Definition 44.2.2 (Canonical Conjugation).label Let $(M, \tau)$ be a tracial von Neumann algebra, then the mapping
\[J: L^{2}(M, \tau) \to L^{2}(M, \tau) \quad \wh x \mapsto \wh{x^*}\]
is a conjugate linear isometry on $L^{2}(M, \tau)$, known as the canonical conjugation.
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