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Báo cáo toán học: "Derivations on certain CSL algebras "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Derivations là một số đại số CSL. | Copyright by INCREST 1984 J. OPERATOR THEORY I Í Í984 145 156 DERIVATIONS ON CERTAIN CSL ALGEBRAS F. G1LFEATHER All derivations from a CSL algebra sđ into itself were shown by E. Christensen to be norm continuous 2 . For nest algebras they are all inner and in fact the nth Hochschild cohomology group íĩtí is trivial for all n and all dual -modules ăẫ which are submodules of 7 . On the other hand .s ja need not be zero even for certain width two CSL algebras the intersection of two nest algebras with mutually commuting nests 6 . However if - d is the finite intersection of nest algebras with mutually commuting and independent nests then -si is trivial 6 . In this note we investigate the obstruction to H sđ sđ 0 when ỉđ is an irreducible CSL algebra containing a purely atomic masa. The irreducible tridiagonal algebra introduced in 4 as well as its finite analogies sdn are shown to be the only obstructions to 0 for this class of CSL algebras. A derivation 5 of sđ into is said to be quasi-inner if there exists bounded operator T e sđ and a possibly unbounded operator s affiliated with the core e l of sđ so that 5 ÔT S ỎK A AR RA . We show that whenever every derivation of jaZ into is quasi-inner then 0 5 0 if and only if .ỉđ is a tridiagonal algebra. If we denote M span then it turns out that j 0 is also equivalent to the condition that for all n Mn is not weakly dense in Z . Moreover if sđ 0 because there exists a non quasi-inner derivation 5 then it is shown that there exists a core projection p so that PsiP is a canonical minimal finite dimensional CSL algebra and 3 I A is not inner. These canonical algebras are isomorphic to certain finite dimensional matrix algebras and are closely related to the tridiagonal CSL algebra. 1. PRELIMINARIES All Hilbert spaces will be separable. In this paper subspace lattices will all be commutative and are assumed to be closed in the strong operator topology. For a subspace lattice we let Pl x denote the core the von Neumann .