Gelfand-Naimark theorem states that every commutative C-* algebra is isometric to $C(M)$, the ring of continuous functions over its spectrum. Is the theorem true for ordinary commutative Banach semisimple algebras, i.e. without *? Every proof that the Gelfand transform is an isometry uses the fact that in C-* algebra $|xx^{*}| = |x|^{2}$, so I wonder whether it is true when we don't have the * and that identity. If it is not isometric, is it isomorphic?(adsbygoogle = window.adsbygoogle || []).push({});

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Question about Gelfand-Naimark

**Physics Forums | Science Articles, Homework Help, Discussion**