c*-algebras unit 6 study guides

Key Concepts and Definitions

• C*-algebra $A$ is a complex Banach algebra with an involution $*$ satisfying $|a^*a| = |a|^2$ for all $a \in A$
• Representation of a C*-algebra $A$ is a *-homomorphism $\pi: A \to B(H)$ where $B(H)$ is the algebra of bounded linear operators on a Hilbert space $H$
  • Representations preserve the algebraic structure and the involution of the C*-algebra
• Irreducible representation has no non-trivial invariant subspaces
• Cyclic vector $\xi \in H$ for a representation $\pi$ satisfies $\overline{\pi(A)\xi} = H$
• Pure state on a C*-algebra $A$ is a positive linear functional $\varphi: A \to \mathbb{C}$ with $|\varphi| = 1$ and $\varphi(a^*a) = |\varphi(a)|^2$ for all $a \in A$
• GNS construction associates a representation to each state on a C*-algebra
• Unitary equivalence of representations $\pi_1$ and $\pi_2$ means there exists a unitary operator $U$ such that $\pi_1(a) = U^*\pi_2(a)U$ for all $a \in A$

Historical Context and Development

• Early 20th century development of quantum mechanics and operator algebras led to the study of representations of C*-algebras
• Gelfand-Naimark theorem (1943) established that every C*-algebra is isometrically -isomorphic to a C-algebra of bounded operators on a Hilbert space
• Segal (1947) introduced the concept of irreducible representations and their connection to pure states
• Gelfand-Naimark-Segal (GNS) construction (1950s) provided a powerful tool for constructing representations from states
  • GNS construction is a key technique in the study of C*-algebras and their representations
• Mackey's work (1950s) on induced representations and the theory of group representations influenced the development of C*-algebra representation theory
• Dixmier (1960s) made significant contributions to the classification of C*-algebras and their representations

Fundamental Theorems and Results

• Gelfand-Naimark theorem every C*-algebra is isometrically -isomorphic to a C-algebra of bounded operators on a Hilbert space
• GNS construction for each state $\varphi$ on a C*-algebra $A$, there exists a representation $\pi_\varphi$ of $A$ on a Hilbert space $H_\varphi$ with a cyclic vector $\xi_\varphi$ such that $\varphi(a) = \langle \pi_\varphi(a)\xi_\varphi, \xi_\varphi \rangle$ for all $a \in A$
  • Pure states correspond to irreducible representations via the GNS construction
• Schur's lemma an intertwining operator between two irreducible representations is either zero or an isomorphism
• Kadison's transitivity theorem for an irreducible representation $\pi$ of a C*-algebra $A$ on a Hilbert space $H$ and vectors $\xi, \eta \in H$, there exists an element $a \in A$ such that $\pi(a)\xi = \eta$
• Fell's absorption principle an irreducible representation is contained in every non-degenerate representation
• SNAG theorem (Stone-Naimark-Ambrose-Godement) characterizes unitary representations of locally compact groups in terms of representations of their group C*-algebras

Representation Types and Properties

• Irreducible representation has no non-trivial invariant subspaces
  • Fundamental building blocks of representation theory
• Cyclic representation has a cyclic vector $\xi$ such that $\overline{\pi(A)\xi} = H$
• Faithful representation $\pi$ is injective, i.e., $\pi(a) = 0$ implies $a = 0$
• Non-degenerate representation $\pi$ satisfies $\overline{\pi(A)H} = H$
• Direct sum of representations $\pi = \oplus_{i \in I} \pi_i$ acts on the direct sum of the corresponding Hilbert spaces $H = \oplus_{i \in I} H_i$
• Tensor product of representations $\pi_1 \otimes \pi_2$ acts on the tensor product of the corresponding Hilbert spaces $H_1 \otimes H_2$
• Unitary equivalence of representations $\pi_1$ and $\pi_2$ means there exists a unitary operator $U$ such that $\pi_1(a) = U^*\pi_2(a)U$ for all $a \in A$
  • Unitary equivalence is an important equivalence relation in representation theory

Applications in Operator Theory

• Representation theory provides a powerful framework for studying operators on Hilbert spaces
• C*-algebras generated by specific operators (Toeplitz operators, Wiener-Hopf operators) can be analyzed using representation-theoretic techniques
• Representation theory is used in the classification of von Neumann algebras (factors) via the study of their representations
  • Type I, II, and III factors are distinguished by the properties of their representations
• Representations of C*-algebras are used in the construction of wavelets and frames in Hilbert spaces
• Arveson's work on completely positive maps and dilation theory relies heavily on representation theory
• Representation theory is a key tool in the study of quantum information theory and quantum computing

Connections to Other Mathematical Fields

• Representation theory of C*-algebras is closely related to the representation theory of locally compact groups via the group C*-algebra construction
• Representations of C*-algebras are used in the study of dynamical systems and crossed product C*-algebras
  • Covariant representations play a crucial role in this context
• K-theory of C*-algebras, which is an important tool in noncommutative geometry, is defined in terms of representations (projections and unitaries)
• Representations of C*-algebras are used in the construction of noncommutative geometric spaces, such as quantum groups and quantum homogeneous spaces
• Connections to operator space theory representations of C*-algebras give rise to completely bounded maps between operator spaces
• Representations of C*-algebras are used in the study of index theory and K-homology

Advanced Topics and Current Research

• Classification of C*-algebras using K-theoretic invariants (Elliott's classification program)
  • Representation theory plays a role in computing these invariants
• Noncommutative geometry and quantum groups heavily rely on the representation theory of C*-algebras
• Representations of C*-algebras in KK-theory and E-theory, which are bivariant versions of K-theory
• Representations of groupoid C*-algebras and their applications to foliation theory and index theory
• Representations of C*-algebras in the study of quantum information theory, quantum entanglement, and quantum error correction
• Representations of C*-algebras in the study of quantum field theory and algebraic quantum field theory
• Representations of infinite-dimensional Lie groups and their C*-algebras, such as loop groups and diffeomorphism groups

Problem-Solving Techniques and Examples

• Constructing representations using the GNS construction
  • Example given a state $\varphi$ on a matrix algebra $M_n(\mathbb{C})$, construct the GNS representation $\pi_\varphi$
• Proving the irreducibility of representations using Schur's lemma
  • Example show that the standard representation of the Cuntz algebra $O_n$ is irreducible
• Classifying representations up to unitary equivalence
  • Example classify all irreducible representations of the C*-algebra of continuous functions on a compact Hausdorff space
• Computing the kernel and image of representations
  • Example compute the kernel of the left regular representation of a group C*-algebra
• Decomposing representations into irreducible components
  • Example decompose the regular representation of a finite group into irreducible components
• Constructing intertwining operators between representations
  • Example construct an intertwining operator between the left and right regular representations of a group C*-algebra
• Applying representation theory to solve problems in operator theory and quantum mechanics
  • Example use representation theory to compute the spectrum of a specific operator arising in quantum mechanics