Failure of amalgamation in Hilbert lattices
Tomasz Kowalski · ANU Open Research (Australian National University) · 2007
We show that Bruns and Harding’s counterexample (see [1]) to amalgamation in orthomodular lattices also works for Hilbert lattices. The argument is based on the example of a non-modular Hilbert lattice devised by von Neumann in a letter to Birkhoff (see [2] for an extensive quotation from that letter). Consider the real sequence space `2 = {f: N+ → R: Σ∞i=1f(i)2 < ∞}, where N+ stands for N \\ {0}. Let 〈en: n ∈ N+ 〉 be the standard orthonormal base of `2, i.e., en(n) = 1 and en(m) = 0 for m 6 = n. It follows from Satz 15 in [3] that there are two unbounded self-adjoint operators X and Y such that dom(X)∩dom(Y) = {0} and moreover Y can be chosen to be the “multiplication by n ” operator Y f = n · f(n) : n ∈ N+〉. Thus, Y is represented over the standard base as the N+ ×N+ matrix with entries yij = i · ei(j). Further, the operators X2 + 2I and Y 2 + 2I are self-adjoint and invertible. Define A = (X2 + 2I)−1 and B = (Y 2 + 2I)−1, where I is the identity operator. The following lemma spells out some properties of A, B and C.