Compositions of conditional expectations, Amemiya-Andô conjecture and paradoxes of thermodynamics (Q2357089)

The present paper deals with the study of conditional expectations. The motivation of the paper is given by the so-called Amemiya-Andô conjecture, which can be explained as follows. If \(H\) is a real Hilbert space, \(x\) is an element of \(H\) and \(P_1,\dots, P_k\) are orthogonal projections in \(H\), consider any sequence \(E_1, E_2,\dots \in \{P_1,\dots, P_k\}\). The question is: is it true that the sequence of iterations \((E_n \dots E_2 \, E_1 \, x)_n\) is convergent? As Amemiya and Andô showed, such a sequence of iterations is weakly convergent. Several counterexamples for the conjecture that it is also convergent in the norm of \(H\) are due to Paszkiewicz, Kopecká and Müller. In this paper, a new counterexample is shown using \(H=L^2(\Omega,\mathcal F,\mu)\) for a measure space \((\Omega,\mathcal F,\mu)\), \(P_1,\dots,P_k\) being conditional expectations, using some recent results connecting orthogonal projections on Hilbert spaces and conditional expectations on \(L^2(\Omega,\mathcal F,\mu)\). After extending the definition of conditional expectation to non-finite measures the author proves in Theorem 20 the following result: There are five conditional expectations \(P_1,\dots,P_5\) on the Lebesgue measure space \((\mathbb R, \text{Borel }(\mathbb R), \lambda)\) -- that is not a probability space --, a function \(f \in L^1(\mathbb R) \cap L^2(\mathbb R)\) and a sequence \(E_1, E_2,\dots \in \{P_1,\dots,P_5\}\) such that the sequence \((E_n\dots E_2 \, E_1 \, x)_n\) diverges in \(L^2(\mathbb R)\). As explained in the paper, the results can also be interpreted in terms of classical thermodynamics in a way that they seem to contradict the second principle of thermodynamics.