The selfadjoint operators of a von Neumann algebra form a conditionally complete lattice

Author:
Milton Philip Olson

Journal:
Proc. Amer. Math. Soc. **28** (1971), 537-544

MSC:
Primary 46.65

DOI:
https://doi.org/10.1090/S0002-9939-1971-0276788-1

MathSciNet review:
0276788

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Abstract: The bounded resolutions of the identity in a von Neumann algebra can be ordered by if . The selfadjoint operators in the algebra are partially ordered by this relation and are shown to form a conditionally complete lattice. The lattice operations are (essentially) defined by for all contained in . This order is called spectral order and agrees with the usual order on commutative subalgebras. For positive operators, is greater than or equal to in spectral order if and only if is greater than or equal to in the usual order for all . Kadison's well-known counterexample is shown to fail. The operator lattice defined by spectral order differs from a vector lattice in the fact that does not imply that .

**[1]**R. Kadison,*Order properties of bounded self-adjoint operators*, Proc. Amer. Math. Soc.**2**(1951), 505-510. MR**13**, 47. MR**0042064 (13:47c)****[2]**B. Lengyel and M. Stone,*Elementary proof of the spectral theorem*, Ann. of Math. (2)**37**(1936), 853-864. MR**1503314****[3]**S. Sherman,*Order in operator algebras*, Amer. J. Math.**73**(1951), 227-232. MR**13**, 47. MR**0042065 (13:47d)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1971-0276788-1

Keywords:
von Neumann algebra,
resolution of the identity,
projection lattice,
conditionally complete lattice,
spectral integral

Article copyright:
© Copyright 1971
American Mathematical Society