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

Abstract

Suppose .A4 is avon Neumann algebra on a Hilbert space H and 2" is any norm closed ideal in M. We extend to this setting the well known fact that the compact operators on a Hilbert space are precisely those whose restrictions to the closed unit ball are weak to norm continuous.The norm closed ideal K~(7~) of compact operators in the von Neumann algebra B(7-/) of all bounded linear operators on a Hilbert space 7"( is determined by its projections : /C(7-() is the norm closure in B(7-/) of the ideal .T'(7-/) of finite rank operators generated by the compact projections (i.e. those with finite dimensional range). The ideal ,T(7" 0 can be characterised in terms of the weak topology of 7-/: it is the set of operators in B(7-/) which are weak to norm continuous. What is rather more interesting is the fact that the ideal /C(7-/) can also be characterised in terms of the weak topology : it consists of those operators whose restriction to the unit ball of 7-[ are weak to norm continuous. This paper is a continuation of [12], where it was shown that every ideal 2" in a yonNeumann algebra M on a Hilbert space 7"[ is determined by a topology t(Z) on 7-/, in the sense that 2-consists of exactly those elements of M which are t(2-) to norm continuous. The set 2-P of projections in 2-determines a 'weak' topology t(ZP), and the ideal corresponding to this topology in the above sense is the ideal generated by 2"P in Ad. 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-. This enables us to use duality arguments to derive the above-mentioned characterisation of t(2-) in the second

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].…”

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

“…Combining these two concepts we will be able to give a topological char~terisation for the closed ideals in a yon Neumann algebra, which is an exact generalisation of the well known result that the ideal of compact operators in a Hilbert space consists precisely of those operators which on the unit ball of the Hilbert space are weak to norm continuous. This is the subject of [1].…”