Topological and bornological characterisations of ideals in von Neumann algebras: I

Abstract

Suppose .M is a von Neumann algebra on a Hilbert space 7-( and 2-is any ideal in 34. We determine a topology t(Z) on 7-/, for which the members of M that are t(g) to norm continuous are exactly those in 2"; and a bornology b(2.) on 7-/such that the elements of M which map the unit ball to an element of b(2.), equivalently those members of 34 that are norm to b(2.) bounded, are exactly those in 2". This is achieved via analogues of the notions of injeetivity and surjectivity in the theory of operator ideals on Banach spaces.In the theory of Banach operator ideals, as found for example in the monograph of Pietseh ([4]), the injective and surjective operator ideals are of interest in that they are essentially those operator ideals which are invariant under enlargement of the codomain, or enlargement of the domain, respectively. (We will not be any more precise than this here, simply referring the reader to [4] w167 4.7.) We will show that ideals in a yon Neumann algebra satisfy properties analogous to certain charaeterisations of injective and surjective operator ideals studied by Stephani ([6], [7] and [8]). As a consequence we show that some of the known properties of injective and surjective operator ideals have analogues in the setting of ideals in yon Neumann algebras. We will show that any ideal 2-can be characterised by means of a topology t(Z) and a bornology b(Z) : the members of fl-I which are t(Z) to norm continuous are exactly the members of 2", and the members of M which are b (Z) to norm bounded are exactly the members of 2". The topological characterisation can be compared with [6] and [8] where it is shown that if an operator ideal is injective then it can be characterised in such a fashion by means of a 'generating topology'. Dual to the notion of 'generating topology' is Stephani's 'generating systems of sets'. In [7] it is shown that if an operator ideal is surjective then it can be characterised in terms of these generating systems of sets. This result can be compared to the bornological characterisation mentioned above.

Topological and bornological characterisations of ideals in von Neumann algebras: II

“…With respect to this pairing, a subset of 7-( is t(I)-equicontinuous if and only it belongs to b(I). Proofs and more information can again be found in [12].…”

“…The bornology b (2-v) determines the ideal ((2-v)), in the sense that b(2-p) = b(((2-P))). For proofs and additional information we refer the reader to [12].…”

“…We show that every norm closed ideal 2-can be characterised in the same way as/C(7-() : 2-consists of exactly those elements of 3,4 whose restrictions to the unit ball of 7-( are t(2-P) to norm continuous. This is done by identifying t(2-) for such ideals as a generalised inductive limit (or mixed) topology.It was also shown in [12] that an ideal 2" in von Neumann algebras can be characterised in terms of a bornology b(2-) which is in a sense dual to the topology t(Z). In the first section of this paper b(2-) is characterised in terms of 2-P for a norm closed ideal 2-.…”

The triangle of operators, topologies, bornologies

“…The explicit construction (with all arrows shown in the diagram) of the (upper) triangle is given in [32], in which several applications to Banach space theory are demonstrated. When the underlying space is a fixed complex Hilbert space, West implements the triangle in the context of operator algebras [25] and provides several applications with Conradie [1] (see Section 2). In this paper, we shall complete the LCS version of the triangle.…”

Operator Ideals Arising from Generating Sequences

“…In [16], we use these concepts in classifying locally convex spaces. They are also used by other authors in operator algebra theory in [13,2].…”