迁移算子稠密性%2cp-Haar酉元及KS-代数.doc

:  , p ? Haar  KS-  , p ? Haar  KS- , : : 2 B(H) , Kadison  B(H)  H  . . Arveson [1] A B(H) , A · , A B(H) . Kadison . [6] U. Haagerup H. Schultz ·  · , Fang, Hadwin Ravichandran  . [4] Π1 . , p ? Haar Kadison-Singer . : , Gi = {e, ui}, u2i = e, i ∈ N ·  , G = ?i∈NGi  {Gi : i ∈ N}  , L(G) . · , R(G) = L(G)′. , {λu1u2 + λu3u4, λu5u6 + λu7u8} R(G) · B(H) , {λu1u2, λu3u4} R(G) · B(H) . , p?Haar ? ? , Mn(C) n ? Haar . , Kadison-Singer CI Kadison-Singer . ,  :  ·  ;  ;  ;  ; Π1  ;  ;  ; ?  ; p ? Haar ; Kadison-Singer ; Kadison-Singer ; .  i Abstract We denote by B(H) the algebra of all bounded linear operators on a Hilbert space H. With regards to the transitive algebra question. Kadison suggested the idea that some self-adjoint maximal abelian subalgebra of B(H) and some elements not in the subalgebra might generate a non-trivial transitive algebra. Arveson proved in [1] that Kadison’s original idea does not work. That is, if A is a transitive subalgebra of B(H) which contains a self-adjoint maximal abelian von Neumann algebra, then A is strong- operator dense in B(H). Inspired by the invariant subspace problem a?liated with a von Neumann algebra [6], Fang, Hadwin and Ravichandran [4] studied the transitive algebra generated by some set of operators in a ?nite von Neumann algebra and its commutant. we study the transitivity in a factor of type Π1. we also study p ? Haar unitary and the problems of Kadison-Singer algebras. This thesis consists of three chapters: In chapter one we study the transitive algebra question by considering the invariant subspace problem relative to von Neumann algebras. Let Gi = {e, ui}, i ∈ N, u2i = e be a cyclic group and its order is two. G = ?i∈NGi. Let L(G) be a von Neumann algebra, L(G)′ = R(G). We prove that the von Neumann algebra generated by {λu1u2 + λu3u4, λu5u6 + λu7u8} and its commutation is strong-operator dense in B(H).We also prove that the algeb