Triangular Algebras and Ideals of Nest Algebras
نویسنده
چکیده
Let %? be a separable Hubert space and !T c 3§(J%f) be an algebra of bounded operators. Say F is triangular if ^ n ^ * is a maximal abelian self-adjoint subalgebra (m.a.s.a.) of 3B{%?) and call this m.a.s.a. the diagonal of J7". A triangular algebra is maximal triangular if it is not properly contained in any triangular algebra. Triangular algebras of operators have been studied for 30 years now, since the seminal paper of Kadison and Singer [6]. In this, they proposed the maximal triangular algebras as infinitedimensional generalizations of the upper triangular matrices and as the non-self-adjoint analogues of the von Neumann algebras. However, general questions on maximal triangular algebras have proved highly intractible and a theory of non-self-adjoint algebras based on these algebras has not developed. Nevertheless, special classes of triangular algebras have provided much of the motivation in the subsequent study of non-self-adjoint algebras, in the areas of nest algebras and CSL algebras (see [3]), algebras connected with ergodic actions [1] and non-self-adjoint subalgebras of certain C*-algebras and von Neumann algebras [10, 11]. Specifically, the study of nest algebras is now well developed and, as we show here, it is now possible to use the techniques of this subject to answer some of the hard questions which arose early in the study of triangular algebras. In particular, it was hoped that, just as Murray and von Neumann essentially reduced the study of von Neumann algebras to the study of factors, analogous "primitive" maximal triangular
منابع مشابه
Derivations on dual triangular Banach algebras
Ideal Connes-amenability of dual Banach algebras was investigated in [17] by A. Minapoor, A. Bodaghi and D. Ebrahimi Bagha. They studied weak∗continuous derivations from dual Banach algebras into their weak∗-closed two- sided ideals. This work considers weak∗-continuous derivations of dual triangular Banach algebras into their weak∗-closed two- sided ideals . We investigate when weak∗continuous...
متن کاملThe Topology of Ideals in Some Triangular Af Algebras
A set of meet-irreducible ideals is described for a class of maximal triangular almost finite algebras. This set forms a topological space under the hull-kernel closure, and there is a one-to-one correspondence between closed sets in this space and ideals in the AF algebra. Some connections with nest representations and nestprimitive ideals are also described.
متن کاملEla Graded Triangular Algebras
The structure of graded triangular algebras T of arbitrary dimension are studied in this paper. This is motivated in part for the important role that triangular algebras play in the study of oriented graphs, upper triangular matrix algebras or nest algebras. It is shown that T decomposes as T = U + ( ∑ i∈I Ti), where U is an R-submodule contained in the 0-homogeneous component and any Ti a well...
متن کاملNew Results on Ideals in $MV$-algebras
In the present paper, by considering the notion of ideals in $MV$-algebras, we study some kinds of ideals in $MV$-algebras and obtain some results on them. For example, we present definition of ultra ideal in $MV$-algebras, and we get some results on it. In fact, by definition of ultra ideals, we present new conditions to have prime ideals, positive implicative ideals and maximal ideals in $MV$...
متن کامل