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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.

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