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.
This paper defines the n‐fold central Haagerup tensor product ⊗i=1ℒhnR of a von Neumann algebra R, and shows that the map θz:⊗i=1ℒhnR→CB(normalRn−1,R) given by θ([a1⊗a2⊗…⊗an])(x1, x2, …,xn−1)=a1x1a2x2…xn−1an is an isometry.
In [1], Proposition 5 has an error. The proposition purports to show that S N , the normalizing semigroup for the nest algebra N , is closed in the strong operator topology. The proof begins with the net T λ in S N converging strongly to T ∈ B(H). It then proceeds to (correctly) establish the first of the two criteria set forth in Theorem 2 needed to conclude that T ∈ S N . The last sentence of the proof then asserts that the second criterion can be established in a similar manner by just replacing all the operators with their adjoints. This is the problem: since the adjoint operation is not strongly continuous, we cannot assume that T * λ converges strongly to T * , which is needed to make the analogous argument work to establish the second criterion.The following replacement for Proposition 5 fixes this, and strengthens the proposition. Since the adjoint operation is weakly continuous, we really do need only establish the first criterion, the second one following by replacing the operators involved with their adjoints.
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