7.2 Quotient C*-algebras and homomorphisms

Quotient C*-algebras are crucial tools for simplifying complex structures. By dividing a C*-algebra by a closed two-sided ideal, we create new algebras that inherit key properties. This process helps us study relationships between different C*-algebras.

Homomorphisms between C*-algebras preserve their structure and properties. These mappings allow us to connect different algebras, revealing similarities and differences. Understanding homomorphisms is key to grasping the relationships between various C*-algebras.

Quotient C-algebras

Definition and Properties of Quotient C-algebras

• Quotient C*-algebra A/I forms when dividing a C*-algebra A by a closed two-sided ideal I
• Consists of equivalence classes of elements in A modulo I
• Inherits algebraic operations from the original C*-algebra A
• Preserves C*-algebra structure, including involution and norm properties
• Canonical projection π: A → A/I maps elements of A to their equivalence classes in A/I
• Kernel of the canonical projection equals the ideal I used to form the quotient
• Surjective *-homomorphism between A and A/I established by the canonical projection

Norm and Completeness in Quotient C-algebras

• Norm in quotient algebra A/I defined as $\|[a]\| = \inf\{\|a + x\| : x \in I\}$ for [a] in A/I
• Quotient norm satisfies $\|[a]\| = \inf\{\|b\| : b \in [a]\}$
• Completeness of A/I inherited from the completeness of A
• Quotient C*-algebra A/I becomes a Banach space under the quotient norm
• C*-identity $\|[a^*a]\| = \|[a]\|^2$ preserved in the quotient C*-algebra

Applications and Examples of Quotient C-algebras

• Used to construct new C*-algebras from existing ones
• Simplifies the study of C*-algebras by factoring out ideals
• Calkin algebra C(H) formed as the quotient of B(H) by K(H) (bounded operators modulo compact operators)
• Continuous functions on a closed subset X of a compact Hausdorff space Y obtained as a quotient C*-algebra C(Y)/I, where I consists of functions vanishing on X
• Irrational rotation algebras constructed as quotients of the universal C*-algebra generated by two unitaries

Homomorphisms and Isomorphisms

Properties and Theorems of C-algebra Homomorphisms

• Homomorphism theorem establishes a unique *-isomorphism between A/ker(φ) and im(φ) for any *-homomorphism φ: A → B
• Isomorphism theorem states that if φ: A → B is a surjective *-homomorphism, then A/ker(φ) is *-isomorphic to B
• -homomorphisms between C-algebras preserve algebraic structure, involution, and norm properties
• Continuity of *-homomorphisms follows automatically from their algebraic properties
• Kernel of a -homomorphism always forms a closed two-sided ideal in the domain C-algebra

Image and Range Properties of C-algebra Homomorphisms

• Image of a -homomorphism φ: A → B forms a C-subalgebra of B
• Closure of the image im(φ) equals the image itself due to continuity of *-homomorphisms
• Range of a -homomorphism inherits C-algebra structure from the codomain
• Surjective *-homomorphisms map the unit ball of A onto the unit ball of B
• Injective *-homomorphisms preserve the norm of elements $\|φ(a)\| = \|a\|$ for all a in A

Applications and Examples of C-algebra Homomorphisms

• Used to study relationships between different C*-algebras
• Gelfand transform provides a -isomorphism between a commutative C-algebra and C(X) for some compact Hausdorff space X
• Representations of C*-algebras on Hilbert spaces defined as *-homomorphisms into B(H)
• GNS construction yields a -representation of a C-algebra from a positive linear functional
• Restriction homomorphism between C(X) and C(Y) for Y a closed subset of X given by f ↦ f|Y