Arveson nests and operator factorization along commutative subspace lattices

Authors:
John Daughtry and Ronald Johns

Journal:
Proc. Amer. Math. Soc. **107** (1989), 943-947

MSC:
Primary 47A68; Secondary 47A15, 47D99

DOI:
https://doi.org/10.1090/S0002-9939-1989-0975636-5

MathSciNet review:
975636

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Abstract | References | Similar Articles | Additional Information

Abstract: Similar commutative subspace lattices (CSL's) are shown to be unitarily equivalent if certain sublattices (which may be taken to be nests!) are unitarily equivalent and a technical condition is satisfied. This result provides a connection between existing results for arbitrary similarities of countable CSL's and similarities of general CSL's by operators near the identity. One consequence is the generalizaton to CSL's of a theorem of David Pitts on the relationship between similarity and unitary equivalence of nests he calls "injective."

**[1]**W. B. Arveson,*Analyticity in operator algebras*, Amer. J. Math.**89**(1967), 578-642. MR**0223899 (36:6946)****[2]**-,*Operator algebras and invariant subspaces*, Ann. of Math.**2**(1974), 433-532. MR**0365167 (51:1420)****[3]**J. Daughtry and B. Dearden,*A test for the existence of Gohberg-Krein representations in terms of multiparameter Wiener processes*, J. Funct. Anal.**63**(1985), 403-411. MR**813207 (87a:60045)****[4]**J. Daughtry,*Factorizations along commutative subspace lattices*, Integral Equations Operator Theory**10**(1987), 290-296. MR**878249 (88c:47032)****[5]**-,*Invariance of projections in the diagonal of a nest algebra*, Proc. Amer. Math. Soc.**102**(1988), 117-120. MR**915727 (88m:47070)****[6]**-,*Conditional expectations and invariant subspaces*, in Contributions to Operator Theory and its Applications, vol. 35, Birkhäuser, Boston, 1988, 23-36. MR**1017664 (90k:47095)****[7]**A. Feintuch and R. Saeks,*System theory: A Hilbert space approach*, Academic Press, New York, 1982. MR**663906 (84e:93003)****[8]**I. Gohberg and M. G. Krein,*Theory and application of Volterra operators in Hilbert space*, A. M. S. Transl. of Math. Mono., vol. 24, 1970. MR**0264447 (41:9041)****[9]**D. R. Larson,*Nest algebras and similarity transformation*, Ann. of Math.**121**(1985), 409-427. MR**794368 (86j:47061)****[10]**David R. Pitts,*Factorization problems for nests: factorization methods and characterizations of the universal factorization property*, J. Funct. Anal.**79**(1988), 57-90. MR**950084 (90a:46160)****[11]**S. Strătilă.*Modular theory in operator algebras*, Abacus Press, Kent, England, 1981.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1989-0975636-5

Keywords:
Commutative subspace lattice algebra,
operator factorization,
nest,
conditional expectations on von Neumann algebras

Article copyright:
© Copyright 1989
American Mathematical Society