Some closed subalgebras of measure algebras and a generalization of P. J. Cohen's theorem II
Jyunji Inoue · Journal of the Mathematical Society of Japan · 1973
From (3.6) and the fact that $ u$ is concentrated on $\eta_{\tau_{1}}^{\tau}(K_{1})$ A $\eta_{\tau_{2}}^{\tau}(K_{2})$ , we obtairn using Proposition 2.1 of [4] that $ u$ belongs to $M(G(\tau_{0}))$ .\tau_{1}))\cap M(G(\tau_{2}))$ .Since the uniqueness of $\tau_{0}$ ia obvious from Theorem 2.5 of [4], this completes the proof of Theorem 3.1.We introduce a partial ordering in $\mathfrak{T}(G(\tau))$ such that if $\tau_{1},$ $\tau_{2}\in \mathfrak{T}(G(\tau))_{r}$ then $\tau_{2}\leqq\tau_{1}$ if and only if $\tau_{1}\subset\tau_{2}$ .COROLLARY $3.2^{1)}$ .$\mathfrak{T}(G(\tau))$ is a lattice under the partial ordering $\leqq$ .PROOF.Let $\tau_{1},$ $\tau_{2}\in \mathfrak{T}(G(\tau))$ .By Theorem 2.8 of [4], there exists $\tau_{3}\in$ $\mathfrak{T}(G(\tau)),$ $1$ .$u$ .$b$ . of $\tau_{1}$ and $\tau_{2}$ such that $L^{1}(G(\tau_{1}))*L^{1}(G(\tau_{2}))\subset L^{1}(G(\tau_{3}))$ .By Theorem 3.1 there exists $\tau_{0}\in \mathfrak{T}(G(\tau)),$ $g$ .$1$ .$b$ . of $\tau_{1}$ and $\tau_{2}$ such thatand this completes the proof.COROLLARY 3.3.Let $\tau_{0}$ , $\tau_{1}$ and $\tau_{2}$ be elements of $\mathfrak{T}(G(\tau))$ such that $M(G(\tau_{1}))\cap M(G(\tau_{2}))=M(G(\tau_{0}))$ .We regard each $\Gamma_{r_{i}}(i=0,1,2)$ as a subgroup $of^{-}$ the semigroup $\Gamma*$ (cf.PropOsitiOn3.2and p. 291 Remark of [4]), then we have $\Gamma_{\tau_{1}}+\Gamma_{\tau_{2}}=\Gamma_{\tau 0}$ .PROOF.In the proof of the Theorem 3.1, we can identify $\Gamma_{\tau_{0}}$ with $\Pi$ andil that we have $\varphi_{\tau_{0}}^{\tau_{1}}(\Gamma_{\tau_{1}})+\varphi_{\tau_{0}^{2}}^{r}(\Gamma_{\tau_{2}})=\eta^{\prime}-1(\varphi_{\tau_{d}}^{\tau_{1}}(\Gamma_{\tau_{1}})+\varphi_{r_{d}}^{\tau_{2}}(\Gamma_{\tau_{2}}))=\Pi$ .Thus we have 1) But $\mathfrak{T}(G(\tau))$ is not generally a $\sigma$ -complete lattice (cf.\S 5 example).