Basic Definitions
A C*-algebra is defined as a complete associative C-algebra A equipped with a norm ã on the C-vector space A and an antilinear map ∗: A → A, denoted as a → a ∗ For any elements a, b in A, the properties of the norm and the antilinear map must hold, establishing the structure and completeness necessary for A to be classified as a C*-algebra.
A (not necessarily complete) norm on A satisfying conditions (1) – (5) is called a
Remark 1.Note that Axioms 1–5 are not independent For instance, Axiom 4 can easily be deduced from Axioms 1,3, and 5.
Example 1.Let (H,(ã,ã)) be a complex Hilbert space, letA=L(H) be the algebra of bounded linear operators on H Let ã be theoperator norm, i.e., a:=sup x∈H x=1 ax.
Institut f¨ur Mathematik, Universit¨at Potsdam, Am Neuen Palais 10, D-14469 Potsdam, Germany e-mail: baer@math.uni-potsdam.de
Institut f¨ur Mathematik, Universit¨at Potsdam, Am Neuen Palais 10, D-14469 Potsdam, Germany e-mail: becker@math.uni-potsdam.de
Leta ∗ be the operator adjoint toa, i.e.,
Axioms 1–4 are easily checked Using Axioms 3 and 4 and the Cauchy–Schwarz inequality we see a 2 = sup x = 1 ax 2 = sup x = 1
Example 2.LetXbe a locally compact Hausdorff space Put
We call C 0 (X) the algebra of continuous functions vanishing at infinity If X is compact, thenA=C 0(X)=C(X) All f ∈C 0(X) are bounded and we may define f:=sup x ∈ X |f(x)|. Moreover let f ∗ (x) := f(x).
Example 3.LetXbe a differentiable manifold Put
C₀∞(X) represents the algebra of smooth functions that vanish at infinity, with its norm and * operation defined similarly to previous examples Although (C₀∞(X), ã, *) adheres to all the axioms of a commutative C* algebra, it is important to note that (A, ã) is not complete However, upon completing this normed vector space, we revert to the familiar context of continuous functions.
Definition 2 A subalgebra A 0 of a C ∗ -algebra A is called a C ∗ -subalgebraif it is a closed subspace and a ∗ ∈ A 0 for all a∈ A 0
AnyC ∗ -subalgebra is aC ∗ -algebra in its own right.
Definition 3 Let S be a subset of a C ∗ -algebra A Then the intersection of all
C ∗ -subalgebras of A containing S is called the C ∗ -subalgebra generated byS.
Definition 4 An element a of a C ∗ -algebra is calledself-adjointif a =a ∗ Remark 2.Like any algebra aC ∗ -algebra Ahas at most one unit 1 Now we have for alla ∈ A
1 ∗ a =(1 ∗ a) ∗∗ =(a ∗ 1 ∗∗ ) ∗ =(a ∗ 1) ∗ =a ∗∗ =a and similarly one seesa1 ∗ =a Thus 1 ∗ is also a unit By uniqueness 1=1 ∗ , i.e., the unit is self-adjoint Moreover,
1 = 1 ∗ 1 = 1 2 , hence1 = 1 or1 =0 In the second case 1=0 and therefore A=0 Hence we may (and will) from now on assume that1 =1.
Example 4.1 In Example 1 the algebra A=L(H) has a unit 1=id H
2 The algebra A =C 0(X) has a unit f ≡1 if and only ifC 0(X) =C(X), i.e., if and only ifX is compact.
LetAbe aC ∗ -algebra with unit 1 We writeA × for the set of invertible elements inA Ifa∈ A × , then alsoa ∗ ∈ A × because a ∗ ã(a − 1 ) ∗ =(a − 1 a) ∗ =1 ∗ =1, and similarly (a − 1 ) ∗ ãa ∗ =1 Hence (a ∗ ) − 1 =(a − 1 ) ∗
Lemma 1 Let A be a C ∗ -algebra Then the maps
The first two maps in normed vector spaces are continuous, which is supported by the triangle inequality and the homogeneity of the norm Additionally, the continuity of multiplication is established for elements a and b in a normed space, where a and b are close to specified values a₀ and b₀, respectively, ensuring that the product ab remains within an ε-neighborhood of a₀b₀.
(c) Continuity of inversion Leta 0 ∈ A × Then we have for all a ∈ A × with a−a < ε a n Eachm∈N 0 can be written uniquely in the formm= pn+q,p, q ∈N 0 , 0≤q ≤n−1 The series
= 1 λ n − 1 q = 0 a λ q ∞ p = 0 a n λ n ã a n the element (λ1 −a) is invertible and thus λ ∈ r A (a). Therefore ρ A (a)≤ inf n∈ N a n 1 n ≤lim inf n →∞ a n 1 n
(d) We showρ A (a)≥lim sup n →∞ a n 1 n We abbreviateρ(a) :=lim sup n →∞ a n 1 n
1= 1 = a n a − n ≤ a n ã a − n would imply 1≤ρ(a)ãρ(a − 1 )=0, which would yield a contradiction Therefore a ∈ A × Thus 0∈ σ A (a) In particular, the spectrum ofais nonempty Hence the
1 C -algebras 7 spectral radiusρ A (a) is bounded from below by 0 and thus ρ(a)=0≤ρ A (a).
Case 2:ρ(a)>0 Ifa n ∈ Aare elements for whichR n :=(1−a n ) − 1 exist, then a n →0 ⇔ R n →1.
This follows from the fact that the map A × → A × ,a → a − 1 is continuous by Lemma 1 Put
We want to show thatS ⊂r A (a) since then there existsλ∈ σ A (a) such that|λ| ≥ ρ(a) and hence ρ A (a)≥ |λ| ≥ρ(a).
Assume in the contrary that S ⊂ r A (a) Letω ∈ Cbe annth root of unity, i.e., ω n =1 Forλ∈ Swe also haveλ/ω k ∈ S⊂r A (a) Hence there exists λ ω k 1−a − 1
R n (a, λ)= 1−a n λ n − 1 for anyλ∈S ⊂r A (a) Moreover forλ∈Swe have
The supremum is finite sincez → (z1−a) − 1 is continuous onr A (a)⊃ Sby part (b) of the proof and since for|z| ≥2ã awe have
Outside the annulusB 2 a (0)−B ρ(a)(0) the expression(z1−a) − 1 is bounded by
1 /aand on the compact annulus it is bounded by continuity Put
R n (a,ρ(a))−R n (a, λ) ≤Cã |ρ(a)−λ| for alln ∈Nand allλ∈S Puttingλ=ρ(a)+ 1 j we obtain
≤ C j for all j∈Nand hence lim sup n→∞
On the other hand we have a n + 1 n+1 1 ≤ a n+1 1 ã a n n+1 1
Hence the sequence a n 1 n n ∈ Nis monotonically nonincreasing and therefore ρ(a)=lim sup k →∞ a k 1 k ≤ a n 1 n for alln∈N.
Thus 1≤ a n /ρ(a) n for alln∈N, in contradiction to (1.2).
(e)The spectrum is nonempty Ifσ(a) = ∅, thenρ A (a) = −∞contradicting ρ A (a)=lim n →∞ a n 1 n ≥0
Definition 6 Let A be a C ∗ -algebra with unit Then a∈ A is called
Remark 4.In particular, self-adjoint elements are normal In a commutative algebra all elements are normal.
Proposition 2 Let A be a C ∗ -algebra with unit and let a,b∈ A Then the following holds:
4 If a is an isometry, thenρ A (a)=1.
6 If a is self-adjoint, thenσ A (a)⊂[−a,a]and moreoverσ A (a 2 )⊂[0,a 2 ].
7 If P(z)is a polynomial with complex coefficients and a∈ A is arbitrary, then σ A
In this article, we demonstrate that a number λ is not part of the spectrum of the operator a if and only if the expression (λ1−a) is invertible This condition is equivalent to stating that (λ1−a) ∗ = λ1−a ∗ is also invertible, which leads to the conclusion that λ does not belong to the spectrum of the adjoint operator a ∗.
To see Assertion 2 letabe invertible Then 0 lies neither in the spectrumσ A (a) ofanor in the spectrumσ A (a − 1 ) ofa − 1 Moreover, we have forλ=0 λ1−a=λa(a − 1 −λ − 1 1) and λ − 1 1−a − 1 =λ − 1 a − 1 (a−λ1).
Henceλ1−ais invertible if and only ifλ − 1 1−a − 1 is invertible.
To show Assertion 3 letabe normal Thena ∗ ais self-adjoint, in particular nor- mal Using theC ∗ -property we obtain inductively a 2 n 2 = (a 2 n ) ∗ a 2 n = (a ∗ ) 2 n a 2 n = (a ∗ a) 2 n
To prove Assertion 4 letabe an isometry Then a n 2 = (a n ) ∗ a n = (a ∗ ) n a n = 1 =1.
1 C -algebras 11 For Assertion 5 letabe unitary On the one hand we have by Assertion 4 σ A (a)⊂ {λ∈C| |λ| ≤1}.
On the other hand we have σ A (a) (1) =σ A (a ∗ )=σ A (a − 1 ) (2) =σ A (a) − 1
To show Assertion 6 letabe self-adjoint We need to showσ A (a)⊂R Letλ∈R withλ − 1 >a Then| −iλ − 1 | =λ − 1 > ρ(a) and hence 1+iλa =iλ(−iλ − 1 +a) is invertible Put
Similarly U U ∗ = 1, i.e.,U is unitary By Assertion 5σ A (U) ⊂ S 1 A simple computation with complex numbers shows that
|(1−iλμ)(1+iλμ) − 1 | =1 ⇔ μ∈R. Thus (1−iλμ)(1+iλμ) − 1 ã1−Uis invertible ifμ∈C\R From
=2iλ(1+iλμ) −1 (a−μ1)(1+iλa) −1 we see thata−μ1 is invertible for allμ∈C\R Thusμ∈r A (a) for allμ∈C\R and henceσ A (a)⊂R The statement aboutσ A (a 2 ) now follows from part 7.
To prove Assertion 7 decompose the polynomialP(z)−λinto linear factors
We insert an algebra elementa ∈ A:
Since the factors in this product commute the product is invertible if and only if all factors are invertible 1 In our case this means λ∈σ A
⇔at least one factor is noninvertible
Ifcis inverse to 1−ab, then (1+bca)ã(1−ba)=1−ba+bc(1−ab)a =1 and (1−ba)ã(1+bca)=1−ba+b(1−ab)ca=1 Hence 1+bcais inverse to
1−ba, which finally yields Assertion 8
Corollary 1 Let (A, ã ,∗) be a C ∗ -algebra with unit Then the norm ã is uniquely determined by A and∗.
Proof Fora∈ Athe elementa ∗ ais self-adjoint and hence a 2 = a ∗ a 2(3) = ρ A (a ∗ a) depends only on Aand∗.
Morphisms
Definition 7 Let A and B be C ∗ -algebras An algebra homomorphism π : A→ B is called∗-morphismif for all a∈ A we have π(a ∗ )=π(a) ∗
A mapπ: A→ A is called∗-automorphismif it is an invertible∗-morphism.
Corollary 2 Let A and B be C ∗ -algebras with unit Each unit-preserving∗-morphism π : A→ B satisfies π(a) ≤ a for all a∈ A In particular,πis continuous.
In algebras with a unit, a product b = a₁ a₂ aₙ of commuting factors is invertible if and only if all individual factors are invertible Specifically, the inverse of b can be expressed as b⁻¹ = a⁻¹ₙ a⁻¹₁ Conversely, if b is invertible, the individual inverses can be determined using the relationship a⁻¹ᵢ = b⁻¹ aⱼ where the factors commute.
1 C -algebras 13 holds and similarlyπ(a − 1 )π(a) = 1 Henceπ(a) ∈ B × withπ(a) − 1 = π(a − 1 ). Now ifλ∈r A (a), then λ1−π(a)=π(λ1−a)∈π(A × )⊂B × , i.e.,λ ∈ r B (π(a)) Hencer A (a) ⊂ r B (π(a)) andσ B (π(a)) ⊂ σ A (a) This implies the inequality ρ B (π(a))≤ρ A (a).
Sinceπ is a∗-morphism anda ∗ a andπ(a) ∗ π(a) are self-adjoint we can estimate the norm as follows: π(a) 2 = π(a) ∗ π(a) =ρ B π(a) ∗ π(a)
Corollary 3 Let A be a C ∗ -algebra with unit Then each unit-preserving
In the context of a polynomial P(z) = ∑(j=0 to n) c_j z^j, where z is a complex variable and an element of an algebra A, we can define P(a) = ∑(j=0 to n) c_j a^j in a straightforward way To extend this concept, we explore how to define f(a) when f is a continuous function and a is a normal element of a C*-algebra A, a process referred to as continuous functional calculus.
Proposition 3 Let A be a C ∗ -algebra with unit Let a ∈ A be normal.
Then there is a unique∗-morphism C(σ A (a))→ A denoted by f → f(a)such that f(a)has the standard meaning in case f is the restriction of a polynomial. Moreover, the following holds:
2 If B is another C ∗ -algebra with unit and π : A → B a unit-preserving
2 Recall from the proof of Corollary 2 that σ B (π(a)) ⊂ σ A (a) Strictly speaking, the statement is π( f (a)) = ( f | σ B (π(a)) )(π(a)).
Proof For any polynomialPwe have thatP(a) is also normal and hence by Propo- sition 2
The map P → P(a) uniquely extends to a linear map from the closure of polynomials in C(σ A (a)) to A Given that polynomials form an algebra with a unit, include complex conjugates, and separate points, the closure encompasses all of C(σ A (a)) as per the Stone–Weierstrass theorem This extension is continuous.
∗-morphism and Assertion 1 follows from (1.3).
Assertion 2 clearly holds if f is a polynomial It then follows for continuous f becauseπ is continuous by Corollary 2.
As to Assertion 3 letλ ∈ σ A (a) Choose polynomials P n such that P n → f in
A × Since the complement of A × is closed we can pass to the limit and we obtain f(a)− f(λ)ã1∈ A ì Hence f(λ)∈σ A (f(a)) This shows f(σ A (a))⊂σ A (f(a)). Conversely, letμ∈ f(σ A (a)) Theng:=(f −μ) − 1 ∈C(σ(a)) Fromg(a)(f(a)− μã1)=(f(a)−μã1)g(a)=1 one sees f(a)−μã1∈ A ì , thusμ∈σ(f(a)).
We expand upon Corollary 3 to address scenarios where π is injective, though not necessarily surjective This extension is not a straightforward implication of Corollary 3, as it is not immediately evident that the image of an a*-morphism is closed, thereby forming a C*-algebra independently.
Proposition 4 Let A and B be C ∗ -algebras with unit Each injective unit-preserving
Proof By Corollary 2 we only have to showπ(a) ≥ a Once we know this inequality for self-adjoint elements it follows for alla∈ Abecause π(a) 2 = π(a) ∗ π(a) = π(a ∗ a) ≥ a ∗ a = a 2
Assume there exists a self-adjoint element a ∈ A such that π(a) < a By Proposition 2, we haveσ A (a)⊂[−a,a] andρ A (a)= a, hencea ∈σ A (a) or−a ∈σ A (a) Similarly,σ B (π(a))⊂[−π(a),π(a)].
Choose a continuous function f : [−a,a] → Rsuch that f vanishes on [−π(a),π(a)] and f(−a)= f(a)=1 From Proposition 3 we conclude π(f(a)) = f(π(a)) = 0 because f|σ B (π(a)) = 0 andf(a) = f C(σ A (a)) ≥ 1.Thus f(a)=0 This contradicts the injectivity ofπ
Remark 5.Any elementa in aC ∗ -algebra Acan be represented as a linear com- binationa =a 1+i a 2 of self-adjoint elements by settinga 1 := 1 2 ã(a+a ∗ ) and a 2:= 2i 1 ã(a−a ∗ ).
Lemma 2 Let a∈ A be a self-adjoint element in a unital C ∗ -algebra A Then the following three statements are equivalent:
Proof Ifa=b 2 for a self-adjoint element, we have by Proposition 3 σ A (a)=σ A (b 2 )= {λ 2 |λ∈σ A (b)} ⊂[0,∞), which proves the implication “1⇒3.”
Ifσ A (a) ⊂ [0,∞), we can define the elementb := √ a using the continuous functional calculus from Proposition 3 We then haveb ∗ = b andb 2 = a, which proves the implication “3⇒1.”
Leta = c ∗ cand supposeσ A (−a) ⊂ [0,∞) By Assertion 8 from Proposition
2, we have σ A (−cc ∗ ) = σ A (−c ∗ c)− {0} ⊂ [0,∞) Writingc = c 1 +i c 2 with self-adjoint elementsc 1,c 2, we findc ∗ c+cc ∗ = 2c 2 1 +2c 2 2 , hencec ∗ c = 2c 2 1 + 2c 2 2 −cc ∗ , which impliesσ A (c ∗ c)⊂ [0,∞) Henceσ A (c ∗ c)= {0}, which implies c ∗ c=a =0.
Now suppose a = c ∗ c for an arbitrary element c ∈ A Since a = c ∗ c is self-adjoint and σ A (a 2 ) ⊂ [0,∞), by the continuous functional calculus from Proposition 3, there exists a unique element|a|:=√ a 2 with σ A (d)= {√ λ|λ∈σ A (a 2 )} ⊂[0,∞).
By the same argument, the elementsa + := 1 2ã(|a| +a) anda − := 1 2ã(|a| −a) are self-adjoint and satisfyσ A (a i )⊂[0,∞) We then havea = a + −a − Further, for the elementd :=ca − , we compute
−d ∗ d= −a − c ∗ ca − = −a − (a + −a − )a − = −a − a + a − +(a − ) 3 =(a − ) 3 , sincea + a − = 1 4(|a| +a)ã(|a| −a)= 1 4(|a| 2 −a 2 )=0 We thus haveσ A (−d ∗ d) σ A ((a − ) 3 )⊂[0,∞), which yieldsd =0 Hencec=0 ora − =0, thusa=a + and σ A (a)=σ A (a + )⊂[0,∞) This proves the implication “1⇒3.”
Definition 8 A self-adjoint element a∈ A is calledpositive, if one and hence all of the properties in Lemma 2 hold.
According to the reasoning presented in the previous proof, any self-adjoint element \( a \in A \) can be expressed as a linear combination \( a = a^+ - a^- \), where \( a^+ \) and \( a^- \) are defined as \( a^+ := \frac{1}{2} \hat{a} (|a| + a) \) and \( a^- := \frac{1}{2} \hat{a} (|a| - a) \), respectively, and satisfy \( a^+ + a^- = 0 \) By integrating this insight with Remark 5, we can deduce that any \(*\)-subalgebra of \( A \) is generated by its positive elements, each having a norm less than or equal to 1.
States and Representations
Let (A, ã ,∗) be aC ∗ -algebra andHa Hilbert space.
Definition 9 A representationof A on H is a ∗-morphism π : A → L(H) A representation is calledfaithful, ifπis injective A subset U ⊂H is calledinvariant under A, if π(A)U := {π(a)ãu|a∈a,u∈U} ⊂U.
A representation is called irreducible, if the only closed vector subspaces of H invariant under A are{0}and H
Remark 7.Letπ λ :A→L(H λ),λ∈Λbe representations ofA Then π λ ∈ Λ πλ: A→L( λ ∈ Λ
H λ), π(a) (x λ)λ ∈ Λ πλ(a)ãx λ λ ∈ Λ, is called thedirect sum representation.
Definition 10 Two representationsπ 1 : A → L(H 1),π 2 : A → L(H 2)are called unitarily equivalent, if there exists a unitary operator U : H 1 → H 2 , such that for every a ∈ A:
Definition 11 A vectorΩ ∈ H is calledcyclicfor a representationπ, if
The commutative C*–algebra A = C(X), consisting of continuous functions on a compact Hausdorff space, naturally represents itself on the Hilbert space H L2(X) through multiplication In this context, the constant function Ω = 1 serves as a cyclic vector, highlighting that continuous functions are dense in L2(X).
Lemma 3 If(H, π)is an irreducible representation, then eitherπ is the zero map or every non-zero vectorΩ∈ H is cyclic forπ.
For any vector Ω in the Hilbert space H, the subspace π(A)Ω remains invariant under the operator A, implying that its closure is either the zero vector or the entire space H If Ω is non-zero, then either π(A)Ω equals zero, indicating that the one-dimensional subspace CãΩ is invariant under A, leading to the conclusion that H equals CãΩ and π equals zero, or there exists an element a in A such that π(a)Ω equals zero, which means that π(A)ãΩ is dense in H.
Definition 12 Astateon a C ∗ -algebra A is a linear functionalτ : A→Cwith
The set of all states on A is denoted by S(A).
Example 7.Let X be a compact Hausdorff space, A = C(X) Let μ be a Borel probability measure on X, i.e., a measure on the Borel sigma algebra of X with
X f dμ is a state For instance, the stateμδ x 0 corresponding to the Dirac measure atx 0is the evaluation atx 0: μδ x 0 (f)= f(x 0).
Example 8.On theC ∗ -algebra A = Mat(n×n;C) of complex matrices, we have the state τ(A) := 1 n ãtr(A).
Example 9.OnA=L(H), a vectorΩ ∈ HwithΩ =1 yields a so-calledvector state τ(A) := AãΩ, Ω.
Proposition 5 Letτ : A → Cbe a state on a C ∗ -algebra A with unit Then we have the following:
1 A×A→C,(a,b)→τ(b ∗ a)is a positive semi-definite, Hermitian sesquilinear form.
Proof It follows immediately from the definitions that the form (a,b) → τ(b ∗ a) is sesquilinear and positive semi-definite To show that it is Hermitian, we set c=aãz+bfor somez∈Cand compute
Imτ(a ∗ b) = −Imτ(b ∗ a), settingz =i, we obtain Reτ(a ∗ b) = Reτ(b ∗ a) Thus τ(a ∗ b)=τ(b ∗ a).
Settingz= − τ(a τ(a ∗ ∗ b) a), (1.4) implies the Cauchy–Schwarz inequality:
0≤ |τ(a ∗ b)| 2 τ(a ∗ a) −|τ(a ∗ b)| 2 τ(a ∗ a) −|τ(a ∗ b)| 2 τ(a ∗ a) +τ(b ∗ b). Since Ahas a unit, we have τ(a ∗ )=τ(a ∗ 1)=τ(1 ∗ a)=τ(a).
To show Assertion 4, we compute
Remark 8.The proof of Assertion 5 shows thatϕ(1)= ϕholds for every positive linear functionalϕ.
Corollary 4 Letτ1, , τ n be states andλ1, , λ n ≥0with n j = 1λ j =1 Then theconvex combinationτ = n j = 1λ n ãτ n is also a state.
Letτ i , i∈Nbe states and defineτ(a) :=lim i→∞ τ i (a)provided the limit exists. Thenτ is a state.
Proof For the convex combinations, we have τ(a ∗ a) n j = 1 λ j
1 C -algebras 19 Similarly, for the pointwise convergence, we have τ(a ∗ a)= lim i →∞τ i (a ∗ a)≥0 and τ(1)= lim i →∞τ i (1)= lim i →∞1=1.
Example 10.Letτ 1 , , τ n be vector states for vectorsΩ 1 , , Ω n ∈ H Then for the stateτ = n j = 1λ j τ j withλ j ≥0, n j = 1λ j =1, we find τ(a) n j = 1 λ j ãτ j (a) n j = 1 λ j ã(aãΩ j , Ω j )=tr(̺ãa).
Here̺∈L(H) is an operator of finite-dimensional range with eigenvectorsΩ j and eigenvaluesλ j
More generally, a positive trace class operator ̺ ∈ L(H) defines a stateτ on
A=L(H) byτ(a) :=tr(̺ãa) States of this form are callednormal.
Lemma 4 Letτ be a state on a C ∗ -algebra A Then the following holds:
Proof 1 Supposeτ(a ∗ a)=0 Then the Cauchy–Schwarz inequality
= 0 ãτ(bb ∗ )=0 impliesτ(ba)=0 The other direction is obvious.
2 If τ(b ∗ b) = 0, thenτ(cb) = 0 for anyc ∈ A, especially forc = b ∗ a ∗ a We thus assumeτ(b ∗ b) > 0 and set ̺(c) := τ(b τ(b ∗ ∗ cb) b) Then ̺ is a positive linear functional with̺ =̺(1)=1 Hence̺is a state, and from Proposition 5 we have̺(a ∗ a)≤ a ∗ a.
From every stateτ on aC ∗ -algebraAwe can construct a representation of Aby making the product (b,a) → τ(b ∗ a) nondegenerate By Assertion 1 in Lemma 4, the null space
N τ := {a∈ A|τ(a ∗ a)=0} is a closed linear subspace ofA By Assertion 2 in Lemma 4,N τ is a left ideal in A.
A/N τ×A/N τ →C, ([a],[b])→τ(b ∗ a) is a well-defined Hermitian scalar product Let H τ be the completion of the pre- Hilbert space A/N τ Then the map πτ : A→L(A/N τ), π τ (a)ã[b] :=[ab] satisfies πτ(a)ã[b] 2 =τ(b ∗ a ∗ ab)≤ a ∗ a ãτ(b ∗ b)= a 2 ã [b] 2 , soπτ(a) ≤ aandπτ ≤1 The mapπτthus extends to a representation πτ : A→L(H τ).
The scalar product induced by ([a],[b])→τ(b ∗ a) onH τ will be denoted byã,ãτ.
Definition 13 Letτbe a state on a C ∗ -algebra A The representation(H τ,ã,ãτ, πτ) constructed above is called the Gelfand–Naimark–Segal representation or GNS representationin short.
Example 11.For A = C(X) with a stateτμ given by a probability measureμas τμ(f) X f dμ, the representation space of the GNS representation is H τ μ L 2 (X, μ).
Remark 9.Letτ be a state on aC ∗ -algebra Awith unit Then we have the follow- ing:
1 The vectorΩτ :=[1]∈ H τis cyclic forπτ, since π τ (A)ãΩ τ =A/N τ ⊂H τ is dense.
2 τ can be represented as a vector state on the GNS representation because τ(a)=τ(1 ∗ a1)= [a1],[1] τ = πτ(a)ãΩτ, Ωτ τ
Definition 14 Let A be a C ∗ -algebra A The direct sum representation τ ∈ S(A) πτ: A→L τ ∈ S(A)
Remark 10.The universal representation is faithful Hence everyC ∗ -algebra A is isomorphic to a subalgebra of the algebra L(H) of bounded linear operators on a Hilbert spaceH.
Definition 15 A stateτon a C ∗ -algebra A is calledpure, if for every positive linear functional̺ : A → Cwith̺(a ∗ a)≤ τ(a ∗ a)∀a ∈ A, there existsλ ∈[0,1]with ̺=λãτ.
Remark 11.A pure stateτ cannot be written as a convex combination of different statesτ 1 =τ2 Ifτ =λãτ1+(1−λ)ãτ2withλ ∈[0,1], thenτ ≥ λãτ1implies λ=0 andτ =τ 2 orλ=1 andτ =τ 1
Example 12.The trace as a state of the algebraA=Mat(n×n;C) (see Example 8) is not pure unlessn =1, namely it can be written as 1 n tr= n i=1
1 n τ i whereτ i is the vector state for theith standard unit vector ofC n
Definition 16 Let S ⊂ Abe a subset of a C ∗ -algebra A The space S ′ := {a ∈
A|[a,s]=0∀a ∈ A,s∈S}is called thecommutantof S Here[a,s] :=as−sa is thecommutatorof a and s.
Remark 12.If S ⊂ A is a∗-invariant subset, i.e., S ∗ := {s ∗ |s ∈ S} ⊂ S, then the commutant S ′ is also∗-invariant.S ′ is closed, since for everys ∈ S, the map
A→ A,a →[a,s] is continuous HenceS ′ is aC ∗ -subalgebra ofA.
Theorem 1 Let(H, π)be a representation of a unital C ∗ -algebra A Then the fol- lowing two statements are equivalent:
Proof Supposeπis irreducible andb∈L(H) commutes with all elements ofπ(A).
By Remark 5, we may writeb=b 1 +i b 2 with self-adjoint elementsb 1 ,b 2 ∈L(H).
To demonstrate that σ_A(b_1) and σ_A(b_2) each consist of a single point, we assume, for contradiction, that σ_A(b_1) contains two distinct numbers, λ and μ We can select functions f and g from C(σ_A(b_1)) such that f(λ) = g(μ) = 1 and f ⊥ g = 0 According to the continuous functional calculus from Proposition 3 in the C*-algebra (π(A))', it follows that f(b_1) ⊥ g(b_1) = (f ⊥ g)(b_1) = 0, leading to f(b_1) and g(b_1) both equating to zero Since g(b_1) commutes with every element of π(A) and π is irreducible, g(b_1) ⊥ H forms an A-invariant, dense subspace of H The fact that f(b_1) vanishes on this subspace necessitates that f(b_1) = 0, which contradicts the isometric nature of the continuous functional calculus from Proposition 3 Therefore, σ_A(b_1) must consist of a single point, indicating that C(σ_A(b_1)) is one-dimensional Given that the continuous functional calculus C(σ_A(b_1)) → A is an isometric embedding with b_1 and id_H in its image, we conclude that b_1 = λ id_H for some λ ∈ C By applying the same reasoning, we find that b_2 and consequently b are also multiples of the identity.
In the context of operator theory, if we consider the equation (π(A)) ′ = Cãid H, and let K be a closed subspace of H that is invariant under the operator A, we can define p as the orthogonal projection from H onto K The invariance property π(A)K ⊂ K implies that p commutes with all operators in π(A), leading to the conclusion that p can be expressed in the form p = λãid H, where λ is a complex number As p is a projection operator, it satisfies the condition p² = p, which results in the equation λ² = λ.
Theorem 2 Letτ be a state on a C ∗ -algebra A Then the following two statements are equivalent:
2 The GNS representation(H τ, πτ)isirreducible, i.e., H τ has no nontrivial closed A-invariant subspace.
Example 13.The GNS representation of the algebraA=Mat(n×n;C) for a vector state for Ω ∈ C n is the standard representation of A on C n , hence irreducible. Therefore, such vector states are pure.
The proof of Theorem 2 begins by considering a pure state τ and a positive element v in L(H) with a norm less than or equal to 1, which commutes with every element in πτ(A) This leads to the definition of a function ̺: A → C, given by ̺(a) = πτ(a)ãvΩτ, which acts as a positive linear functional on A It satisfies the inequality ̺(a* a) ≤ τ(a* a) for all a in A, implying that ̺ can be expressed as ̺ = λãτ for some λ in the interval [0, 1] Consequently, for any arbitrary elements a and b in A, we derive the relation in the pre-Hilbert space A/Nτ: vã(a + Nτ), (b + Nτ)τ = vãπτ(a)Ωτ, πτ(b)Ωττ.
This impliesv=λid H τ, sinceA/N τ is dense inH τ By Proposition 1, we conclude thatπ τ is irreducible.
Now suppose that πτ is irreducible Let̺ be a positive linear functional on A such that̺(a ∗ a)≤τ(a ∗ a) for alla ∈ A Then the pairing
The expression (a+N τ ,b+N τ )→̺(b ∗ a) defines a positive semi-definite, Hermitian sesquilinear form on A/N τ This form is majorized by ã and ã τ, allowing it to extend to an inner product ã,ã ̺ on the Hilbert space H τ Consequently, there exists a bounded positive operator m∈L(H) such that the relationship x,y ̺ = x,my τ holds for all x,y∈ H τ.
0≤̺(a ∗ a)= πτ(a)Ω,mπτ(a)Ω τ ≤τ(a ∗ a)= πτ(a)Ω, πτ(a)Ω τ yieldsm ≤1 For everya,b,c∈ A, we have
By Theorem 1,mis a multiple of the identity and thus̺is a multiple of the state τ This shows thatτ is pure
Lemma 5 In a unital C ∗ -algebra A, every state is a pointwise limit of convex com- binations of pure states.
According to Corollary 4, convex combinations of pointwise limits of states are indeed states, establishing that S(A) forms a bounded closed convex set under the topology of pointwise convergence The Banach–Alaoglu theorem indicates that S(A) is a compact subset within the closed unit ball of the dual space of A, also in the topology of pointwise convergence Furthermore, the Krein–Milman theorem asserts that S(A) is the closed convex hull of its extreme points, which, as noted in Remark 11, include all pure states.
To demonstrate that all extreme points in S(A) are pure, consider an extreme point τ ∈ S(A) and a positive linear functional ̺ on A such that ̺(a ∗ a) ≤ τ(a ∗ a) for all a ∈ A, with the assumption that τ = ̺ = 0 By setting t = ̺ ∈ (0,1), we derive τ = tã ̺ + (1−t)ã (τ−̺), leading to τ − ̺ = τ(1) − ̺(1) = τ − ̺ = 1 − t, as noted in Remark 8 Consequently, we find that ̺/̺ (τ − ̺)/τ − ̺ = τ, confirming that τ is indeed an extreme point of S(A) Thus, we conclude that ̺ = tτ, which establishes that τ is a pure state.
The restriction of a pure state to a subalgebra may not retain its purity For instance, consider A as Mat(4×4;C), where the vector state τ corresponding to the unit vector Ω = (1, 0, 0, 1) is pure When embedding the subalgebra B = Mat(2×2;C) into A through the mapping b → b0, the resulting state may lose its pure characteristics.
The restriction ofτ toByields the trace state ofBwhich is not pure.
The converse situation can occur where τ1 represents the vector state of A as (1,0,0,0) and τ2 as (0,0,1,0) In this case, τ = 1/2τ1 + 1/2τ2 is not a pure state for A; however, it does restrict to a pure state for B, specifically the vector state (1,0).
Product States
This section focuses on states within the tensor product of C∗-algebras, highlighting the nonuniqueness of norms that transform the algebraic tensor product into a C∗-algebra In contrast, the norm that converts the algebraic tensor product of Hilbert spaces into a pre-Hilbert space is unique Consequently, it is logical to explore norms on algebras through the lens of norms on representation spaces We will utilize the finest norm topology that enables the algebraic tensor product of C∗-algebras to function as a C∗-algebra, assuming that the C∗-algebras under consideration are unital.
Remark 14.Let (H,ã,ã H ) and (K,ã,ã K ) be Hilbert spaces Then there is a unique inner productã,ãon the algebraic tensor product ofHandK such that x⊗y,x ′ ⊗y ′ = x,x ′ H ã y,y ′ K ∀x,x ′ ∈ H,y,y ′ ∈K.
The tensor product of Hilbert spaces, denoted as H⊗K, is formed by completing the algebraic tensor product with respect to a specific inner product Additionally, for bounded operators a ∈ L(H) and b ∈ L(K), there exists a unique operator a⊗b ∈ L(H⊗K).
Given two C ∗ -algebras (A, ã A ,∗) and (B, ã B ,∗), we want to construct
C*-norms can be applied to the algebraic tensor product A⊗B through universal representations By defining a C-antilinear map ∗ : A⊗B → A⊗B, where (a⊗b) ∗ := a ∗ ⊗ b ∗ for homogeneous elements and extending it bilinearly, the algebraic tensor product is transformed into an involutive algebra.
Lemma 6 Let(H, ϕ)and(K, ψ)be representations of A and B, respectively Then there is a unique∗-homomorphismπ :A⊗B→L(H⊗K)such that π(a⊗b)=ϕ(a)⊗ψ(b) ∀a ∈ A,b∈ B.
Moreover, if the representationsϕandψare faithful, then so isπ.
Proof The map A×B →L(H⊗K), (a,b)→ ϕ(a)⊗ψ(b) is bilinear and thus yields a unique linear mapπ : A⊗B →L(H⊗K) as claimed, which is indeed a
∗-morphism If bothϕandψare injective andz ∈ A⊗Bsatisfiesπ(z)=0, then by writingz= n j = 1 a j ⊗b j with linearly independentb j , we concludeϕ(a j )=0 for j =0, ,n Hencea j =0 for j =1, ,nand thusz=0
By this lemma, it is natural to make use of the universal representation from Definition 14 to obtain aC ∗ -norm on the algebraic tensor product.
In the context of C*-algebras, let (A, ã A , ∗) and (B, ã B , ∗) represent two C*-algebras equipped with universal representations π A : A → L(H) and π B : B → L(K) The injective C*-norm ã ι on the algebraic tensor product is defined by c ι := π(c), where π : A⊗B → L(H⊗K) is the unique ∗-morphism induced by the product of the representations π A and π B, as established in Lemma 6 The process involves completing the algebraic tensor product in accordance with this norm.
C ∗ -norm ã ι is called theinjectiveC ∗ -tensor productand is denoted by A⊗ ι B.
The injective morphism π created through universal representations establishes a valid C*-norm on A⊗B Additionally, an alternative C*-norm on A⊗B can be formed by taking the supremum of all existing C*-norms According to Remark 2, this process preserves the unit.
∗-morphismπfrom the algebraic tensor productA⊗Bto aC ∗ -algebraC satisfies π(x) ≤ x γ with respect to anyC ∗ -norm ã γ on A⊗ B This yields the following characterization of the maximalC ∗ -norm onA⊗B:
Definition 18 Let(A, ã A ,∗)and (B, ã B ,∗)be C ∗ -algebras Theprojective
C ∗ -norm ã π on the algebraic tensor product A⊗B is defined by c π :=inf
The completion of A⊗B with respect to the C ∗ -norm ã π is called theprojective
C ∗ -tensor productand is denoted by A⊗ π B.
Remark 15.The projectiveC ∗ -norm ã π satisfiesa⊗b π = a A ã b B for all a ∈ A,b ∈ B Clearly, any otherC ∗ -norm ã γ on A⊗B satisfiesc γ ≤ c π for allc∈ A⊗B, hence the projectiveC ∗ -norm is maximal among allC ∗ -norms on
A⊗B One can show that the injectiveC ∗ -norm ã ιon the other hand is minimal among allC ∗ -norms onA⊗B.
The projectiveC ∗ -tensor product has the following universal property.
Lemma 7 Let A, B, and C be C ∗ -algebras and letϕ : A → C andψ : B → C be∗-morphisms such thatϕ(a)andψ(b)commute for all a∈ A, b∈ B Then there exists a unique∗-morphismπ :A⊗ π B→C such that π(a⊗b)=ϕ(a)ãψ(b) ∀a ∈ A,b∈ B (1.5)
The bilinear map A × B → C, defined by (a, b) → ϕ(a)ãψ(b), induces a unique linear map π: A⊗B → C that satisfies the given conditions This map is identified as a *-morphism Furthermore, the map ã γ: A⊗B → R, given by c → π(c) γ, establishes a C* norm, ensuring that c γ ≤ c π for all c in A⊗B Consequently, π is continuous with respect to the projective C* norm, allowing it to uniquely extend from the dense subset A⊗B to the projective C* tensor product A⊗ π B.
Now we study states on the projectiveC ∗ -tensor product A⊗ π B Taking linear functionalsμ: A→Candν:B→C, setting
(μ⊗ν)(a⊗b) :=μ(a)ãν(b) on homogeneous elements and extending bilinearly, we obtain a linear functional on
A⊗B In the projectiveC ∗ -norm, we haveμ⊗ν π = μ A ã ν B Furthermore, for the homogeneous elementsa⊗b, we have
Hence the functionalμ⊗ν :A⊗B →Cis positive, ifμandνare.
Definition 19 Let A and B be C ∗ -algebras and letμ ∈ S(A)andν ∈ S(B)be states The unique extension ofμ⊗νto a state on the projective C ∗ -tensor product
Since AandB have a unit, we can restrict a stateτ ∈ S(A⊗B) to one of the factors by setting τ A (a) :=τ(a⊗1) ∀a∈ A τ B (b) :=τ(1⊗b) ∀b∈ B
For any two states μ in S(A) and ν in S(B), there exists a state τ in S(A⊗π B) such that τ A = μ and τ B = ν, represented as the product state τ = μ⊗ν Consequently, τ can be expressed as τ = τ A ⊗ τ B Therefore, measuring an observable in the state τ of A⊗π B yields the product of measurements from the states τ A and τ B.
In general this is not the case, so we set the following.
Definition 20 A stateτ ∈ S(A⊗ π B)is calledcorrelated, if there exists a∈ A and b∈ B such thatτ(a⊗b)=τ A (a)ãτ B (b).
Definition 21 A stateτ ∈S(A⊗π B)is calleddecomposable, if it is the pointwise limit of convex combinations of product states A state τ ∈ S(A⊗π B)is called entangled, if it is not decomposable.
In the literature, the weak-* limit refers to the pointwise limit of linear functionals Consequently, the set of decomposable states can be identified as the weak-* closure of the convex hull formed by the product states.
A pure state on A⊗π B cannot be expressed as a convex combination of different states or as a pointwise limit of such combinations Therefore, a pure state is considered decomposable if and only if it is a product state.
The set of decomposable states forms a convex subset within the realm of positive linear functionals on the projective C∗-tensor product A⊗ π B Researchers aim to characterize this convex set through specific inequalities Although a complete characterization remains elusive, a notable inequality has been derived from Bell's work in the late 1950s concerning the Einstein–Podolsky–Rosen paradox Consequently, these inequalities are frequently known as generalized Bell's inequalities.
Lemma 8 Let A and B be C ∗ -algebras and letτ be a decomposable state on the projective C ∗ -tensor product A⊗ π B Then
|τ(a⊗(b−b ′ ))| + |τ(a ′ ⊗(b+b ′ ))| ≤2 (1.6) holds for all self-adjoint elements a,a ′ ∈ A, b,b ′ ∈ B of norm≤1.
Proof For a product stateτ =μ⊗ν, we have τ(a⊗(b−b ′ ))=μ(a)ãν(b)−μ(a)ãν(b ′ )
By assumption,|μ(a)|,|μ(a ′ )|,|ν(b)|,|ν(b ′ )| ≤1, so we have
Hence, (1.6) holds for all product states Ifτ is a convex combination of product states,τ = n j = 1λ j μ j ⊗ν j , we obtain
Taking pointwise limits of convex combinations, the inequality holds by continuity
Example 15.Let A = B = Mat(2×2;C) be matrix algebras Lete 1,e 2 be the standard basis ofC 2 On A⊗B, we have the Bell stateτ, which is the vector state with the vector Ω := 1
It is easy to see that the Bell state is entangled For instance, the observables a 1 0
Thus the stateτ violates Bell’s inequality and is therefore entangled by Lemma 8.
Bell's inequalities are fundamental principles that apply universally to classical systems, highlighting the absence of entangled states The presence of entangled states serves as a defining feature of quantum systems Specifically, if one of the observable algebras is abelian, such as in classical systems, entangled states do not exist in the combined system A⊗π B.
Proposition 6 Let A and B be C ∗ -algebras with unit If A or B is abelian, then all states on the projective C ∗ -tensor product A⊗π B are decomposable.
According to Theorem 2, it is sufficient to demonstrate that every pure state τ on A⊗π B is a product state We assert that the equation τ(x y) = τ(x)ãτ(y) is valid for all x in A⊗π B and all y in Z(A⊗π B), where Z(A⊗π B) represents the center of A⊗π B.
Z(A⊗ π B) is spanned by its positive elements of norm≤1, it suffices to prove the claim forypositive, i.e.,y=z 2 for a self-adjointz∈ Z(A⊗ π B), withy ≤1 If τ(y)=0, the Cauchy–Schwarz inequality
=τ(x yx ∗ )ãτ(y) implies τ(x y) = 0 If τ(y) = 1, then τ(1−y) = 0; thus 0 = τ(x(1− y)) τ(x)ãτ(y)−τ(x y).
Since y∈ Z(A⊗π B), we haveτ 1 (x ∗ x)= τ(y) 1 ãτ(x ∗ x y)= τ(y) 1 ãτ((zx) ∗ zx)≥0. Similarly,τ 2 (x ∗ x)= 1 − 1 τ(y) ãτ(x ∗ x(1−y))≥0, since τ(x ∗ x y)=τ(x ∗ z ∗ zx)≤ z ∗ z ãτ(x ∗ x)≤τ(x ∗ x).
Clearly,τ 1 (1) = τ 2 (1) = 1; henceτ 1 andτ 2 are states on A⊗π B Since τ is a pure state by assumption, we concludeτ = τ1 = τ2 Henceτ1(x) =τ2(x) for all x∈ A⊗π B, which yieldsτ(x y)=τ(y)ãτ(x).
Now ifAis abelian, thenA⊗ π {1} ⊂Z(A⊗ π B) As we have seen, every pure stateτ on A⊗ π Bsatisfies τ(a⊗b)=τ((a⊗1)ã(1⊗b))=τ A (a)ãτ B (b) ∀a∈ A,b∈ B.
Weyl Systems
This section presents Weyl systems and CCR representations, which formalize the canonical commutator relations from quantum field theory in an exponentiated form The key outcome is Theorem 3, which establishes that for every symplectic vector space, there exists an essentially unique CCR representation Our methodology aligns with concepts from reference [7], and an alternative proof of this result is also available.
Let (V, ω) be asymplectic vector space, i.e.,V is a real vector space of finite or infinite dimension andω:V ×V →Ris an antisymmetric bilinear map such that ω(φ, ψ)=0 for allψ ∈V impliesφ=0.
Definition 22 AWeyl systemof(V, ω)consists of a C ∗ -algebra A with unit and a map W :V → A such that for allφ, ψ∈V we have
Condition (iii) indicates that W serves as a representation of the additive group V in A, adjusted by the "twisting factor" e^(-iω(φ,ψ)/2) It is important to note that V lacks a specified topology, which means that continuity is not a requirement for W In fact, we will demonstrate that this holds true even in certain cases.
V is finite dimensional and soV carries a canonical topologyW will in general not be continuous.
In this article, we develop a Weyl system for a general symplectic vector space (V, ω) We define the Hilbert space H as L²(V, C), which comprises square-integrable complex-valued functions on V, adhering to the counting measure Specifically, H includes functions F: V → C that are zero almost everywhere, except at countably many points.
The Hermitian product onHis given by
Let A :=L(H) be theC ∗ -algebra of bounded linear operators on H as in Exam- ple 1 We define the mapW :V → Aby
Obviously, W(φ) is a bounded linear operator on H for anyφ ∈ V andW(0) id H =1 We check (ii) by making the substitutionχ=φ+ψ:
=(F,W(−φ)G) L 2 HenceW(φ) ∗ =W(−φ) To check (iii) we compute
ThusW(φ)W(ψ)=e − iω(φ,ψ)/2 W(φ+ψ) Let CCR(V, ω) be theC ∗ -subalgebra of L(H) generated by the elementsW(φ),φ∈V Then CCR(V, ω) together with the mapW forms a Weyl system for (V, ω).
Proposition 7 Let (A,W)be a Weyl system of a symplectic vector space (V, ω).
3 the algebra A is not separable unless V = {0},
4 the family{W(φ)} φ ∈ V is linearly independent.
W(φ)W(φ) ∗ =1 we see thatW(φ) is unitary.
To show Assertion 2 letφ, ψ∈V withφ=ψ For arbitraryχ∈V we have
The real number ω(χ, φ − ψ) spans all of R as χ varies over V, indicating that the spectrum of W(φ − ψ) is U(1)-invariant According to Assertion 5 of Proposition 2, this spectrum is confined within S¹, and Proposition 1 confirms its nonemptiness Consequently, we conclude that σ A (W(φ − ψ)) = S¹, which leads to the result that σ A (e^(iω(ψ,φ)/2) W(φ − ψ)) also equals S¹.
Thus σ A (e iω(ψ,φ)/2 W(φ−ψ)−1) is the circle of radius 1 centered at −1 Now Assertion 3 of Proposition 2 says e iω(ψ,φ)/2 W(φ−ψ)−1 =ρ A e iω(ψ,φ)/2 W(φ−ψ)−1
This shows part 2 Assertion 3 now follows directly since the balls of radius 1 cen- tered at W(φ),φ ∈ V form an uncountable collection of mutually disjoint open subsets.
We show Assertion 4 Let φ j ∈ V, j = 1, ,n be pairwise different and let n j = 1α j W(φ j ) = 0 We showα1 = ã ã ã = α n = 0 by induction onn The case n =1 is trivial by Assertion 1 Without loss of generality assumeα n =0 Hence
−α j α n e − iω( − φ n ,φ j )/2 W(φ j −φ n ) n − 1 j = 1 β j W(φ j −φ n ), where we have putβ j := − α α n j e iω(φ n ,φ j )/2 For an arbitraryψ∈V we obtain
From n − 1 j= 1 β j W(φ j −φ n ) n − 1 j= 1 β j e − iω(ψ,φ j − φ n ) W(φ j −φ n ) we conclude by the induction hypothesis β j =β j e − iω(ψ,φ j − φ n ) for all j=1, ,n−1 If someβ j =0, thene − iω(ψ,φ j − φ n ) =1, hence ω(ψ, φ j −φ n )=0 for allψ∈V Sinceωis nondegenerateφ j −φ n =0, a contradiction Therefore all β j and thus allα j are zero, a contradiction
In a Weyl system (A,W) of the symplectic vector space (V, ω), the linear span of W(φ) for φ in V is closed under multiplication and the adjoint operation This span is denoted as W(V) ⊂ A If another Weyl system (A′, W′) exists for the same symplectic vector space, a unique linear map π: W(V) → W′(V) can be established, defined by π(W(φ)) = W′(φ) This map acts as a bijection on the bases {W(φ)} and {W′(φ)}, making π a linear isomorphism and a ∗-isomorphism due to the properties of Weyl systems Consequently, a unique ∗-isomorphism exists that ensures the commutativity of the associated diagram.
Remark 18.OnW(V)we can define the norm φ a φ W(φ)
This norm is not aC ∗ -norm but for everyC ∗ -norm ã 0 onW(V)we have by the triangle inequality and by Assertion 1 of Proposition 7 a 0≤ a 1 (1.7) for alla ∈ W(V).
Lemma 9 Let(A,W)be a Weyl system of a symplectic vector space(V, ω) Then a max :=sup{a 0 | ã 0 is a C ∗ -norm onW(V)} defines a C ∗ -norm onW(V).
The C∗-norm defined on A restricts to W(V), ensuring the supremum is not derived from an empty set Additionally, the estimate (1.7) confirms that this supremum is finite The characteristics of a C∗-norm are straightforward to verify, with the triangle inequality being evident from the relation a + b = max{sup{a + b | a ≥ 0}} for a C∗-norm on W(V).
≤sup{a 0 + b 0 | ã 0 is aC ∗ –norm onW(V)}
≤sup{a 0 | ã 0 is aC ∗ –norm onW(V)} +sup{b 0 | ã 0 is aC ∗ –norm onW(V)}
The other properties are shown similarly
Lemma 10 Let(A,W)be a Weyl system of a symplectic vector space(V, ω) Then the completionW(V) max ofW(V)with respect to ã max issimple, i.e., it has no nontrivial closed two-sided∗-ideals.
Proof By Remark 17 we may assume that (A,W) is the Weyl system constructed in Example 16 In particular,W(V)carries theC ∗ -norm ã Op, the operator norm given byW(V) ⊂L(H) whereH=L 2 (V,C).
Let I ⊂ W(V) max be a closed two-sided ∗-ideal ThenI 0 := I ∩CãW(0) is a (complex) vector subspace inCãW(0) = Cã1 ∼= Cand thus I 0 = {0}or
I 0 =CãW(0) If I 0 =CãW(0), then I contains 1 and thereforeI = W(V) max Hence we may assumeI 0= {0}.
Now we look at the projection map
We check that P extends to a bounded operator onW(V) max Letδ0 ∈ L 2 (V,C) denote the function given by δ0(0) = 1 and δ0(φ) = 0 otherwise For a φ a φ W(φ) andψ∈V we have
P(a) max = a 0 W(0) max = |a 0| = |(δ0,aãδ0) L 2 | ≤ a Op ≤ a max , which shows that Pextends to a bounded operator onW(V) max
Now leta ∈I ⊂ W(V) max Fixǫ >0 We write a=a 0 W(0)+ n j = 1 a j W(φ j )+r, where theφ j =0 are pairwise different and the remainder termrsatisfiesr max < ǫ For anyψ∈V we have
I ∋W(ψ)a W(−ψ)=a 0 W(0)+ n j = 1 a j e − iω(ψ,φ j ) W(φ j )+r(ψ), wherer(ψ) max = W(ψ)r W(−ψ) max ≤ r max < ǫ If we chooseψ1andψ2 such thate − iω(ψ 1 ,φ n ) = −e − iω(ψ 2 ,φ n ) , then adding the two elements a 0 W(0)+ n j = 1 a j e − iω(ψ 1 ,φ j ) W(φ j )+r(ψ1)∈ I a 0 W(0)+ n j = 1 a j e − iω(ψ 2 ,φ j ) W(φ j )+r(ψ2)∈ I yields a 0 W(0)+ n− 1 j = 1 a ′ j W(φ j )+r 1 ∈ I,
1 C -algebras 35 wherer 1 max = r(ψ 1 ) + 2 r(ψ 2 ) max < ǫ + 2 ǫ =ǫ Repeating this procedure we eventu- ally get a 0 W(0)+r n ∈I, wherer n max< ǫ Sinceǫis arbitrary and Iis closed we conclude
Fora φ a φ W(φ) ∈ I and arbitraryψ ∈ V we haveW(ψ)a ∈ I as well, hence P(W(ψ)a) = 0 This meansa − ψ = 0 for all ψ, thusa = 0 This shows
Definition 23 A Weyl system(A,W)of a symplectic vector space(V, ω)is called aCCR representationof(V, ω)if A is generated as a C ∗ -algebra by the elements
W(φ),φ∈V In this case we call A aCCR-algebraof(V, ω).
Of course, for any Weyl system (A,W) we can simply replace A by the
C ∗ -subalgebra generated by the elements W(φ), φ ∈ V, and we obtain a CCR representation.
Existence of Weyl systems, and hence CCR representations, has been established in Example 16 Uniqueness also holds in the appropriate sense.
Theorem 3 Let(V, ω)be a symplectic vector space and let(A 1,W 1)and(A 2,W 2) be two CCR representations of(V, ω).
Then there exists a unique∗-isomorphismπ: A 1→ A 2 such that the diagram
To demonstrate that the ∗-isomorphism π: W 1(V) → W 2(V) extends to an isometry (A 1, ã 1) → (A 2, ã 2), we note that the pullback of the norm ã 2 on A 2 to W 1(V) through π is a C∗-norm This implies that for all a ∈ W 1(V), the inequality π(a)² ≤ a_max holds, indicating that π can be extended to a ∗-morphism.
W 1(V) max → A 2 By Lemma 10 the kernel ofπ is trivial, henceπ is injective. Proposition 4 implies thatπ : (W 1(V) max , ã max )→(A 2, ã 2 ) is an isometry.
In the special case (A 1, ã 1 ) =(A 2, ã 2 ) whereπis the identity this yields ãmax= ã1 Thus for arbitraryA 2the mapπextends to an isometry (A 1,ã1)→
From now on we will call CCR(V, ω) as defined in Example 16theCCR-algebra of (V, ω).
Corollary 5 CCR-algebras of symplectic vector spaces are simple, i.e., all unit- preserving∗-morphisms to other C ∗ -algebras are injective.
Proof Direct consequence of Corollary 2 and Lemma 10
Corollary 6 Let(V 1, ω1)and(V 2, ω2)be two symplectic vector spaces and let S :
V 1 → V 2 be asymplectic linear map, i.e.,ω2(Sφ,Sψ) =ω1(φ, ψ)for allφ, ψ ∈
Then there exists a unique injective ∗-morphism CCR(S) : CCR(V 1, ω 1 ) → CCR(V 2, ω 2 )such that the diagram
Proof One immediately sees that (CCR(V 2, ω 2 ),W 2 ◦ S) is a Weyl system of
From uniqueness of the map CCR(S) we conclude that CCR(id V )=idCCR(V,ω) and CCR(S 2◦S 1) = CCR(S 2)◦CCR(S 1) In other words, we have constructed a functor
The category CCR, defined as SymplVec→C ∗ Alg, encompasses symplectic vector spaces as its objects and symplectic linear maps as its morphisms Specifically, a morphism A: (V 1, ω1)→(V 2, ω2) is characterized by the condition A ∗ ω2 = ω1, ensuring the preservation of the symplectic structure In this context, C ∗ Alg represents the category where the objects are C*-algebras, highlighting the interplay between symplectic geometry and functional analysis.
C ∗ -algebras and whose morphisms are injective unit-preserving ∗–morphisms. Observe that symplectic linear maps are automatically injective.
In the caseV 1=V 2the induced∗-automorphisms CCR(S) are calledBogoliubov transformationin the physics literature.
1 B¨ar, C., Ginoux, N., Pf¨affle, F.: Wave equations on Lorentzian manifolds and quantization. EMS Publishing House, Z¨urich (2007) 1
2 Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics I. Springer, Berlin Heidelberg (2002) 1
3 Davidson, K.: C -algebras by example AMS, Providence (1997) 1
4 Dixmier, J.: Les C*-alg`ebres et leurs repr´esentations, 2nd edition, Gauthier-Villars ´ Editeur, Paris (1969) 1
5 Murphy, G.: C*-algebras and operator theory Academic Press, Boston (1990) 1
6 Takesaki, M.: Theory of Operator Algebra I, Springer, Berlin, Heidelberg, New York (2002) 1
7 Manuceau, J.: C*-alg`ebre de relations de commutation Ann Inst H Poincar´e Sect A (N.S.)
8 Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics II. Springer, Berlin Heidelberg 2002 29
9 Baez, F.: Bell’s inequality for C ∗ -Algebras Lett Math Phys 13(2), 135–136 (1987) 26
In this chapter some basic notions from Lorentzian geometry will be reviewed.
In particular causality relations will be explained, Cauchy hypersurfaces and the concept of global hyperbolic manifolds will be introduced Finally the structure of globally hyperbolic manifolds will be discussed.
More comprehensive introductions can be found in [1] and [2].
Preliminaries on Minkowski Space
LetV be ann-dimensional real vector space ALorentzian scalar productonV is a nondegenerate symmetric bilinear formã,ãof index 1 This means one can find a basise 1 , ,e n ofV such that e i ,e j ⎧⎨
The Minkowski product is a fundamental example of a Lorentzian scalar product in R^n, defined as x,y = -x_1 y_1 + x_2 y_2 + + x_n y_n This relationship implies that any n-dimensional vector space equipped with a Lorentzian scalar product (V, ·, ·) is isometric to Minkowski space (R^n, ·, ·_0).
We denote the quadratic form associated withã,ãby γ :V →R, γ(X) := −X,X.
A vector X ∈ V \ {0}is calledtimelikeifγ(X) > 0,lightlikeifγ(X) = 0 and
X=0,causalif timelike or lightlike, andspacelikeifγ(X)>0 orX =0.
Institut f¨ur Mathematik, Universit¨at Potsdam, Am Neuen Palais 10, D-14469 Potsdam, Germany e-mail: pfaeffle@math.uni-potsdam.de
Forn ≥2 the setI(0) of timelike vectors consists of two connected components.
We choose a time-orientationon V by picking one of these two connected com- ponents Denote this component by I + (0) and call its elementsfuture-directed We put J + (0) := I + (0),C + (0) := ∂I + (0), I − (0) := −I + (0), J − (0) := −J + (0), and
C − (0) := −C + (0) Causal vectors in J + (0) (or in J − (0)) are calledfuture-directed
In a vector space V, when a positive number α > 0 is applied to a Lorentzian scalar product ã, the resulting product αã maintains the same properties Specifically, a vector X in V is considered timelike under the original product ã if and only if it remains timelike under the scaled product αã Similarly, the classifications of lightlike, causal, and spacelike vectors are consistent between both scalar products, demonstrating their equivalence in these respects.
Hence, for both Lorentzian scalar products one gets the same setI(0) If dim(V)≥
2 and we choose identical time-orientations forã,ãandαã ã,ã, the setsI ± (0),
J ± (0),C ± (0) are determined independently whether formed with respect toã,ã orαã ã,ã.
Lorentzian Manifolds
ALorentzian manifoldis a pair (M,g) whereM is ann-dimensional smooth man- ifold and g is a Lorentzian metric, i.e., g associates with each point p ∈ M a Lorentzian scalar productg p on the tangent spaceT p M.
For a function \( p \) to depend smoothly on \( p \), it is essential that, for any selection of local coordinates \( x = (x_1, \ldots, x_n) \) mapping from an open subset \( U \subset M \) to another open subset \( V \subset \mathbb{R}^n \), the functions \( g_{ij} : V \to \mathbb{R} \), defined by \( g_{ij} = g\left(\frac{\partial x}{\partial i}, \frac{\partial x}{\partial j}\right) \), must be smooth for all indices \( i, j = 1, \ldots, n \) Here, \( \frac{\partial x}{\partial i} \) and \( \frac{\partial x}{\partial j} \) represent the standard coordinate vector fields.
(see Fig 2.2) With respect to these coordinates one writesg i, j g i j d x i ⊗d x j or shortlyg i, j g i j d x i d x j
Next we will give some prominent examples for Lorentzian manifolds.
Example 1.In cartesian coordinates (x 1, ,x n ) on R n the Minkowski metric is defined byg Mi nk = −(d x 1) 2 +(d x 2) 2 + ã ã ã +(d x n ) 2 This turns Minkowski space into a Lorentzian manifold.
Of course, the restriction ofg Mi nk to any open subsetU⊂R n yields a Lorentzian metric onUas well.
Example 2.Consider the unit circle S 1 ⊂ R 2 with its standard metric (dθ) 2 The
Lorentzian cylinderis given by M = S 1 ×Rtogether with the Lorentzian metric g= −(dθ) 2 +(d x) 2
In a connected Riemannian manifold \( (N,h) \) with an open interval \( I \subset \mathbb{R} \), for any \( t \in I \) and \( p \in N \), the tangent space is identified as \( T(t, p)(I \times N) = T_t I \oplus T_p N \) For a smooth positive function \( f: I \to (0, \infty) \), the Lorentzian metric \( g = -dt^2 + f(t)^2 d\theta^2 \) on \( I \times M \) is defined such that for any vectors \( \xi_1, \xi_2 \in T(t, p)(I \times N) \), one can express \( \xi^i = \alpha_i \frac{d}{dt} \).
⊕ζ i withα i ∈ Randζ i ∈ T p N,i = 1,2, and one has g(ξ1, ξ2) −α 1 ãα 2 + f(t) 2 ãh(ζ1, ζ 2 ) Such a Lorentzian metricgis called awarped product metric(Fig 2.3).
This article discusses Robertson–Walker spacetimes, emphasizing the need for the pair (N,h) to be complete and exhibit constant curvature Notably, Friedmann cosmological models fall into this category Within the framework of general relativity, these models are essential for exploring phenomena such as the big bang, the expansion of the universe, and the cosmological redshift.
5 and 6] or [1, Chap 12] A special case of this isdeSitter spacetimewhereI =R,
N =S n − 1 ,his the canonical metric ofS n − 1 of constant sectional curvature 1, and f(t)=cosh(t).
Example 4.For a fixed positive numberm > 0 one considers the Schwarzschild function h: (0,∞)→Rgiven byh(r)=1− 2m r This function has a pole atr =0 and one hash(2m)=0 On bothP I = {(r,t) ∈R 2 |r >2m}andP I I = {(r,t)∈
In the context of Lorentzian metrics defined by \( g = -h(r) dt \otimes dt + \frac{1}{h(r)} dr \otimes dr \) for \( 0 < r < 2m \), the structures are referred to as the Schwarzschild half-plane \((P I, g)\) and the Schwarzschild strip \((P II, g)\) A tangent vector \( \alpha \frac{\partial}{\partial t} + \beta \frac{\partial}{\partial r} \) is considered timelike if \( \alpha^2 > h(r) \frac{1}{2} \beta^2 \) This relationship allows for a visual representation of the set of timelike vectors in the tangent spaces \( T_{(r,t)} P I \) and \( T_{(r,t)} P II \), as depicted in Figure 2.4.
The apparent singularity of the Lorentzian metric g at r = 2m is not as significant as it appears, as it can be resolved through a change of coordinates, such as utilizing Kruskal coordinates.
In the study of static rotationally symmetric black holes with mass m, the exterior and interior are analyzed using the metrics (P I ,g) and (P II ,g) as referenced in [1, Chap 13] This analysis involves the two-dimensional sphere S², equipped with its inherent Riemannian metric By employing the structures N = P I × S² and B = P II × S², a Lorentzian metric is derived for both configurations.
−h(r)ãdt⊗dt+ 1 h(r)ãdr⊗dr+r 2 ãcan S 2
Equipped with this metric, N is calledSchwarzschild exterior spacetime and B Schwarzschild black hole, both of massm.
Example 5.Let S n − 1 = {(x 1, ,x n ) ∈ R n |(x 1) 2 + ã ã ã + (x n ) 2 = 1} be the n-dimensional sphere equipped with its natural Riemannian metric can S n−1 The restriction of this metric to S + n − 1 = {(x 1, ,x n ) ∈ S n − 1 |x n > 0}is denoted by can S n−1
+ Then, onR×S + n − 1 one defines a Lorentzian metric by g Ad S = 1 (x n ) 2 ã −dt 2 +can S n−1
The definition of the n-dimensional anti-de Sitter spacetime, denoted as \( m \) and \( 2m \), is expressed as \( (R \times S + n - 1, g_{Ad S}) \) While this definition slightly differs from that presented in [1, Chap 8, p 228f.], it can be demonstrated that both definitions are equivalent, as shown in [3, Chap 3.5., p 95ff.].
According to Remark 1, a tangent vector in R×S^(n−1) is classified as timelike, lightlike, or spacelike in relation to the g Ad S metric if and only if it holds the same classification under the Lorentzian metric −dt² + can S^(n−1).
In general relativity one is interested in four-dimensional anti-deSitter spacetime because it provides a vacuum solution of Einstein’s field equation with cosmological constantΛ= −3; see [1, Chap 14, Example 41].
Time-Orientation and Causality Relations
In a Lorentzian manifold of dimension n ≥ 2, denoted as (M,g), the set of timelike vectors in the tangent space TpM at each point p ∈ M is divided into two connected components These components cannot be distinguished intrinsically, meaning they are not uniquely identifiable based on internal properties A time-orientation on M is established by selecting one of these connected components, denoted as I+(0) ⊂ TpM, in a manner that varies continuously with the point p.
A time-orientation (Fig 2.5) is given by a continuous timelike vector fieldτ on
Mwhich takes values in these chosen connected components:τ(p)∈I + (0)⊂T p M for eachp∈ M.
A Lorentzian manifold (M, g) is considered time-orientable if there exists a continuous timelike vector field τ on M When a Lorentzian manifold (M, g) is paired with such a vector field τ, it is termed time-oriented In this context, connected Lorentzian manifolds that are time-oriented will be referred to as spacetimes.
Time-orientability is a concept that relies on the Lorentzian metric, distinguishing it from orientability, which is solely determined by the topology of the underlying manifold.
Fig 2.6 Example for orientable and time-orientable manifold
The examples from Section 2.2 illustrate that all the Lorentzian manifolds discussed are time-orientable, as depicted in Figure 2.7 In Minkowski space, timelike vector fields can be represented using the partial derivative ∂x/∂.
In a Lorentzian cylinder, the tangent space at each point \( p \in M \) is denoted as \( T_p M \), equipped with the Lorentzian scalar product \( g_p \) and a time-orientation determined by the lightlike vector \( \tau(p) \) This framework is applicable across various spacetimes, including the warped product in Example 3, Schwarzschild exterior spacetime, Schwarzschild black hole, and anti-de Sitter spacetime, where the derivatives \( \partial_\theta, \partial_t, \partial_r \), and \( \partial_t \) are utilized, respectively Within \( (T_p M, g_p) \), key concepts such as timelike, lightlike, causal, and spacelike vectors, as well as future-directed vectors, are defined, following the explanations provided in Section 2.1.
A continuous piecewise C^1-curve in a manifold M is classified based on the nature of its tangent vectors It is termed timelike, lightlike, causal, spacelike, future-directed, or past-directed depending on whether all its tangent vectors exhibit the corresponding characteristics.
Fig 2.7 Lorentzian manifold which is orientable, but not time-orientable
Fig 2.8 Lorentzian manifold which is not orientable, but time-orientable
Thecausality relationsonMare defined as follows: Let p,q ∈ M, then one has p≪q :⇐⇒there is a future-directed timelike curve inM fromptoq, p 0 thatc(t) ∈ I + (0) for allt ∈ (0, ε) Hence P(c(t)) is timelike and future-directed fort ∈(0, ε).
Forξ =c(t),ζ1=2ξ = −gradγ(ξ) andζ2 =(d/dt)c(t) the Gauss lemma gives d dt (γ◦c) (t)= −g p (ζ1, ζ2)= −g exp p (ξ )
If there weret 1∈(0,b] withγ(c(t 1))=0, w.l.o.g lett 1be the smallest value in
(0,b] withγ(c(t 1))=0, then one could find at 0∈(0,t 1) with
On the other hand, having chosent 1 minimally implies thatP(c(t 0)) is timelike and future-directed Together with ˙c(t 0)∈ I + M (c(t 0)) this yieldsg exp p (ξ )
In the interval (0, b], the function γ(c(t)) remains positive for all t, indicating that the continuous curve c does not exit the initially defined connected component of I(0) This conclusion holds true under the assumption that c is smooth For a comprehensive proof applicable to more general cases, refer to [1, Chap 5, Lemma 33].
Definition 5 A domainΩ ⊂M is calledgeodesically starshapedwith respect to a fixed point p∈Ωif there exists an open subsetΩ ′ ⊂T p M, starshaped with respect to0, such that the Riemannian exponential mapexp x mapsΩ ′ diffeomorphically ontoΩ.
One calls a domain Ω ⊂ M geodesically convex (or simply convex) if it is geodesically starshaped with respect to all of its points.
Every point on a Lorentzian manifold has a convex neighborhood, as detailed in [1, Chap 5, Prop 7] Additionally, any open covering of a Lorentzian manifold can be refined into a collection of convex open subsets, according to [1, Chap 5, Lemma 10].
Sometimes sets that are geodesically starshaped with respect to a point pare useful to get relations between objects defined in the tangentT p M and objects defined on
M For the moment this will be illustrated by the following lemma.
Lemma 3 Let M a spacetime and p∈ M Let the domainΩ ⊂ M be a geodesi- cally starshaped with respect to p (Fig 2.15) LetΩ ′ be an open neighborhood of
0in T p M such thatΩ ′ is starshaped with respect to0andexp p | Ω ′ :Ω ′ −→Ω is a diffeomorphism Then one has
Proof We will only prove the equationI + Ω (p)=exp p
Fig 2.15 Ω is geodesically starshaped with respect to p
For any point \( q \) in the future-directed causal set \( I^+_\Omega(p) \), it is possible to identify a future-directed timelike curve \( c: [0,b] \to \Omega \) connecting \( p \) to \( q \) By defining the curve \( c \) in the tangent space \( T_p M \) as \( c = \exp_p^{-1} \circ c \), we can apply Lemma 2 to establish that \( c(t) \in I^+(0) \) for \( 0 < t \leq b \) Notably, we find that \( \exp_p^{-1}(q) = c(b) \in I^+(0) \), which demonstrates that \( I^+_\Omega(p) \) is a subset of \( \exp_p \).
Causality Condition and Global Hyperbolicity
In general relativity worldlines of particles are modeled by causal curves If now the spacetime is compact something strange happens.
Proposition 2 If the spacetime M is compact, there exists a closed timelike curve in M.
Proof The family{I + M (p)} p ∈ M is an open covering ofM By compactness one has
M =I + M (p 1 )∪ ã ã ã ∪I + M (p k ) for suitably chosen p 1 , ,p k ∈ M We can assume that p 1 ∈ I + M (p 2 )∪ ã ã ã ∪I + M (p k ), otherwise p 1 ∈ I + M (p m ) for anm≥2 and hence
I + M (p 1 ) ⊂ I + M (p m ) and we can omit I + M (p 1 ) in the finite covering Therefore we can assume p 1∈ I + M (p 1), and there is a timelike future-directed curve starting and ending in p 1
In spacetimes featuring timelike loops, paradoxes can arise from potential time travel to the past, a concept often explored in science fiction To maintain physical plausibility, compact spacetimes are typically excluded, necessitating the adherence to causality or strong causality conditions in reasonable spacetimes.
Definition 8 A spacetime is said to satisfy the causality condition if it does not contain any closed causal curve.
A spacetime M meets the strong causality condition if it lacks almost closed causal curves Specifically, for any point p in M and any open neighborhood U around p, there exists a smaller open neighborhood V within U such that all causal curves originating and terminating in V remain entirely within U.
Obviously, the strong causality condition implies the causality condition.
Example 9.In Minkowski space (R n ,g Mi nk ) the strong causality condition holds. One can prove this as follows: LetUbe an open neighborhood ofp=(p 1, ,p n )∈
R n For anyδ > 0 denote the open cube with center p and edges of length 2δ by
The set \( W \) is defined as \( W = (p_1 - \delta, p_1 + \delta) \times (p_n - \delta, p_n + \delta) \) There exists an \( \epsilon > 0 \) such that \( W^{2\epsilon} \subset U \), allowing us to set \( V = W_{\epsilon} \) It is observed that any causal curve \( c = (c_1, \ldots, c_n) \) in \( \mathbb{R}^n \) satisfies the inequality \( (\dot{c}_1)^2 \geq (\dot{c}_2)^2 + \ldots + (\dot{c}_n)^2 \) with \( (\dot{c}_1)^2 > 0 \) Therefore, we can conclude that any causal curve that begins and ends within \( V = W_{\epsilon} \) cannot exit the region \( W^{2\epsilon} \subset U \).
In a spacetime characterized by an open connected subset Ω of a manifold M that meets the strong causality condition, the absence of (almost) closed causal curves in M guarantees that such curves will also be absent in Ω Consequently, this ensures that Ω upholds the strong causality condition as well.
Example 10.In the Lorentzian cylinderS 1 ×Rthe causality condition is violated.
If one unwraps S 1 ×Ras in Example 8 it can be easily seen that there are closed timelike curves (Fig 2.20).
Fig 2.20 Closed timelike curve c in Lorentzian cylinder Identify! c
In the spacetime M, derived from the Lorentzian cylinder by removing two spacelike half-lines G1 and G2, the causality condition is satisfied, yet the strong causality condition is not upheld Specifically, for any point on the short lightlike curve and any sufficiently small neighborhood around that point, it is possible to find a causal curve that begins and ends within this neighborhood but does not remain entirely within it.
Definition 9 A spacetime M is called aglobally hyperbolic manifold if it satisfies the strong causality condition and if for allp,q ∈ Mthe intersectionJ + M (p)∩J − M (q) is compact.
Global hyperbolicity, introduced by J Leray, refers to a significant class of manifolds that exhibit desirable properties for wave equations These globally hyperbolic manifolds enable a robust global solution theory, as detailed in Chapter 3.
In Minkowski space (R^n, g_Mink), the sets J^+_R^n(p) and J^-_R^n(q) are both closed for any points p, q in R^n Additionally, the intersection of these sets, J^+_R^n(p) ∩ J^-_R^n(q), is bounded in terms of the Euclidean norm, making it compact As previously established in Example 9, Minkowski space satisfies the strong causality condition, confirming that it is globally hyperbolic.
The Lorentzian cylinder M = S¹ × R fails to meet the strong causality condition, indicating it is not globally hyperbolic Additionally, it does not satisfy the compactness condition outlined in Definition 9, as there exists an intersection between J⁺ₘ(p) and J⁻ₘ(q) for any points p, q in M.
Example 14.Consider the subsetΩ =R×(0,1) of two-dimensional Minkowski space (R 2 ,g Mi nk ) By Remark 5 the strong causality condition holds forΩ, but there
Fig 2.21 Causality condition holds but strong causality condition is violated
Fig 2.22 J + Ω ( p) ∩ J − Ω (q) is not always compact in the strip Ω = R × (0, 1) p q Ω are points p,q ∈Ωfor which the intersection J + Ω (p)∩J − Ω (q) is not compact, see Fig 2.22.
The n-dimensional anti-deSitter spacetime (R × S^(n−1), g_AdS) lacks global hyperbolicity, as illustrated in Figure 2.23 Similar to Example 5, a curve in M = R × S^(n−1) is considered causal with respect to the metric g_AdS if and only if it is also causal under the Lorentzian metric -dt² + can S^(n−1).
+ Hence for bothg Ad S and−dt 2 +can S n−1
+ one gets the same causal futures and pasts A similar picture as in Example 14 then shows that for p,q ∈ Mthe intersectionJ + M (p)∩J − M (q) need not be compact.
In general one does not know much about causal futures and pasts in spacetime. For globally hyperbolic manifold one has the following lemma (see [1, Chap 14, Lemma 22]).
Fig 2.23 J + M (p) ∩ J − M (q ) is not compact in anti-deSitter spacetime q p
Fig 2.24 For K is closed J + M (K ) need not be open
Lemma 4 In any globally hyperbolic manifold M the relation “≤” is closed, i.e., whenever one has convergent sequences p i → p and q i →q in M with p i ≤q i for all i , then one also has p≤q.
Therefore in globally hyperbolic manifolds for any p ∈ M and any compact set
K ⊂M one has thatJ ± M (p) and J ± M (K) are closed.
In two-dimensional Minkowski space, a curve K can be closed as a subset yet still result in a non-closed causal future, denoted as J ± M (K) For instance, as illustrated in Fig 2.24, curve K is asymptotic to a lightlike line, leading to its causal future J + M (K) being represented as an open half-plane that is bounded by this lightlike line.
Cauchy Hypersurfaces
We recall that a piecewise C 1 -curve in M is calledinextendible, if no piecewise
C 1 -reparametrization of the curve can be continuously extended beyond any of the end points of the parameter interval.
Definition 10 A subset S of a connected time-oriented Lorentzian manifold is calledachronal (or acausal) if and only if each timelike (or causal, respectively) curve meets S at most once.
A subset S of a connected time-oriented Lorentzian manifold is aCauchy hyper- surfaceif each inextendible timelike curve in M meets S at exactly one point.
Every acausal subset is achronal, but not all achronal sets are acausal Achronal spacelike hypersurfaces are always acausal, and any Cauchy hypersurface qualifies as achronal Cauchy hypersurfaces are closed topological surfaces intersected by every inextendible causal curve at least once Additionally, any two Cauchy hypersurfaces within a manifold M are homeomorphic, and their causal future and past are compact, as demonstrated in relevant literature.
Example 16.In Minkowski space (R n ,g Mi nk ) consider a spacelike hyperplane A 1 ,hyperbolic spaces A 2 = {x =(x 1, ,x n )| x,x = −1 andx 1 > 0}and A 3 {x =(x 1, ,x n )| x,x =0,x 1 ≥0} Then all A 1,A 2, and A 3are achronal, but onlyA 1is a Cauchy hypersurface; see Fig 2.25.
A 2 inextendible timelike curve which avoids both
In the context of a connected Riemannian manifold (N,h) and an open interval I ⊂ R, a smooth function f: I → (0,∞) leads to the warped product metric g = -dt² + f(t)ãh on M = I × N The hypersurface {t₀} × N is a Cauchy hypersurface in the manifold (M,g) for any t₀ ∈ I if and only if the Riemannian manifold (N,h) is complete.
In particular, in any Robertson–Walker spacetime one can find a Cauchy hyper- surface.
In the context of Schwarzschild spacetime, let N represent the exterior Schwarzschild spacetime and B denote the Schwarzschild black hole, both having the same mass m For any fixed time t₀ within the range of the Cauchy hypersurface of N, the hypersurface is defined as (2m, ∞) × {t₀} × S² Conversely, in the Schwarzschild black hole B, a Cauchy hypersurface is formed by {r₀} × R × S², where r₀ is constrained to the interval 0 < r₀ < 2m.
Definition 11 TheCauchy development(Fig 2.26) of a subset S of a spacetime M is the set D(S)of points of M through which every inextendible causal curve in M meets S, i.e.,
D(S)= p∈ Mevery inextendible causal curve passing through p meets S!
Remark 6.It follows from the definition that D(D(S)) = D(S) for every subset
Of course, ifSis achronal, then every inextendible causal curve inMmeetsSat most once The Cauchy developmentD(S) of everyacausalhypersurfaceSis open, see [1, Chap 14, Lemma 43].
IfS⊂Mis a Cauchy hypersurface, then obviouslyD(S)=M.
For a proof of the following proposition, see [1, Chap 14, Thm 38].
Proposition 3 For any achronal subset A ⊂ M the interior int(D(A)) of the Cauchy development is globally hyperbolic (if nonempty).
From this we conclude that a spacetime is globally hyperbolic if it possesses aCauchy hypersurface In view of Examples 17 and 18, this shows that Robertson–
Walker spacetimes, Schwarzschild exterior spacetime, and Schwarzschild black hole are all globally hyperbolic.
The following theorem is very powerful and describes the structure of glob- ally hyperbolic manifolds explicitly: they are foliated bysmooth spacelikeCauchy hypersurfaces.
Theorem 1 Let M be a connected time-oriented Lorentzian manifold Then the fol- lowing are equivalent:
(2) There exists a Cauchy hypersurface in M.
(3) M is isometric toR×S with metric−βdt 2 +g t whereβ is a smooth positive function, g t is a Riemannian metric on S depending smoothly on t ∈ Rand each{t} ×S is a smooth spacelike Cauchy hypersurface in M.
The key aspect of this theorem is that statement (1) leads to statement (3), a result demonstrated by A Bernal and M Sánchez, referencing R Geroch's work Earlier mentions of this relationship can be found in additional sources Furthermore, the implication that (3) leads to (2) is straightforward, as detailed in Proposition 3.
In every globally hyperbolic Lorentzian manifold M, there exists a smooth function h: M → R This function has a gradient that is past-directed timelike at every point, ensuring that its level sets are spacelike Cauchy hypersurfaces.
Proof Definehto be the compositiont◦ΦwhereΦ: M →R×Sis the isometry given in Theorem 1 andt :R×S→Ris the projection onto the first factor
Such a functionhon a globally hyperbolic Lorentzian manifold is called aCauchy time function Note that a Cauchy time function is strictly monotonically increasing along any future-directed causal curve.
We conclude with an enhanced form of Theorem 1, due to A Bernal and
Theorem 2 Let M be a globally hyperbolic manifold and S be a spacelike smooth
Cauchy hypersurface in M Then there exists a Cauchy time function h : M → R such that S=h − 1 ({0})
Any given smooth spacelike Cauchy hypersurface in a (necessarily globally hyper- bolic) spacetime is therefore the leaf of a foliation by smooth spacelike Cauchy hypersurfaces.
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2 Wald, R.M.: General Relativity University of Chicago Press, Chicago (1984) 39, 41, 57
3 B¨ar, C., Ginoux, N., Pf¨affle, F.: Wave equations on Lorentzian manifolds and quantization.
4 Friedlander, F.: The wave equation on a curved space-time Cambridge University Press,
5 Leray, J.: Hyperbolic Differential Equations Mimeographed Lecture Notes, Princeton (1953) 53
6 Bernal, A.N., S´anchez, M.: Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes Commun Math Phys 257, 43 (2005) 57
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Cauchy time functions Lett Math Phys 77, 183 (2006) 57
Introduction
This chapter explores linear wave equations on Lorentzian manifolds, beginning with the physical principles that underpin these equations Specifically, we examine the propagation of waves in three-dimensional space, represented by the function u(t, x), which indicates the wave height at the spatial coordinates x = (x₁, x₂, x₃) in R³ and at time t in R.
∂x 2 j is the so-called d’Alembert operator on the four-dimensional Minkowski spaceR 4 =R×R 3
The solutions to the wave equation reveal that the operator is linear, indicating that its kernel forms a vector space The functions defined as (t,x)→cos(nt) cos(nx 1), where n varies over the integers, are included in the kernel, demonstrating its infinite dimensionality However, when the height and speed of the wave are specified at a fixed time, the resulting solution is uniquely determined.
This article explores wave equations linked to generalized d’Alembert operators on Lorentzian manifolds, focusing on the local and global existence of solutions and their relevance to quantization theory The initial section clarifies the concept of generalized d’Alembert operators and highlights the importance of fundamental solutions for differential operators In particular, it discusses how fundamental solutions for the d’Alembert operator in Minkowski space can be derived from Riesz distributions, with advanced and retarded solutions defined by their support in the causal future or past of the origin.
In the context of Lorentzian manifolds, there is no global analogue of Riesz distributions; however, normal coordinates allow for the transport of Riesz distributions from a point's tangent space to its neighborhood While these distributions do not directly yield local fundamental solutions for the classical d’Alembert operator, a linear combination of an infinite number of them can formally solve the wave equation By applying a cutoff function to correct the formal series, a local fundamental solution can be achieved, albeit with an error term Utilizing general methods of functional analysis, this error term can be eliminated, leading to the construction of true local fundamental solutions for any generalized d’Alembert operator, which are closely related to the original formal series.
The global aspect of the theory focuses on globally hyperbolic spacetimes, analogous to complete Riemannian manifolds in a Lorentzian context, as constructing global fundamental solutions on arbitrary spacetimes is illusory In this framework, global fundamental solutions for generalized d’Alembert operators are derived from the Cauchy problem associated with these operators The discussion includes the uniqueness of fundamental solutions and the construction of local and global solutions to the Cauchy problem, ultimately leading to the derivation of global fundamental solutions It is essential to note that the local existence of fundamental solutions plays a crucial role in this global construction, as it provides local solutions to the inhomogeneous wave equation.
This article concludes by presenting the concept of advanced and retarded Green’s operators linked to generalized d’Alembert operators, highlighting their fundamental properties and close connection to fundamental solutions Green’s operators serve as the foundation for the local approach to quantization, as their existence, combined with specific assumptions about spacetime, leads to the direct construction of a C*-algebra in a functorial manner For further details, refer to the final chapter.
This chapter serves as an introductory guide for first- and second-year university students, focusing on key results and concepts while omitting most proofs It specifically addresses scalar operators, though the findings can be generalized to include d’Alembert operators on vector bundles For a comprehensive understanding of the topic and additional references, we recommend consulting [1], which forms the foundation of this survey.