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Here the definition of Banach star algebra is given as Banach algebra with an involution. In the book by Murphy for example, it is given as Banach algebra with an involution plus the condition that $\|a\|=\|a^\ast\|$.

My question is: is the condition necessary or does it follow from the other conditions?

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  • $\begingroup$ @MikeMiller No the condition for a C star algebra is $\|a^\ast a \| = \|a\|^2$. These are not the same. $\endgroup$ – user167889 Sep 10 '14 at 5:27
  • $\begingroup$ You're quite right, I had an inequality backwards in my "proof". Comment deleted. $\endgroup$ – user98602 Sep 10 '14 at 5:35
  • $\begingroup$ @MikeMiller I think I found the answer here: they are two non-equivalent definitions but mostly the involution is isometric. I will wait some more time before deleting the question maybe someone will post an enlightening answer. $\endgroup$ – user167889 Sep 10 '14 at 6:34
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The property does not follow from the others, but is useful and satisfied by enough examples to make it often worth assuming. Aside from C*-algebras, $L^1$ algebras on locally compact groups satisfy this definition. One of the useful consequences of the property is that it implies that $*$-representations of Banach $*$-algebras on Hilbert space are (weakly) contractive.

You can consider various norms on $M_2(\mathbb C)$ with the usual operations to see that the other axioms can hold without this one.

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