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

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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)