14.3 Operator algebras and C*-algebras

Operator algebras and C*-algebras are powerful tools in functional analysis. They extend the concept of algebras to spaces of bounded linear operators, providing a rich framework for studying quantum mechanics and noncommutative geometry.

The Gelfand-Naimark Theorem is a cornerstone, showing that every C*-algebra can be realized as an algebra of operators on a Hilbert space. This connection bridges abstract algebra and concrete operator theory, opening doors to various applications in physics and mathematics.

Operator Algebras and C*-Algebras

Operator and C*-algebras

• Operator algebras are subalgebras of the algebra of bounded linear operators on a Hilbert space that are closed under the operator norm topology
  • The algebra of all bounded linear operators on a Hilbert space and the algebra of all compact operators on a Hilbert space are examples of operator algebras
• C*-algebras are operator algebras that are closed under the involution operation (adjoint) and satisfy the C*-identity: $\|A^*A\| = \|A\|^2$
  • Examples include the algebra of all bounded linear operators on a Hilbert space, the algebra of continuous functions on a compact Hausdorff space with pointwise addition and multiplication and the supremum norm, and the group C*-algebra of a locally compact group

Gelfand-Naimark Theorem

• The Gelfand-Naimark Theorem states that every C*-algebra is -isomorphic to a C-algebra of bounded operators on some Hilbert space
  • The proof involves constructing a Hilbert space using the positive linear functionals on the C*-algebra, defining a representation of the C*-algebra on this Hilbert space using the GNS construction, and showing that this representation is faithful and preserves the involution and norm
• The theorem implies that every abstract C*-algebra can be concretely realized as an operator algebra and that C*-algebras provide a natural framework for studying noncommutative topology and geometry

Spectral properties in C*-algebras

• The spectrum $\sigma(a)$ of an element $a$ in a C*-algebra $A$ is the set of all $\lambda \in \mathbb{C}$ such that $a - \lambda 1$ is not invertible in $A$
  • The spectrum is always a non-empty compact subset of $\mathbb{C}$ and the spectral radius of $a$ equals the radius of the smallest disk containing $\sigma(a)$
• The spectral theorem for normal elements states that for a normal element $a$ in a C*-algebra, there exists a unique projection-valued measure $E$ on the Borel sets of $\sigma(a)$ such that $a = \int_{\sigma(a)} \lambda dE(\lambda)$
  • This implies that normal elements can be diagonalized in a suitable sense and allows for the construction of new elements from normal elements using continuous functions via the functional calculus

Applications of C*-algebras

• In quantum mechanics, observables are represented by self-adjoint elements in a C*-algebra, while states are represented by positive linear functionals of norm 1 on the C*-algebra
  • The Gelfand-Naimark Theorem ensures that these abstract observables can be realized as concrete operators on a Hilbert space
• C*-algebras provide a natural framework for studying quantum statistical mechanics, with the KMS condition characterizing equilibrium states in terms of a condition on the C*-algebra of observables
• In noncommutative geometry, C*-algebras are used to generalize the concepts of topology and geometry to noncommutative spaces
  • The Connes-Kasparov conjecture relates the K-theory of C*-algebras to the geometry of the underlying noncommutative space