KK(A,B)×KK(B,C)→KK(A,C), generalizing the composition of morphisms in the category of C*-algebras
The Kasparov product is constructed using the notion of Kasparov modules, which are generalized morphisms between C*-algebras
The associativity of the Kasparov product allows for the composition of multiple extensions and the study of their structural properties
The Universal Coefficient Theorem in KK-theory relates the KK-group to the K-theory and K-homology of the C*-algebras involved
The theorem states that there is a short exact sequence 0→Ext(K∗(A),K∗(B))→KK(A,B)→Hom(K∗(A),K∗(B))→0
The Ext group captures the obstruction to lifting homomorphisms between K-theory groups to KK-elements
The Hom group consists of the homomorphisms between the K-theory groups that can be lifted to KK-elements
The Universal Coefficient Theorem simplifies the computation of KK-groups by reducing it to the computation of K-theory, K-homology, and the Ext and Hom groups
In many cases, the K-theory and K-homology of C*-algebras are easier to compute than the KK-groups directly
The Ext and Hom groups can often be computed using homological algebra techniques
Examples of applications of the Universal Coefficient Theorem include:
Computing the KK-groups of commutative C*-algebras, which can be expressed in terms of the K-theory and K-homology of their spectrum
Studying the KK-theory of , where the Universal Coefficient Theorem relates the KK-groups of the crossed product to the equivariant K-theory and K-homology of the underlying
KK-Theory of Crossed Products
Crossed Products and Their K-Theory
Crossed product algebras are C*-algebras constructed from a dynamical system, consisting of a C*-algebra A and a group action of a locally compact group G on A
The crossed product algebra A⋊G captures the structure of the dynamical system and its associated operator algebras
Examples of crossed product algebras include the irrational rotation algebra Aθ (crossed product of C(T) by Z) and the group C*-algebra C∗(G) (crossed product of C by G)
KK-theory provides a powerful tool for studying the K-theory of crossed product algebras, particularly in the case of
Amenability of the group G ensures that the full and reduced crossed products coincide, simplifying the study of their K-theory
The and the are key tools in computing the K-theory of crossed products
The Connes-Thom isomorphism in KK-theory relates the K-theory of a crossed product algebra to the K-theory of the original algebra and the group
The isomorphism states that K∗(A⋊αRn)≅K∗+n(A), where α is an action of Rn on A
This result allows for the computation of the K-theory of crossed products by Rn in terms of the K-theory of the original algebra
The Pimsner-Voiculescu exact sequence in KK-theory provides a powerful tool for computing the K-theory of crossed products by free groups
The exact sequence relates the K-theory of A⋊αFn to the K-theory of A and the action of the generators of Fn on K∗(A)
This result has important applications in the study of group C*-algebras and the K-theory of
Connes-Thom Isomorphism and Pimsner-Voiculescu Exact Sequence
The Connes-Thom isomorphism in KK-theory relates the K-theory of a crossed product algebra A⋊αRn to the K-theory of the original algebra A
The isomorphism states that K∗(A⋊αRn)≅K∗+n(A), where α is an action of Rn on A
The shift in the degree of the K-theory corresponds to the dimension of the group Rn
The Connes-Thom isomorphism is a powerful tool for computing the K-theory of crossed products by Rn
It reduces the computation of the K-theory of the crossed product to the computation of the K-theory of the original algebra
Examples of applications include the computation of the K-theory of the irrational rotation algebra Aθ and the Heisenberg C*-algebra
The Pimsner-Voiculescu exact sequence in KK-theory relates the K-theory of a crossed product algebra A⋊αFn to the K-theory of A and the action of the generators of Fn on K∗(A)
The exact sequence has the form ⋯→K∗(A)1−α∗K∗(A)→K∗(A⋊αFn)→K∗−1(A)1−α∗K∗−1(A)→⋯
The map α∗ is the induced action of the generators of Fn on the K-theory of A
The Pimsner-Voiculescu exact sequence is a key tool in computing the K-theory of crossed products by free groups
It reduces the computation of the K-theory of the crossed product to the computation of the K-theory of A and the action of the generators on K∗(A)
Examples of applications include the computation of the K-theory of the noncommutative tori and the free group C*-algebras
KK-Theory and the Baum-Connes Conjecture
The Baum-Connes Conjecture and Group C*-Algebras
The is a central problem in the study of group C*-algebras, relating the K-theory of the reduced group C*-algebra to the of the classifying space for proper actions
The conjecture states that the μ:K∗G(EG)→K∗(Cr∗(G)) is an isomorphism, where EG is the classifying space for proper actions of G
The equivariant K-homology K∗G(EG) captures the topology of the group G and its proper actions
KK-theory plays a crucial role in the formulation and study of the Baum-Connes conjecture
The assembly map μ is defined using the Kasparov product in KK-theory, making KK-theory a natural framework for studying the conjecture
The Baum-Connes conjecture can be reformulated in terms of the existence of a certain element in the KK-group KKG(C0(EG),C)
The Baum-Connes conjecture has important implications for the structure and classification of group C*-algebras
The conjecture implies that the K-theory of the reduced group C*-algebra is completely determined by the equivariant topology of the group
In particular, the conjecture implies that the K-theory of the reduced C*-algebra of a torsion-free discrete group is a homotopy invariant of the group
The Baum-Connes conjecture has been verified for a large class of groups
The conjecture holds for all amenable groups, hyperbolic groups, and certain classes of discrete subgroups of Lie groups
The proof of the conjecture for these classes of groups often involves techniques from KK-theory and geometric group theory
Counterexamples and Complexity
Counterexamples to the Baum-Connes conjecture have been constructed using techniques from KK-theory and
The first counterexamples were constructed by Higson, Lafforgue, and Skandalis using the notion of and the
These counterexamples show that the assembly map μ is not always an isomorphism, even for discrete groups
The existence of counterexamples highlights the complexity and richness of the Baum-Connes conjecture
The conjecture is closely related to deep problems in topology, geometry, and group theory
The study of the Baum-Connes conjecture has led to the development of new techniques in KK-theory and noncommutative geometry
Despite the existence of counterexamples, the Baum-Connes conjecture remains a central problem in the study of group C*-algebras
The conjecture has been verified for a large class of groups, and the techniques developed in the study of the conjecture have found applications in other areas of mathematics
The Baum-Connes conjecture is closely related to other important conjectures in topology and geometry, such as the Novikov conjecture and the Borel conjecture
The study of the Baum-Connes conjecture and its counterexamples is an active area of research in KK-theory and noncommutative geometry
New techniques and approaches are being developed to study the conjecture and its relationship to other problems in mathematics
The resolution of the Baum-Connes conjecture for all groups remains a major open problem in the field
KK-Theory vs Noncommutative Geometry
Noncommutative Geometry and KK-Theory
Noncommutative geometry is a generalization of classical geometry that studies "spaces" described by noncommutative algebras, such as C*-algebras and von Neumann algebras
In noncommutative geometry, the algebra of functions on a space is replaced by a noncommutative algebra, and geometric concepts are reformulated in terms of this algebra
Examples of noncommutative spaces include , noncommutative tori, and the standard model of particle physics
KK-theory provides a natural framework for studying the K-theory and K-homology of noncommutative spaces
The K-theory and K-homology of noncommutative spaces play a crucial role in the development of noncommutative geometry
KK-theory allows for the generalization of important concepts from classical geometry, such as the and the , to the noncommutative setting
The Connes-Skandalis index theorem is a fundamental result in noncommutative geometry that relates the index of an elliptic operator on a noncommutative space to a certain element in the KK-group
The theorem generalizes the classical Atiyah-Singer index theorem to the noncommutative setting
The index of an elliptic operator on a noncommutative space is defined using the notion of , which encode the geometric information of the noncommutative space
KK-theory and noncommutative geometry have important applications in mathematical physics
Noncommutative spaces naturally arise in the study of quantum field theories and string theory
KK-theory provides a framework for studying the geometry and topology of these noncommutative spaces and their physical implications
Applications and Future Directions
KK-theory and noncommutative geometry have found applications in various areas of mathematics and physics
In topology, KK-theory has been used to study the K-theory of C*-algebras arising from foliations and group actions
In geometry, noncommutative geometry has been used to study the geometry of quantum groups and noncommutative manifolds
In physics, noncommutative geometry has been used to study the standard model of particle physics and the geometry of spacetime in quantum gravity theories
The study of KK-theory and noncommutative geometry is an active and rapidly developing area of research
New techniques and approaches are being developed to study the geometry and topology of noncommutative spaces
The relationship between KK-theory, noncommutative geometry, and other areas of mathematics, such as homotopy theory and category theory, is being explored
Some current research directions in KK-theory and noncommutative geometry include:
The study of the Baum-Connes conjecture and its relationship to the geometry of group C*-algebras
The development of a theory of noncommutative vector bundles and their relationship to KK-theory
The study of the geometry of quantum groups and their homogeneous spaces using techniques from
Key Terms to Review (25)
Amenable Groups: Amenable groups are a class of groups that possess a certain property making them compatible with averaging processes, which allows for the existence of invariant means on bounded functions. This property is important in various areas of mathematics, especially in operator algebras and noncommutative geometry, as it connects group theory with functional analysis and the study of representations.
Assembly map: The assembly map is a fundamental concept in K-theory that connects algebraic K-theory with topological K-theory, specifically providing a way to assemble homotopy-theoretic data into algebraic invariants. This map plays a crucial role in understanding how certain topological spaces relate to operator algebras, thereby bridging the gap between geometry and algebra in the context of noncommutative geometry.
Baum-Connes Conjecture: The Baum-Connes Conjecture is a significant hypothesis in the realm of K-Theory and operator algebras that proposes a connection between K-Theory of topological spaces and the K-Theory of C*-algebras associated with groups. This conjecture provides a framework for understanding how K-Theory can be applied to geometric problems, particularly in the context of noncommutative geometry and KK-Theory, helping to establish relationships between various mathematical structures.
Bifunctor: A bifunctor is a mathematical structure that maps two categories to a set, taking an object from each category and producing a new object in a way that respects the morphisms of both categories. This means it can be thought of as a functor that operates in two dimensions, allowing for more complex interactions between categories, particularly in the context of operator algebras and noncommutative geometry.
C*-algebra: A c*-algebra is a type of algebra that consists of complex-valued continuous functions on a topological space, equipped with an involution operation and a norm satisfying specific properties. This structure allows for the study of both algebraic and analytic aspects of operators on Hilbert spaces, making it essential in the context of operator algebras and noncommutative geometry, where these algebras serve as the framework for understanding noncommutative spaces.
Chern character: The Chern character is an important topological invariant associated with complex vector bundles, which provides a connection between K-theory and cohomology. It captures information about the curvature of the vector bundle and its underlying geometric structure, serving as a bridge in various applications, from fixed point theorems to differential geometry.
Connes-Thom Isomorphism: The Connes-Thom isomorphism is a key result in noncommutative geometry that establishes an isomorphism between the K-theory of the space of compact operators on a Hilbert space and the K-theory of the space of continuous functions vanishing at infinity on a non-commutative space. This result connects the topological properties of noncommutative spaces to operator algebras, revealing deep relationships between geometry and functional analysis.
Crossed product algebras: Crossed product algebras are a type of algebra that arise from the interaction between a group and a Banach algebra, particularly in the context of noncommutative geometry. They generalize the concept of group algebras by incorporating additional structure that reflects how the group acts on the algebra, making them essential in studying operator algebras and various applications in mathematical physics.
Equivariant k-homology: Equivariant k-homology is a variant of k-homology that incorporates a group action, allowing for the study of topological spaces with symmetry. This concept extends traditional k-homology by considering how actions from a group, like a Lie group or finite group, influence the structure and properties of spaces in noncommutative geometry and operator algebras. It plays a crucial role in connecting topology with algebraic structures, helping to analyze spaces equipped with symmetries in these advanced mathematical contexts.
Expanders: Expanders are a class of graph structures that exhibit strong connectivity properties and are used to improve the performance of algorithms, particularly in areas like computer science and mathematics. They have the ability to spread information quickly across a network, making them useful in various applications, including operator algebras and noncommutative geometry, where they help understand the interplay between algebraic structures and geometric properties.
Extensions: In the context of K-Theory, extensions refer to the process of extending a mathematical object or structure, such as a vector bundle or an algebra, in a way that incorporates additional features or complexities. This concept is crucial in understanding how simpler objects can be related or transformed into more complex structures, which has significant implications in operator algebras and noncommutative geometry.
Index theorem: The index theorem is a fundamental result in mathematics that connects the analytical properties of differential operators on manifolds to topological features of those manifolds. It reveals how the geometry and topology of a space can influence the solutions to differential equations, particularly concerning elliptic operators, and has profound implications in various mathematical fields, including geometry, topology, and even theoretical physics.
K-homology: K-homology is a cohomological theory that assigns a sequence of abelian groups to a topological space, reflecting the space's structure and properties. It serves as a dual theory to K-theory, allowing for the classification of vector bundles and providing insights into both geometric and analytical aspects of the space.
K-Theory: K-Theory is a branch of mathematics that studies vector bundles and their generalizations through the construction of K-groups, which provide a way to classify and understand vector bundles up to isomorphism. It connects various areas of mathematics, including topology, algebra, and geometry, offering insights into fixed point theorems, quantum field theory, and even string theory.
Kazhdan Property (T): Kazhdan Property (T) is a property of groups that states every unitary representation of the group on a Hilbert space has a non-zero invariant vector if the representation is not nearly trivial. This property implies strong rigidity in the group and plays a crucial role in various areas such as operator algebras and noncommutative geometry. It connects to spectral theory, representation theory, and has implications for the classification of C*-algebras.
Kk-group: The kk-group is a fundamental construction in K-theory that provides a way to classify and analyze morphisms between C*-algebras and their related topological spaces. It plays a vital role in connecting algebraic structures with topological invariants, serving as a bridge between the worlds of operator algebras and noncommutative geometry. This structure helps in studying the homotopy theory of C*-algebras and their representations, making it essential for understanding deeper concepts in these areas.
KK-Theory: KK-Theory is a homological algebra framework that provides a way to classify and study the K-theory of C*-algebras and their relationships. It serves as a bridge between topological K-theory and the algebraic structures of operator algebras, allowing for an understanding of morphisms between these algebras and their associated K-groups. This connection has profound implications in areas such as noncommutative geometry and the study of operator algebras.
N. higson: The term n. higson refers to an influential figure in the field of K-Theory, specifically known for contributions related to KK-Theory and noncommutative geometry. His work has played a significant role in connecting various mathematical concepts, particularly in how K-Theory can be applied to operator algebras and the study of noncommutative spaces. This integration has opened new pathways for research and applications in mathematics and theoretical physics.
Noncommutative geometry: Noncommutative geometry is a branch of mathematics that generalizes geometric concepts to spaces where the coordinates do not commute, often reflecting the structures found in quantum mechanics and operator algebras. This field connects algebraic and geometric methods, allowing for the study of spaces that are 'noncommutative' in nature, such as those arising in quantum physics.
Noncommutative Tori: Noncommutative tori are mathematical objects that arise in the study of noncommutative geometry and operator algebras, specifically characterized by their algebraic structure that resembles that of a torus but does not commute. They can be understood as deformations of the algebra of continuous functions on a standard torus, allowing for a richer structure that captures the behavior of quantum mechanics and other areas where classical intuition fails.
Pimsner-Voiculescu Exact Sequence: The Pimsner-Voiculescu exact sequence is a fundamental tool in K-theory, particularly within the realm of operator algebras and noncommutative geometry. It provides a way to compute the K-theory groups of certain types of C*-algebras by relating them to simpler ones through an exact sequence of groups. This sequence is crucial for understanding how algebraic structures interact and helps reveal deep relationships between different mathematical objects in these fields.
Quantum groups: Quantum groups are mathematical structures that generalize the concept of groups in the context of noncommutative geometry and representation theory. They arise in the study of symmetries of quantum systems and provide a framework for understanding algebraic structures that behave differently from classical groups, especially under deformation and quantization processes. Quantum groups play a significant role in operator algebras, allowing for a new perspective on algebraic objects and their representations.
Six-term exact sequence: A six-term exact sequence is a specific type of sequence in homological algebra that involves six modules and a series of morphisms between them, maintaining the property that the image of one morphism equals the kernel of the next. This structure is significant in algebraic K-theory and helps understand relationships between different spaces or structures, particularly in contexts involving operator algebras and noncommutative geometry.
Spectral triples: Spectral triples are mathematical structures that arise in noncommutative geometry, consisting of an algebra, a Hilbert space, and a self-adjoint operator that encodes geometric information. They provide a framework to study spaces that are not necessarily smooth or traditional, linking algebraic properties to geometric intuition in a way that is particularly useful in the context of operator algebras and noncommutative geometry.
Universal Coefficient Theorem: The Universal Coefficient Theorem provides a powerful way to relate homology and cohomology theories, giving a method to compute cohomology groups using homology groups and certain Ext and Tor functors. This theorem highlights how various algebraic structures can connect topological features, allowing for a deeper understanding of spaces within the realms of algebraic topology, operator algebras, and noncommutative geometry.