Ideal structure and factorization properties of the regular kernel operators
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Ideal structure and factorization properties of the regular kernel operators A. Blanco1 Received: 14 February 2019 / Accepted: 5 December 2019 © The Author(s) 2019
Abstract We consider the structure of the lattice of (order and algebra) ideals of the band of regular kernel operators on L p -spaces. We show, in particular, that for any L p (μ) space, with μ σ -finite and 1 < p < ∞, the norm-closure of the ideal of finite-rank operators on L p (μ), is the only non-trivial proper closed (order and algebra) ideal of this band. Key to our results in the L p setting is the fact that every regular kernel operator on an L p (μ) space (μ and p as before) factors with regular factors through p . We show that a similar but weaker factorization property, where p is replaced by some reflexive purely atomic Banach lattice, characterizes the regular kernel operators from a reflexive Banach lattice with weak order unit to a KB-space with weak order unit. Keywords Algebra ideal · Banach lattice · Kernel operator · Order ideal · Regular operator Mathematics Subject Classification Primary 46B28 · 47B07 · 47L10; Secondary 46B42 · 47B65
1 Introduction Let X be a Banach lattice and let Lr (X ) be the space of all regular maps (i.e., linear combinations of positive maps) from X to itself. It is well known that Lr (X ), endowed with the regular norm and with composition as product, is a Banach algebra and an ordered vector space whose positive cone is closed under multiplication. There has been interest in understanding the structure of the lattice of closed ideals of Lr (X ), where by ideal, here and henceforth, we shall mean an order and two-sided algebra ideal. For instance, if X = p (1 < p < ∞) then Ar ( p ) (:= the closure of
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A. Blanco [email protected] Mathematical Sciences Research Centre, Queen’s University Belfast, Belfast, UK
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A. Blanco
the ideal of finite-rank operators in Lr ( p )) is the only non-trivial proper closed ideal of Lr ( p ) [3,4]. If X = c0 or 1 , then Lr (X ) = B(X ) (:= the Banach algebra of all bounded operators on X ) and the same conclusion follows from a classical result of Gohberg, Markus and Feldman. It is also known that if X = L p [0, 1] (1 < p < ∞) then the structure of the lattice of closed ideals of Lr (X ) significantly departs from that of the lattice of closed (two-sided) algebra ideals of B(X ), e.g., for every 1 < p < ∞, there are in Lr (L p [0, 1]) at least four distinct well-documented closed ideals apart from the trivial ones, namely, Ar (L p [0, 1]), Ir (L p [0, 1]) (see below for definitions), the span of the positive compact operators and the AM-compact operators (this can be easily deduced, for instance, from [22, Theorem 125.5], [17, Theorem 3.3] and [20, Theorem 3.4]). In this note, our main concern will be with the band Ir (X ) of all kernel (or integral) operators in Lr (X ). When X is order continuous, the latter is a closed ideal (in the sense of the note) of Lr (X ). If X is in addition atomic then Ir (X ) = Lr (X ) and one can think of the resu
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