zbMATH — the first resource for mathematics

On the products of quantum logics. (English) Zbl 0542.03037
A quantum logic is a pair (L,M), where L is an orthomodular \(\sigma\)- lattice and M is a set of states on L which is strong for L, i.e., \(a\not\perp b\) (a,\(b\in L) \Rightarrow \exists m\in M:\quad m(a)=1,\quad m(b)\neq 0.\) In addition, the Jauch-Piron property is supposed, i.e., \(m(a_ i)=1\) for all \(i\in N\) iff \(m(\bigwedge_{i\in N}a_ i)=1\) for all \(m\in M\). For \(S\subset M\) put \(\bar S=\{m\in M:s(a)=1\) for all \(s\in S \Rightarrow \quad m(a)=1\}\) (\(a\in L)\). The following definition is introduced: Definition. We say that \((L,M)_{\alpha,\beta}\) is the tensor product of the quantum logics \((L_ i,M_ i)\), \(i=1,2\), if \((i)\quad \alpha:L_ 1\times L_ 2\to L, \beta:M_ 1\times M_ 2\to M, \beta(m_ 1,m_ 2)(\alpha(a_ 1,a_ 2))=m_ 1(a_ 1)m_ 2(a_ 2)\) for any \(m_ i\in M_ i\), \(a_ i\in L_ i\), \(i=1,2\); \((ii)\quad \{m\in M:m(a)=1\}=\{\beta(m_ 1,m_ 2):\beta(m_ 1,m_ 2)(a)=1\}^-\) for elements \(a\in L\) of the form \(a=\bigwedge_{k}(a_ i^ k,a_ 2^ k), a_ i^ k\in L_ 1, a_ 2^ k\in L_ 2,\) \(k\in N\), and \(a=(a_ 1,1)^{\perp},\) resp. \(a=(1,a_ 2)^{\perp}, a_ 1\in L_ i, i=1,2\); \((iii)\quad \alpha [L_ 1\times L_ 2]\) generates L; \((iv)\quad \beta [M_ 1\times M_ 2]^-=M.\)
It is shown that this definition agrees with the definition of free orthodistributive product of orthomodular \(\sigma\)-lattices introduced by T. Matolcsi [Acta Sci. Math. 37, 263-272 (1975; Zbl 0342.06005)], and includes the product of Hilbert space quantum logics for real or complex Hilbert space, separable and with the dimension at least three.

03G12 Quantum logic
46C99 Inner product spaces and their generalizations, Hilbert spaces
Zbl 0342.06005