Spectral theory Banach algebras Definition A character on A is a non-zero multiplicative linear functional A → C . Observation A character ϕ : A → C is automatically continuous, in fact � ϕ � = 1 . Proof: As ϕ is non-zero, we have 0 � = ϕ ( 1 ) = ϕ ( 1 ) 2 , hence ϕ ( 1 ) = 1 . If x were to satisfy | ϕ ( x ) | > � x � , then ϕ ( x ) − x is invertible by the Neumann series trick. However, it lies in the kernel of ϕ , which yields a contradiction. Definition For commutative A , we define its spectrum (aka character space) as ˆ A = { characters ϕ : A → C } . Due to the Banach-Alaoglu theorem, we see that the topology of pointwise convergence turns ˆ A into a compact Hausdorff space. Gábor Szabó (KU Leuven) C*-algebras November 2018 7 / 50
Spectral theory Banach algebras Observation If J ⊂ A is a maximal ideal in a (unital) Banach algebra, then J is closed. If A is commutative, then A/J ∼ = C as a Banach algebra. Proof: Part 1: Since the invertibles are open, there are no non-trivial dense ideals in A . So J is a proper ideal, hence J = J by maximality. Gábor Szabó (KU Leuven) C*-algebras November 2018 8 / 50
Spectral theory Banach algebras Observation If J ⊂ A is a maximal ideal in a (unital) Banach algebra, then J is closed. If A is commutative, then A/J ∼ = C as a Banach algebra. Proof: Part 1: Since the invertibles are open, there are no non-trivial dense ideals in A . So J is a proper ideal, hence J = J by maximality. Part 2: The quotient is a Banach algebra in which every non-zero element is invertible. If it has a non-scalar element x ∈ A/J , then λ − x � = 0 is invertible for all λ ∈ C , which is a contradiction to σ ( x ) � = ∅ . Gábor Szabó (KU Leuven) C*-algebras November 2018 8 / 50
Spectral theory Banach algebras Observation If J ⊂ A is a maximal ideal in a (unital) Banach algebra, then J is closed. If A is commutative, then A/J ∼ = C as a Banach algebra. Proof: Part 1: Since the invertibles are open, there are no non-trivial dense ideals in A . So J is a proper ideal, hence J = J by maximality. Part 2: The quotient is a Banach algebra in which every non-zero element is invertible. If it has a non-scalar element x ∈ A/J , then λ − x � = 0 is invertible for all λ ∈ C , which is a contradiction to σ ( x ) � = ∅ . Observation For commutative A , the assignment ϕ �→ ker ϕ is a 1-1 correspondence between ˆ A and maximal ideals in A . Proof: Clearly the kernel of a character is a maximal ideal as it has codimension 1 in A . Since we have ϕ ( 1 ) = 1 for every ϕ ∈ ˆ A and A = C 1 + ker ϕ , every character is uniquely determined by its kernel. Conversely, if J ⊂ A is a maximal ideal, then A/J ∼ = C , so the quotient map gives us a character. Gábor Szabó (KU Leuven) C*-algebras November 2018 8 / 50
Spectral theory Banach algebras A is still commutative. Theorem Let x ∈ A . Then � � ϕ ( x ) | ϕ ∈ ˆ σ ( x ) = A . Proof: Let λ ∈ C . If λ = ϕ ( x ) , then λ − x ∈ ker( ϕ ) , so λ − x is not invertible. Conversely, if λ − x is not invertible, then it is inside a (proper) maximal ideal. By the previous observation, this means ( λ − x ) ∈ ker ϕ for some ϕ ∈ ˆ A , or λ = ϕ ( x ) . Gábor Szabó (KU Leuven) C*-algebras November 2018 9 / 50
Spectral theory Banach algebras A is still commutative. Theorem Let x ∈ A . Then � � ϕ ( x ) | ϕ ∈ ˆ σ ( x ) = A . Proof: Let λ ∈ C . If λ = ϕ ( x ) , then λ − x ∈ ker( ϕ ) , so λ − x is not invertible. Conversely, if λ − x is not invertible, then it is inside a (proper) maximal ideal. By the previous observation, this means ( λ − x ) ∈ ker ϕ for some ϕ ∈ ˆ A , or λ = ϕ ( x ) . Theorem (Spectral radius formula) For any Banach algebra A and x ∈ A , one has � n � x n � . r ( x ) = lim n →∞ Proof: The “ ≤ ” part follows easily from the above (for A commutative). The “ ≥ ” part is another clever application of complex analysis. Gábor Szabó (KU Leuven) C*-algebras November 2018 9 / 50
Spectral theory Banach algebras For commutative A , consider the usual embedding → A ∗∗ , ι : A ֒ − ι ( x )( f ) = f ( x ) . Since every element of A ∗∗ is a continuous function on ˆ A ⊂ A ∗ in a natural way, we have a restriction mapping A ∗∗ → C ( ˆ A ) . The composition of these two maps yields: Gábor Szabó (KU Leuven) C*-algebras November 2018 10 / 50
Spectral theory Banach algebras For commutative A , consider the usual embedding → A ∗∗ , ι : A ֒ − ι ( x )( f ) = f ( x ) . Since every element of A ∗∗ is a continuous function on ˆ A ⊂ A ∗ in a natural way, we have a restriction mapping A ∗∗ → C ( ˆ A ) . The composition of these two maps yields: Definition (Gelfand transform) The Gelfand transform is the unital homomorphism A → C ( ˆ A ) , x �→ ˆ x given by ˆ x ( ϕ ) = ϕ ( x ) . Gábor Szabó (KU Leuven) C*-algebras November 2018 10 / 50
Spectral theory Banach algebras For commutative A , consider the usual embedding → A ∗∗ , ι : A ֒ − ι ( x )( f ) = f ( x ) . Since every element of A ∗∗ is a continuous function on ˆ A ⊂ A ∗ in a natural way, we have a restriction mapping A ∗∗ → C ( ˆ A ) . The composition of these two maps yields: Definition (Gelfand transform) The Gelfand transform is the unital homomorphism A → C ( ˆ A ) , x �→ ˆ x given by ˆ x ( ϕ ) = ϕ ( x ) . Observation The Gelfand transform is norm-contractive. In fact, for x ∈ A we have x ( ˆ ˆ A ) = σ ( x ) and hence � ˆ x � = r ( x ) ≤ � x � for all x ∈ A . Gábor Szabó (KU Leuven) C*-algebras November 2018 10 / 50
Spectral theory C ∗ -algebras Definition Let A be a unital C ∗ -algebra. An element x ∈ A is 1 normal, if x ∗ x = xx ∗ . 2 self-adjoint, if x = x ∗ . 3 positive, if x = y ∗ y for some y ∈ A . Write x ≥ 0 . 4 a unitary, if x ∗ x = xx ∗ = 1 . Gábor Szabó (KU Leuven) C*-algebras November 2018 11 / 50
� � � � Spectral theory C ∗ -algebras Definition Let A be a unital C ∗ -algebra. An element x ∈ A is 1 normal, if x ∗ x = xx ∗ . positive self-adjoint 2 self-adjoint, if x = x ∗ . 3 positive, if x = y ∗ y for some y ∈ A . normal Write x ≥ 0 . 4 a unitary, if x ∗ x = xx ∗ = 1 . unitary Gábor Szabó (KU Leuven) C*-algebras November 2018 11 / 50
� � � � Spectral theory C ∗ -algebras Definition Let A be a unital C ∗ -algebra. An element x ∈ A is 1 normal, if x ∗ x = xx ∗ . positive self-adjoint 2 self-adjoint, if x = x ∗ . 3 positive, if x = y ∗ y for some y ∈ A . normal Write x ≥ 0 . 4 a unitary, if x ∗ x = xx ∗ = 1 . unitary Observation Any element x ∈ A can be written as x = x 1 + ix 2 for the self-adjoint x 1 = x + x ∗ x 2 = x − x ∗ elements , . 2 2 i Gábor Szabó (KU Leuven) C*-algebras November 2018 11 / 50
� � � � Spectral theory C ∗ -algebras Definition Let A be a unital C ∗ -algebra. An element x ∈ A is 1 normal, if x ∗ x = xx ∗ . positive self-adjoint 2 self-adjoint, if x = x ∗ . 3 positive, if x = y ∗ y for some y ∈ A . normal Write x ≥ 0 . 4 a unitary, if x ∗ x = xx ∗ = 1 . unitary Observation Any element x ∈ A can be written as x = x 1 + ix 2 for the self-adjoint x 1 = x + x ∗ x 2 = x − x ∗ elements , . 2 2 i Observation If x ∈ A is self-adjoint, then it follows for all t ∈ R that � x + it � 2 = � ( x − it )( x + it ) � = � x 2 + t 2 � ≤ � x � 2 + t 2 . Gábor Szabó (KU Leuven) C*-algebras November 2018 11 / 50
Spectral theory C ∗ -algebras Proposition If x ∈ A is self-adjoint, then σ ( x ) ⊂ R . Proof: Step 1: The spectrum of x inside A is the same as the spectrum of x inside its bicommutant A ∩ { x } ′′ . 1 As x is self-adjoint, this is a commutative C ∗ -algebra. So assume A is commutative. 1 This holds in any Banach algebra. Gábor Szabó (KU Leuven) C*-algebras November 2018 12 / 50
Spectral theory C ∗ -algebras Proposition If x ∈ A is self-adjoint, then σ ( x ) ⊂ R . Proof: Step 1: The spectrum of x inside A is the same as the spectrum of x inside its bicommutant A ∩ { x } ′′ . 1 As x is self-adjoint, this is a commutative C ∗ -algebra. So assume A is commutative. Step 2: For ϕ ∈ ˆ A , we get | ϕ ( x ) + it | 2 = | ϕ ( x + it ) 2 | ≤ � x � 2 + t 2 , t ∈ R . But this is only possible for ϕ ( x ) ∈ R , as the left-hand expression will otherwise outgrow the right one as t → ( ± ) ∞ . 2 1 This holds in any Banach algebra. 2 Notice: this works for any ϕ ∈ A ∗ with � ϕ � = � ϕ ( 1 ) � = 1 ! Gábor Szabó (KU Leuven) C*-algebras November 2018 12 / 50
Spectral theory C ∗ -algebras Proposition Let A be a commutative C ∗ -algebra. Then every character ϕ ∈ ˆ A is ∗ -preserving, i.e., it satisfies ϕ ( x ∗ ) = ϕ ( x ) for all x ∈ A . Proof: Write x = x 1 + ix 2 as before and use the above for ϕ ( x ∗ ) = ϕ ( x 1 − ix 2 ) = ϕ ( x 1 ) − iϕ ( x 2 ) = ϕ ( x 1 ) + iϕ ( x 2 ) = ϕ ( x ) . Gábor Szabó (KU Leuven) C*-algebras November 2018 13 / 50
Spectral theory C ∗ -algebras Proposition Let A be a commutative C ∗ -algebra. Then every character ϕ ∈ ˆ A is ∗ -preserving, i.e., it satisfies ϕ ( x ∗ ) = ϕ ( x ) for all x ∈ A . Proof: Write x = x 1 + ix 2 as before and use the above for ϕ ( x ∗ ) = ϕ ( x 1 − ix 2 ) = ϕ ( x 1 ) − iϕ ( x 2 ) = ϕ ( x 1 ) + iϕ ( x 2 ) = ϕ ( x ) . Corollary For a commutative C ∗ -algebra A , the Gelfand transform A → C ( ˆ A ) , ˆ x ( ϕ ) = ϕ ( x ) is a ∗ -homomorphism. Gábor Szabó (KU Leuven) C*-algebras November 2018 13 / 50
Spectral theory C ∗ -algebras Let A be a C ∗ -algebra and B ⊆ A a C ∗ -subalgebra. Observation An element x ∈ A is invertible if and only if x ∗ x and xx ∗ are invertible. Gábor Szabó (KU Leuven) C*-algebras November 2018 14 / 50
Spectral theory C ∗ -algebras Let A be a C ∗ -algebra and B ⊆ A a C ∗ -subalgebra. Observation An element x ∈ A is invertible if and only if x ∗ x and xx ∗ are invertible. Observation An element x ∈ B is invertible in B if and only if it is invertible in A . Proof: By the above we may assume x = x ∗ . We know σ B ( x ) ⊂ R , so n →∞ x n = x + i − → x is a sequence of invertibles in B . We know n n � − 1 implies that x is invertible in B . � x n − x � < � x − 1 Gábor Szabó (KU Leuven) C*-algebras November 2018 14 / 50
Spectral theory C ∗ -algebras Let A be a C ∗ -algebra and B ⊆ A a C ∗ -subalgebra. Observation An element x ∈ A is invertible if and only if x ∗ x and xx ∗ are invertible. Observation An element x ∈ B is invertible in B if and only if it is invertible in A . Proof: By the above we may assume x = x ∗ . We know σ B ( x ) ⊂ R , so n →∞ x n = x + i − → x is a sequence of invertibles in B . We know n n � − 1 implies that x is invertible in B . So if x is not � x n − x � < � x − 1 invertible in B , then � x − 1 n � → ∞ . Since inversion is norm-continuous on the invertibles in any Banach algebra, it follows that x cannot be invertible in A . Gábor Szabó (KU Leuven) C*-algebras November 2018 14 / 50
Spectral theory C ∗ -algebras Let A be a C ∗ -algebra and B ⊆ A a C ∗ -subalgebra. Observation An element x ∈ A is invertible if and only if x ∗ x and xx ∗ are invertible. Observation An element x ∈ B is invertible in B if and only if it is invertible in A . Proof: By the above we may assume x = x ∗ . We know σ B ( x ) ⊂ R , so n →∞ x n = x + i − → x is a sequence of invertibles in B . We know n n � − 1 implies that x is invertible in B . So if x is not � x n − x � < � x − 1 invertible in B , then � x − 1 n � → ∞ . Since inversion is norm-continuous on the invertibles in any Banach algebra, it follows that x cannot be invertible in A . Corollary We have σ B ( x ) = σ A ( x ) for all x ∈ B . 3 3 This often fails for inclusions of Banach algebras! Gábor Szabó (KU Leuven) C*-algebras November 2018 14 / 50
Spectral theory C ∗ -algebras Let A be a C ∗ -algebra. Observation x ∈ A is normal if and only if C ∗ ( x, 1 ) ⊆ A is commutative. In this case the spectrum of C ∗ ( x, 1 ) is homeomorphic to σ ( x ) . Gábor Szabó (KU Leuven) C*-algebras November 2018 15 / 50
Spectral theory C ∗ -algebras Let A be a C ∗ -algebra. Observation x ∈ A is normal if and only if C ∗ ( x, 1 ) ⊆ A is commutative. In this case the spectrum of C ∗ ( x, 1 ) is homeomorphic to σ ( x ) . Proposition For a normal element x ∈ A , we have r ( x ) = � x � . Proof: Observe from the C ∗ -identity that � x � 4 = � x ∗ x � 2 = � x ∗ xx ∗ x � = � ( x 2 ) ∗ x 2 � = � x 2 � 2 . By induction, we get � x 2 n � = � x � 2 n . By the spectral radius formula, we have � 2 n � x 2 n � = � x � . r ( x ) = lim n →∞ Gábor Szabó (KU Leuven) C*-algebras November 2018 15 / 50
Spectral theory C ∗ -algebras Let A be a C ∗ -algebra. Observation x ∈ A is normal if and only if C ∗ ( x, 1 ) ⊆ A is commutative. In this case the spectrum of C ∗ ( x, 1 ) is homeomorphic to σ ( x ) . Proposition For a normal element x ∈ A , we have r ( x ) = � x � . Proof: Observe from the C ∗ -identity that � x � 4 = � x ∗ x � 2 = � x ∗ xx ∗ x � = � ( x 2 ) ∗ x 2 � = � x 2 � 2 . By induction, we get � x 2 n � = � x � 2 n . By the spectral radius formula, we have � 2 n � x 2 n � = � x � . r ( x ) = lim n →∞ Corollary � � x ∗ x � = � r ( x ∗ x ) . For all x ∈ A , we have � x � = Gábor Szabó (KU Leuven) C*-algebras November 2018 15 / 50
Spectral theory Gelfand-Naimark theorem Theorem (Gelfand–Naimark) For a commutative C ∗ -algebra A , the Gelfand transform A → C ( ˆ A ) , ˆ x ( ϕ ) = ϕ ( x ) is an isometric ∗ -isomorphism. Proof: We have already seen that it is a ∗ -homomorphism. Gábor Szabó (KU Leuven) C*-algebras November 2018 16 / 50
Spectral theory Gelfand-Naimark theorem Theorem (Gelfand–Naimark) For a commutative C ∗ -algebra A , the Gelfand transform A → C ( ˆ A ) , ˆ x ( ϕ ) = ϕ ( x ) is an isometric ∗ -isomorphism. Proof: We have already seen that it is a ∗ -homomorphism. As every element x ∈ A is normal, we have � x � = r ( x ) = � ˆ x � , hence the Gelfand transform is isometric. Gábor Szabó (KU Leuven) C*-algebras November 2018 16 / 50
Spectral theory Gelfand-Naimark theorem Theorem (Gelfand–Naimark) For a commutative C ∗ -algebra A , the Gelfand transform A → C ( ˆ A ) , ˆ x ( ϕ ) = ϕ ( x ) is an isometric ∗ -isomorphism. Proof: We have already seen that it is a ∗ -homomorphism. As every element x ∈ A is normal, we have � x � = r ( x ) = � ˆ x � , hence the Gelfand transform is isometric. For surjectivity, observe that the image of A in C ( ˆ A ) is a closed unital self-adjoint subalgebra, and which separates points. By the Stone–Weierstrass theorem , it follows that it is all of C ( ˆ A ) . Gábor Szabó (KU Leuven) C*-algebras November 2018 16 / 50
Spectral theory Functional calculus Observation Let x ∈ A be a normal element in a C ∗ -algebra. Let A x = C ∗ ( x, 1 ) be the commutative C ∗ -subalgebra generated by x . Then ˆ A x ∼ = σ ( x ) by observing that for every λ ∈ σ ( x ) there is a unique ϕ ∈ ˆ A x with ϕ ( x ) = λ . x ∈ C ( ˆ Under this identification ˆ A x ) becomes the identity map on σ ( x ) . Gábor Szabó (KU Leuven) C*-algebras November 2018 17 / 50
Spectral theory Functional calculus Observation Let x ∈ A be a normal element in a C ∗ -algebra. Let A x = C ∗ ( x, 1 ) be the commutative C ∗ -subalgebra generated by x . Then ˆ A x ∼ = σ ( x ) by observing that for every λ ∈ σ ( x ) there is a unique ϕ ∈ ˆ A x with ϕ ( x ) = λ . x ∈ C ( ˆ Under this identification ˆ A x ) becomes the identity map on σ ( x ) . Theorem (functional calculus) Let x ∈ A be a normal element in a (unital) C ∗ -algebra. There exists a unique (isometric) ∗ -homomorphism C ( σ ( x )) → A, f �→ f ( x ) that sends id σ ( x ) to x . Proof: Take the inverse of the Gelfand transform A x ) ∼ A x → C ( ˆ = C ( σ ( x )) . Gábor Szabó (KU Leuven) C*-algebras November 2018 17 / 50
Spectral theory Applications Theorem An element x ∈ A is positive if and only if x is normal and σ ( x ) ⊆ R ≥ 0 . Proof: If the latter is true, then y = √ x satisfies y ∗ y = y 2 = x . So x is positive. The “only if” part is much trickier. Gábor Szabó (KU Leuven) C*-algebras November 2018 18 / 50
Spectral theory Applications Theorem An element x ∈ A is positive if and only if x is normal and σ ( x ) ⊆ R ≥ 0 . Proof: If the latter is true, then y = √ x satisfies y ∗ y = y 2 = x . So x is positive. The “only if” part is much trickier. Observation x = x ∗ ∈ A is positive if and only if � ≤ r for some (or all) r ≥ � x � . � � r − x � Gábor Szabó (KU Leuven) C*-algebras November 2018 18 / 50
Spectral theory Applications Theorem An element x ∈ A is positive if and only if x is normal and σ ( x ) ⊆ R ≥ 0 . Proof: If the latter is true, then y = √ x satisfies y ∗ y = y 2 = x . So x is positive. The “only if” part is much trickier. Observation x = x ∗ ∈ A is positive if and only if � ≤ r for some (or all) r ≥ � x � . � r − x � � Corollary For a, b ∈ A positive, the sum a + b is positive. Proof: Apply the triangle inequality: We have � a + b � ≤ � a � + � b � and � ≤ � + � ≤ � a � + � b � . � � � � � � � ( � a � + � b � ) − ( a + b ) � � a � − a � � b � − b Gábor Szabó (KU Leuven) C*-algebras November 2018 18 / 50
Spectral theory Applications Theorem Every algebraic (unital) ∗ -homomorphism ψ : A → B between (unital) C ∗ -algebras is contractive, and hence continuous. 4 Proof: It is clear that σ ( ψ ( x )) ⊆ σ ( x ) for all x ∈ A . By the spectral characterization of the norm, it follows that � ψ ( x ) � 2 = r ( ψ ( x ∗ x )) ≤ r ( x ∗ x ) = � x � 2 . 4 This generalizes to the non-unital case as well! Gábor Szabó (KU Leuven) C*-algebras November 2018 19 / 50
Spectral theory Applications Theorem Every algebraic (unital) ∗ -homomorphism ψ : A → B between (unital) C ∗ -algebras is contractive, and hence continuous. 4 Proof: It is clear that σ ( ψ ( x )) ⊆ σ ( x ) for all x ∈ A . By the spectral characterization of the norm, it follows that � ψ ( x ) � 2 = r ( ψ ( x ∗ x )) ≤ r ( x ∗ x ) = � x � 2 . Observation For x ∈ A normal and f ∈ C ( σ ( x )) , we have ψ ( f ( x )) = f ( ψ ( x )) . Proof: Clear for f ∈ { *-polynomials } . The general case follows by continuity of the assignments [ f �→ f ( x )] and [ f �→ f ( ψ ( x ))] and the Weierstrass approximation theorem. 4 This generalizes to the non-unital case as well! Gábor Szabó (KU Leuven) C*-algebras November 2018 19 / 50
Spectral theory Applications Theorem Every injective ∗ -homomorphism ψ : A → B is isometric. Proof: By the C ∗ -identity, it suffices to show � ψ ( x ) � = � x � for positive x ∈ A . Suppose we have � ψ ( x ) � < � x � . Choose a non-zero continuous function f : σ ( x ) → R ≥ 0 with f ( λ ) = 0 for λ ≤ � ψ ( x ) � . Gábor Szabó (KU Leuven) C*-algebras November 2018 20 / 50
Spectral theory Applications Theorem Every injective ∗ -homomorphism ψ : A → B is isometric. Proof: By the C ∗ -identity, it suffices to show � ψ ( x ) � = � x � for positive x ∈ A . Suppose we have � ψ ( x ) � < � x � . Choose a non-zero continuous function f : σ ( x ) → R ≥ 0 with f ( λ ) = 0 for λ ≤ � ψ ( x ) � . 1 f � ϕ ( x ) � � x � Gábor Szabó (KU Leuven) C*-algebras November 2018 20 / 50
Spectral theory Applications Theorem Every injective ∗ -homomorphism ψ : A → B is isometric. Proof: By the C ∗ -identity, it suffices to show � ψ ( x ) � = � x � for positive x ∈ A . Suppose we have � ψ ( x ) � < � x � . Choose a non-zero continuous function f : σ ( x ) → R ≥ 0 with f ( λ ) = 0 for λ ≤ � ψ ( x ) � . Then f ( x ) � = 0 , but 1 ψ ( f ( x )) = f ( ψ ( x )) = 0 , f which means ψ is not injective. � ϕ ( x ) � � x � Gábor Szabó (KU Leuven) C*-algebras November 2018 20 / 50
Representation theory Definition Let A be a C ∗ -algebra. A representation (on a Hilbert space H ) is a ∗ -homomorphism π : A → B ( H ) . Gábor Szabó (KU Leuven) C*-algebras November 2018 21 / 50
Representation theory Definition Let A be a C ∗ -algebra. A representation (on a Hilbert space H ) is a ∗ -homomorphism π : A → B ( H ) . It is said to be 1 faithful, if it is injective. Gábor Szabó (KU Leuven) C*-algebras November 2018 21 / 50
Representation theory Definition Let A be a C ∗ -algebra. A representation (on a Hilbert space H ) is a ∗ -homomorphism π : A → B ( H ) . It is said to be 1 faithful, if it is injective. 2 non-degenerate if span π ( A ) H = H . Gábor Szabó (KU Leuven) C*-algebras November 2018 21 / 50
Representation theory Definition Let A be a C ∗ -algebra. A representation (on a Hilbert space H ) is a ∗ -homomorphism π : A → B ( H ) . It is said to be 1 faithful, if it is injective. 2 non-degenerate if span π ( A ) H = H . 3 cyclic, if there exists a vector ξ ∈ H with π ( A ) ξ = H . For � ξ � = 1 , we say that ( π, H , ξ ) is a cyclic representation. Gábor Szabó (KU Leuven) C*-algebras November 2018 21 / 50
Representation theory Definition Let A be a C ∗ -algebra. A representation (on a Hilbert space H ) is a ∗ -homomorphism π : A → B ( H ) . It is said to be 1 faithful, if it is injective. 2 non-degenerate if span π ( A ) H = H . 3 cyclic, if there exists a vector ξ ∈ H with π ( A ) ξ = H . For � ξ � = 1 , we say that ( π, H , ξ ) is a cyclic representation. 4 irreducible, if π ( A ) ξ = H for all 0 � = ξ ∈ H . Gábor Szabó (KU Leuven) C*-algebras November 2018 21 / 50
Representation theory Positive functionals Let A be a C ∗ -algebra. Definition A functional ϕ : A → C is called positive, if ϕ ( a ) ≥ 0 whenever a ≥ 0 . Gábor Szabó (KU Leuven) C*-algebras November 2018 22 / 50
Representation theory Positive functionals Let A be a C ∗ -algebra. Definition A functional ϕ : A → C is called positive, if ϕ ( a ) ≥ 0 whenever a ≥ 0 . Observation Every positive functional ϕ : A → C is continuous. Proof: Suppose not. By functional calculus, every element x ∈ A can be written as a linear combination of at most four positive elements x = ( x + 1 − x − 1 ) + i ( x + 2 − x − 2 ) with norms � x + 1 � , � x − 1 � , � x + 2 � , � x − 2 � ≤ � x � . So ϕ is unbounded on the positive elements. Gábor Szabó (KU Leuven) C*-algebras November 2018 22 / 50
Representation theory Positive functionals Let A be a C ∗ -algebra. Definition A functional ϕ : A → C is called positive, if ϕ ( a ) ≥ 0 whenever a ≥ 0 . Observation Every positive functional ϕ : A → C is continuous. Proof: Suppose not. By functional calculus, every element x ∈ A can be written as a linear combination of at most four positive elements x = ( x + 1 − x − 1 ) + i ( x + 2 − x − 2 ) with norms � x + 1 � , � x − 1 � , � x + 2 � , � x − 2 � ≤ � x � . So ϕ is unbounded on the positive elements. Given n ≥ 1 , one may choose a n ≥ 0 with � a n � = 1 and ϕ ( a n ) ≥ n 2 n . Then a = � ∞ n =1 2 − n a n is a positive element in A . By positivity of ϕ , we have ϕ ( a ) ≥ ϕ (2 − n a n ) ≥ n for all n , a contradiction. Gábor Szabó (KU Leuven) C*-algebras November 2018 22 / 50
Representation theory Positive functionals Observation For a positive functional ϕ : A → C , we have ϕ ( x ∗ ) = ϕ ( x ) . Gábor Szabó (KU Leuven) C*-algebras November 2018 23 / 50
Representation theory Positive functionals Observation For a positive functional ϕ : A → C , we have ϕ ( x ∗ ) = ϕ ( x ) . Corollary For a positive functional ϕ , the assignment ( x, y ) �→ ϕ ( y ∗ x ) defines a positive semi-definite, anti-symmetric, sesqui-linear form. In particular, it is subject to the Cauchy–Schwarz inequality | ϕ ( y ∗ x ) | 2 ≤ ϕ ( x ∗ x ) ϕ ( y ∗ y ) . Gábor Szabó (KU Leuven) C*-algebras November 2018 23 / 50
Representation theory Positive functionals Theorem Let A be a unital C ∗ -algebra. A linear functional ϕ : A → C is positive if and only if � ϕ � = ϕ ( 1 ) . Proof: For the “only if” part, observe for � y � ≤ 1 that | ϕ ( y ) | 2 = | ϕ ( 1 y ) | 2 ≤ ϕ ( 1 ) ϕ ( y ∗ y ) ≤ ϕ ( 1 ) � ϕ � . Taking the supremum over all such y yields � ϕ � = ϕ ( 1 ) . Gábor Szabó (KU Leuven) C*-algebras November 2018 24 / 50
Representation theory Positive functionals Theorem Let A be a unital C ∗ -algebra. A linear functional ϕ : A → C is positive if and only if � ϕ � = ϕ ( 1 ) . Proof: For the “only if” part, observe for � y � ≤ 1 that | ϕ ( y ) | 2 = | ϕ ( 1 y ) | 2 ≤ ϕ ( 1 ) ϕ ( y ∗ y ) ≤ ϕ ( 1 ) � ϕ � . Taking the supremum over all such y yields � ϕ � = ϕ ( 1 ) . For the “if” part, suppose ϕ ( 1 ) = 1 = � ϕ � . Let a ≥ 0 with � a � ≤ 1 . Repeating an argument we have used for characters, we know ϕ ( a ) ∈ R . Gábor Szabó (KU Leuven) C*-algebras November 2018 24 / 50
Representation theory Positive functionals Theorem Let A be a unital C ∗ -algebra. A linear functional ϕ : A → C is positive if and only if � ϕ � = ϕ ( 1 ) . Proof: For the “only if” part, observe for � y � ≤ 1 that | ϕ ( y ) | 2 = | ϕ ( 1 y ) | 2 ≤ ϕ ( 1 ) ϕ ( y ∗ y ) ≤ ϕ ( 1 ) � ϕ � . Taking the supremum over all such y yields � ϕ � = ϕ ( 1 ) . For the “if” part, suppose ϕ ( 1 ) = 1 = � ϕ � . Let a ≥ 0 with � a � ≤ 1 . Repeating an argument we have used for characters, we know ϕ ( a ) ∈ R . We have � 1 − a � ≤ 1 . If ϕ ( a ) < 0 , then it would necessarily follow that ϕ ( 1 − a ) = 1 − ϕ ( a ) > 1 , which contradicts � ϕ � = 1 . Hence ϕ ( a ) ≥ 0 . Since a was arbitrary, it follows that ϕ is positive. Gábor Szabó (KU Leuven) C*-algebras November 2018 24 / 50
Representation theory Positive functionals Corollary For an inclusion of (unital) C ∗ -algebras B ⊆ A , every positive functional on B extends to a positive functional on A . Proof: Use Hahn–Banach and the previous slide. Gábor Szabó (KU Leuven) C*-algebras November 2018 25 / 50
Representation theory Positive functionals Corollary For an inclusion of (unital) C ∗ -algebras B ⊆ A , every positive functional on B extends to a positive functional on A . Proof: Use Hahn–Banach and the previous slide. Definition A state on a C ∗ -algebra is a positive functional with norm one. Gábor Szabó (KU Leuven) C*-algebras November 2018 25 / 50
Representation theory Positive functionals Corollary For an inclusion of (unital) C ∗ -algebras B ⊆ A , every positive functional on B extends to a positive functional on A . Proof: Use Hahn–Banach and the previous slide. Definition A state on a C ∗ -algebra is a positive functional with norm one. Observation For x ∈ A normal, there is a state ϕ with � x � = | ϕ ( x ) | . Proof: Pick λ 0 ∈ σ ( x ) with | λ 0 | = � x � . We know A x = C ∗ ( x, 1 ) ∼ = C ( σ ( x )) so that x �→ id . The evaluation map f �→ f ( λ 0 ) corresponds to a state on A x with the desired property. Extend it to a state ϕ on A . Gábor Szabó (KU Leuven) C*-algebras November 2018 25 / 50
Representation theory Interlude: Order on self-adjoints Let A be a C ∗ -algebra. Definition For self-adjoint elements a, b ∈ A , write a ≤ b if b − a is positive. Gábor Szabó (KU Leuven) C*-algebras November 2018 26 / 50
Representation theory Interlude: Order on self-adjoints Let A be a C ∗ -algebra. Definition For self-adjoint elements a, b ∈ A , write a ≤ b if b − a is positive. Observation The order “ ≤ ” is compatible with sums. For all self-adjoint a ∈ A , we have a ≤ � a � . If a ≤ b and x ∈ A is any element, then x ∗ ax ≤ x ∗ bx . Gábor Szabó (KU Leuven) C*-algebras November 2018 26 / 50
Representation theory Interlude: Order on self-adjoints Let A be a C ∗ -algebra. Definition For self-adjoint elements a, b ∈ A , write a ≤ b if b − a is positive. Observation The order “ ≤ ” is compatible with sums. For all self-adjoint a ∈ A , we have a ≤ � a � . If a ≤ b and x ∈ A is any element, then x ∗ ax ≤ x ∗ bx . For proving the last part, write b − a = c ∗ c . Then x ∗ bx − x ∗ ax = x ∗ ( b − a ) x = x ∗ c ∗ cx = ( cx ) ∗ cx ≥ 0 . Gábor Szabó (KU Leuven) C*-algebras November 2018 26 / 50
Representation theory States and representations Given a state ϕ on A , we have observed that ( x, y ) �→ ϕ ( y ∗ x ) forms a positive semi-definite, anti-symmetric, sesqui-linear form. Observation For all a, x ∈ A , we have ϕ ( x ∗ a ∗ ax ) ≤ � a � 2 ϕ ( x ∗ x ) . The null space N ϕ = { x ∈ A | ϕ ( x ∗ x ) = 0 } is a closed left ideal in A . Gábor Szabó (KU Leuven) C*-algebras November 2018 27 / 50
Representation theory States and representations Given a state ϕ on A , we have observed that ( x, y ) �→ ϕ ( y ∗ x ) forms a positive semi-definite, anti-symmetric, sesqui-linear form. Observation For all a, x ∈ A , we have ϕ ( x ∗ a ∗ ax ) ≤ � a � 2 ϕ ( x ∗ x ) . The null space N ϕ = { x ∈ A | ϕ ( x ∗ x ) = 0 } is a closed left ideal in A . Observation The quotient H ϕ = A/N ϕ carries the inner product � [ x ] | [ y ] � ϕ = ϕ ( y ∗ x ) , and the left A -module structure satisfies � [ ax ] � ϕ ≤ � a � · � [ x ] � ϕ for all a, x ∈ A . Gábor Szabó (KU Leuven) C*-algebras November 2018 27 / 50
Representation theory GNS construction Definition (Gelfand–Naimark–Segal construction) For a state ϕ on a C ∗ -algebra A , let H ϕ be the Hilbert space completion �·� ϕ . Then H ϕ carries a unique left A -module structure which H ϕ = H ϕ extends the one on H ϕ and is continuous in H ϕ . This gives us a representation π ϕ : A → B ( H ϕ ) via π ϕ ( a )([ x ]) = [ ax ] for all a, x ∈ A . Gábor Szabó (KU Leuven) C*-algebras November 2018 28 / 50
Representation theory GNS construction Definition (Gelfand–Naimark–Segal construction) For a state ϕ on a C ∗ -algebra A , let H ϕ be the Hilbert space completion �·� ϕ . Then H ϕ carries a unique left A -module structure which H ϕ = H ϕ extends the one on H ϕ and is continuous in H ϕ . This gives us a representation π ϕ : A → B ( H ϕ ) via π ϕ ( a )([ x ]) = [ ax ] for all a, x ∈ A . The only non-tautological part is that π ϕ is compatible with adjoints. For this we observe � = � [ x ] | [ a ∗ y ] � ϕ , � [ ax ] | [ y ] � ϕ = ϕ ( y ∗ ax ) = ϕ � ( a ∗ y ) ∗ x which forces π ϕ ( a ) ∗ = π ϕ ( a ∗ ) . Gábor Szabó (KU Leuven) C*-algebras November 2018 28 / 50
Representation theory GNS construction Definition (Gelfand–Naimark–Segal construction) For a state ϕ on a C ∗ -algebra A , let H ϕ be the Hilbert space completion �·� ϕ . Then H ϕ carries a unique left A -module structure which H ϕ = H ϕ extends the one on H ϕ and is continuous in H ϕ . This gives us a representation π ϕ : A → B ( H ϕ ) via π ϕ ( a )([ x ]) = [ ax ] for all a, x ∈ A . The only non-tautological part is that π ϕ is compatible with adjoints. For this we observe � = � [ x ] | [ a ∗ y ] � ϕ , � [ ax ] | [ y ] � ϕ = ϕ ( y ∗ ax ) = ϕ � ( a ∗ y ) ∗ x which forces π ϕ ( a ) ∗ = π ϕ ( a ∗ ) . Definition In the (unital) situation above, set ξ ϕ = [ 1 ] ∈ H ϕ . Then � ξ ϕ � = 1 as we have assumed ϕ to be a state. Gábor Szabó (KU Leuven) C*-algebras November 2018 28 / 50
Representation theory GNS construction Theorem (GNS) The assignment ϕ �→ ( π ϕ , H ϕ , ξ ϕ ) is a 1-1 correspondence between states on A and cyclic representations modulo unitary equivalence. Proof: Let us only check that ( π ϕ , H ϕ , ξ ϕ ) is cyclic. Indeed, π ϕ ( A ) ξ ϕ = π ϕ ( A )([ 1 ]) = [ A ] = H ϕ ⊆ H ϕ , which is dense by definition. Gábor Szabó (KU Leuven) C*-algebras November 2018 29 / 50
Representation theory GNS construction Theorem (GNS) The assignment ϕ �→ ( π ϕ , H ϕ , ξ ϕ ) is a 1-1 correspondence between states on A and cyclic representations modulo unitary equivalence. Proof: Let us only check that ( π ϕ , H ϕ , ξ ϕ ) is cyclic. Indeed, π ϕ ( A ) ξ ϕ = π ϕ ( A )([ 1 ]) = [ A ] = H ϕ ⊆ H ϕ , which is dense by definition. Theorem (Gelfand–Naimark) Every abstract C ∗ -algebra A is a concrete C ∗ -algebra. In particular, there exists a faithful representation π : A → H on some Hilbert space. 5 Proof: For x ∈ A , find ϕ x with � ϕ x ( x ∗ x ) � = � x � 2 . Then form the cyclic representation ( π ϕ x , H ϕ x , ξ ϕ x ) . 5 If A is separable, we may choose H to be separable! Gábor Szabó (KU Leuven) C*-algebras November 2018 29 / 50
Representation theory GNS construction Theorem (GNS) The assignment ϕ �→ ( π ϕ , H ϕ , ξ ϕ ) is a 1-1 correspondence between states on A and cyclic representations modulo unitary equivalence. Proof: Let us only check that ( π ϕ , H ϕ , ξ ϕ ) is cyclic. Indeed, π ϕ ( A ) ξ ϕ = π ϕ ( A )([ 1 ]) = [ A ] = H ϕ ⊆ H ϕ , which is dense by definition. Theorem (Gelfand–Naimark) Every abstract C ∗ -algebra A is a concrete C ∗ -algebra. In particular, there exists a faithful representation π : A → H on some Hilbert space. 5 Proof: For x ∈ A , find ϕ x with � ϕ x ( x ∗ x ) � = � x � 2 . Then form the cyclic representation ( π ϕ x , H ϕ x , ξ ϕ x ) . We claim that the direct sum � � � � π := π ϕ x : A → B H ϕ x x ∈ A x ∈ A does it. Indeed, given any x � = 0 we have � π ( x ) � 2 ≥ � π ( x ) ξ ϕ x � 2 = � [ x ] | [ x ] � ϕ x = ϕ x ( x ∗ x ) = � x � 2 . 5 If A is separable, we may choose H to be separable! Gábor Szabó (KU Leuven) C*-algebras November 2018 29 / 50
Examples Let us now discuss noncommutative examples of C ∗ -algebras: Example The set of C -valued n × n matrices, denoted M n , becomes a C ∗ -algebra. By linear algebra, M n ∼ = B ( C n ) . Gábor Szabó (KU Leuven) C*-algebras November 2018 30 / 50
Examples Let us now discuss noncommutative examples of C ∗ -algebras: Example The set of C -valued n × n matrices, denoted M n , becomes a C ∗ -algebra. By linear algebra, M n ∼ = B ( C n ) . Example For numbers n 1 , . . . , n k ≥ 1 , the C ∗ -algebra A = M n 1 ⊕ M n 2 ⊕ · · · ⊕ M n k has finite ( C -linear) dimension. Gábor Szabó (KU Leuven) C*-algebras November 2018 30 / 50
Examples Let us now discuss noncommutative examples of C ∗ -algebras: Example The set of C -valued n × n matrices, denoted M n , becomes a C ∗ -algebra. By linear algebra, M n ∼ = B ( C n ) . Example For numbers n 1 , . . . , n k ≥ 1 , the C ∗ -algebra A = M n 1 ⊕ M n 2 ⊕ · · · ⊕ M n k has finite ( C -linear) dimension. Theorem Every finite-dimensional C ∗ -algebras has this form. Gábor Szabó (KU Leuven) C*-algebras November 2018 30 / 50
Examples Recall A linear map between Banach spaces T : A → B is called compact, if T · A �·�≤ 1 ⊆ B is compact. Gábor Szabó (KU Leuven) C*-algebras November 2018 31 / 50
Examples Recall A linear map between Banach spaces T : A → B is called compact, if T · A �·�≤ 1 ⊆ B is compact. Observation Compact operators are bounded. The composition of a compact operator with a bounded operator is compact. Gábor Szabó (KU Leuven) C*-algebras November 2018 31 / 50
Examples Recall A linear map between Banach spaces T : A → B is called compact, if T · A �·�≤ 1 ⊆ B is compact. Observation Compact operators are bounded. The composition of a compact operator with a bounded operator is compact. Example For a Hilbert space H , the set of compact operators K ( H ) ⊆ B ( H ) forms a norm-closed, ∗ -closed, two-sided ideal. If dim( H ) = ∞ , then it is a proper ideal and a non-unital C ∗ -algebra. Gábor Szabó (KU Leuven) C*-algebras November 2018 31 / 50
Examples Universal C ∗ -algebras Notation (ad-hoc!) Let G be a countable set, and let P be a family of (noncommutative) ∗ -polynomials in finitely many variables in G and coefficients in C . We shall understand a relation R as a collection of formulas of the form � p ( G ) � ≤ λ p , p ∈ P , λ p ≥ 0 . A representation of ( G | R ) is a map π : G → A into a C ∗ -algebra under which the relation becomes true. Gábor Szabó (KU Leuven) C*-algebras November 2018 32 / 50
Examples Universal C ∗ -algebras Notation (ad-hoc!) Let G be a countable set, and let P be a family of (noncommutative) ∗ -polynomials in finitely many variables in G and coefficients in C . We shall understand a relation R as a collection of formulas of the form � p ( G ) � ≤ λ p , p ∈ P , λ p ≥ 0 . A representation of ( G | R ) is a map π : G → A into a C ∗ -algebra under which the relation becomes true. Example The expression xyx ∗ − z 2 for x, y, z ∈ G is a noncommutative ∗ -polynomial. The relation could mean � xyx ∗ − z 2 � ≤ 1 . Gábor Szabó (KU Leuven) C*-algebras November 2018 32 / 50
Examples Universal C ∗ -algebras Definition A representation π u of ( G | R ) into a C ∗ -algebra B is called universal, if 1 B = C ∗ ( π u ( G )) . Gábor Szabó (KU Leuven) C*-algebras November 2018 33 / 50
Examples Universal C ∗ -algebras Definition A representation π u of ( G | R ) into a C ∗ -algebra B is called universal, if 1 B = C ∗ ( π u ( G )) . 2 whenever π : G → A is a representation of ( G | R ) into another C ∗ -algebra, there exists a ∗ -homomorphism ϕ : B → A such that ϕ ◦ π u = π . Gábor Szabó (KU Leuven) C*-algebras November 2018 33 / 50
Examples Universal C ∗ -algebras Definition A representation π u of ( G | R ) into a C ∗ -algebra B is called universal, if 1 B = C ∗ ( π u ( G )) . 2 whenever π : G → A is a representation of ( G | R ) into another C ∗ -algebra, there exists a ∗ -homomorphism ϕ : B → A such that ϕ ◦ π u = π . Observation Up to isomorphism, a C ∗ -algebra B as above is unique. One writes B = C ∗ ( G | R ) and calls it the universal C ∗ -algebra for ( G | R ) . Gábor Szabó (KU Leuven) C*-algebras November 2018 33 / 50
Examples Universal C ∗ -algebras Example Given n ≥ 1 , one can express M n as the universal C ∗ -algebra generated by { e i,j } n i,j =1 subject to the relations e ∗ e ij e kl = δ jk e il , ij = e ji . Gábor Szabó (KU Leuven) C*-algebras November 2018 34 / 50
Examples Universal C ∗ -algebras Example Given n ≥ 1 , one can express M n as the universal C ∗ -algebra generated by { e i,j } n i,j =1 subject to the relations e ∗ e ij e kl = δ jk e il , ij = e ji . Example Let H be a separable, infinite-dimensional Hilbert space. Then one can express K ( H ) as the universal C ∗ -algebra generated by { e i,j } i,j ∈ N subject to the relations e ∗ e ij e kl = δ jk e il , ij = e ji . Gábor Szabó (KU Leuven) C*-algebras November 2018 34 / 50
Examples Universal C ∗ -algebras Example Given n ≥ 1 , one can express M n as the universal C ∗ -algebra generated by { e i,j } n i,j =1 subject to the relations e ∗ e ij e kl = δ jk e il , ij = e ji . Example Let H be a separable, infinite-dimensional Hilbert space. Then one can express K ( H ) as the universal C ∗ -algebra generated by { e i,j } i,j ∈ N subject to the relations e ∗ e ij e kl = δ jk e il , ij = e ji . (Here e ij represents a rank-one operator sending the i -th vector in an ONB to the j -th vector.) Gábor Szabó (KU Leuven) C*-algebras November 2018 34 / 50
Examples Universal C ∗ -algebras Definition A relation R on a set G is compact if for every x ∈ G sup {� π ( x ) � | π : G → A representation of ( G | R ) } < ∞ . Gábor Szabó (KU Leuven) C*-algebras November 2018 35 / 50
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