Proofs and complements: beyond the real line

This companion follows Part II through strongly commuting families and the construction of their joint spectral measure, then normal operators, manifold-valued spectral measures, other outcome spaces, normal states and geometric commutators. The final two sections justify the post's further perspectives on distance and quantization. It uses the spectral calculus proved in the Part I companion. The normal spectral theorem uses a short closed-graph argument given below for its positive square-root reduction. Mackey's full classification and the exploratory geometric questions are cited background, not further theorems proved here.

Each operator includes its domain. Inner products are conjugate-linear in the first argument and linear in the second. Section and theorem numbers are local to this companion; references to another companion name its part explicitly.

1Strongly commuting families and their joint spectral measure

Commutation of full observables

The diagonal example at the start of the post is the atomic case of the theorem below. There the coordinate-axis projections satisfy \(P_1 P_2 = 0\) and \(P_1 + P_2 = I\). Assigning them to \((1, 2)\) and \((- 1, 3)\) defines a normalized PVM with two atoms. Its coordinate integrals are the displayed matrices, and multiplying finite spectral sums gives the displayed joint calculus. Its marginals are obtained by inverse image under the coordinate maps. The theorem below proves the same assertions without an atomic assumption.

Let \(A, B\) be self-adjoint. We first justify the equivalences used in the post. Strong commutation means \(E_A (\Delta) E_B (\Gamma) = E_B (\Gamma) E_A (\Delta)\) for all Borel sets. Simple spectral integrals and bounded convergence show that this implies commutation of \(f (A)\) and \(g (B)\) for every bounded Borel \(f, g\). In particular all nonreal resolvents, and all members of the two unitary groups, commute.

Conversely suppose all nonreal resolvents commute. The algebra generated by \((x - i)^{- 1}\) and \((x + i)^{- 1}\) is uniformly dense in \(C_0 (\mathbb{R})\), by Stone–Weierstrass. Thus the two continuous calculi commute. Adjoin constants and fix a bounded operator commuting with one calculus. The bounded Borel functions whose spectral integrals commute with it are closed under uniformly bounded pointwise limits, because such spectral integrals converge strongly. The functional monotone-class theorem therefore gives all bounded Borel functions. Applying this argument successively to the two calculi gives commuting spectral projections.

For the unitary criterion, use a bounded operator \(T\) commuting with \(e^{itA}\) for every real \(t\). For \(h, k \in H\), the finite complex measures \(\langle h, TE_A (\cdot) k \rangle\) and \(\langle h, E_A (\cdot) Tk \rangle\) have equal Fourier transforms. Uniqueness of the Fourier transform of finite measures makes them equal, so \(T\) commutes with every spectral projection of \(A\). First take \(T = e^{isB}\), then repeat the argument for \(B\) with \(T = E_A (\Delta)\). This proves strong commutation from the unitary-group criterion.

These conditions imply the weak form identity on the full intersection \(D (A) \cap D (B)\). For example, the joint cutoffs in the proof below approximate each vector together with its \(A\)- and \(B\)-images; the identity holds on each bounded joint spectral subspace and passes to the limit. In contrast, the periodic/antiperiodic momentum example satisfies both algebraic and form commutation on \(C_c^{\infty} (0, 1)\) without strongly commuting. The graph norm of either momentum is the \(H^1\) norm. The closure of that test space is \(H_0^1 (0, 1)\), whose functions vanish at both endpoints, a proper subspace of either self-adjoint boundary domain. Thus the test space is not an operator core.

For completeness, let \(S_{\theta} = - id / dx\) on \(D (S_{\theta}) = \{ u \in H^1 (0, 1) : u (1) = e^{i \theta} u (0) \}\). Integration by parts shows that a vector in the adjoint domain has a weak derivative in \(L^2\), hence lies in \(H^1\); the boundary term vanishes for every \(u \in D (S_{\theta})\) exactly when the same boundary condition holds for the adjoint vector. Thus \(S_{\theta}\) is self-adjoint. Multiplication by \(e^{i \theta x}\) carries the Fourier basis \(e^{2 \pi ikx}\) to a complete eigenbasis, which verifies the stated periodic and antiperiodic spectral projections directly.

The failure of this particular test space to be a core is not a general converse criterion. Algebraic commutation on a common invariant core need not imply the bounded commutation identities proved above; Nelson's Analytic vectors contains the classical counterexample. That construction is cited as a further warning; the criteria proved here are commutation of resolvents, spectral projections or full unitary groups.

Constructing the measure from the coordinate projections

Theorem 1. [Joint spectral measure of a finite family] Let \(A_1, \ldots, A_d\) be pairwise strongly commuting self-adjoint operators, with spectral measures \(E_j\). There is a unique normalized PVM \(E\) on \(\mathbb{R}^d\) such that

\(\displaystyle E (\Delta_1 \times \cdots \times \Delta_d) = E_1 (\Delta_1) \cdots E_d (\Delta_d)\)

for all Borel sets \(\Delta_j\). It gives a nondegenerate representation \(J : C_0 (\mathbb{R}^d) \rightarrow B (H)\) satisfying

\(\displaystyle J (f_1 \otimes \cdots \otimes f_d) = f_1 (A_1) \cdots f_d (A_d) .\)

The coordinate integrals are exactly the given operators, including domains. Every real-valued Borel function \(h\), finite at every point, defines a self-adjoint joint spectral integral with domain \(\left\{ u : \int | h |^2 d \mu_u < \infty \right\}\), where \(\mu_u (F) = \langle u, E (F) u \rangle\). Conversely, the coordinate operators of a normalized PVM on \(\mathbb{R}^d\) are pairwise strongly commuting.

Proof. For a rectangle \(R = \Delta_1 \times \cdots \times \Delta_d\), put \(P (R) = \prod_{j = 1}^d E_j (\Delta_j)\). Commutation makes this product self-adjoint and idempotent, hence an orthogonal projection. Its range is the intersection of the coordinate ranges: the product fixes every vector in that intersection, and each factor fixes the product's range. Also \(P (R) P (S) = P (R \cap S)\). In particular disjoint rectangles give orthogonal projections.

Partition each real coordinate into finitely many disjoint Borel sets, including the complement of any bounded region being considered. The resulting grid partitions \(\mathbb{R}^d\) into finitely many rectangles \(R_{\alpha}\). Their projections are orthogonal and sum to \(I\), by distributivity and normalization of the \(E_j\). For a function constant on these cells define

Refining a coordinate partition subdivides its spectral projection into the corresponding finite sum. Two grids have a common refinement, so this definition is independent of the representation of \(s\). Refinement also proves linearity, multiplication and preservation of complex conjugation. Orthogonality gives, for every \(u \in H\),

Compactly supported continuous functions can be approximated uniformly by such step functions: enclose the support in a larger box, divide the box into sufficiently small half-open rectangular cells, sample the function on each cell, and put zero outside. Uniform continuity controls the error, including the cells near the support's boundary. Continuous functions vanishing at infinity are then approximated by first cutting them off. The norm estimate defines a unique contractive map \(J : C_0 (\mathbb{R}^d) \rightarrow B (H)\). Taking uniform limits in the identities for \(J_0\) shows that \(J\) is a \(\ast\)-homomorphism. Applying the same approximation to each one-variable factor proves the tensor-product formula in the statement.

Choose continuous compactly supported functions \(0 \leqslant \chi_n \leqslant 1\) on \(\mathbb{R}\) tending pointwise to one. The one-dimensional calculus gives \(\chi_n (A_j) \rightarrow I\) strongly. Their finite products also tend strongly to \(I\): expand the difference of the products as a telescoping sum and use that every factor has norm at most one. Thus \(J\) is nondegenerate.

Apply the representation–PVM theorem of the Part I companion to the locally compact, second-countable space \(\mathbb{R}^d\). Specifically, Riesz–Markov represents each positive functional \(f \mapsto \langle u, J (f) u \rangle\) by a finite Radon measure. Polarization produces its operator-valued measure; multiplicativity, extended from continuous functions by bounded monotone approximation, makes its values projections. Nondegeneracy supplies total projection \(I\). This gives a PVM \(E\) with \(J (f) = \int fdE\). No joint measure was used in constructing \(J\).

To identify its marginals, fix \(j\) and \(h \in C_0 (\mathbb{R})\). Apply the tensor formula to \(h (x_j) \prod_{k \neq j} \chi_n (x_k)\). The operator side tends strongly to \(h (A_j)\). Bounded convergence for the PVM integral makes the other side tend strongly to \(\int h (x_j) dE (x)\). Equality for all such \(h\), scalar measure uniqueness and polarization identify the marginal with \(E_j\). A rectangle is the intersection of its coordinate cylinders; multiplication of the corresponding spectral projections now gives exactly the asserted rectangle formula.

For uniqueness, two candidate PVMs give finite scalar measures \(\langle u, E (\cdot) u \rangle\) agreeing on every Borel rectangle. Rectangles form a generating pi-system containing the whole space. The pi-lambda theorem makes these scalar measures equal on all Borel sets, and polarization makes the PVMs equal. The marginal identity recovers each \(A_j\) with its original spectral domain. Integrating a real Borel \(h\) that is finite at every point uses the real spectral-integral theorem of Part I, or its proof with cutoffs \(1_{| h | \leqslant n}\); it gives exactly the stated domain and self-adjointness. Finally, starting with any joint PVM, its coordinate projections are the projections of cylinders. They commute because all values of a PVM do. This proves the converse.\(\Box\)

The examples in the post follow directly from uniqueness. Coordinate multiplication on \(L^2 (\mathbb{R}^d, dx)\) has \(E (F) u = 1_F u\), because this PVM has the required rectangle projections. Its scalar law is \(| u |^2 dx\). For the pair \((A, A)\), push \(E_A\) forward by \(x \mapsto (x, x)\); on a rectangle it gives \(E_A (\Delta \cap \Gamma)\), which is the required product. Thus the joint law is concentrated on the diagonal, and compatibility alone does not imply independence. Likewise the pushforward under a Borel map \(F\) is a PVM since inverse images preserve intersections, complements and countable disjoint unions; the spectral substitution rule follows first for indicators, then bounded simple functions, and finally by spectral truncation.

The common coordinate domain is described by \(\int | x |^2 d \mu_u (x) < \infty\), because \(| x |^2 = \sum_{j = 1}^d x_j^2\). The subspaces \(E ([- n, n]^d) H\) lie in every coordinate domain and have dense union. For a vector in the common domain, these cutoffs converge together with every coordinate image, by dominated convergence. This justifies the joint-cutoff argument in the commutation discussion.

Addition in the joint calculus

Theorem 2. [Strongly commuting sums] If all spectral projections of \(A = A^{\ast}\) commute with all those of \(B = B^{\ast}\), the algebraic sum on \(D (A) \cap D (B)\) is essentially self-adjoint. Its closure is the joint spectral integral of \(x + y\).

Proof. Theorem 1 constructs the joint PVM \(E\) from the spectral projections of \(A, B\). Integrating the real function \(x + y\) against it gives a self-adjoint operator \(C\). On \(D (A) \cap D (B)\), joint truncation and the inequality \(|x + y|^2 \le 2 x^2 + 2 y^2\) prove \(Cu = Au + Bu\).

For \(u \in D (C)\), let \(u_n = E ([- n, n]^2) u\). Then \(u_n \in D (A) \cap D (B)\), and both \(u_n \to u\) and \(Cu_n \to Cu\) by dominated convergence with \(1 + |x + y|^2\). The restricted sum is therefore a core restriction of \(C\). This proves the claim and also explains why the closure may have a larger domain than the original intersection. For any bounded Borel \(f\), uniqueness of the spectral calculus gives \(f (C) = \int f (x + y) dE (x, y)\). Taking \(f (s) = e^{its}\) and multiplying bounded joint spectral integrals gives \(e^{itC} = e^{itA} e^{itB}\). For \(B = - A\), the spectral sum is the everywhere-defined zero operator; this verifies the example \(B = - A\) in the post.\(\Box\)

The unitary representation of the additive group

The equality follows by multiplication in the bounded Borel calculus. That multiplication also gives \(U (s + t) = U (s) U (t)\) and \(U (t)^{\ast} = U (- t)\). For every vector \(u\),

by dominated convergence, with dominating constant four. Thus this is a strongly continuous unitary representation of \(\mathbb{R}^d\). Conversely, let \(U\) be such a representation. Its restrictions to the coordinate axes are strongly continuous unitary groups, so Stone's theorem gives self-adjoint generators \(A_j\). These groups commute by the group law. The equivalence proved above shows that their generators are pairwise strongly commuting. Theorem 1 constructs their joint PVM, whose representation agrees with \(U\) because every group element is a sum of coordinate-axis elements. Uniqueness follows from Stone's theorem and uniqueness of the joint PVM.

One can also recover the measure directly from the scalar Fourier tests \(\langle u, U (t) u \rangle\). For completeness, if a finite complex measure \(\nu\) on \(\mathbb{R}^d\) has zero Fourier transform, convolve it with a centered Gaussian density \(g_{\varepsilon}\) of covariance \(\varepsilon I\). The Gaussian Fourier inversion formula and Fubini apply since the Gaussian Fourier transform is integrable and \(\nu\) has finite total variation. They give \(g_{\varepsilon} \ast \nu = 0\). Hence for \(f \in C_c (\mathbb{R}^d)\), \(\int (f \ast g_{\varepsilon}) d \nu = 0\). The Gaussian approximate identity converges uniformly on such \(f\), so \(\int fd \nu = 0\). Riesz–Markov uniqueness gives \(\nu = 0\). Apply this to the difference of two scalar spectral measures and polarize. This proves the Fourier characterization used in the post without presupposing its coordinate operators.

2Normal operators and the complex plane

Lemma 3. [Closed-graph square-root lemma] If \(S\) is closed and densely defined, then \(T = S^{\ast} S\) is positive self-adjoint, \(D (T^{1 / 2}) = D (S)\), and \(\| T^{1 / 2} u \| = \| S u \|\) on this domain.

Proof. The graph of \(S\) is closed in \(H \oplus H\), and its orthogonal complement consists of \((- S^{\ast} v, v)\) with \(v \in D (S^{\ast})\). Decompose \((y, 0)\) along these two closed subspaces. The second component gives \(v = - S u\); hence \(u \in D (S^{\ast} S)\) and \((I + S^{\ast} S) u = y\). Thus \(I + T\) is onto. It is injective and its inverse has norm at most one, since \(\langle u, (I + T) u \rangle \geqslant \| u \|^2\). That everywhere-defined inverse is symmetric, hence self-adjoint; its injectivity and dense range imply that \(T\) is self-adjoint.

Set \(Q_n = n (n + T)^{- 1}\). The spectral theorem gives \(Q_n \rightarrow I\) strongly and \(Q_n H \subseteq D (T)\). Applying the preceding argument to \(S^{\ast}\) also makes \(SS^{\ast}\) positive self-adjoint. For \(u \in D (S)\) the resolvent equation, applied to \(w = (n + T)^{- 1} u\), gives \(S Q_n u = n (n + SS^{\ast})^{- 1} S u\). The right side tends to \(S u\) by the spectral theorem. On \(D (T)\) the identity is \(\| S v \|^2 = \langle v, T v \rangle = \| T^{1 / 2} v \|^2\). Therefore \(T^{1 / 2} Q_n u\) is Cauchy when \(u \in D (S)\); closedness of \(T^{1 / 2}\) gives \(u \in D (T^{1 / 2})\) and the asserted norm identity. Conversely, if \(u \in D (T^{1 / 2})\), the same bounded functions of \(T\) converge in the graph norm of \(T^{1 / 2}\). The norm identity on \(D (T)\) makes \(S Q_n u\) Cauchy, and closedness of \(S\) gives \(u \in D (S)\).\(\Box\)

Theorem 4. [The normal spectral theorem and bounded transform] Let \(N\) be densely defined and closed. The following conditions are equivalent: \(N^{\ast} N = NN^{\ast}\), including domains; and \(D (N) = D (N^{\ast})\) with \(\|Nu\| = \|N^{\ast} u\|\) there. Under these conditions there is a unique PVM \(E\) on \(\mathbb{C}\), with \(E (\mathbb{C}) = I\), such that

\(\displaystyle D (N) = \left\{ u : \int_{\mathbb{C}} |z|^2 \hspace{0.17em} d \langle u, E (z) u \rangle < \infty \right\}, \qquad N = \int_{\mathbb{C}} z \hspace{0.17em} dE (z) .\)

Conversely every such PVM defines a normal operator. These objects also correspond to nondegenerate representations of \(C_0 (\mathbb{C})\) and to bounded normal contractions \(T\) with \(\ker (I - T^{\ast} T) = \{0\}\). The last correspondence is

\(\displaystyle T = N (I + N^{\ast} N)^{- 1 / 2} .\)

\(\displaystyle D (N) = \text{Ran} (I - T^{\ast} T)^{1 / 2} .\)

\(\displaystyle N = T (I - T^{\ast} T)^{- 1 / 2} .\)

Proof. The preceding closed-graph lemma applied to \(N\) gives \(P = (N^{\ast} N)^{1 / 2}\) with \(D (P) = D (N)\) and \(\| P u \| = \| N u \|\) there. Applied to \(N^{\ast}\), it shows that the two conditions in the theorem are equivalent: equal positive operators have the same square-root domains and norms; conversely equality of the two domains and squared norms gives equality of their closed quadratic forms by polarization. An operator \(S^{\ast} S\) is recovered from that form: \(u \in D (S^{\ast} S)\) exactly when \(v \mapsto \langle S u, S v \rangle\) on \(D (S)\) is represented by a vector of \(H\), namely \(S^{\ast} S u\). Thus the positive operators agree.

Define \(V (Pu) = Nu\) on \(\text{Ran} P\). This is well-defined and isometric by the norm identity. It extends to an isometry from \((\ker N)^{\perp}\) onto \(\overline{\text{Ran} N} = (\ker N^{\ast})^{\perp}\). Normality gives \(\ker N = \ker N^{\ast}\), so extending this isometry by the identity on \(\ker N\) yields a unitary \(V\) on \(H\). Thus \(N = VP\) on \(D (P)\), and \(N^{\ast} = PV^{\ast}\) with its natural domain. In particular \(NN^{\ast} = VP^2 V^{\ast}\). Normality says \(VP^2 V^{\ast} = P^2\). The uniqueness of the nonnegative self-adjoint square root gives \(VPV^{\ast} = P\). It also follows that \(V\) commutes with the bounded calculus of \(P\): it commutes with its resolvents, hence with the continuous calculus and, by bounded pointwise limits, with its spectral projections.

Put \(Q = (I + P^2)^{- 1 / 2}\) and \(T = VPQ\). The positive calculus of \(P\) shows that \(PQ\) is bounded by one, that \(Q\) is injective, and that \(\text{Ran} Q = D (P)\). For the last equality, \(Qv \in D (P)\) because \(t^2 / (1 + t^2) \leq 1\); conversely \(u \in D (P)\) has \(v = (I + P^2)^{1 / 2} u \in H\) and \(Qv = u\). Commutation with \(V\) gives

Consequently \(T\) is a normal contraction and \(\ker (I - T^{\ast} T) = \{0\}\).

Let \(F\) be its bounded spectral measure on the closed unit disc. The spectral projection \(F (\{|w| = 1\})\) is the kernel projection of \(I - T^{\ast} T\), hence is zero. The map \(g (w) = w / \sqrt{1 - |w|^2}\) is a homeomorphism from the open disc to \(\mathbb{C}\). Define \(E (B) = F (g^{- 1} (B))\); the discarded boundary has zero projection, so \(E (\mathbb{C}) = I\). Its scalar quadratic measures are finite Radon measures, as all finite Borel measures on these second-countable locally compact spaces are Radon.

We record the domain argument for complex spectral integrals. Given any PVM on \(\mathbb{C}\), let \(E_R = E (\{|z| \leq R\})\) and \(N_R = \int z 1_{\{|z| \leq R\}} \hspace{0.17em} dE\). Orthogonality and the bounded calculus give \(\|N_R u\|^2 = \int_{|z| \leq R} |z|^2 \hspace{0.17em} d \mu_u\). Thus \(N_R u\) converges exactly on the asserted domain, which is dense since \(E_R \to I\) strongly. Its limit defines \(N\) there. If \(u_n \to u\) and \(Nu_n \to v\), then \(N_R u = E_R v\) for every \(R\). Hence \(\int_{|z| \leq R} |z|^2 \hspace{0.17em} d \mu_u \leq \|v\|^2\); monotone convergence gives \(u \in D (N)\) and \(Nu = v\). This proves closedness.

Let \(\bar{N}\) denote the integral of \(\bar{z}\) in this paragraph only. Bounded truncation and convergence give \(\langle Nu, v \rangle = \langle u, \bar{N} v \rangle\) on the common domain, so \(\bar{N} \subset N^{\ast}\). If \(v \in D (N^{\ast})\) and \(N^{\ast} v = w\), testing against \(E_R u\) gives \(\left( \int \bar{z} 1_{|z| \leq R} \hspace{0.17em} dE \right) v = E_R w\). The same norm estimate forces \(v \in D (\bar{N})\), proving equality. The scalar measure of \(Nu\) is \(|z|^2 \mu_u\) (test first on bounded sets and pass to the limit). It follows that

The fourth-moment condition implies the second-moment condition since \(|z|^2 \leq 1 + |z|^4\). Thus every complex spectral integral is normal.

Apply this construction to the pushforward of \(F\). For its resulting operator \(N_0 = g (T)\), write \(S = (I - T^{\ast} T)^{1 / 2}\). The domain condition for \(g (T)\) is equivalent to integrability of \((1 - |w|^2)^{- 1}\), because \(|g (w) |^2 = (1 - |w|^2)^{- 1} - 1\). The bounded spectral calculus therefore gives \(D (N_0) = \text{Ran} S\) and \(N_0 S = T\): one direction follows by applying \(S\) to any vector, and the other by applying the spectral integral of \((1 - |w|^2)^{- 1 / 2}\) to a vector in \(D (N_0)\). In the construction from \(N\), \(S = Q\) and \(NQ = T\), so the domains and actions of \(N_0\) and \(N\) coincide.

Conversely the same argument starts from any normal contraction with the stated kernel condition and constructs \(N\). The identity \((I + |g (w) |^2)^{- 1 / 2} = \sqrt{1 - |w|^2}\) proves that its bounded transform is precisely \(T\). If another PVM represented the same \(N\), the already proved product formula would make its integral of \(z / \sqrt{1 + |z|^2}\) the same \(T\). Uniqueness of the bounded normal spectral measure and injectivity of this scalar map force the two PVMs to agree. Finally the representation–PVM theorem in the Part I companion, whose proof applies to any second-countable locally compact space, identifies these measures with the nondegenerate representations of \(C_0 (\mathbb{C})\).\(\Box\)

Proposition 5. [Real and imaginary parts and two-parameter evolution] For the preceding \(N\), the spectral integrals \(A = \int \text{Re} z \hspace{0.17em} dE\) and \(B = \int \text{Im} z \hspace{0.17em} dE\) are strongly commuting self-adjoint operators, \(D (N) = D (A) \cap D (B)\), and \(N = A + iB\) on that domain. Moreover \(D (N)\) is a core for both \(A\) and \(B\); consequently the closures of \((N + N^{\ast}) / 2\) and \((N - N^{\ast}) / (2 i)\) are \(A\) and \(B\). Conversely every strongly commuting self-adjoint pair arises this way. The corresponding strongly continuous unitary representation is

\(\displaystyle U (s, t) = \int e^{i (s \text{Re} z + t \text{Im} z)} \hspace{0.17em} dE (z) = e^{isA} e^{itB} .\)

Every strongly continuous unitary representation of \(\mathbb{R}^2\) is obtained uniquely in this manner.

Proof. The real spectral-integral theorem makes \(A, B\) self-adjoint. Their projections are \(E (\{z : \text{Re} z \in C\})\) and \(E (\{z : \text{Im} z \in D\})\), so they commute. The domain identity follows from \(|z|^2 = (\text{Re} z)^2 + (\text{Im} z)^2\); truncation then gives the asserted sum. If \(u \in D (A)\), put \(u_R = E_R u\). These vectors belong to \(D (N)\), converge to \(u\), and \(Au_R \to Au\) by dominated convergence of the squared real coordinate. This proves the core assertion for \(A\); the proof for \(B\) uses the imaginary coordinate.

Conversely, Theorem 1 constructs the joint PVM of any strongly commuting self-adjoint pair \(A, B\). Push this measure forward by \((x, y) \mapsto x + iy\) and apply Theorem 4. Its coordinate integrals and their domains agree with \(A, B\); the identity \(| x + iy |^2 = x^2 + y^2\) proves that the normal integral is \(A + iB\) on precisely their domain intersection.

The assertion about unitary representations is the case \(d = 2\) of the result proved in Section 1. Under the same identification of the plane, its characters are \(e^{i (s \Re z + t \Im z)}\). Thus the joint Fourier integral gives the displayed formula, and the converse and uniqueness follow from the coordinate-axis Stone generators, as proved there.\(\Box\)

On \(L^2 (\mathbb{C}, d^2 z)\) multiplication by \(z\) has the PVM of multiplication by set indicators, so it is normal. For any \(z_0\) and \(r > 0\), the normalized indicator of the disc \(|z - z_0 | < r\) belongs to its domain and satisfies \(\|(N - z_0) u_r \| \leq r\). A bounded inverse would imply \(1 \leq \|(N - z_0)^{- 1} \| r\), impossible as \(r \downarrow 0\). Hence its spectrum is all of \(\mathbb{C}\). For \(t \ne 0\) the multiplier \(e^{itz}\) has modulus \(e^{- t \text{Im} z}\) and is unbounded; unit vectors supported in discs tending in the appropriate imaginary direction show this directly. These examples justify the two limitations on resolvents and one-parameter exponentials stated in Section 2 of the post.

3Manifold-valued observables and other outcome spaces

Proposition 6. [Changing the outcome space] Let \(M\) be a second-countable Hausdorff manifold. Normalized PVMs on \(M\) correspond bijectively to nondegenerate representations of \(C_0 (M)\). For such a PVM, every real-valued Borel function \(h\) defines a self-adjoint spectral integral with domain \(\{u : \int |h|^2 \hspace{0.17em} d \mu_u < \infty\}\). A Borel map \(F : M \to L\), where \(L\) is another such manifold, gives a PVM \(E_F (B) = E (F^{- 1} (B))\) and the substitution rule \(g (E_F) = (g \circ F) (E)\), including equality of the maximal domains for unbounded real Borel \(g\).

Proof. A second-countable manifold is locally compact and admits an increasing sequence of compactly supported continuous cutoffs tending to one. Continuous functions separate points and generate its Borel sigma-algebra. These are precisely the ingredients used in the representation–PVM proof in the Part I companion: Riesz–Markov produces finite regular scalar measures, polarization produces the operator measures, multiplication follows by the functional monotone-class theorem, and nondegeneracy gives total mass \(I\). This proves the first assertion with the same construction and uniqueness, without a coordinate choice.

Inverse images preserve complements and disjoint countable unions, so \(E_F\) is a normalized PVM. The substitution formula holds for indicator functions, then simple functions, and then bounded Borel functions by uniform simple approximation. For unbounded real \(g\), the scalar pushforward identity gives \(\int |g|^2 \hspace{0.17em} d \mu^F_u = \int |g \circ F|^2 \hspace{0.17em} d \mu_u\), so the domains coincide. Apply the bounded identity to \(g 1_{|g| \leq n}\) and pass to the defining spectral-integral limit. Taking \(L =\mathbb{R}\) and \(F = h\) gives the self-adjoint assertion.

If \(F\) is continuous, \(g \circ F\) is bounded continuous for \(g \in C_0 (L)\), but need not vanish at infinity. Its integral is already available in the bounded Borel extension. For example the constant map from a noncompact \(M\) pulls a function nonzero at its image point back to a nonzero constant. If \(F\) is proper, the sets \(\{|g \circ F| \geq \varepsilon\}\) are compact, proving that the pullback then lies in \(C_0 (M)\). Thus no properness hypothesis is silently needed for the substitution rule.\(\Box\)

Proposition 7. [Coordinates on the circle, sphere and an embedded manifold] A PVM on \(S^1\) is equivalently a unitary operator. A PVM on \(S^2\) is equivalently a commuting triple of bounded self-adjoint operators \(X_1, X_2, X_3\) with \(\sum_a X_a^2 = I\). Given a proper embedding \(\iota : M \to \mathbb{R}^k\), a PVM on \(M\) is equivalently a strongly commuting self-adjoint \(k\)-tuple whose joint PVM is concentrated on \(\iota (M)\).

Proof. On the circle, integration of \(z\) gives a unitary since \(\bar{z} z = 1\); the bounded spectral theorem recovers its PVM, and density of polynomials in \(z, \bar{z}\) gives uniqueness. On the sphere the three coordinate integrals commute, are bounded and self-adjoint, and satisfy the stated relation. Conversely, a commuting triple of bounded self-adjoint operators is strongly commuting. Theorem 1 gives its joint PVM \(E\). Its coordinate marginals are supported in bounded intervals, so \(E\) is concentrated in a bounded box. Let \(p (x) = x_1^2 + x_2^2 + x_3^2 - 1\). The assumed relation and multiplication in the bounded calculus on that box give \(p (E) = 0\). Thus \(\int | p (x) |^2 d \mu_u (x) = \| p (E) u \|^2 = 0\) for every \(u\). Positivity gives zero scalar mass to \(| p | \geqslant 1 / n\) for every positive integer \(n\), hence to the complement of the sphere. Polarization gives the same assertion for \(E\). Restrict this measure to \(S^2\). Its coordinate integrals are exactly the given operators. Coordinate polynomials separate points on \(S^2\) and are closed under conjugation, so Stone–Weierstrass and scalar measure uniqueness prove uniqueness.

For the last assertion, push the PVM forward by \(\iota\) and integrate the real coordinate functions. Their spectral projections commute because they all belong to the same PVM. Conversely a finite strongly commuting tuple has the joint PVM constructed in Theorem 1. The embedding is a homeomorphism onto its image, which is closed by properness. A joint PVM concentrated there can therefore be transported back by \(\iota^{- 1}\). This reverses both constructions.\(\Box\)

Other outcome spaces and joint observables

The representation–PVM theorem applies without change to every second-countable locally compact Hausdorff space: the proof uses Riesz–Markov, a compact exhaustion and continuous functions generating the Borel sets, not smooth coordinates. The multiplication model on \(L^2 (M, \nu)\) is immediate: multiplication by \(1_B\) gives its projections, and \(\langle u, E (B) u \rangle = \int_B |u|^2 d \nu\). For a Hermitian vector bundle use the fibre inner product in the same formula; multiplication by a scalar function still has norm at most its supremum norm.

For a countable discrete space, \(P_x = E (\{x\})\) are orthogonal and sum strongly to \(I\). Conversely such a family defines \(E (B) = \sum_{x \in B} P_x\): orthogonality gives norm convergence on each vector since \(\sum_x \|P_x u\|^2 = \|u\|^2\). The same identity proves strong convergence of \(\sum_x f (x) P_x\) for bounded \(f\), multiplicativity, and the asserted bounded calculus. On an arbitrary measurable outcome space, a normalized PVM gives a unital bounded Borel calculus by simple approximation. Conversely a unital homomorphism with bounded pointwise weak continuity gives \(E (B) = j (1_B)\); orthogonality and continuity of indicators of increasing unions give a PVM, as in the Part I construction.

Here is the local-compactness point for an infinite-dimensional Hilbert space \(X\). If a continuous function \(f\) vanishing at infinity were nonzero at \(x\), the compact set \(\{|f| \geq |f (x) | / 2\}\) would contain an open neighbourhood of \(x\). It would therefore contain a closed ball of positive radius, which would be compact. But if \((e_n)\) is an orthonormal sequence, the points \(x + re_n\) of such a ball have mutual distance \(r \sqrt{2}\), so admit no convergent subsequence. This contradiction proves \(C_0 (X) = 0\). No analogous argument prevents normalized PVMs on its Borel sets.

Proposition 8. [Joint sharp observables and operations on outcomes] Let \(E_1, E_2\) be normalized PVMs on second-countable locally compact Hausdorff spaces \(X, Y\), acting on the same Hilbert space. They are marginals of a joint PVM on \(X \times Y\) if and only if \(E_1 (B) E_2 (C) = E_2 (C) E_1 (B)\) for all Borel sets \(B, C\). The joint PVM is unique and satisfies \(E (B \times C) = E_1 (B) E_2 (C)\). A Borel map from \(X \times Y\) to another such space can then be applied by pushforward. This includes group multiplication when \(X = Y = G\) is a second-countable locally compact group.

Proof. For a joint PVM, multiply the projections of the cylinder sets \(B \times Y\) and \(X \times C\). Their intersection is \(B \times C\), which proves necessity and the rectangle formula. Scalar measure uniqueness on the sigma-algebra generated by rectangles proves uniqueness.

For sufficiency the two bounded calculi commute, first for simple functions and then by strong limits. Let \(\mathcal{A}\) be the commutative unital \(C^{\ast}\)-algebra generated by \(I\) and their \(C_0\) calculi in \(B (H)\). It is \(C (K)\) for its compact character space \(K\), with a spectral measure \(P\) on \(K\). A character restricted to either \(C_0\) calculus is either zero or evaluation at a point of the corresponding outcome space: this follows from the locally compact Gelfand theorem, even if that calculus is not faithful. Write the zero restriction as the extra point \(\infty\) of \(X^+\) or \(Y^+\); for a compact space use an extra isolated point. The resulting maps \(a : K \to X^+\) and \(b : K \to Y^+\) are continuous, since all their pulled-back continuous functions are Gelfand transforms of the generating elements.

Push \(P\) forward under \((a, b)\). Its marginal on \(X^+\) integrates every \(f \in C_0 (X)\) to the first calculus. If \(\chi_n \uparrow 1\) are continuous compactly supported cutoffs on \(X\), nondegeneracy and bounded convergence give \(P (a^{- 1} (X)) = \lim_n j_1 (\chi_n) = I\) strongly. Thus the marginal has zero mass at \(\infty\); similarly for \(Y\). The measure is concentrated on \(X \times Y\) and has the required marginals by uniqueness of the scalar representing measures. This constructs the joint PVM. The already proved substitution rule gives the pushforward statement, with no properness assumption on the Borel map.\(\Box\)

4Observables, normal states and probability laws

This section supplies the operator-algebra argument for Section 4 of Part II. Here the von Neumann algebra contains the bounded tests of each observable, while the normal state turns its spectral projections into probabilities.

Proposition 9. [Affiliated self-adjoint observables and laws] For \(A = A^{\ast}\) and a von Neumann algebra \(\mathcal{M} \subset B (H)\), the following are equivalent: \(A\) commutes, with domain invariance, with all unitaries of \(\mathcal{M}'\); its spectral projections lie in \(\mathcal{M}\); its bounded Borel calculus lies in \(\mathcal{M}\); its resolvents lie in \(\mathcal{M}\). In a normal state \(\omega\), \(\omega (E_A (\cdot))\) is a probability measure and \(\omega (f (A)) = \int f \hspace{0.17em} d \mu\).

Proof. Domain commutation passes to bounded inverses \(R_A (z)\), and the bicommutant theorem places these inverses in \(\mathcal{M}\). Resolvent generation and strong closure place the \(C_0\) calculus and then the Borel calculus in \(\mathcal{M}\). Conversely spectral projections in \(\mathcal{M}\) commute with \(\mathcal{M}'\); their truncations show that commuting unitaries preserve the spectral domain and commute with \(A\). The projections and bounded calculus implications are immediate from integration. Normality of the state applied to increasing sums of disjoint projections gives countable additivity and total mass one. The integral identity holds first for simple functions and then by bounded convergence. For a single observable the law is classical; a joint projection-valued measure for two observables would make their marginal projections commute. Noncommuting marginal projections therefore preclude such a joint measure.\(\Box\)

Ordered tests are not determined by marginal laws

Proposition 10. [A matrix example of ordered expectations] Let

\(\displaystyle A = \sigma_1 = \left( \begin{array}{cc} 0 & 1\\ 1 & 0 \end{array} \right), \qquad B = \sigma_2 = \left( \begin{array}{cc} 0 & - i\\ i & 0 \end{array} \right), \qquad u = \binom{1}{0} .\)

In the state \(\omega (T) = \langle u, T u \rangle\), both spectral laws equal \((\delta_{- 1} + \delta_1) / 2\), whereas \(\omega (AB) = i\) and \(\omega (BA) = - i\). The pair \((A, - B)\) has the same two marginal laws and a different mixed expectation.

Proof. Direct multiplication gives \(A^2 = B^2 = I\) and \(A = A^{\ast}, B = B^{\ast}\). Thus their spectral projections for the values \(\pm 1\) are \((I \pm A) / 2\) and \((I \pm B) / 2\). The diagonal matrix entries \(\langle u, A u \rangle\) and \(\langle u, B u \rangle\) are zero, so each projection has probability one half. Also \(AB = i \text{diag} (1, - 1)\) and \(BA = - i \text{diag} (1, - 1)\), giving the asserted expectations. Negating \(B\) interchanges its two equally weighted spectral projections and changes the sign of \(AB\). This proves that the marginal laws do not determine the ordered expectation.

For any state, positivity implies \(\omega (T^{\ast}) = \overline{\omega (T)}\). For completeness, the expectation of a bounded self-adjoint operator is real, since after adding a sufficiently large multiple of \(I\) it becomes positive. Decompose an arbitrary \(T\) into its real and imaginary self-adjoint parts to obtain the identity. Thus for real bounded tests \(f, g\), the expectations of \(j_1 (f) j_2 (g)\) and \(j_2 (g) j_1 (f)\) are complex conjugates. All finite ordered words are well-defined bounded operators, since \(\| j_{k_1} (f_1) \cdots j_{k_m} (f_m) \| \leqslant \prod_{r = 1}^m \| f_r \|_{\infty}\). Their state expectations constitute the joint bounded-test distribution.\(\Box\)

Continuous-test laws and spectral-event probabilities

Let \(j : C_0 (\mathbb{R}) \rightarrow \mathcal{M}\) be a nondegenerate representation and \(\omega\) any state. The positive functional \(\ell = \omega \circ j\) has norm at most one. Riesz–Markov gives a finite positive measure \(\mu\) with \(\mu (\mathbb{R}) = \| \ell \|\). For increasing compactly supported cutoffs \(\chi_n \uparrow 1\), monotone convergence of this measure gives

Hence full mass is exactly the cutoff condition stated in the post. If the state is normal, \(j (\chi_n) \uparrow I\) strongly and normality makes the limit one. The preceding proposition then identifies this measure with \(B \mapsto \omega (E (B))\). The following example shows why full mass alone does not give that identification.

Proposition 11. [Full mass does not imply spectral normality] On \(H = L^2 (0, 1)\) let \(A\) multiply by \(t\). There is a state \(\omega\) on \(B (H)\) such that \(\omega (f (A)) = f (0)\) for every bounded continuous real-line test \(f\), although \(\omega (E_A (\{ 0 \})) = 0\). Its restriction to \(C_0 (\mathbb{R})\) has norm one and Riesz law \(\delta_0\), but the state is not normal.

Proof. Put \(u_n = \sqrt{n} 1_{(0, 1 / n)}\). These are unit vectors. Fix a free ultrafilter \(\mathcal{U}\) on the positive integers, and define

Each bounded complex sequence has a unique ultrafilter limit in a compact disc; such limits preserve addition, scalar multiplication and nonnegativity, as follows by intersecting the sets on which each sequence lies near its limit. Thus \(\omega\) is linear and positive, \(\omega (I) = 1\), and \(| \omega (T) | \leqslant \| T \|\). It is a state. If \(f\) is bounded and continuous, then

Indeed the difference is bounded by \(\sup_{0 \leqslant t \leqslant 1 / n} | f (t) - f (0) |\). The ultrafilter limit agrees with the ordinary limit. A continuous compactly supported cutoff equal to one on \([0, 1]\) satisfies \(f (A) = I\); thus the restriction has norm one. Uniqueness in Riesz–Markov identifies its law as \(\delta_0\).

The spectral projections of \(A\) multiply by indicators of Borel subsets of \((0, 1)\), so \(E_A (\{ 0 \}) = 0\). To see failure of normality directly, let \(P_k = E_A ((0, 1 / k))\). These projections decrease strongly to zero by dominated convergence, but \(\omega (P_k) = 1\) for every \(k\), since \(P_k u_n = u_n\) whenever \(n \geqslant k\). Equivalently \(I - P_k \uparrow I\) while all their expectations are zero. This contradicts normality.\(\Box\)

5The geometric commutator and the distance formula

Proposition 12. [Compact resolvent and vanishing-at-infinity functions] For self-adjoint \(D\), compactness of one nonreal resolvent is equivalent to compactness of every \(f (D)\), \(f \in C_0 (\mathbb{R})\).

Proof. One compact resolvent and its adjoint generate only compact operators under sums, products and norm limits without a constant term. Resolvent density therefore gives all \(C_0\) multipliers. The converse follows by choosing \(f (x) = (x - i)^{- 1}\).\(\Box\)

Theorem 13. [The commutative spectral distance] On a compact connected Riemannian spin manifold without boundary, let \(D\) be its Dirac operator on square-integrable spinors. For real smooth \(f\), \([D, f]\) is bounded with norm \(\sup |df|\), and

\(\displaystyle d (p, q) = \sup \{|f (p) - f (q) | : f \in C^{\infty} (M, \mathbb{R}), \|[D, f]\| \le 1\} .\)

Proof. The geometric input is the local expression \(D = - i \sum_j c (e^j) \nabla_{e_j}\) for an orthonormal coframe, with Clifford multiplication satisfying \(c (\xi)^{\ast} c (\xi) = | \xi |^2 I\) for real covectors (the factor \(i\) moves with conventions). The Leibniz rule gives \([D, f] = - ic (df)\). Its pointwise operator norm is \(|df|\); the norm of this multiplication operator on \(L^2\) is its essential supremum, equal to its supremum by continuity. Multiplication by smooth \(f\) preserves the \(H^1\) domain.

For clarity, that is indeed the self-adjoint domain. Integration by parts on a boundaryless manifold makes \(D\) symmetric. The elliptic energy estimate \(\|u\|_{H^1} \le C (\|Du\|_2 +\|u\|_2)\) follows by integrating the identity \(D^2 = \nabla^{\ast} \nabla + \text{Scal} / 4\) and using bounded curvature on the compact manifold. If \(u \in D (D^{\ast})\), then distributionally \(Du \in L^2\). Local smoothing and a partition of unity give smooth \(u_{\varepsilon} \to u\) with \(Du_{\varepsilon}\) bounded in \(L^2\): the commutator of a smooth-coefficient first-order operator with convolution is uniformly \(L^2\) bounded, as its kernel contains \((a (x) - a (y)) \partial \eta_{\varepsilon} (x - y)\) and bounded coefficient-derivative terms, whose integral bounds are uniform by \(|a (x) - a (y) | \le C |x - y|\). The energy estimate bounds \(u_{\varepsilon}\) in \(H^1\); weak compactness gives \(u \in H^1\). Conversely every \(H^1\) spinor is in the adjoint domain by smooth approximation. Thus \(D = D^{\ast}\) on \(H^1\). The compact embedding \(H^1 \hookrightarrow L^2\) makes its resolvent compact.

If \(\sup |df| \le 1\), integration along any piecewise smooth path from \(p\) to \(q\) gives \(|f (p) - f (q) | \le\) its length. Infimizing proves one inequality. For the reverse, let \(g (x) = d (x, q)\), a one-Lipschitz function. We detail the smoothing step. Cover \(M\) by finitely many sufficiently small coordinate balls on which metric norms differ from fixed Euclidean norms by a factor at most \(1 + \eta\), and choose a subordinate smooth partition \(\chi_j\). Mollify \(g\) on slightly larger balls, obtaining \(g_j\) uniformly within \(\delta\) of \(g\) and with \(|dg_j | \le (1 + \eta)^2\). For \(g_{\delta} = \sum \chi_j g_j\),

because \(\sum d \chi_j = 0\). Thus \(|dg_{\delta} | \le (1 + \eta)^2 + \delta \sum \|d \chi_j \|_{\infty}\). Choose first \(\eta\), then \(\delta\), to make this at most \(1 + \varepsilon\), while \(\|g_{\delta} - g\|_{\infty} \to 0\). The admissible functions \(g_{\delta} / (1 + \varepsilon)\) have differences tending to \(d (p, q)\), proving equality. This proof also identifies the geometric information absent from the calculus of \(D\) alone: its commutators use the separately specified multiplication algebra.\(\Box\)

6Sharp observables and Poisson brackets

This section justifies the obstruction in the final section of the post. Let \(\mathcal{A} \subseteq C_0 (M)\) be a complex algebra of smooth tests, stable under conjugation and the Poisson bracket. For \(\hbar > 0\) tending to zero, consider linear maps \(Q_{\hbar} : \mathcal{A} \rightarrow B (H_{\hbar})\) that respect adjoints. The operator norms of the multiplication and Dirac defects in the post are well-defined, even if the Hilbert spaces vary. The norm correspondence \(\| Q_{\hbar} (f) \| \rightarrow \| f \|_{\infty}\) prevents the classical limit from losing a nonzero test. These conditions describe requirements on a proposed family, not an assertion that one exists on every symplectic manifold.

Proposition 14. [An exact sharp calculus cannot quantize a nonzero bracket] Suppose all the \(Q_{\hbar}\) are \(\ast\)-homomorphisms, and assume both norm correspondence for every test and the Dirac condition for every pair. Then \(\{ f, g \} = 0\) for all \(f, g \in \mathcal{A}\).

Proof. Pointwise products in \(\mathcal{A}\) commute, so \([Q_{\hbar} (f), Q_{\hbar} (g)] = Q_{\hbar} (fg - gf) = 0\). The Dirac condition therefore says \(\| Q_{\hbar} (\{ f, g \}) \| \rightarrow 0\). Since \(\{ f, g \} \in \mathcal{A}\), norm correspondence gives the same limit as \(\| \{ f, g \} \|_{\infty}\). Hence the bracket is zero. In fact this proof requires only that the image of each \(Q_{\hbar}\) commute, not its full multiplicativity.\(\Box\)

The obstruction applies to the full compactly supported smooth test algebra on a positive-dimensional symplectic manifold. In a Darboux chart centred at \(p\), choose canonical coordinates \(q_1, p_1\) with \(\{ q_1, p_1 \} = 1\). Multiply each by a smooth cutoff compactly supported in the chart and equal to one near \(p\), then extend them by zero. The resulting \(f, g \in C_c^{\infty} (M)\) have \(\{ f, g \} (p) = 1\), because the cutoff derivatives vanish there. Thus their bracket is nonzero, and the proposition rules out a family of sharp calculi with those classical-limit properties.

Positivity of a quantization map is a different condition: it says \(Q_{\hbar} (f) \geqslant 0\) for nonnegative tests, and does not force commuting images. Actual constructions, and the continuous-field structure used in strict quantization, require further work; see the Landsman references below. No additional existence theorem is used here.