TAILIEUCHUNG - Báo cáo toán học: On factorial states of operator algebras

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Trên trạng thái thừa của các đại số nhà điều hành. | J. OPERATOR THEORY J3 1985 131 142 Copyright by INCREST 1985 ON FACTORIAL STATES OF OPERATOR ALGEBRAS. II c. J. K. BATTY and R. J. ARCHBOLD 1. INTRODUCTION It was shown in 7 19 that the set P X of pure states of a unital c -algebra A other than A. C is weak dense in the state space S y4 if and only if A is both prime and antiliminal. More recently it was shown in 2 that the set F J. of factorial states is weak dense in S T if and only if A is prime. Comparison of these results suggests that weak density of PỌ4. in F X is related to antiliminality. On the other hand abelian c -algebras for which P l F A must also be taken into consideration. It will be shown in Theorem that these are essentially the only two cases P l is weak dense in F l if and only if A is an antiliminal extension of an abelian c -algebra. Approximate factorial extensions of factorial states on c -subalgebras were also considered in 2 by means of the fact that the set FMG4 of type I factorial states is always weak dense in F J . It will be shown in Section 5 how the arguments of 2 can be simplified by considering only the set Ff A of states p for which TtfiA is a finite type I factor. Both F 00 4 and Ff f are described simply in terms of pure states in Section 2. In Section 4 it is shown by methods parallel to those of Section 3 that the states for which nv AỴ is a type I factor of bounded degree are weak dense in F X if and only if A is an antiliminal extension of a subhomogeneous c -algebra. Standard definitions and properties of c -algebras as described in 6 will often be used without comment. Throughout A will be a c -algebra with or without unit whose spectrum A is equipped with the Jacobson topology. The equivalence class in A of an irreducible representation 71 will be denoted by 7r . For a subset E of A E will be the weak ơ A zí closure of E in A . The sets of all all pure and all factorial states of A will be denoted by S X P A and F 4 respectively. For a state p will be the Hilbert .

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• báo cáo của tạp chí Journal of Operator Theory
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