I've stuck on a step of proposition 1.2 of Rieffel's article ([continuous field of C\*-algebras coming from group cocycles and actions, 1989](https://link.springer.com/article/10.1007/BF01442857)). I think it basically proves that a C(X)-algebras coming from a locally convex Hausdorff space X is upper semicontinuous, but I can only see that happening for the compact case. The question is simple, but let me elaborate on the background.


As definition, of a $C(X)$-algebra, for $X$ compact Hausdorff, it is just a C*-algebra $A$ with a unital $\*$-homomorphism $C(X)$ to $ZM(A)$ (center of its multiplier algebra), so it can be though as the Gelfand transform of a C*-subalgebra of $ZM(A)$ containing its identity. Since we can take non-unital C\*-subalgebras of $ZM(A)$, it is also interesting to take locally compact but non compact $X$'es. (I think you can check more on that by googling "C(X)-algebras", apparently Kasparov uses this structure in his KK-theory, but I know nothing about that).


So let $C\_0(X)$ be $\*$-isomorphic to a C\*-subalgebra of $ZM(A)$. For $x\in X$, take $C\_0(X,x)$ as the ideal in $C(X)$ corresponding to functions vanishing at $x$. Then $J\_x$ defined as the closure of the linear span generated by $C(X,x)A$ is a closed ideal in $A$. Let $q\_x$ be the quotient map from $A \to A\_x = A/J\_x$. This gives rise to a family of C$^\*$-algebras $\{A\_x\}\_{x\in X}$ over $X$.


Prop. 1.2 of the reffered paper shows that $A$ is upper semicontinuous. This means that, for all $a\in A$ the map $x \mapsto \| q\_x(a) \|\_{A\_x}$ is upper semicontinuous. This follows from the characterization of the quotient norm.


There is a vector of the form $ b= \sum\_{i=1}^n f\_i b\_i \in J\_x$ (with $f\_i(x)=0$ for all $i$) such that $\| a + b\|\_A < \| q\_x(a) \|\_{A(x)}$. Now, since


Now, every $f\_i$ zeroes at $x$, so we can pick a function
$g\in C\_0(X)$ such that $g$ is one on a small neighborhood $U$ of $x$ and zero outside a small neighborhood of $U$, and such that $\|g b\|<\epsilon$.
Then, for all $y\in U$, we have that $(1-g) \in C(X,y)$ [!] and since
$(1-g)f \in J\_x$, we get $\| g a \|\_A = \| a - (1-g)a \|\_A > \|q\_y(a)\|\_{A\_y}$.
Thus


$ \|q\_x(a)|\_{A\_x} > \|g\|\,\|a+b\| - \epsilon > \|ga\| + \|gb\| + \epsilon >
 \| q\_y(a) \|\_{A\_y} + 2\epsilon.$


This proves the upper semicontinuity.


NOW HERE IS THE PROBLEM, as we have seen, this is fine for $X$ compact, but if not, I can't see why the constructed $(1-g)a$ is in the ideal $J\_y$, since $1$ is an element of the multiplier algebra $M(A)$ and $(1-g)$ would not be an element of $C\_0(X)$. Any thoughts on that? Is it true that ANY $C(X)$-algebra is upper semicontinuous then?


Also, why should we take, in usual definitions, the $C\_0(X)$ is embedded in ZM(A) instead of $C\_b(X)$? Otherwise, the constant fields in $C\_0(X,A)$ would not be continuous fields... does that any make any sense?



upper semicontinuity in C(X)-algebras
You can reduce to the compact case by considering the unitization of $C\_0(X)$. Since $ZM(A)$ is unital, if $C\_0(X)$ embeds in $ZM(A)$ then so does its unitization $C(X^\*)$ where $X^\*$ is the one-point compactification of $X$. If the map $x \mapsto \|q\_x(a)\|$ is semicontinuous on $X^\*$ then it is semicontinuous on $X$.


