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Projection-valued measure

In , particularly in , a projection-valued measure (PVM) on a (X, \Sigma) with values in the orthogonal projections on a separable H is a map E: \Sigma \to \Proj(H) such that E(\emptyset) = 0, E(X) = I_H (the identity on H), E(E \cap F) = E(E)E(F) for all E, F \in \Sigma, and for any countable collection of pairwise \{E_n\}_{n=1}^\infty \subset \Sigma, E\left(\bigcup_{n=1}^\infty E_n\right) = \sum_{n=1}^\infty E(E_n) in the strong topology. This structure generalizes classical measures by assigning to each measurable set a projection rather than a scalar, enabling the of functions against E to yield bounded operators on H. Projection-valued measures play a central role in the spectral theorem for self-adjoint and normal operators on Hilbert spaces, providing a functional calculus that decomposes such operators as integrals with respect to a unique PVM. Specifically, for a bounded self-adjoint operator A on H, there exists a unique PVM E on the Borel \sigma-algebra of \mathbb{R} (or a compact interval containing the spectrum \sigma(A)) such that A = \int \lambda \, dE(\lambda), where the integral is defined via the sesquilinear forms \langle Av, w \rangle = \int \lambda \, d\langle Ev, w \rangle for v, w \in H. This decomposition extends to unbounded self-adjoint operators and normal operators, allowing the construction of functions of A as f(A) = \int f(\lambda) \, dE(\lambda) for suitable Borel measurable f, with the spectrum \sigma(A) identified as the essential range of the measure induced by E. Beyond pure mathematics, projection-valued measures are foundational in , where they model the of observables—self-adjoint operators representing physical quantities like or —via the projection postulate and . Key properties include the positivity of E(E) for all E \in \Sigma, the of projections for , and the fact that scalar measures \langle E(E)v, w \rangle are complex measures on \Sigma, ensuring the framework aligns with probability interpretations in . These measures also facilitate multiplicity theory and direct integral decompositions of Hilbert spaces, as developed in classical texts on .

Basics

Definition

In , a projection-valued measure is a that generalizes scalar measures to operators on a , providing a framework for decompositions of operators. It is defined on a (X, \Sigma), where X is a set and \Sigma is a \sigma-algebra of subsets of X, and takes values in the orthogonal projections on a separable complex H. Orthogonal projections on H are bounded operators P satisfying P^2 = P and P^* = P, which project onto closed subspaces of H. The separability of H ensures that the space has a countable , facilitating the construction and analysis of such measures. Formally, a projection-valued measure is a map \pi: \Sigma \to \mathcal{P}(H), where \mathcal{P}(H) denotes the set of orthogonal projections on H, satisfying the following axioms: \pi(\emptyset) = 0 (the zero operator), \pi(X) = I_H (the operator on H), countable additivity in the strong operator topology for disjoint unions—that is, if \{E_n\}_{n=1}^\infty \subset \Sigma are pairwise disjoint and \bigcup_{n=1}^\infty E_n = E \in \Sigma, then \pi(E) = \sum_{n=1}^\infty \pi(E_n)—and multiplicativity \pi(E \cap F) = \pi(E) \pi(F) for all E, F \in \Sigma. The multiplicativity condition implies orthogonality of projections for : if E \cap F = \emptyset, then \pi(E) \pi(F) = 0. These properties ensure that \pi behaves analogously to a classical measure but in the setting. For any \xi, \eta \in H, the map \mu_{\xi, \eta}: \Sigma \to \mathbb{C} defined by \mu_{\xi, \eta}(E) = \langle \xi \mid \pi(E) \eta \rangle is a complex-valued measure on (X, \Sigma), with total variation bounded by \|\xi\| \|\eta\|. When \xi = \eta, \mu_{\xi, \xi} is a positive finite measure with \mu_{\xi, \xi}(X) = \|\xi\|^2. This scalarization connects the operator-valued structure to classical measure theory. The with respect to \pi is defined for bounded measurable functions f: X \to \mathbb{C}. For a f = \sum_{k=1}^n c_k \chi_{E_k} with disjoint E_k \in \Sigma and c_k \in \mathbb{C}, the is the \int_X f \, d\pi := \sum_{k=1}^n c_k \pi(E_k). This extends by uniform approximation to all bounded measurable f, yielding a on H such that \langle \xi \mid \left( \int_X f \, d\pi \right) \eta \rangle = \int_X f \, d\mu_{\xi, \eta} for all \xi, \eta \in H. This construction allows representation of via over measures.

Historical Development

The concept of projection-valued measures emerged in the early as part of the rigorous and in s. introduced the foundational ideas in his 1932 monograph Mathematische Grundlagen der Quantenmechanik, where he developed spectral resolutions for operators using continuous geometries, laying the groundwork for representing operators via integrals against projections. This work addressed the need for a precise framework to handle observables in , linking them to measures taking values in orthogonal projections on . Concurrently, Marshall Stone advanced the theory through his 1932 theorem on one-parameter unitary groups, which established a correspondence between such groups and operators, incorporating projection measures to resolve the . Stone's result connected the of to spectral decompositions, influencing the understanding of and providing an early link to projection-valued measures in the context of unitary representations. In the mid-20th century, mathematicians like Jacques Dixmier formalized projection-valued measures within the framework of operator algebras during the 1950s, establishing them as essential tools in for von Neumann algebras. Dixmier's contributions in works such as Les algèbres d'opérateurs dans l'espace hilbertien (1957) integrated these measures into the broader structure of W*-algebras, emphasizing their role in resolving normal operators. These developments were influenced by advancements in measure theory, particularly extensions of the Radon-Nikodym theorem to operator-valued settings in the 1940s and 1950s, as explored in the integration theory for vector measures by Dunford and Schwartz. Key publications include von Neumann's 1932 text and subsequent refinements in the 1970s by Michael Reed and in their series, which synthesized and expanded the spectral applications of projection-valued measures.

Properties and Constructions

Key Properties

Projection-valued measures exhibit several fundamental algebraic and analytic properties that underpin their role in . Central to their structure is the multiplicativity property: for any E, the \pi(E) satisfies \pi(E)^2 = \pi(E) and \pi(E)^* = \pi(E), confirming that each \pi(E) is a idempotent on the H. This idempotence ensures that \pi(E) acts as an orthogonal onto the \pi(E)H, while self-adjointness preserves the inner product structure of H. A key algebraic feature is orthogonality for disjoint sets. If \{E_n\} is a countable collection of pairwise disjoint Borel sets, then \pi\left(\bigcup_n E_n\right) = \sum_n \pi(E_n) in the strong operator topology, where the sum converges strongly to the projection onto the closed span of the ranges of the \pi(E_n). Moreover, for any two disjoint Borel sets E_1 and E_2, \pi(E_1)\pi(E_2) = 0, reflecting the mutual orthogonality of the corresponding projections. This additivity extends to finite disjoint unions and is essential for decomposing the identity operator across partitions of the measure space. Analytically, projection-valued measures are characterized by strong . For an increasing of Borel sets E_n \uparrow E, the projections satisfy \pi(E_n) \xi \to \pi(E) \xi strongly for every \xi \in H, meaning the holds in the norm topology on H. This property ensures that the measure is countably additive in a topological , facilitating limits in expressions. The family \{\pi(E)\} thus forms a of the , satisfying \int_X d\pi(\lambda) = I_H in the of integrals over the space X, where the integral represents the strong limit of Riemann sums involving simple functions. Finally, uniqueness holds in the context of the : for a given bounded on H, the associated projection-valued measure is unique up to equivalence of the underlying . This uniqueness arises from the canonical construction via the and is crucial for the representation of operators.

Constructions and Examples

One fundamental construction of a projection-valued measure (PVM) arises in the context of multiplication operators on Hilbert spaces of square-integrable functions. Consider a measure space (X, \mathcal{B}, \mu) where \mathcal{B} is the \sigma-algebra of Borel sets and \mu is a \sigma-finite measure. On the Hilbert space L^2(X, \mu), define the map \pi: \mathcal{B} \to \mathcal{B}(L^2(X, \mu)) by \pi(E) f = \chi_E f for f \in L^2(X, \mu), where \chi_E is the characteristic (indicator) function of E \in \mathcal{B}. This \pi yields a PVM on (X, \mathcal{B}), as each \pi(E) is the orthogonal projection onto the closed subspace \{f \in L^2(X, \mu) : \operatorname{supp} f \subseteq E\}. In finite-dimensional settings, PVMs can be constructed using Dirac measures and orthonormal bases. For a finite-dimensional \mathbb{C}^n equipped with the discrete \sigma-algebra on \{1, \dots, n\}, the Dirac measures at points i = 1, \dots, n correspond to rank-one orthogonal projections |e_i\rangle\langle e_i|, where \{e_i\} is an . The map assigning to each \{i\} the projection |e_i\rangle\langle e_i| extends to a PVM on the power set \sigma-algebra, satisfying the required and additivity for disjoint unions. This construction illustrates how PVMs reduce to resolutions of the identity in the discrete spectral case for self-adjoint matrices. Another construction derives from unitary representations of abelian s. For an abelian G with left , the left regular representation \lambda: G \to U(L^2(G)) defined by \lambda(g) f(h) = f(g^{-1} h) induces a PVM on the Pontryagin dual group \hat{G} (the group of characters). Specifically, there exists a unique regular PVM P on the Borel \sigma-algebra of \hat{G} such that \lambda(g) = \int_{\hat{G}} \chi(g) \, dP(\chi) for \chi \in \hat{G}. This PVM captures the decomposition of the regular representation into irreducible components via the . A concrete example of spectral projections occurs for a multiplication operator M_g on L^2(X, \mu), where g: X \to \mathbb{R} is a measurable function bounding the spectrum. The associated PVM \pi has \pi((a, b]) as the orthogonal projection onto the subspace \{f \in L^2(X, \mu) : g(x) \in (a, b] \ \mu\text{-a.e.}\}, implemented by \pi((a, b]) f = \chi_{g^{-1}((a,b])} f. Intervals generate the Borel \sigma-algebra on \sigma(M_g), and this \pi resolves the identity for M_g = \int \lambda \, d\pi(\lambda). These constructions satisfy the PVM axioms: each \pi(E) is an orthogonal projection (idempotent and ), \pi(\emptyset) = 0, \pi(X) = I, and for disjoint E_n, strong additivity holds as \sum \pi(E_n) \xi \to \pi(\cup E_n) \xi for all \xi, since characteristic functions additively decompose: \chi_{\cup E_n} = \sum \chi_{E_n} on disjoint supports, preserving L^2-norms. Orthogonality follows from disjoint supports implying \pi(E) \pi(F) = 0 for E \cap F = \emptyset.

Spectral Theory Connections

Spectral Theorem

The provides a fundamental decomposition for operators in terms of projection-valued measures (PVMs). Specifically, for every bounded A on a separable H, there exists a unique (up to equivalence) PVM \pi^A defined on the Borel \sigma-algebra of its \sigma(A) \subseteq \mathbb{R} such that A = \int_{\sigma(A)} \lambda \, d\pi^A(\lambda), where the integral is taken in the strong operator topology, meaning it converges strongly to A on H. This representation expresses A as a "continuous sum" of scalar multiples of orthogonal projections, weighted by the spectral measure. This decomposition enables a functional calculus for A: for any bounded Borel measurable function f: \sigma(A) \to \mathbb{C}, the operator f(A) is defined by f(A) = \int_{\sigma(A)} f(\lambda) \, d\pi^A(\lambda), again in the strong sense, yielding a bounded self-adjoint (or normal) operator on H. For unbounded self-adjoint operators, the theorem extends analogously, but the domain of f(A) is restricted to vectors \psi \in H such that \int_{\sigma(A)} |f(\lambda)|^2 \, d\|\pi^A(\lambda)\psi\|^2 < \infty, ensuring f(A) is densely defined. A sketch of the proof relies on the multiplication operator representation: the spectral theorem establishes a unitary equivalence between (H, A) and (L^2(\sigma(A), \mu), M_\lambda), where \mu is a scalar measure induced by \pi^A and M_\lambda denotes multiplication by the identity function \lambda on L^2(\sigma(A), \mu); the PVM \pi^A then corresponds to the canonical projections onto characteristic functions of Borel sets in this model space. John von Neumann originally formulated the spectral theorem for unbounded self-adjoint operators in his 1932 treatise on the mathematical foundations of , providing the rigorous measure-theoretic framework that underpins modern operator theory. As a concrete example, consider a diagonal matrix A = \operatorname{diag}(\lambda_1, \lambda_2, \dots, \lambda_n) on \mathbb{C}^n with distinct eigenvalues \lambda_k and corresponding orthonormal eigenvectors e_k. Here, the associated PVM is \pi^A(\{\lambda_k\}) = |e_k\rangle\langle e_k|, the rank-one projection onto the eigenspace spanned by e_k, and A = \sum_k \lambda_k \pi^A(\{\lambda_k\}), recovering the standard diagonalization.

Direct Integrals

A direct integral provides a framework for decomposing a Hilbert space into a measurable family of Hilbert spaces indexed by a measure space. Specifically, given a measure space (X, \Sigma, \mu), a measurable family of Hilbert spaces \{H_x\}_{x \in X} consists of Hilbert spaces H_x such that there exists a countable collection of vectors generating each H_x with measurable coordinate functions. The direct integral Hilbert space is then H = \int_X^\oplus H_x \, d\mu(x), comprising equivalence classes of measurable sections \xi = (\xi_x)_{x \in X} with \xi_x \in H_x satisfying \int_X \|\xi_x\|^2 \, d\mu(x) < \infty, equipped with the inner product \langle \xi, \eta \rangle = \int_X \langle \xi_x, \eta_x \rangle_{H_x} \, d\mu(x). Operators on H act fiberwise if they are decomposable, meaning they map sections to sections by applying operators on each fiber almost everywhere. Projection-valued measures (PVMs) admit a canonical representation via direct integrals, revealing their structure as multiplication operators on fiber spaces. A fundamental representation theorem states that every PVM \pi on a standard Borel space (X, \Sigma) with measure \mu is unitarily equivalent to the multiplication PVM on the direct integral \int_X^\oplus L^2(\Sigma_x, \pi_x) \, d\mu(x), where \Sigma_x denotes the local \sigma-algebra consisting of sets E \subset X such that \{y \in X \mid (x,y) \in \mathrm{graph}(E)\} is measurable for \mu-almost every x, and \pi_x is the scalar measure on \Sigma_x obtained from the disintegration of \pi with respect to \mu. This equivalence preserves the spectral properties of the PVM and facilitates analysis in the fiberwise setting. In this construction, the projection \pi(E) corresponding to a measurable set E \in \Sigma acts on the direct integral by multiplication with the characteristic function \chi_E on each fiber L^2(\Sigma_x, \pi_x), i.e., (\pi(E) \xi)_x = \chi_E(y) \xi_x(y) for sections \xi, ensuring orthogonality for disjoint sets and additivity in the strong operator topology. This fiberwise multiplication preserves the key properties of PVMs, such as resolution of the identity \int_X \pi(dx) = I in the appropriate sense. The direct integral representation offers significant advantages by allowing the decomposition of operators affiliated with the PVM into measurable fields of operators on the fibers, which simplifies the study of infinite-dimensional systems by reducing them to families of finite- or lower-dimensional problems parameterized measurably over X. This approach is particularly useful for handling continuous spectra and non-trivial multiplicities without relying on explicit bases. A representative example arises in the spectral analysis of the position operator Q on L^2(\mathbb{R}), where the associated PVM can be realized via the Fourier transform as a direct integral over momentum space fibers. Specifically, the unitary Fourier map interchanges the position representation with a direct integral \int_\mathbb{R}^\oplus \mathbb{C} \, dp/(2\pi), where the PVM projections act by modulating the momentum fibers according to position sets, enabling the computation of spectral integrals like expectation values in the momentum basis.

Applications

In Quantum Mechanics

In quantum mechanics, physical observables are modeled as self-adjoint operators on a Hilbert space, each associated with a projection-valued measure (PVM) via the spectral theorem. For an observable represented by a self-adjoint operator A, there exists a unique PVM \pi on the Borel \sigma-algebra of \mathbb{R} such that A = \int_{\mathbb{R}} \lambda \, d\pi(\lambda), where the integral is understood in the strong operator topology, and the values \lambda in the spectrum of A correspond to the possible outcomes of a measurement of the observable. This construction ensures that the PVM encodes the resolution of the identity for A, with \pi(E) being the orthogonal projection onto the subspace spanned by generalized eigenvectors of A with eigenvalues in the Borel set E \subseteq \mathbb{R}. The Born rule, which governs the probabilistic interpretation of quantum measurements, is directly formulated using PVMs. For a normalized state vector |\phi\rangle in the Hilbert space and a Borel set E, the probability P(E) of obtaining a measurement outcome in E is given by P(E) = \langle \phi | \pi(E) | \phi \rangle. Upon measurement yielding an outcome in E with P(E) > 0, the state collapses to the normalized post-measurement state \frac{\pi(E) |\phi\rangle}{\sqrt{P(E)}}, which projects |\phi\rangle onto the corresponding to \pi(E). This process integrates the collapse postulate into the formalism, updating the according to the while preserving . A concrete example is the in the L^2(\mathbb{R}), where the PVM \pi for the Q (multiplication by the x) assigns to each B \subseteq \mathbb{R} the projection \pi(B) that multiplies wavefunctions by the \chi_B, thereby projecting onto the of square-integrable functions supported in B. For an such as (a, b], \pi((a, b]) thus isolates the component of the wavefunction within that interval, aligning outcomes with values in (a, b]. This measurement framework using PVMs was first rigorously formalized by in his 1932 work, where spectral projections provided the mathematical basis for describing quantum observables and the associated measurement process. For scenarios involving inefficient or indirect measurements, the PVM formalism generalizes to positive operator-valued measures (POVMs).

In Functional Analysis and Beyond

In , projection-valued measures (PVMs) arise in the spectral analysis of Koopman operators associated with measure-preserving . For a (\Phi_t)_{t \in \mathbb{R}} on a (M, \mu), the Koopman operator U_t f = f \circ \Phi_t acts unitarily on the L^2(M, \mu). By the , there exists a PVM E on the Borel \sigma-algebra of \mathbb{R} such that U_t = \int_{\mathbb{R}} e^{i \lambda t} \, dE(\lambda), decomposing the unitary dynamics into its spectral components and facilitating the study of ergodic properties like mixing and recurrence. In , PVMs play a key role in decomposing the of s via the Peter-Weyl theorem. For a G, the L^2(G) decomposes as a \bigoplus_{\rho \in \hat{G}} \mathrm{HS}(V_\rho), where \hat{G} is the set of irreducible unitary representations, \mathrm{HS}(V_\rho) denotes the space of Hilbert-Schmidt operators on the representation space V_\rho, and the multiplicity equals \dim V_\rho. The corresponding PVM projects onto the isotypic components E_\rho corresponding to each \rho, with projection operators given by integrals over characters: P_\rho f(g) = (\dim V_\rho) \int_G \overline{\chi_\rho(h)} f(gh^{-1}) \, dh. This decomposition resolves self-adjoint operators on L^2(G), such as the Laplacian, into irreducible spectral parts. Projection-valued processes extend PVMs to time-dependent settings, modeling operator-valued martingales in infinite-dimensional Hilbert spaces for stochastic analysis. In non-commutative probability spaces, a projection-valued process (E_t)_{t \geq 0} satisfies E_t E_s = E_{\min(t,s)} and generates a filtration of conditional expectations, allowing integrals \int f(s) \, dE_s for operator-valued functions f. This framework underpins quantum stochastic calculus, where such processes define martingale differences for infinite-dimensional evolutions, as in the integration of Hilbert space-valued measures against resolutions of the identity derived from self-adjoint operators. As of 2025, PVMs underpin quantum error-correcting codes through of . In codes like the surface code, the H = -\sum_i A_i - \sum_j B_j is a sum of commuting Pauli projectors A_i, B_j onto stabilizer eigenspaces, and its yields a PVM E on the spectrum \{ -k : 0 \leq k \leq n \} such that H = \int \lambda \, dE(\lambda), identifying ground-state degeneracy and excitation gaps critical for . This decomposition quantifies error syndromes via eigenvalue projections, enabling decoding algorithms that correct errors below the by resolving low-lying spectral subspaces. A representative example is the as a PVM on group \mathbb{T} = \mathbb{R}/2\pi\mathbb{Z}, resolving the Laplacian \Delta = -d^2/d\theta^2 on L^2(\mathbb{T}). The eigenfunctions e_n(\theta) = (2\pi)^{-1/2} e^{in\theta} satisfy \Delta e_n = n^2 e_n, and the PVM E projects onto finite-dimensional eigenspaces: E((n-1/2, n+1/2]) = |e_n\rangle\langle e_n| for n \in \mathbb{Z}, so \Delta = \sum_{n \in \mathbb{Z}} n^2 E(\{n\}). The \mathcal{F} f(n) = \langle f, e_n \rangle diagonalizes \Delta in this basis, illustrating PVMs in classical .

Generalizations

Positive Operator-Valued Measures

A positive operator-valued measure (POVM) is a function E from a \sigma-algebra \Sigma on a measurable space to the set of positive semi-definite operators on a H, satisfying E(\emptyset) = 0 and the normalization condition \sum_{i} E(E_i) = I_H for any countable partition \{E_i\}_{i \in I} of the space into measurable sets E_i \in \Sigma, where I_H is the identity operator on H. Unlike projection-valued measures, the operators E(A) in a POVM are not required to be orthogonal projections, meaning they need not be idempotent (E(A)^2 = E(A)) or mutually orthogonal (E(A) E(B) = 0 for disjoint A, B). This generalization was introduced to provide an operational framework for quantum measurements beyond ideal projective cases. Every projection-valued measure is a special case of a , since projections are positive semi-definite and satisfy the required properties when orthogonal. However, the converse does not hold; POVMs encompass a broader class of measurements where the effects E(A) represent non-projective outcomes, allowing for probabilities p(A) = \langle \psi | E(A) | \psi \rangle (or \operatorname{tr}(\rho E(A)) for mixed states \rho) that capture realistic, non-ideal quantum processes without collapsing the state orthogonally. This relaxation enables the description of measurements that cannot be represented by a single projective basis. A representative example arises in for modeling an inefficient detector, where detection \eta < 1 accounts for losses. Here, the projective is given by a projection-valued measure \pi with projectors \pi(E) onto number subspaces for outcomes E \in \Sigma. For an on/off (Geiger-mode) detector, the corresponding POVM elements are the no-click E_{\mathrm{no-click}} = \sum_{n=0}^\infty (1-\eta)^n |n\rangle \langle n| and the click E_{\mathrm{click}} = I_H - E_{\mathrm{no-click}}. This model, derived from a beam-splitter of losses with input on the reflected port followed by an detector, accurately describes experimental with imperfect . POVMs offer key advantages in quantum information theory by enabling the mathematical description of general quantum measurements, including those involving ancillary systems or indirect schemes that cannot be realized projectively on the system alone, thus expanding the toolkit for tasks like state discrimination and channel estimation.

Further Extensions

In non-commutative settings, PVMs generalize to projection-valued states, which are completely orthoadditive maps from the projections of one to those of another, preserving the unit projection. Their structure is given by P \mapsto U_1 (P \otimes I_1) U_1^* \oplus U_2 (P \otimes I_2) U_2^*, where U_1 is unitary and U_2 is antiunitary. Vector-valued generalizations of PVMs take values in Banach spaces of operators, employing Pettis integration to handle the lack of a Bochner integral in non-reflexive spaces. A vector projection family \{\mu_\Lambda \mid \Lambda \in X^*\} on a Banach space X assigns to each continuous linear functional \Lambda a complex measure \mu_\Lambda, satisfying linearity in \Lambda and setwise convergence. The integral \int f \, d\mu for f \in L^1(\mu) is defined in the second dual X^{**} via \int f \, d\mu(\Lambda) = \int f \, d\mu_\Lambda, ensuring the measure satisfies monotone and dominated convergence theorems. When X is reflexive, integrals lie in X, extending classical PVMs to operator-valued settings in B(X). As of 2025, recent developments link PVMs to non-commutative probability, particularly probability theory, through measures in algebras. In a tracial ( \mathcal{A}, \tau ), for a normal element a \in \mathcal{A}, the distribution \mu_a is obtained from the projection-valued measure \nu_a via \mu_a(E) = \tau[\nu_a(E)], aligning with the non-commutative theorem. This connection enables the study of random variables, where Brown measures extend supports for non-normal elements, revealing asymptotic behaviors in large random matrices and products. In C*-dynamical systems, crossed products by group actions on operator algebras yield PVMs that capture invariant structures. For a consisting of a C*-algebra \mathcal{A} and a group of automorphisms, the crossed product \mathcal{A} \rtimes G incorporates projections resolving data from the action; if the associated measure E is projection-valued, the to the crossed product is a *-isomorphism, preserving the measure's properties under the group dynamics.