A structure theorem for truncations on an Archimedean vector lattice

Abstract

Let X be an Archimedean vector lattice and \(X_{+}\) denote the positive cone of X. A unary operation \(\varpi \) on \(X_{+}\) is called a truncation on X if

$$\begin{aligned} x\wedge \varpi \left( y\right) =\varpi \left( x\right) \wedge y\quad \text {for all }x,y\in X_{+}. \end{aligned}$$

Let \(X^{u}\) denote the universal completion of X with a distinguished weak element \(e>0.\) It is shown that a unary operation \(\varpi \) on \(X_{+}\) is a truncation on X if and only if there exists an element \(u\in X^{u}\) and a component p of e such that

$$\begin{aligned} p\wedge u=0\quad \text {and}\quad \varpi \left( x\right) =px+u\wedge x\ \text {for all }x\in X_{+}. \end{aligned}$$

Here, px is the product of p and x with respect to the unique lattice-ordered multiplication in \(X^{u}\) having e as identity. As an example of illustration, if \(\varpi \) is a truncation on some \(L_{p}\left( {\mu } \right) \)-space then there exists a measurable set A and a function \(u\in L_{0}\left( {\mu } \right) \) vanishing on A such that \(\varpi \left( x\right) =1_{A}x+u\wedge x\) for all \(x\in L_{p}\left( {\mu } \right) .\)