Weak CCRs and CARs and inductive limits of families of groups and algebras (Q1580172)

Let \(V\) be a real linear space equipped with a nondegenerate antisymmetric form \(\tau\). The canonical commutation relations are defined as a family of unitary operators \(\{U(x), x\in V\}\), acting in some Hilbert space, with \(U(x+ y)= \exp(i\tau(x,y)) U(x)U(y)\), \(x,y\in V\), such that the mapping \(s\to U(sx)U(sy)\), \(s\subset\{\mathbb{R}\}\) is weakly continuous. If \(\tau(x,y)= 0\) then, obviously, \(U(x+ y)= U(x)U(y)\). A natural question is whether the converse is true. The author states a second problem concerning the canonical commutation relations and another one in connection with the canonical anticommutation relations. Using some inductive limit results (whose proof is not given), the author discusses the answers to these questions.