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Recently I have been reading about star algebras. In particular, $C^\ast$-algebras. It seems that the condition $\|a^\ast a\| = \|a\|^2$ is quite strong and much is known about $C^\ast$-algebras.

I was wondering if it makes sense to consider the class of algebras satisfying the condition

$$ \|a^{-1}\| = \|a\|^{-1}$$

or, similarly/alternatively the condition

$$\|a^{-1}\|=\|a\|$$

DOes either one of these two make for an interesting structure to be studied?

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    The point of the condition is not that someone decided it would be fun to write down random conditions and see what happens. The point of the condition is that it's a nice condition that's necessary for a C-algebra to embed into the C-algebra of bounded operators on a Hilbert space, and by doing more work you can show that it's in fact sufficient. C-algebras are in fact precisely the (norm) closed -algebras of bounded operators on Hilbert spaces. The axioms are just a convenient way of capturing precisely this class of objects. – Qiaochu Yuan Sep 10 '14 at 07:21
  • @QiaochuYuan Yes but my question still remains. Unless you see the reason why the conditions above are not interesting? My main concern is whether I have come up with something non-sensical without noticing (or whether there is no obvious reason why either of the two condition cannot make for an interesting theory). –  Sep 10 '14 at 07:26
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    Both conditions are way too strong. In the case of continuous functions $C(X)$ on a compact Hausdorff space, the first condition implies that $X$ is a point. The second condition is just never satisfied (take $a = 2$). – Qiaochu Yuan Sep 10 '14 at 07:30

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Both conditions are way too strong. In the case of continuous functions $C(X)$ on a compact Hausdorff space, the first condition implies that $X$ is a point. The second condition is just never satisfied (take $a=2$). – Qiaochu Yuan yesterday

If the first property holds for all invertible operators in the algebra of bounded operators on a Banach space, then the space is one dimensional. If it holds for a single element, that element is a scalar multiple of an isometry.

Jonas Meyer
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