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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: $||x^{*}x||=||x|{|}^2\mathrm{\forall }x\in A$:

$||x||=\underset{||y||\le 1}{\sup }\{||xy||\}$ for all $x\in A$.

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

$\underset{||y||\le 1}{\sup }\{||xy||\}\le \underset{||y||\le 1}{\sup }\{||x||\cdot ||y||\}\le ||x||$

But I can't seem to do the other.

Cheers,

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

Are you allowed to use $\Vert x\Vert =\Vert x^{*}\Vert$? If so, try $y=x^{*}/\Vert x\Vert$. Then $\Vert y\Vert =1$ and $\Vert xy\Vert =\Vert x\Vert$, 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)