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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
Zbl 0342.06005