🎵C*-algebras Unit 13 – C*–algebras in Noncommutative Geometry

Key Concepts and Definitions

• C*-algebras combine the algebraic structure of an associative algebra over the complex numbers with the topological structure of a Banach space
• Involution operation * satisfies properties similar to complex conjugation and transposition of matrices
• Norm on a C*-algebra is compatible with the involution and multiplication, satisfying the C*-identity: $\|a^*a\| = \|a\|^2$
• Commutative C*-algebras are precisely the algebras of continuous functions vanishing at infinity on locally compact Hausdorff spaces
• States on a C*-algebra are positive linear functionals of norm one, generalizing the notion of probability measures
• Spectrum of an element consists of complex numbers $\lambda$ for which the element $a - \lambda 1$ is not invertible
• Positive elements are self-adjoint elements with non-negative spectrum, forming a cone in the algebra

Historical Context and Development

• C*-algebras emerged in the 1940s through the work of Gelfand and Naimark on normed involutive algebras
• Motivated by the study of group representations and the foundations of quantum mechanics
• Von Neumann algebras, a special class of C*-algebras with additional structure, were introduced earlier by von Neumann in the 1930s
• C*-algebras provide a unified framework for studying various areas of mathematics, including operator theory, harmonic analysis, and topological dynamics
• Development of C*-algebras was influenced by the work of Dixmier, Kadison, Kaplansky, and others in the 1950s and 1960s
• Noncommutative geometry, initiated by Connes in the 1980s, relies heavily on the theory of C*-algebras
• C*-algebras continue to be an active area of research with connections to various branches of mathematics and physics

Fundamental Theorems and Properties

• Gelfand-Naimark Theorem characterizes commutative C*-algebras as algebras of continuous functions on locally compact Hausdorff spaces
• Gelfand-Naimark-Segal (GNS) construction associates a Hilbert space representation to each state on a C*-algebra
• Functional calculus allows continuous functions to be applied to normal elements, generalizing the spectral theorem for self-adjoint operators
• Continuous functional calculus extends to non-normal elements using the holomorphic functional calculus
• Spectral radius formula expresses the spectral radius of an element as the limit of the nth root of the norm of its nth power
• C*-algebras are amenable if and only if they are nuclear, a property related to the existence of finite-dimensional approximations
• Tensor products of C*-algebras can be defined in several ways (minimal, maximal, spatial), leading to different C*-norms on the algebraic tensor product

Algebraic Structure and Operations

• C*-algebras are complex algebras equipped with an involution * satisfying $(ab)^* = b^*a^*$ and $(a^*)^* = a$
• Multiplication is associative and distributes over addition, with the involution being conjugate-linear
• Self-adjoint elements are those satisfying $a^* = a$, forming a real subspace of the algebra
• Unitary elements satisfy $u^*u = uu^* = 1$ and form a group under multiplication
• Projections are self-adjoint idempotents ($p^2 = p = p^*$) and correspond to closed subspaces of a Hilbert space representation
• Positive elements form a cone, closed under addition and multiplication by positive scalars
  • Square root of a positive element is unique and positive
  • Positive elements can be written as $a^*a$ for some $a$ in the algebra
• Ideals in a C*-algebra are self-adjoint (closed under involution) and norm-closed
  • Quotient of a C*-algebra by a closed two-sided ideal is again a C*-algebra

Topological Aspects

• C*-algebras are Banach spaces with respect to the C*-norm, which satisfies $\|ab\| \leq \|a\| \|b\|$ and $\|a^*a\| = \|a\|^2$
• Norm topology makes the involution and multiplication continuous, and the group of unitary elements a topological group
• Spectrum of an element is a non-empty compact subset of the complex plane
  • Spectral radius of an element is the supremum of the absolute values of the elements in its spectrum
• Resolvent set of an element consists of complex numbers for which the element $a - \lambda 1$ is invertible
  • Resolvent function $(\lambda - a)^{-1}$ is analytic on the resolvent set and encodes information about the spectrum
• Continuous functions on the spectrum of a normal element can be continuously extended to the algebra using the functional calculus
• States on a C*-algebra form a convex weak*-compact subset of the dual space, with extreme points corresponding to pure states
• Irreducible representations of a C*-algebra on Hilbert spaces are in bijective correspondence with its pure states via the GNS construction

Representation Theory

• Representations of a C*-algebra are *-homomorphisms into the algebra of bounded operators on a Hilbert space
• Irreducible representations are those with no non-trivial closed invariant subspaces
• GNS construction associates an irreducible representation to each pure state on the algebra
  • Cyclic vector in the GNS space corresponds to the original pure state
• Unitary equivalence of representations is captured by the notion of similarity of their associated pure states
• Direct sums and tensor products of representations can be defined, corresponding to operations on the underlying Hilbert spaces
• Fell topology on the space of irreducible representations generalizes the Jacobson topology on the primitive ideal space
• Decomposition theory aims to express a general representation as a direct integral of irreducible ones
  • Type I C*-algebras are those for which this decomposition is always possible

Applications in Noncommutative Geometry

• Noncommutative geometry studies "spaces" described by noncommutative C*-algebras, generalizing classical notions from geometry and topology
• Commutative C*-algebras correspond to ordinary topological spaces via the Gelfand-Naimark theorem
• K-theory of C*-algebras provides a powerful invariant, generalizing topological K-theory
  • K_0 group is built from projections in matrix algebras over the C*-algebra
  • K_1 group is constructed using unitary elements in matrix algebras over the C*-algebra
• Cyclic cohomology of a C*-algebra is the noncommutative analog of de Rham cohomology, related to K-theory via the Chern character
• Noncommutative differential geometry studies analogs of smooth structures, differential forms, and Riemannian metrics in the setting of C*-algebras
• Spectral triples, consisting of a C*-algebra, a Hilbert space, and an unbounded self-adjoint operator (Dirac operator), encode geometric information
• Applications to mathematical physics, including quantum field theory and string theory, where spacetime is modeled by noncommutative C*-algebras

Advanced Topics and Current Research

• Classification of C*-algebras aims to characterize them up to *-isomorphism using invariants such as K-theory and traces
  • Elliott classification program has been successful for simple, separable, nuclear C*-algebras with certain regularity properties
• Kirchberg algebras are simple, separable, nuclear, and purely infinite C*-algebras, playing a key role in classification
• Crossed product C*-algebras are constructed from dynamical systems, such as group actions on C*-algebras or topological spaces
  • Provide a rich source of examples and are used to study the original dynamical system
• Quantum groups are noncommutative analogs of groups, described by certain C*-algebras with additional structure (coproduct, counit, antipode)
• KK-theory is a bivariant version of K-theory for C*-algebras, introduced by Kasparov, with applications to index theory and classification
• Noncommutative geometry of foliations studies leaf spaces of foliations using associated C*-algebras (foliation algebras, holonomy groupoid algebras)
• Interactions with other areas of mathematics, such as dynamical systems, ergodic theory, and coarse geometry, continue to inspire new developments in C*-algebra theory