5.2 The GNS construction revisited

The GNS construction revisits a key concept in C*-algebras, showing how to represent abstract algebras as concrete operators on Hilbert spaces. This process starts with a positive linear functional and builds a representation that captures the algebra's structure.

Understanding the GNS construction is crucial for grasping how abstract algebraic properties translate to concrete mathematical objects. It bridges the gap between algebraic and analytic approaches, forming a cornerstone of C*-algebra theory.

GNS Representation

Construction and Representation

• GNS construction creates a representation of C*-algebra on Hilbert space
• Representation maps elements of C*-algebra to bounded linear operators on Hilbert space
• Process begins with positive linear functional on C*-algebra
• Constructs Hilbert space from algebra elements modulo null space of functional
• Defines representation using left multiplication by algebra elements

Hilbert Space and Completion

• Initial vector space consists of equivalence classes of algebra elements
• Inner product defined using positive linear functional
• Completion of vector space yields full Hilbert space
• Completion process adds limit points to make space topologically complete
• Resulting Hilbert space contains dense subspace of original algebra elements

Cyclic Vector and Inner Product

Cyclic Vector Properties

• Cyclic vector generates entire Hilbert space under action of representation
• Typically denoted as unit vector corresponding to identity element of algebra
• Spans dense subspace when acted upon by all elements of represented algebra
• Crucial for connecting abstract algebra to concrete Hilbert space operators
• Enables reconstruction of positive linear functional from representation

Inner Product and Quotient Space

• Inner product defined on equivalence classes of algebra elements
• Formula for inner product: $\langle [a], [b] \rangle = \phi(b^*a)$ where $\phi$ is positive linear functional
• Well-defined due to properties of positive linear functionals
• Quotient space formed by modding out null space of sesquilinear form
• Resulting space satisfies Hilbert space axioms after completion

Algebraic Structures

Kernel and Null Space

• Kernel of GNS construction consists of elements mapped to zero operator
• Corresponds to null space of positive linear functional
• Characterized by elements $a$ satisfying $\phi(a^*a) = 0$
• Forms two-sided ideal in original C*-algebra
• Quotient by kernel yields faithful representation on Hilbert space

Left Ideal Properties

• Left ideal generated by elements annihilated by positive linear functional
• Consists of linear combinations of elements of form $ba$ where $\phi(a^*a) = 0$
• Closed under left multiplication by arbitrary algebra elements
• Plays key role in defining equivalence classes for GNS construction
• Relationship between left ideal and two-sided kernel important for understanding representation structure