Talk:Operator K-theory

This whole article could be a great deal clearer

For example, this paragraph under "Overview" could hardly be more unclear:

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"Operator K-theory is a generalization of topological K-theory, defined by means of vector bundles on locally compact Hausdorff spaces. Here, a vector bundle over a topological space X is associated to a projection in the C* algebra of matrix-valued—that is, $M_{n}(\mathbb {C} )$-valued—continuous functions over X. Also, it is known that isomorphism of vector bundles translates to Murray-von Neumann equivalence of the associated projection in K ⊗ C(X), where K is the compact operators on a separable Hilbert space."

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In the first sentence, the word "defined" might refer to topological K-theory, but then again it might refer instead to operator K-theory; there is no way to be sure.

As if that were not bad enough, in the second sentence the first word "Here" is even less clear than the word "defined" was.

The phrase "associated to" in the second sentence is never explained. How does this association work? Not even a hint?

Oh, and I almost forgot to mention that the phrase "Murray-von Neumann equivalence" is neither explained nor linked to anything at all — so many readers will be unable to understand this entire paragraph.

(Sure, the reader might be able to guess what some things mean. But no reader should ever have to do that with Wikipedia. And it would be hard to be sure if they have guessed right.2600:1700:E1C0:F340:B120:AEAD:CE8A:82CC (talk) 05:09, 6 October 2018 (UTC)