Definition 44.2.3 (Left/Right Regular Representations).label Let $(M, \tau)$ be a tracial von Neumann algebra and $x \in M$, then
\[L_{x}: L^{2}(M, \tau) \to L^{2}(M, \tau) \quad L_{x}\wh y = \wh{xy}\]
and
\[R_{x}: L^{2}(M, \tau) \to L^{2}(M, \tau) \quad R_{x}\wh y = \wh{yx}\]
are the left/right multiplication maps by $x$, respectively. The mapping
\[\lambda: M \to B(L^{2}(M, \tau)) \quad x \mapsto L_{x}\]
is the left regular representation of $M$, with $L(M) = \lambda(M)$ being the algebra of left multiplications. Similarly,
\[\rho: M^{op}\to B(L^{2}(M, \tau)) \quad x \mapsto R_{x}\]
is the right regular representation of $M^{op}$, with $R(M) = \rho(M)$ being the algebra of right multiplications.
Post a Comment