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Some applications of freeness with amalgamation. (English) Zbl 0926.46046
Using the construction of \(C^\ast\)-algebras which are generated from a \(B\)-probability space and creation operators that are \(ast\)-free with amalgamation over \(B\) the author proves that the fundamental group of a separable \(II_1\) factor \(M\) is contained in the fundamental group of the free product of \(M\) with \(L({\mathbb F}_\infty)\), the group \(W^\ast\)-algebra of the free group \({\mathbb F}_\infty\). He further proves that the \(W^\ast\)-algebra \(\Gamma(L^2({\mathbb R};{\mathbb R}),U_t)''\) generated by the left regular representation \(U_t\) of \({\mathbb R}\) on \(L^2({\mathbb R};{\mathbb R})\) is a \(III_1\) factor with core isomorphic to \(L(\mathbb F)_\infty)\times B(H)\).

MSC:
46L35 Classifications of \(C^*\)-algebras
46L53 Noncommutative probability and statistics
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