Wikipedia:Reference desk/Archives/Mathematics/2014 October 18

= October 18 =

Norm identity for a normed *-algebra with B*-identity
Hi,

I'm trying to show a claim for a normed *-algebra A that is not necessarily unital or commutative but is non-zero and satisfies the B*-identity: $$||x^*x||=||x||^2\;\forall x \in A$$:


 * $$||x||=\sup_{||y||\leq 1}\left\{||xy||\right\}$$ for all $$x\in A$$.

One direction is trivial; the required property of a normed *-algebra gives:


 * $$\sup_{||y||\leq 1}\left\{||xy||\right\}\leq \sup_{||y||\leq 1}\left\{||x||\cdot ||y||\right\}\leq ||x||$$

But I can't seem to do the other.

Cheers,

Neuroxic (talk) 14:32, 18 October 2014 (UTC)


 * Are you allowed to use $$\|x\|=\|x^*\|$$? If so, try $$y=x^*/\|x\|$$.  Then $$\|y\|=1$$ and $$\|xy\| = \|x\|$$, which gives the opposite inequality.   Sławomir Biały  (talk) 21:13, 18 October 2014 (UTC)
 * Yes, one can derive that that B* condition implies the involution is isometric. Many thanks!
 * Neuroxic (talk) 22:45, 18 October 2014 (UTC)