Bill Page wrote:
>
> On Tue, Feb 6, 2018 at 8:28 AM, Waldek Hebisch <[email protected]>
> wrote:
> > Prof. Dr. Johannes Grabmeier wrote:
> >>
> >> So my remark shoud better read:
> >>
> >> All general functions as leftRecip and rightRecip and associator are
> >> superfluous as
> >> soon as we have an associative structure,
> >
> > That is more complicated, for example Cuntz algebra is associative,
> > but leftRecip is different than rightRecip and both are different
> > than recip.
>
> Sorry, it seemed trivial. Can you give an example, a reference or explain?
>
> Def: rightRecip(a) * a = 1
> Def: a * leftRecip(a) = 1
I would write:
a * rightRecip(a) = 1
but need to check what our code implements. Docsting "conveniently"
does not explan what left and right means...
> Associative: (rightRecip(a) * a) * leftRecip(a) = rightRecip(a) * (a *
> leftRecip(a))
>
> --> 1 * leftRecip(a) = rightRecip(a) * 1
> --> leftRecip(a) = rightRecip(a)
>
> No?
I should have written leftRecip and rightRecip _as functions_
are different. In particular there are element where left
inverse exists but right inverse does not exist (that is gives
"failed"). In Cuntz algebra (see for example Wikipedia
page) you have elements ai and bi (bi is ai^*, but this is irrelevant
for us) such that:
ai*bi = 1
\sum_{i=1}^n bi*ai = 1
We also have bi*ai \ne 0 for each i, so for each i we have
bi*ai \ne 1. Your argument shows that when both left
and right inverse exist, then thay are equal. So the equalities
above for n > 1 shows that bi has one-sided inverse (which I
would call left inverse) but does not have inverse from the other side.
--
Waldek Hebisch
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