Thanks again! Very helpful. I had requested the article through interlibrary loan, but it takes a bit to process.

Todd, I know about Lawvere's article because I've seen it referenced almost everywhere, but I can't seem to get a hold of it: I feel out of the loop. Is there a particular place you know of that I could find it? Also, I second unknown's request about the article by Kreisel :)

@Todd: Thanks for the answer! Do you know anything about foundations using the category of categories? I've read that this is another alternative, but that it is problematic. However, I've had the toughest time finding anything that actually explains in detail using the category of categories as a foundation. Does your answer still hold true for this approach to foundations?

@Yemon: I'm afraid I don't know anything about spectral triples(other than what wikipedia tells me), but thanks for the suggestion nevertheless. I'm sorry my question was not very precise and therefore frustrating. I guess some properties don't translate well. Thanks again to everybody for all the suggestions!

Okay, so after I had some time on my hands, I was finally able to play around with this question again after Jonas' revised answer. In short, I didn't find anything else. Of course, if our manifold is compact, then referring to Jonas' answer, we can always find finitely many such ideals which are *-isomorphic to $C_0(R^n)$. Then our entire algebra will be the sum of these ideals. I couldn't find much else other than that. And of course Jonas is right in that it brings up another question about interesting properties of $C_0(R^n)$.

@Yemon: That's a good observation, I had forgotten about that. Thanks. I think this may give me a better starting point for investigating my question.

@Qiachu: Thanks for the suggestion. I took a look at that blog entry, and it was really interesting. It seems like there should be some sort of connection between these partitions of unity and my question, but so far I haven't come up with anything relevant. I also agree with Jonas' comment that we'll need something more than just the existence of partitions of unity, since we have them in any compact Hausdorff space.

@Jonas: Thanks for the paper, it was interesting. While it fits my requirements of an algebraic property (well, C^*-algebraic, but we have no other choice in that), I think I'm going to wait and see what other interesting things crop up :).

@Yemon: I phrased the question vaguely somewhat on purpose, for the chance that there was some cool and fairly well-known result I was not aware of. Now that it seems like there is not, I agree with you that it would be good to rephrase it and ask with a more specific intent.

@Steven and Martin: thanks for the references. I thought I remembered similar questions, but I couldn't find them for the life of me. I feel like it may be a bit simpler to get some condition on the C^*-algebra of continuous functions to C rather than the R-algebra of smooth functions to R simply because of the Gel'fand transform, but this may be a misplaced suspicion.

@Martin: yes, that's right.

The underlying Heyting lattice structure in a topos is mainly interesting because it gives a way to observe the internal logic of a topos. These categories that Francois has listed may not give alternative subobject lattices, but the underlying logics in them are non-standard, which I believe is closer the heart of my question.