# Banach 28

This lecture (banach28 ; don’t mind the tile) was given on 10/30/09

### Excerpts

definitions

1. def of $L^\infty(T_\mu)$ given $T_\mu$ and $L^\infty(\mu)$
2. When two Borel measures are unitarily equivalent; $\mu << \nu << \mu$.  Question here.

Results

1. all op in $L^\infty(T_\mu)$ (think its incorrect in notes) commute with $T_\mu, T_mu^*$.
2. Cor. $L^\infty(T_\mu)$ is max. abel. subalg of $\mathscr{L}(L^2(\mu))$
3. Addendum: (???) a hilbert spac. op commutes also with $T^*$.
4. The reducing subspace of $T_\mu$ are the subspaces $\chi_E(T_\mu)L^2(\mu)$ for $E$ a Borel set.
5. $T_\mu \sim T_\nu$ iff $\mu \sim \nu$.  Here $\sim$ means unitarily equivalent.
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