ON CONNES SPECTRUM ＄￥Gamma＄ OF A TENSOR PRODUCT OF ACTIONS ON VON NEUMANN ALGEBRAS

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"Let ＄￥alpha^{j}(j-1,2)＄ be actions of a locally compact abelian group ＄G＄ on von Neumann algebras ＄￥vee￥nearrow￥swarrow_{j}＄ satisfying ＄￥alpha_{j}(￥ovalbox{￥tt￥small REJECT}_{j})^{￥prime}￥cap(_{c_{￥wedge}}￥ovalbox{￥tt￥small REJECT}_{j}￥times￥alpha jG)=￥tau_{￥ovalbox{￥tt￥small REJECT}_{j^{￥times}￥alpha^{j^{G}}}}^{p},＄ . ＄1f＄＄(￥alpha^{1}￥otimes￥alpha^{2})_{t}=￥alpha_{t}^{1}￥otimes￥alpha_{t}^{2}＄ , then ＄￥Gamma(￥alpha^{1}￥otimes￥alpha^{2})＄ is the set of all ＄p￥in G＄ such that ＄￥alpha_{p}^{1}￥otimes￥ell＄ is trivial on the fixed point algebra of the center of ＄(￥ovalbox{￥tt￥small REJECT}_{1}￥times￥alpha^{1}G)￥otimes^{-}(￥leftarrow￥ovalbox{￥tt￥small REJECT}_{2}￥times￥alpha^{2}G)＄ with respect to the action ＄￥hat{￥alpha}_{p}=￥hat{￥alpha}_{p}^{1}￥otimes￥hat{￥alpha}_{-p}^{2}＄ . Let ＄￥beta^{j}(j=1,2)＄ be ergodic actions of ＄G＄ on von Neumann algebras ＄￥vee￥phi_{j}^{￥prime}＄ . If ＄H^{2}(G, ￥mathbb{T})=￥{0￥}＄ and both ＄￥beta^{1}＄ and ＄￥beta^{2}＄ have invariant faithful normal states, then ＄(￥leftrightarrow.r_{1}￥otimes_{c}￥Lambda_{2}^{￥wedge})^{￥beta}-＄ is abelian, where ＄￥beta_{t}=￥beta_{t}^{1}￥otimes￥beta_{-t}^{2}＄ ."
Yokohama City Universityの論文
1978-00-00