Normalizers of nest algebras

Abstract

Abstract. For a nest N with associated nest algebra A N , we define S N , the normalizer of A N . We develop a characterization of elements of S N based on certain order homomorphisms of N into itself. This characterization enables us to prove several structure theorems.A normalizer of a subalgebra A of B(H) can be defined as the set of operators T such that T * AT ⊆ A and T AT * ⊆ A. Normalizers of diagonal algebras (which are typically defined so as to comprise only partial isometries) have played an important role in the study of certain limit algebras [P]. In this paper, we examine normalizers of nest algebras.Theorem 2, the main theorem of this paper, establishes a characterization of an element of the normalizer of a nest algebra in terms of certain order homomorphisms of the nest into itself. We show that the normalizer is strongly closed, and that the order homomorphisms defined in Theorem 2 are related to the order homomorphisms defined in [EP]. We also develop a simplified characterization in the special case where N is continuous. The latter part of the paper examines the theory of finite rank operators. Theorem 12 establishes that every finite rank element of the normalizer is a sum of rank one elements in the normalizer.We first recall some basic concepts of the theory of nests and nest algebras, which can be found in greater detail in [D].For H a Hilbert space, a nest N is defined to be a complete totally ordered lattice of (self-adjoint) projections. Where there is no possibility of confusion, we identify a projection with its range so that for N ∈ N , the statement "x ∈ H such that N x = x" is shortened to "x ∈ N ". We actually make use of the identification in its strongest form: [D, Theorem 2.13] states that a nest of subspaces with the order topology is homeomorphic to the corresponding nest of projections with the strong operator topology.For M, N ∈ N , M ≤ N, the interval (M, N ) refers to the set {L ∈ N : M < L < N}. N − M is called an interval projection. We define N − to be ∨{M ∈ N : M < N}, and N + to be ∧{M ∈ N : M > N}. Since N is a complete lattice, both N − and N + lie in N . If N − = N, we say that N − is the immediate predecessor of N. If N + = N , we say that N + is the immediate successor of N . If N −N − is nonzero, then it is a minimal projection, or atom, in the core of N , the von Neumann algebra generated by N . A nest N is said to be continuous if it contains no atoms; it is said to be purely atomic if its atoms span H.

Normalizers of Operator Algebras and Reflexivity

“…For example, we show that every normalizer is the norm-limit of linear combinations of normalizing partial isometries, and every compact normalizer is the limit of finite rank normalizers. We also show that the sum of two normalizers of CSL algebras is again a normalizer only when both are contained in a single reflexive masa bimodule consisting of normalizers, and obtain generalizations of the results of Coates [3] on normalizers of nest algebras.…”

“…In this case Proposition 3.4 and Theorem 4.1 immediately yield the next corollary. The last statement was proved by Coates [3] for the case of nest algebras. We would like to conclude this paper with a discussion of the behaviour of the set of semi-normalizers (normalizers) with respect to addition.…”

“…In [3] it is shown that the set of normalizers of a nest algebra into itself is closed in the weak operator topology. We show that, for arbitrary reflexive algebras A and B, the sets SN (B, A) and N (B, A) are closed in the strong operator topology, but not always in the weak operator topology; nevertheless, Coates' result remains valid whenever the algebras A and B are strongly reflexive.…”

Normalisers of operator algebras and tensor product formulas

“…Normalisers of tensor products of von Neumann algebras were considered in [4], [12], [26] and [27]. The study of the normalisers of non-self-adjoint operator algebras, namely of nest algebras, was initiated in the 1990s (see [1], [5] and [7]). In [17] the notion of a normaliser was generalised and studied in the context of reflexive algebras, a non-self-adjoint generalisation of von Neumann algebras.…”