# Equivalence of categories ($c^*$ algebras <-> topological spaces)

I try to use a littlebit category theory to have a better overview of the results in the theory of $c^*$-algebras, but I really have to read an introduction to category theory because I know almost nothing about it.

My first question is: What are the isomorphisms in the category of commutative, unital $c^*$-algebras $c^*com_1$: unital $*$-isomorphisms between objects in $c^*com_1$ or unital $*$-isomorphisms between objects in $c^*com_1$ which are isometric?

Let KTOP be the category of compact Hausdorff spaces. Gelfand-Naimark and the homeomorphism $X\to \Gamma_{C(X)}, x\mapsto \sigma_x(f)=f(x)$ give us that the categories $(c^*com_1)^{op}$ and $KTOP$ are equivalent.

Now, there is a version of Gelfand-Naimark for nonunital $c^*$-algebras:
Gelfand-Naimark: Let $V$ be a commutative $c^*$-algebra, then the Gelfand representation $^\wedge: V\to C_0(\Gamma_{V}),\; v\mapsto \hat{v}\;\;$ $\hat{v}(f)=f(v)$ is an isometric $*$-isomorphism.
Therefore I first thought that the categories $(c^*com)^{op}$ (dual of the category of commutative $c^*$-algebras) is equivalent to $LKTOP$ (category of localcompact Hausdorff spaces), but I read elsewhere that this is false. I have read that $(c^*com)^{op}$ is equivalent to $Cpt_{\cdot}$, the category of pointet compact spaces.

Could you explain me why this what I expacted, that the categories $(c^*com)^{op}$ and $LKTOP$ are equivalent, is not true?

Regards.

• See the discussion at math.stackexchange.com/questions/170984/…. – Qiaochu Yuan May 8 '15 at 17:39
• oh, thank you! the discussion seems to answer my second question. – banach-c May 8 '15 at 17:44
• And the isomorphsms in $(c^*com_1)^{op}$, they are not isometric in general, is it correct? – banach-c May 8 '15 at 20:11
• The algebraic structure of a unital C*-algebra determines its norm (in terms of spectral radius), so an isomorphism of unital C*-algebras, as *-algebras, is automatically an isometry. – Qiaochu Yuan May 9 '15 at 1:09
• In fact a map of unital C*-algebras has norm at most $1$; that is, it is a weak contraction. And an isomorphism in the category of metric spaces with weak contractions is an isometry. – Qiaochu Yuan May 9 '15 at 5:03