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In mathematics, a commutation theorem explicitly identifies the commutant of a specific von Neumann algebra acting on a Hilbert space in the presence of a trace. The first such result was proved by F.J. 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. ==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'' * (''ab''Ω, Ω) = (''ba''Ω, 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 ''a''Ω = 0 for ''a'' in ''M'', then ''aMΩ= (0), and hence ''a'' = 0. It follows that the map : for ''a'' in ''M'' defines a conjugate-linear isometry of ''H'' with square the identity ''J''2 = ''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 : The commutation theorem of Murray and von Neumann states that : One of the easiest ways to see this is to introduce ''K'', the closure of the real subspace ''M''sa Ω, where ''M''sa denotes the self-adjoint elements in ''M''. It follows that : an orthogonal direct sum for the real part of inner product. This is just the real orthogonal decomposition for the ±1 eigenspaces of ''J''. On the other hand for ''a'' in ''M''sa and ''b'' in ''Msa, the inner product (''ab''Ω, 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 : follows by reversing the roles of ''M'' and ''M. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Commutation theorem」の詳細全文を読む スポンサード リンク
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