# upper semicontinuity in C(X)-algebras

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). 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?

• The spaces $C_0(X,x)$, $C_0(X)$, and $C(X,x)$ above are all the same, aren't they? – Pietro Majer Jun 18 '12 at 4:39
• Hello Mr. Majer, thank you for your comment. Seems like I wasn't clear enough in the definitions. For any point $x$ of $X$, I call $C_0(X,x)$ the (closed ideal) set of (complex valued) function $f\in C_0(X)$ such that $f(x)=0$ (at the particularly defined $x$). This is, of course, not equal to $C_0(X)$. If we take $C(X,x)$ in the same fashion, as the set of functions $f\in C(X)$ such that $f(x)=0$, then $C_0(X,x)$ and $C(X,x)$ are equal iff $X$ is compact. Thank you. – Yul Otani Jun 18 '12 at 7:17

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 $X$.
• I guess people usually stay away from $C_b(X)$ because it's so big (e.g., nonseparable). Remember that $C_b(X)$ is the same as $C(\beta X)$, the continuous functions on the Stone-Cech compactification of $X$. – Nik Weaver Jun 18 '12 at 14:33
• Hello! Thank you again for the comment. I have finally given it some look! I have some considerations. I think that if we evaluate the $x\mapsto \|q_x(a)\|$ defined on $X^*$, the problem is that the ideals and quotients are different to those considering $C_0(X)$ (also, the extension of the injection may not be unique if $C_0(X)$ is degenerate in $ZM(A)$). However, we can simply pick $g$ in $C(X^*)$ instead of in $C_0(X)$, such that all the wanted properties hold, since we only need that $1-g$ to be in $C_0(X)$ (we take the unital extension of the injection $C(X^*)$ in $ZM(A)$). – Yul Otani Jul 22 '12 at 20:56