Quantum physics on a general Hilbert space

In this chapter we generalize the results of Chapter 2 to infinite-dimensional Hilbert spaces. So let H be a Hilbert space and let B(H) be the set of all bounded operators on H. Here a notable point is that linear operators on finite-dimensional Hilbert spaces are automatically bounded, whereas in general they are not. Thus we impose boundedness as an extra requirement, beyond linearity. This is very convenient, because as in the finite-dimensional case, B(H) is a C*-algebra, cf. §C.1. At the same time, assuming boundedness involves no loss of generality whatsoever, since we can alway replace closed unbounded operators by bounded ones through the bounded transform, as explained in §B.21. Nonetheless, even the relatively easy setting of bounded operators leads to some technical complications we have to deal with.

The proof for density operators is analogous.
Defining the mean value a ψ of a with respect to the Born measure μ ψ by and similarly for ρ, using Theorem 4.3.2 we easily obtain a ψ = ψ, aψ ; (4.11) a ρ = Tr (ρa). (4.12) As an important special case, suppose that σ (a) = σ p (a) (i.e., each λ ∈ σ (a) is an eigenvalue); this always happens if H is finite-dimensional. Eq. (A.57) then gives where e λ is the projection onto the eigenspace H λ = {ψ ∈ H | aψ = λ ψ}. Thus μ ψ (λ ) = e λ ψ 2 , (4.13) and using the notation P ψ (a = λ ) for μ ψ (λ ), eq. (4.11) just becomes a ψ = ∑ λ ∈σ (a) λ · P ψ (a = λ ). (4.14) It is customary to extend the Born measure on σ (a) ⊂ R to a (probability) measure μ ψ on all of R by simply stipulating that μ ψ (Δ ) = μ ψ (Δ ∩ σ (a)); (4.15) we will often assume this and omit the prime. This obviously implies that μ ψ (Δ ) = 0 for any Borel set Δ ⊂ R disjoint from σ (a); in particular, if σ (a) is discrete, then μ ψ is concentrated on the eigenvalues λ of a, in that μ ψ (λ ). (4.16) To state an interesting property of the Born measure we need Hausdorff's solution to the relevant special case of the famous Hamburger Moment Problem: Theorem 4.5. If K ⊂ R is compact, then any finite measure μ on K is determined by its moments α n = K dμ(x) x n . (4.17) Using f (x) = x n in (4.6), we therefore obtain: Corollary 4.6. The Born measure μ ψ is determined by its moments α n = ψ, a n ψ . (4.18) More precisely, we need to be sure that numbers (α n ) of the kind (4.18) are the moments of some (probability) measure. This follows from the spectral theorem by running the above argument backwards, but one may also use the general solution of the Hamburger Moment Problem, which we here state without proof: Theorem 4.7. A sequence of real numbers (α n ) forms the moments of some measure μ on R iff for all N ∈ N and (β 1 ,..., β N ) ∈ C N one has ∑ N n,m=0 β nβ m α n+m ≥ 0. Furthermore, if there are constants C and D such that |α n | ≤ CD n n!, then μ is uniquely determined by its moments (α n ).
These conditions are easily checked from (4.18).
If a is unbounded, but still assumed to be self-adjoint (in the sense appropriate for unbounded operators, cf. Definition B.70), the spectrum σ (a) remains real (see Theorem B.93) but it is no longer compact. Nonetheless, the Born measure on σ (a) may be constructed in almost exactly the same way as in the bounded case, this time invoking Corollary B.21 and Theorem B.158 instead of Theorems 4.2 and B.94, respectively. Corollary 4.4 then holds almost verbatim for the unbounded case: Corollary 4.8. Let H be a Hilbert space, let a * = a, and let ψ ∈ H be a unit vector. There exists a unique probability measure μ ψ on the spectrum σ (a) such that (4.19) Also, eqs. (4.7) and (4.9) hold, as does (4.8), with f ∈ C 0 (σ (a)).
There is no need to worry about domains, since even if a is unbounded, f (a) is bounded for f ∈ C b (σ (a)), and hence also for f ∈ C 0 (σ (a)).
The physical relevance of the Born measure is given by the Born rule: If an observable a is measured in a state ρ, then the probability P ρ (a ∈ Δ ) that the outcome lies in Δ ⊂ R is given by the Born measure μ ρ defined by a and ρ, i.e., (4.20) As in the finite-dimensional case, the Born measure may be generalized to families (a 1 ,..., a n ) of commuting self-adjoint operators. Assuming these are bounded, the C*-algebra C * (a 1 ,..., a n ) is defined in the obvious way, i.e., as the smallest C*algebra containing each a i , or, equivalently, as the norm-closure of the algebra of all finite polynomials in the (a 1 ,..., a n ). This C*-algebra is commutative, as a simple approximation argument shows: polynomials in the a i obviously commute, and this property extends to the closure by continuity of multiplication. However, even in the bounded case, the correct notion of a joint spectrum is not obvious. In order to motivate the following definition, it helps to recall Definition 1.4, Theorem C.24, and especially the last sentence before the proof of the latter, making the point that the spectrum σ (a) of a single (bounded) self-adjoint operator coincides with the image of the Gelfand spectrum Σ (C * (a)) in C under the map ω → ω(a).
To justify this definition, we note that: • For n = 1, this definition reproduces the usual spectrum, cf. Theorem C.24.
• For n > 1 and dim(H) < ∞, we recover the joint spectrum of Definition A.16.
• For n > 1 and dim(H) = ∞, Weyl's Theorem B.91 generalizes in the obvious way: we have λ ∈ σ (a) iff there exists a sequence (ψ k ) of unit vectors in H with for each i = 1,..., n. The proof is similar.
One way to see the second claim is to use Proposition C.14 joined with the observation that, as in the case of A = B(H) for finite-dimensional H, any pure state on a finite-dimensional C*-algebra A ⊂ B(H) is a vector state (2.42), too. To see this, we first specialize Theorem C.133 to the finite-dimensional case (where the proof becomes elementary), so that each state on C * (a) takes the form (2.33). Subsequently, we use the spectral decomposition (2.6), and use the definition of purity: Then ω υ i = ω for each i, so that ω is a vector state, say ω(b) = ψ, bψ where ψ is one of the υ i . Once we know this, suppose λ = (λ 1 ,..., λ n ) ∈ σ (a), with λ i = ω(a i ).
Multiplicativity of ω implies that for any finite polynomial in n real variables we have ψ, p(a)ψ = p(λ ), which easily gives a i ψ = λ i ψ for each i; for example, take p(x) = (x i − λ i ) 2 , so that the previous equation gives (a i − λ i )ψ 2 = 0. Conversely, if λ is a joint eigenvalue of a, then by definition there exists a joint eigenvector ψ whose vector state ω(b) = ψ, bψ on C * (a) is multiplicative.
Using this (perhaps contrived) notion of a joint spectrum, Theorem 2.19 now holds by construction also if dim(H) = ∞, where the pertinent isomorphism f → f (a) is given as in the single operator case, that is, by starting with polynomials and using a continuity argument to pass to arbitrary continuous functions.
Theorem 2.18 and Corollary 4.4 then generalize to: Theorem 4.10. Let H be a Hilbert space, let a = (a 1 ,..., a n ) be a finite family of commuting bounded self-adjoint operators, and let ψ ∈ H be a unit vector. There exists a unique probability measure μ ψ on the joint spectrum σ (a) such that or, equivalently, for special Borel sets where the e Δ i = 1 Δ i (a i ) are the pertinent spectral projections (which commute).
Similarly for density operators instead of pure states. If (some of) the operators a i are unbounded, we use the trick of §B.21 and pass to their bounded transforms b i , see Theorem B.152. We say that the b i commute iff the corresponding bounded operators b i do; this is equivalent to commutativity of all spectral projections of the a i . We then define, in self-explanatory notation, (4.25) This leads to Born measures on σ (a) defined either as in (4.23), with f ∈ C(σ (a)) replaced by f ∈ C 0 (σ (a)), cf. (4.19), or as in (4.24).
, 1] n and hence σ (a) = R n . For a measurable region Δ ⊂ R n we then have Pauli's famous formula (4.27) for finding the particle in the region Δ , given that the system is in a pure state ψ.

Density operators and normal states
For example, let (υ i ) be a basis of H with associated one-dimensional projections If ω is assumed to be a state, then the additivity condition (4.28) implies (4.30) or, equivalently, using Definition B.6 etc. as well as the notation e F ≡ ∑ i∈F e i , lim F ω(e F ) = 1. (4.31) If H is separable, any orthogonal family (e i ) of projections is necessarily countable, and (4.28) is analogous to the countable additivity condition defining a measure. Proof. First, eq. (2.33) implies (4.28). To see this, take the trace with respect to some basis (υ j ) of H that is adapted to the family (e i ) in the sense that for each j, either e i υ j = υ j (i.e., υ j ∈ e i H) for one value of i, or e i υ j = υ j for all i. Then where the sum ∑ j is over those j for which υ j ∈ K ≡ ∨ i e i H. On the other hand, since the basis is adapted, we have where J F consists of those j for which υ j ∈ ∑ i∈F e i H. This gives (4.28). Conversely, assume ω is normal. For the e i in (4.28) we now take the projections (4.29) determined by some basis (υ i ). For each a ∈ B(H) we then have (4.32) Indeed, using Cauchy-Schwarz for the positive semi-definite form (a, b) = ω(a * b), as in (C.197), and using ∑ i e i = 1 H and hence ω(a) = ω(∑ i e i a) we have Since ω(e F ) + ω(e F c ) = ω(1 H ) = 1, eq. (4.31) gives lim F ω(e F c ) = 0, so that (4.33) gives (4.32). For each finite F ⊂ I, the operator e F a has finite rank and hence is compact. According to Theorem B.146, the restriction of ω : B(H) → C to the C*-algebra B 0 (H) of compact operators on H is induced by a trace-class operator ρ, which (from the requirement that ω be a state) must be a density operator. Hence ω(e F a) = Tr (ρe F a), and we finally have (4.34) To derive the final equality, we rewrite Tr (ρe F a) = Tr (e F aρ), cf. (A.78) and Proposition B.144, note that aρ ∈ B 1 (H), as shown in Corollary B.147, and observe that for To see this, simply compute the trace in the basis (υ i ) defining the projections e i through (4.29), so that Tr (e F b) = ∑ i∈F υ i , bυ i , and note that by Definition B.6, Finally, suppose ω(a) = Tr (ρ 1 a) = Tr (ρ 2 a) for each a ∈ B(H) and hence for each a ∈ B 0 (H). It follows from (B.476) that Tr (ρa) = 0 for all a ∈ B 0 (H) iff ρ = 0. Hence ρ 1 = ρ 2 , i.e., a normal state ω uniquely determines a density operator ρ.
If ω is normal, we may therefore use the spectral resolution (2.6) of the corresponding density operator ρ, i.e., ρ = ∑ i p i |υ i υ i |, where (υ i ) is some basis of H consisting of eigenvectors of ρ (which exists because ρ is compact and self-adjoint), and the corrsponding eigenvalues satisfy p i ≥ 0 and ∑ i p i = 1; see the explanation after Definition B.148. Computing the trace in the same basis gives We may characterize normality in a number of other ways. First note that because of the duality B 1 (H) * ∼ = B(H) of Theorem B.146, cf. (B.477), we may equip B(H) with the w * -topology in its role as the dual of the trace-class operators B 1 (H), see §B.9; this means that a λ → a iff Tr (ρa λ ) → Tr (ρa) for each ρ ∈ B 1 (H), or, equivalently, for each ρ ∈ D(H), since each trace-class operator is a linear combination of at most four density operators, as follows from Lemma C.53 with (C.8) -(C.9). The w * -topology on B(H), seen as the dual of B 1 (H), is called the σ -weak topology. By Proposition B.46, the σ -weakly continuous linear functionals ϕ on B(H) are just those given by ϕ(a) = Tr (ρb) for some trace-class operator b ∈ B 1 (H).
Secondly, B(H) is monotone complete, in the sense that each net (a λ ) of positive operators that is bounded (i.e., 0 ≤ a λ ≤ c · 1 H for some c > 0 and all λ ∈ Λ ) and increasing (in that a λ ≤ a λ whenever λ ≤ λ ) has a supremum a with respect to the standard ordering ≤ on B(H) + , which supremum coincides with the strong limit of the net (i.e., lim λ a λ ψ = aψ for each ψ ∈ H); the proof is the same as for Proposition B.98, and also here we write a λ a to describe this entire situation.
Proof. We have seen 1 ↔ 3 ↔ 4, and 2 → 1 is obvious, so establishing 3 → 2 would complete the proof. To this effect, we first note that because the sum (4.35) is convergent, for ε > 0 we may find a finite subset F ⊂ I for which ∑ i / Consequently, for such λ , This shows that lim λ |Tr (ρ(a λ − a))| = 0, so that assumption 3 implies no. 2.
We denote the normal state space of B(H), i.e., the set of all normal states on B(H) by S n (B(H)). It is easy to see from Definition B.148 that S n (B(H)) is a convex (but not necessarily compact!) subset of the total state space S(B(H)).
Corollary 4.14. The relation ω(a) = Tr (ρa) induces an isomorphism of convex sets (i.e., ω ↔ ρ). Furthermore, for the corresponding pure states we have i.e., any pure state ω on B 0 (H), as well as any normal pure state on B(H), is given by The proof of (4.38) is practically the same as in the finite-dimensional case. From Theorem B.146 we obtain another characterization of S n (B(H)) and hence of D(H): in the sense that any (pure) state ω on B 0 (H) has a unique normal extension to a (pure) state ω on B(H), given by the same density operator ρ that yields ω.
It can be shown that any state ω ∈ S(B(H)) has a convex decomposition where t ∈ [0, 1], ω n is a normal state, and ω s is called a singular state. In particular, since for t ∈ (0, 1) the state ω is mixed, a pure state is either normal or singular. Singular states are not as aberrant as the terminology may suggest: such states are routinely used in the physics literature and are typically denoted by |λ , where λ lies in the continuous spectrum of some self-adjoint operator (that has to be maximal for this notation to even begin to make sense, see §4.3 below). Examples of such "improper eigenstates" are |x and |p , which many physicists regard as idealizations. However, mathematically such states are at least defined, namely as singular pure states on B(H). The key to the existence of such states lies in Proposition C.15 and its proof, which should be reviewed now; we only need the case a * = a.
Proposition 4.16. Let a = a * ∈ B(H) have non-empty continuous spectrum, so that there is some λ ∈ σ (a) that is not an eigenvalue of a. Then ω λ ( f (a)) = f (λ ) defines a pure state on A = C * (a), whose extension to B(H) by any pure state is singular.
Proof. Normal pure states on B(H) take the form ω ψ (b) = ψ, bψ , where ψ ∈ H is a unit vector and b ∈ B(H). We know from Proposition C.14 that ω λ is multiplicative on C * (a). However, if some multiplicative state ω on C * (a) has the form ω = ω ψ , then ψ must be eigenvector of a; cf. the proof of Proposition 2.3.

The Kadison-Singer Conjecture
To obtain deeper insight into singular pure states, and as a matter of independent interest, we return to the Kadison-Singer problem, cf. §2.6. Recall that this problem asks if some abelian unital C*-algebra A ⊂ B(H) has the Kadison-Singer property, stating that a pure state ω A on A has a unique pure extension ω to B(H). Here the issue is uniqueness rather than existence, since at least one such extension exists: since A is necessarily unital (with 1 A = 1 H ) and ω A is a state on A, so that in particular (4.42) (4.43) The inverse of this map is simply the pullback of the inclusion A → B, i.e., ω B ∈ P(B) defines ω A ∈ P(A) by restriction, so that we have a bijection P(A) ∼ = P(B), ω A ↔ ω B . Since for any pair of C*-algebras A ⊆ B the pullback S(B) → S(A) is continuous (in the pertinent w * -topology), the map ω B → ω A is continuous. As in Lemma C.20, this implies that it is in fact a homeomorphism, so that A ∼ = B through the inclusion A → B. This gives A = B, and hence A is maximal.
Maximality of A implies A = A, so that A is a von Neumann algebra, sharing the unit of B(H). To see the relevance of singular states for the Kadison-Singer problem, we first settle the normal case. We know what it means for a state on B(H) to be normal (cf. Definition 4.11 and Corollary 4.13); for arbitrary von Neumann algebras A ⊂ B(H) the situation is exactly the same: we define normality by (4.28) and characterize it by the equivalent properties in Corollary 4.13, where the σ -weak topology on A may be defined either as the one inherited from B(H), or, more intrinsically, and the w * -topology from the duality A = A * * , where the Banach space A * is the so-called predual of A, e.g., ∞ * ∼ = 1 and L ∞ (0, 1) * = L 1 (0, 1), cf. §B.9.
Theorem 4.18. Let H be a separable Hilbert space and let ω A be a normal pure state on a maximal commutative unital C*-algebra A in B(H). Then ω A has a unique extension to a state ω on B(H), which is necessarily pure and normal.
Proof. As noted after (4.41), a pure state on B(H) is either normal or singular. The possibility that ω A is normal whereas ω is singular is excluded by Corollary 4.13.3, so ω must be normal and hence given by a density operator. The proof of uniqueness is then the same as in the finite-dimensional case, cf. Theorem 2.21.
We now recall the classification of maximal maximal abelian * -algebras (and hence of maximal abelian von Neumann algebras) A in B(H) up to unitary equivalence (cf. Theorem B.118). This classification is the relevant one for the Kadison-Singer problem, since, as is easily seen, A ⊂ B(H) has the Kadison-Singer property iff uAu −1 ⊂ B(uH) has it. The uniqueness of the finite-dimensional case will be lost:
This classification sheds some more light on Theorem 4.18. Since L ∞ (0, 1) has no pure normal states and D n (C) has been dealt with in Theorem 2.21, the interesting case is ∞ . Using Corollary 4.13.3 (or the analysis below), it is easy to check that the normal pure states on ∞ are given by ω A ( f ) = f (x) for some x ∈ N; these are vector state of the kind ω A ( f ) = ψ, m f ψ with ψ = δ x , or, in other words, they are given by ω A ( f ) = Tr (ρm f ) with ρ = |δ x δ x |. We now invoke a fairly deep result:

Proposition 4.20. A pure state ω on B(H) is singular iff one (and hence all) of the following equivalent conditions is satisfied:
• ω(a) = 0 for each a ∈ B 0 (H); • ω(e) = 0 for each one-dimensional projection e; • ∑ i ω(e i ) = 0 for the projections e i = |υ i υ i | defined by some basis (υ i ).
One direction is easy: a normal pure state certainly does not satisfy the condition in question. For example, given (2.42) one may take a = |ψ ψ|, which as a onedimensional projection lies in B 0 (H), so that ω ψ (a) = 1. We omit the other direction of the proof. We conclude from this proposition that a pure singular state on B( 2 ) cannot restrict to a normal pure state on ∞ , which reconfirms Theorem 4.18.
We now study the Kadison-Singer property for each of the three cases in Theorem 4.19 (where the third will be an easy corollary of the first and the second). Since the proofs of the first two cases are formidable, we just sketch the argument.
The statement about ∞ is the Kadison-Singer Conjecture, which dates from 1959 but was only proved in 2013. The first claim (which was already known to Kadison and Singer themselves) is equally remarkable, however, as is the contrast between the two parts of Theorem 4.21. In particular, Dirac's notation |λ may be ambiguous.
The key to the proof of the first claim lies in the choice of a total countable family of normal states on L ∞ (0, 1), from which all pure states may be constructed by a limiting operation. Here we call a (countable) family (ω n ) n∈N of states on some C*-algebra A total if, for any self-adjoint a ∈ A, the conditions ω n (a) ≥ 0 for each n imply a ≥ 0 (the converse is trivial). For example, the well-known Haar basis (h n ) of L 2 (0, 1) provides such a family. The functions forming this basis are defined via some bijection β between the set of pairs (k, l) and N, e.g., β (k, l) = k + 2 l , by (4.44) Basic analysis then shows that the Haar functions h n form a basis of L 2 (0, 1) and that the associated vector states ω n on L ∞ (0, 1) form a total set, where obviously The relevance of total sets to the conjecture is explained by the following lemma. where co(T ) − is the w * -closure of the convex hull of T in A * or in S(A).

Proof. The inclusion co(T ) − ⊆ S(A) is obvious, since T ⊆ S(A) and S(A)
is a compact (and hence a closed) convex set. To prove the converse inclusion, suppose a = a * ∈ A and s ∈ R are such that ω(a) ≥ s for each ω ∈ T . Then ω(a − s · 1 A ) ≥ 0 and hence ω(a) ≥ s for each ω ∈ S(A). Using Theorem B.43 (of Hahn-Banach type), this property would lead to a contradiction if S(A) were not contained in co(T ) − . The second claim, which is the one we will use, follows from the first through a corollary of the Krein-Milman Theorem B.50, stating that if T ⊂ K is any subset of a compact convex set K such that K = co(T ) − , then ∂ e K ⊆ T − . This corollary may be proved (by contradiction) from Theorem B.43 in a similar way.
Our next aim is to get rid of the closure in (4.49). The Haar basis yields a map h : N → S(L ∞ (0, 1)); (4.50) n → ω n , (4.51) with image T , i.e., the set of Haar states. Since S(A) is a compact Hausdorff space (in its w * -topology), the universal property (B.135) of theČech-Stone compactification β N of N implies that h extends (uniquely) to a continuous map whose image is compact and hence closed (since β N is compact).
We now pass to the (even) more difficult case of ∞ ⊂ B( 2 ). Although this will not be used in the proof, it gives some insight to know which states on ∞ we are actually talking about, i.e., the singular pure states, and compare this with (4.53).
Theorem 4.24. There is a bijective correspondence between states ω d on ∞ and finitely additive probability measures μ on N, where: 1. ω d is normal iff μ is countably additive (and hence is a probability measure). 2. ω d is pure iff μ corresponds to some ultrafilter U on N, in which case: ω d is normal iff U is principal (and hence singular iff U is free).
This follows from case no. 5 in §B.9, notably eqs. (B.153) -(B.154). In other words, the pure states ω d on ∞ are given by ultrafilters U on N through the analogy with (4.53) is even clearer if we write f (n) = δ n , m f δ n ≡ ω n ( f ). If U = U n is a principal ultrafilter, n ∈ N, we thus recover the normal pure states We now show that that ∞ has the Kadison-Singer property, making ω (U) the only extension of ω (U) d . The proof relies on an extremely difficult lemma from linear algebra (formerly known as a paving conjecture). We first define a linear map D : M n (C) → D n (C) by D(a) ii = a ii , i = 1,..., n, and D(a) i j = 0 whenever i = j. |ω(e i ae j )| 2 ≤ ω(a * e i a)ω(e j ). (4.71) Since ω(e i ) = ω d (e i ) and ω d is a pure state (and hence is multiplicative), we have ω(e i ) ∈ {0, 1}, since e i is a projection. Moreover, in view of (4.68) and the normalization ω(1 H ) = 1, there must be exactly one value of i = 1,..., l, say i = i 0 , such that ω(e i 0 ) = 1, and ω(e i ) = 0 for all i = i 0 . Eqs. (4.70) -(4.71) therefore imply that ω(e i ae j ) = 0 iff i = j = i 0 . Using (4.68) once more, we see that ω(a) = ∑ i, j ω(e i ae j ) = ω(e i 0 ae i 0 ), so that |ω(a)| ≤ ω e i 0 ae i 0 ≤ 1 · ε a by (4.66). Letting ε → 0, we proved: provided that ω extends ω d , as before. This shows that ω is determined by ω d and hence is unique, completing the proof (sketch) of Theorem 4.21.

Gleason's Theorem in arbitrary dimension
To a large extent the thrust and difficulty of the proof of Gleason's Theorem 2.28 already lies in its finite-dimensional version, but some care is needed in the general case, and also Corollary 2.29 needs to be refined. A major point here is that Definition 2.23 has no unambiguous generalization to arbitrary Hilbert spaces. Proof. The proof of part 1 is practically the same as in finite dimension, except for the fact that in the proof of Lemma 2.33 the reference to Proposition A.23 should be replaced by Proposition B.79, upon which one obtains a bounded positive operator ρ for which (2.123) holds. The normalization condition (2.110) then yields Tr (ρ) = 1 if the trace is taken over any basis of H, and since ρ is positive this implies ρ ∈ B 1 (H), see §B.20 (complete additivity of P is just necessary to relate it to p). Unfortunately, the proof of part 2 exceeds the scope of this book (see Notes).
In infinite dimension, Corollary 2.29 becomes more complicated, too; for one thing, Definition 2.26 of a quasi-state bifurcates into two possibilities. The one given still makes perfect sense and is natural from the point of view of Bohrification; to avoid confusion we call a map ω : B(H) → C satisfying the conditions in Definition 2.26 a strong quasi-state. In the context of Gleason's Theorem, a slightly different notion is appropriate: a weak quasi-state on B(H) satisfies Definition 2.26, except that linearity is only required on commutative C*-algebras in B(H) of the form C * (a), where a = a * ∈ B(H) (these are singly generated). Since commutative unital C*-subalgebras of B(H) are not necessarily singly generated, and a specific counterexample exists, weak quasi-states are not necessarily strong quasi-states.

Proposition 4.30. The map ω → ω |P(H) gives a bijective correspondence between weak quasi-states ω on B(H) and finitely additive probability measures on P(H).
Proof. For some finite family (e 1 ,..., e n ) of mutually orthogonal projections on H, add e 0 = 1 H − ∑ j e j if necessary and let a = ∑ n j=0 λ j e j , with all λ j ∈ R different. Then σ (a) = {λ 0 ,..., λ n }, so that C * (a) ∼ = C(σ (a) ∼ = C n+1 (cf. Theorem B.94) coincides with the linear span of the projections e j . If ω is a weak quasi-state, then it is linear on C * (a) and hence also on the e j , so that ω |P(H) is finitely additive.
Conversely, let μ be a finitely additive probability measure on P(H). If a = a * ∈ B(H) is given, using the notation (B.328) we symbolically define ω on a by (4.79) More precisely, for any ε > 0 we use Corollary B.104 to define ω ε (a) = ∑ n i=1 λ i μ(e A i ) and let ω(a) = lim ε→0 ω ε (a); it follows from Lemma B.103 (or the theory underlying the Riemann-Stieltjes integral (4.79)) that this limit exists. Now let b, c ∈ C * (a), so that b = f (a) and c = g(a) for certain f , g ∈ C(σ (a)), and b + c = ( f + g)(a), cf. Theorem B.94. By (B.325) we therefore have Since this holds for every ε > 0, letting ε → 0 we obtain ω(b + c) = ω(b) + ω(c), making ω linear on C * (a). It is clear that the quasi-state ω thus obtained, on restriction to P(H) reproduces μ, making the map ω → ω |P(H)  Another corollary of Gleason's Theorem is the Kochen-Specker Theorem, which we will explain in detail in Chapter 6, where it will also be proved in a different way.
Cf. Definitions 6.1 and 6.3. To see that these conditions are equivalent to those stated in Theorem 4.32 (despite the impression that linearity on all commuting self-adjoint operators seems stronger than linearity on each C * (a)), extend ω to ω : B(H) → C by complex linearity, as in Definition 2.26.1, and note that dispersion-freeness implies positivity and hence continuity on each subalgebra C * (a) (cf. Theorem C.52 and Lemma C.4). We then see that the two conditions just stated imply that ω is multiplicative on C * (a), and hence pure, see Proposition C.14, which conversely implies that pure states on C * (a) are dispersion-free. We now prove Theorem 4.32.