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Let $A$ be a C*-algebra and let $f: \mathbb{R}^n \to \mathbb{R}^n$ be the linear map which permutes the coordinates via a permutation $\sigma$. There is an induced map $K_0(C_0(\mathbb{R}^n) \otimes A) \to K_0(C_0(\mathbb{R}^n) \otimes A)$, and it is left as an exercise in several textbooks to show that this map corresponds to multiplication by $(-1)^{|\sigma|}$. I'm sure this is not difficult, but I'm having trouble; here is my progress.

Since permutations are products of transpositions, we reduce to the case $n = 2$ and $f$ simply swaps the coordinates of $\mathbb{R}^2$. $f$ is homotopic to the map $g: \mathbb{R}^2 \to \mathbb{R}^2$ given by $g(x,y) = (-x, y)$ via a simple rotation homotopy, so it suffices to show that the map $h: \mathbb{R} \to \mathbb{R}$ given by $h(x) = -x$ induces the inversion automorphism on $K_0(C_0(\mathbb{R}) \otimes A) \cong K_1(A)$. Sadly, I don't see how to do this. I'm sure I'll hate myself when I learn how.

Thanks in advance for the help!

Added: I should have mentioned at the outset that I am asking this question to understand a certain proof of the Bott periodicity theorem, and so I'm looking for a fairly elementary argument.

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Hint: Using the Künneth Theorem for tensor products (see Blackadars books for instance) it suffices to show that $x\mapsto -x$ induces multiplication by $-1$ on $K_1(C_0(\mathbb R))\cong\mathbb Z$. Now you can just check that a generator is mapped to its inverse.

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    Sorry I have neglected this for so long. My only objection to this answer is that it appears to assume Bott periodicity (correct me if I'm wrong), while my purpose in pursuing this exercise is to understand a certain proof of Bott periodicity. Can you think of an elementary argument?2011-07-01