# Commutation theorem for traces
In mathematics, a **commutation theorem for traces** is a result in the theory of von Neumann algebras that explicitly identifies the commutant of a von Neumann algebra acting on a [Hilbert space](https://www.edgepedia.org/hilbert-space) in the presence of a trace. The semifinite version says that, for a von Neumann algebra \( \mathcal{A} \) equipped with a semifinite trace \( \operatorname{tr} \), the algebra coincides with the commutant of \( \mathcal{A}^{\mathrm{op}} \)—that is, the same algebra endowed with multiplication taken in the opposite order—when the two act on the Hilbert space of operators satisfying \( \operatorname{tr}(a^{*}a)<\infty \), one through left and the other through right multiplication.[1](https://scoap3-prod-backend.s3.cern.ch/media/files/88983/10.1007/JHEP10%282024%29063_a.pdf)
| Key fact | Detail |
|---|---|
| Statement | A von Neumann algebra \( \mathcal{A} \) with semifinite trace \( \operatorname{tr} \) equals the commutant of \( \mathcal{A}^{\mathrm{op}} \) acting on \( \{ a \in \mathcal{A} : \operatorname{tr}(a^{*}a) < \infty \} \) by left and right multiplication[1](https://scoap3-prod-backend.s3.cern.ch/media/files/88983/10.1007/JHEP10%282024%29063_a.pdf) |
| Inner product | The Hilbert space carries \( \langle a \mid a \rangle = \operatorname{tr}(a^{*}a) \), with the finite-trace operators forming a dense subspace[1](https://scoap3-prod-backend.s3.cern.ch/media/files/88983/10.1007/JHEP10%282024%29063_a.pdf) |
| Tracial-vector form | If \( \xi \) is a cyclic tracial vector for \( M \) and \( J \) is the closure of \( m\xi \mapsto m^{*}\xi \), then \( JMJ = M' \)[2](https://math.berkeley.edu/~ltomczak/notes/Spring2024/VNA_Notes.pdf) |
| Algebraic setting | A von Neumann algebra is semifinite when it admits an exact normal semifinite trace; type III algebras admit no nonzero normal semifinite traces[3](https://encyclopediaofmath.org/wiki/Von_Neumann_algebra) |
| Applications | The crossed product of a type III algebra with its modular automorphism group is semifinite, and the commutation theorem is used to construct gauge-invariant algebras in gauge theory and gravity[4](https://arxiv.org/abs/2407.01695) |
## What the theorem says
A von Neumann algebra \( M \) acting on a Hilbert space \( H \) has a commutant \( M' \), the set of bounded operators on \( H \) that commute with every element of \( M \). The double commutant theorem of von Neumann identifies \( M'' \) with \( M \) itself, but it does not describe \( M' \) concretely.
The semifinite version proceeds through the GNS-type Hilbert space built from the trace. Given \( \mathcal{A} \) with semifinite trace \( \operatorname{tr} \), the operators \( a \) with \( \operatorname{tr}(a^{*}a) < \infty \) form a dense subspace \( H \) with inner product \( \langle a \mid a \rangle = \operatorname{tr}(a^{*}a) \). The algebra \( \mathcal{A} \) acts on \( H \) by left multiplication, \( \hat{b}_{L} \mid a \rangle = \mid ba \rangle \), and its opposite algebra \( \mathcal{A}^{\mathrm{op}} \) acts by right multiplication, \( \hat{b}_{R} \mid a \rangle = \mid ab \rangle \). The commutation theorem for semifinite traces states that \( \mathcal{A} \) is exactly the commutant of \( \mathcal{A}^{\mathrm{op}} \) in this representation.[1](https://scoap3-prod-backend.s3.cern.ch/media/files/88983/10.1007/JHEP10%282024%29063_a.pdf)
**Tracial vector form.** The same statement appears in a form closer to modular theory. Let \( M \) act on \( H \) with a cyclic tracial vector \( \xi \), meaning \( \langle x\xi, \xi \rangle \) is a trace on \( M \). Let \( J \) be the closure of the antilinear map \( m\xi \mapsto m^{*}\xi \). Then \( J \) is anti-unitary with \( J^{2} = I \), and \( JMJ = M' \); equivalently, \( JMJ \) is the representation of \( M^{\mathrm{op}} \) by right multiplication. A von Neumann algebra in such a representation is said to be in standard form.[2](https://math.berkeley.edu/~ltomczak/notes/Spring2024/VNA_Notes.pdf) The proof rests on the trace identity: for \( x, y \in M' \), the vector \( \Omega \) gives \( \langle xy\Omega, \Omega \rangle = \langle yx\Omega, \Omega \rangle \), so \( x \mapsto \langle x\Omega, \Omega \rangle \) is a trace on \( M' \) as well, and the argument can be run on both sides.[5](https://math.berkeley.edu/~vfr/MATH20909/VonNeumann2009.pdf)
## Historical development
The theorem originates in the series of papers on rings of operators by Francis J. Murray and [John von Neumann](https://www.edgepedia.org/john-von-neumann), whose fourth paper, "On rings of operators IV", appeared in the [Annals of Mathematics](https://www.edgepedia.org/annals-of-mathematics) in 1943 (volume 44, pages 716–808).[6](https://netlib.org/bibnet/authors/v/von-neumann-john.html)
The semifinite formulation extends the finite case. Semifinite traces are required to handle the situation in which a transformation \( T \) (or a group action \( \Gamma \)) preserves only an infinite measure equivalent to the reference measure. Tomita's commutant theorem, a related result, states that for von Neumann algebras \( M \subseteq B(H) \) and \( N \subseteq B(K) \) one has \( (M \otimes N)' = M' \otimes N' \); its statement is simple while its proof is difficult.[7](https://assets.cambridge.org/97805217/19193/excerpt/9780521719193_excerpt.pdf)
## Examples
**Group von Neumann algebras.** For a discrete group \( \Gamma \), the left regular representation on \( \ell^{2}(\Gamma) \) generates a von Neumann algebra whose commutant is the von Neumann algebra generated by the right regular representation; the modular conjugation implements the passage, since \( Ju_{\gamma}J e_{\gamma'} = e_{\gamma'\gamma^{-1}} \).[5](https://math.berkeley.edu/~vfr/MATH20909/VonNeumann2009.pdf) The standard \( \Gamma \)-trace on this algebra is \( \operatorname{tr}_{\Gamma}(a) = \langle (e)a, e \rangle \), where \( e \) is the unit element viewed as a vector in \( \ell^{2}(\Gamma) \); left invariance gives the trace property \( \operatorname{tr}_{\Gamma}(ab) = \operatorname{tr}_{\Gamma}(ba) \), and the trace is faithful.[8](https://math.berkeley.edu/~teichner/Papers/COT1.pdf)
**Type II₁ factors.** In the standard form \( L^{2}(M, \operatorname{tr}) \) of a type II₁ factor \( M \), the commutant of the left action of \( M \) is the *-algebra of right multiplication operators. In particular, the commutant of a type II₁ factor on \( L^{2}(M) \) is again a type II₁ factor. This fails on an arbitrary Hilbert space: for \( M \otimes 1 \) on \( L^{2}(M) \otimes H \) with \( H \) infinite dimensional, the commutant is \( JMJ \otimes B(H) \), infinite matrices over \( JMJ \).[5](https://math.berkeley.edu/~vfr/MATH20909/VonNeumann2009.pdf)
## Why semifiniteness matters
The trace hypothesis separates the algebra types. A von Neumann algebra is called semifinite when it admits an exact normal semifinite trace, while type III algebras admit no nonzero normal semifinite traces at all.[3](https://encyclopediaofmath.org/wiki/Von_Neumann_algebra) On a semifinite factor, all nonzero semifinite normal traces are proportional, which restricts how the trace can be chosen.[9](https://link.springer.com/article/10.1007/s00220-024-05180-7) The theorem therefore applies to type I and type II algebras, including the II₁ and II∞ factors, but not directly to type III.
## Applications
The theorem is a working tool in mathematical physics. In quantum field theory and quantum gravity, the crossed product \( M \rtimes \mathbb{R} \) of a type III algebra by its modular automorphism group is semifinite, and the crossed product presents a natural method for implementing constraints through the commutation theorem, which makes it useful for constructing gauge-invariant algebras; under suitable conditions an associated trace computes physical expectation values.[4](https://arxiv.org/abs/2407.01695) In the algebraic treatment of gravitational path integrals and boundary degrees of freedom, the commutation theorem for semifinite traces is invoked to identify left and right algebras acting on a diagonal Hilbert space as mutual commutants.[1](https://scoap3-prod-backend.s3.cern.ch/media/files/88983/10.1007/JHEP10%282024%29063_a.pdf) In the study of algebras and states in JT gravity, the identification of a von Neumann algebra with semifinite trace \( \operatorname{Tr} \) and its opposite algebra as commutants of each other is likewise the content of the theorem.[10](https://vkalvakotamath.github.io/papers/Algebras%20and%20States%20in%20JT%20Gravity.pdf)
## References
1. [Commutation theorem for semifinite traces as used in algebraic quantum field theory, JHEP 10 (2024) 063](https://scoap3-prod-backend.s3.cern.ch/media/files/88983/10.1007/JHEP10%282024%29063_a.pdf)
2. [L. Tomczak, Von Neumann Algebras lecture notes, Theorem 16.2 (JMJ = M′), UC Berkeley, Spring 2024](https://math.berkeley.edu/~ltomczak/notes/Spring2024/VNA_Notes.pdf)
3. [Von Neumann algebra, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Von_Neumann_algebra)
4. [S. Ali Ahmad, M. S. Klinger, S. Lin, Semifinite von Neumann algebras in gauge theory and gravity, arXiv:2407.01695](https://arxiv.org/abs/2407.01695)
5. [Von Neumann algebras lecture notes, Theorem 9.1.6, UC Berkeley](https://math.berkeley.edu/~vfr/MATH20909/VonNeumann2009.pdf)
6. [BibTeX bibliography of John von Neumann, entry Murray:1943:ROI](https://netlib.org/bibnet/authors/v/von-neumann-john.html)
7. [Finite von Neumann Algebras and Masas, Cambridge excerpt, Theorem 2.2.4 (Tomita's commutant theorem)](https://assets.cambridge.org/97805217/19193/excerpt/9780521719193_excerpt.pdf)
8. [T. Cochran, K. Orr, P. Teichner, Annals of Mathematics 157 (2003) 433–519, group von Neumann algebra and Γ-trace](https://math.berkeley.edu/~teichner/Papers/COT1.pdf)
9. [Quantum Reference Frames, Measurement Schemes and the Type of Local Algebras in Quantum Field Theory, Communications in Mathematical Physics (2024)](https://link.springer.com/article/10.1007/s00220-024-05180-7)
10. [Algebras and States in JT Gravity](https://vkalvakotamath.github.io/papers/Algebras%20and%20States%20in%20JT%20Gravity.pdf)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Operator algebras › Von Neumann algebras › Weights, traces, and noncommutative integration*
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