Isometries on conservative subalgebras of bounded sequences

Abstract:

Using extreme point techniques, we show that if $A$ is a closed subalgebra of the bounded sequences which contain $c$, then any linear isometry of $A$ onto itself is a permutation up to a modulus one multiplication. If the subalgebra $A$ is generated by an ideal, then a permutation $P$ maps $A$ onto itself if and only if $P$ maps $\mu$-null sets to $\mu$-null sets where $\mu$ is a 0,1-valued finitely additive measure associated with the ideal. In particular, if $T$ is a nonnegative regular summability method, we characterize the isometries which map the bounded strongly $T$-summable sequences onto themselves and give a concrete sufficient condition for a permutation to map the bounded strongly Cesaro summable sequences onto themselves.