Are commutative C*-algebras really dual to locally compact Hausdorff spaces?

It is true that the category of locally compact Hausdorff spaces is equivalent to the category of commutative $C^*$-algebras . . . with appropriately chosen morphisms.

Let $A$ and $B$ be commutative $C^*$-algebras. Then, a morphism from $A$ to $B$ is defined to be a nondegenerate homomorphism of $^*$-algebras from $\phi :A\rightarrow M(B)$, where $M(B)$ is the multiplier algebra of $B$. Here, nondegeneracy means that the the span of $\left\{ \pi (a)b:a\in A,b\in M(B)\right\}$ is dense in $M(B)$. Note that you need a bit of machinery to even make this into a category because it is not obvious a priori that composition makes sense. Nevertheless, it does work out. Proposition 1 on pg. 11 and Theorem 2 on pg. 12 of Superstrings, Geometry, Topology, and $C^*$-algebras (in fact this chapter is on the arXiv) respectively show that this forms a category and that the dual of this category is equivalent to the category of locally compact Hausdorff spaces.

So, for the sake of having an answer written down to this question: as t.b. says in the comments, we simply don't have this duality as stated.

The following categories are contravariantly equivalent:

• locally compact Hausdorff spaces with proper continuous maps
• commutative C$^*$-algebras with non-degenerate $*$-homomorphisms

Here, a $*$-homomorphism $f : A \to B$ is non-degenerate if the following equivalent conditions are satisfied:

1. The ideal generated by the set-theoretic image of $f$ is dense in $B$.
2. For every approximative unit $(u_i)$ in $A$ its image $f(u_i)$ is an approximative unit in $B$.
3. For some approximative unit $(u_i)$ in $A$ its image $f(u_i)$ is an approximative unit in $B$.

I don't think that multiplier algebras are necessary ...

It was recognised in the 50's that if one wants a duality theory for more general spaces than compact ones, then one has to go beyond the category of Banach spaces. In the context of a linear duality for locally compact spaces (generalising the Riesz representation theorem), R.C. Buck introduced the strict topology. This was later extended to the completely regular case by several authors---for example, by using the techniques of mixed topologies and Saks spaces which had been developed by the polish school. In this context, Gelfand-Naimark duality can be also extended and in the book Saks spaces and applications to functional analysis, a class of so-called Saks algebras (see eg these notes) was identified as the dual to the category of locally compact spaces. In the language of category theory, this provides a concrete representation of the opposite category to that of locally compact spaces (with continuous mappings as morphisms) extending the celebrated duality between compact spaces and commutative $C^*$-algebras with unit.