The multi-form generalised Gleason theorem
The authors prove the following theorem: Theorem. Let \(X\) be a Banach space. Let \(k\) be a positive integer. For \(j=1,2,\dots,k\) let \(A_j\) be a von Neumann algebra with no direct summand of Type \(I_2\). Let \(m\) be an \(X\)-valued \(k\)-measure on \(P(A_1)\times P(A_2)\times \cdots \times P(A_k)\) whose values are bounded in norm by a constant \(K\). Then there exists a unique \(f\) in \text{Lin}\((A_1, A_2,\dots,A_k;X)\) such that \(f(p_1, p_2,\dots,p_k)=m(p_1, p_2,\dots,p_k)\) for all projections \(p_1, p_2,\dots,p_k\) in, respectively \(A_1, A_2,\dots,A_k\). Also \(\|f\|\leq 4^kK\). Here \text{Lin}\((A_1, A_2,\dots,A_k;X)\) denotes the Banach space of all \(X\)-valued \(k\)-linear forms. It is claimed that the result is a necessary tool for the study of decoherence in quantum theory.
Banach space of all \(X\)-valued \(k\)-linear forms
scientific article; zbMATH DE number 1269792