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I wonder if there would be an example of a $C^*$ algebra obtained in this way you mentioned whose all elements are zero divisor?Please see this question as a possible related post physicsoverflow.org/45602/…
@GeraldEdgar I still do not understand your comment. I belive that the integral you mentioned is always finite if $x\in \ell_p$ for all $p\in [1,2]$. so the finiteness condition you mentioned is redundant.