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other two Murray-von Neumann types of infinite unitary representations in that it alone would retain (like the finite-dimensional case which it includes) a certain uniqueness of decomposition into reps- uniqueness but not necessarily discreteness since it may involve either a continuous-discrete sum or a discrete-continuous sum. "For physics we may add the following comments: The concept of multiplicity-freeness enables us to de- fine quite generally the conditions on a set of obseryv- ables such that, when the observables are measured, the quantum mechanical state is completely specified. In other words the concept of 'multiplicity freeness' permits us to state more concisely a general defini- tion which Jauch (1960, 1961) has given of 'a com- plete set of commuting observables.' This definition applies equally well to operators with continuous as with discrete spectra-in contrast with the usual definition of a complete commuting set, requiring that there occur only nondegenerate common eigen- values. Suppose L is the set of all bounded functions generated by a set of commuting observables. Then L is the smallest 'weakly closed' algebra2 containing the original set. By a theorem of von Neumann (1929) any such 'smallest weakly closed algebra' is equal to its double commutant,
Reviews of Modern Physics – American Physical Society (APS)
Published: Jul 1, 1962
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