# Commutation theorem for traces

The first such result was proved by Francis Joseph Murray and John von Neumann in the 1930s and applies to the von Neumann algebra generated by a discrete group or by the dynamical system associated with a measurable transformation preserving a probability measure.

Another important application is in the theory of unitary representations of unimodular locally compact groups, where the theory has been applied to the regular representation and other closely related representations. In particular this framework led to an abstract version of the Plancherel theorem for unimodular locally compact groups due to Irving Segal and Forrest Stinespring and an abstract Plancherel theorem for spherical functions associated with a Gelfand pair due to Roger Godement. Their work was put in final form in the 1950s by Jacques Dixmier as part of the theory of Hilbert algebras.

It was not until the late 1960s, prompted partly by results in algebraic quantum field theory and quantum statistical mechanics due to the school of Rudolf Haag, that the more general non-tracial Tomita–Takesaki theory was developed, heralding a new era in the theory of von Neumann algebras.

Contents 1 Commutation theorem for finite traces 1.1 Examples 2 Commutation theorem for semifinite traces 3 Hilbert algebras 3.1 Definition 3.2 Examples 3.3 Properties 4 See also 5 Notes 6 References Commutation theorem for finite traces Let H be a Hilbert space and M a von Neumann algebra on H with a unit vector Ω such that M Ω is dense in H M ' Ω is dense in H, where M ' denotes the commutant of M (abΩ, Ω) = (baΩ, Ω) for all a, b in M.

The vector Ω is called a cyclic-separating trace vector. It is called a trace vector because the last condition means that the matrix coefficient corresponding to Ω defines a tracial state on M. It is called cyclic since Ω generates H as a topological M-module. It is called separating because if aΩ = 0 for a in M, then aM'Ω= (0), and hence a = 0.

It follows that the map {displaystyle JaOmega =a^{*}Omega } for a in M defines a conjugate-linear isometry of H with square the identity, J2 = I. The operator J is usually called the modular conjugation operator.

It is immediately verified that JMJ and M commute on the subspace M Ω, so that[1] {displaystyle JMJsubseteq M^{prime }.} The commutation theorem of Murray and von Neumann states that {displaystyle JMJ=M^{prime }} One of the easiest ways to see this[2] is to introduce K, the closure of the real subspace Msa Ω, where Msa denotes the self-adjoint elements in M. It follows that {displaystyle H=Koplus iK,} an orthogonal direct sum for the real part of the inner product. This is just the real orthogonal decomposition for the ±1 eigenspaces of J. On the other hand for a in Msa and b in M'sa, the inner product (abΩ, Ω) is real, because ab is self-adjoint. Hence K is unaltered if M is replaced by M '.

In particular Ω is a trace vector for M' and J is unaltered if M is replaced by M '. So the opposite inclusion {displaystyle JM^{prime }Jsubseteq M} follows by reversing the roles of M and M'.

Examples One of the simplest cases of the commutation theorem, where it can easily be seen directly, is that of a finite group Γ acting on the finite-dimensional inner product space {displaystyle ell ^{2}(Gamma )} by the left and right regular representations λ and ρ. These unitary representations are given by the formulas {displaystyle (lambda (g)f)(x)=f(g^{-1}x),,,(rho (g)f)(x)=f(xg)} for f in {displaystyle ell ^{2}(Gamma )} and the commutation theorem implies that {displaystyle lambda (Gamma )^{prime prime }=rho (Gamma )^{prime },,,rho (Gamma )^{prime prime }=lambda (Gamma )^{prime }.} The operator J is given by the formula {displaystyle Jf(g)={overline {f(g^{-1})}}.} Exactly the same results remain true if Γ is allowed to be any countable discrete group.[3] The von Neumann algebra λ(Γ)' ' is usually called the group von Neumann algebra of Γ. Another important example is provided by a probability space (X, μ). The Abelian von Neumann algebra A = L∞(X, μ) acts by multiplication operators on H = L2(X, μ) and the constant function 1 is a cyclic-separating trace vector. It follows that {displaystyle A'=A,} so that A is a maximal Abelian subalgebra of B(H), the von Neumann algebra of all bounded operators on H. The third class of examples combines the above two. Coming from ergodic theory, it was one of von Neumann's original motivations for studying von Neumann algebras. Let (X, μ) be a probability space and let Γ be a countable discrete group of measure-preserving transformations of (X, μ). The group therefore acts unitarily on the Hilbert space H = L2(X, μ) according to the formula {displaystyle U_{g}f(x)=f(g^{-1}x),} for f in H and normalises the Abelian von Neumann algebra A = L∞(X, μ). Let {displaystyle H_{1}=Hotimes ell ^{2}(Gamma ),} a tensor product of Hilbert spaces.[4] The group–measure space construction or crossed product von Neumann algebra {displaystyle M=Artimes Gamma } is defined to be the von Neumann algebra on H1 generated by the algebra {displaystyle Aotimes I} and the normalising operators {displaystyle U_{g}otimes lambda (g)} .[5] The vector {displaystyle Omega =1otimes delta _{1}} is a cyclic-separating trace vector. Moreover the modular conjugation operator J and commutant M ' can be explicitly identified.

One of the most important cases of the group–measure space construction is when Γ is the group of integers Z, i.e. the case of a single invertible measurable transformation T. Here T must preserve the probability measure μ. Semifinite traces are required to handle the case when T (or more generally Γ) only preserves an infinite equivalent measure; and the full force of the Tomita–Takesaki theory is required when there is no invariant measure in the equivalence class, even though the equivalence class of the measure is preserved by T (or Γ).[6][7] Commutation theorem for semifinite traces Let M be a von Neumann algebra and M+ the set of positive operators in M. By definition,[3] a semifinite trace (or sometimes just trace) on M is a functional τ from M+ into [0, ∞] such that {displaystyle tau (lambda a+mu b)=lambda tau (a)+mu tau (b)} for a, b in M+ and λ, μ ≥ 0 (semilinearity); {displaystyle tau left(uau^{*}right)=tau (a)} for a in M+ and u a unitary operator in M (unitary invariance); τ is completely additive on orthogonal families of projections in M (normality); each projection in M is as orthogonal direct sum of projections with finite trace (semifiniteness).

If in addition τ is non-zero on every non-zero projection, then τ is called a faithful trace.

If τ is a faithful trace on M, let H = L2(M, τ) be the Hilbert space completion of the inner product space {displaystyle M_{0}=left{ain Mmid tau left(a^{*}aright)

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