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What are some good sources of examples (and/or the simplest example) for:

Pairs of automorphisms $\phi,\psi:A\to A$ over the same base $C^*$-algebra $A$
with non-stably isomorphic crossed products, i.e. $$ (A\rtimes_{\phi}\mathbb{Z})\otimes\mathbb{K}\ncong(A\rtimes_{\psi}\mathbb{Z})\otimes\mathbb{K}? $$

Specifically it would be nice to know from my side what such actions look like on the algebra.

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    $\begingroup$ Stable isomorphism is a very strong equivalence relation! So it is very rare that different automorphisms give you stably isomorphic crossed products. A good example of crossed products by $\mathbb Z$ is the irrational rotation algebras $\mathcal A_\theta = C(\mathbb T) \rtimes_{r(\theta)} \mathbb Z$ for $\theta \in [0,1] \setminus \mathbb Q$, where the automorphism is induced by rotation by $2\pi \theta$. Then $\mathcal A_{\theta_1}$ and $\mathcal A_{\theta_2}$ are (stably) isomorphic if and only if $\theta_1 = \theta_2$ or $\theta_1 = 1-\theta_2$. $\endgroup$
    – Jamie Gabe
    Commented Dec 7, 2021 at 15:11
  • $\begingroup$ Right, I see, thanks Jamie also for the nice example! :) Do you maybe wanna have it converted to an answer? I could accept it then. $\endgroup$
    – C-star-W-star
    Commented Dec 7, 2021 at 15:18

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