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For example,

  • For what $C^*$ algebras $A$ is unitary equivalence the same as mvn equivalence for projections.
  • For what $C^*$ algebras $A$ is unitary equivalence the same as homotopy equivalence for projections.
  • For what $C^*$ algebras $A$ is unitary equivalence the same as approximate unitary equivalence

Of course it's already known that $p \sim_h q \implies p \sim_u q \implies p \sim_{mvn} q$, so I suppose my question is when do we have implications in the other direction. One thing I've already noticed is that for (some nontrivial) classes of $C^*$ algebras there are so few projections that this line of questioning is meaningless.

So far I've seen remarkably few results contrasting the different types of projections, and the only well-known result I've seen is that if $A$ is unital then $p \sim_u q \iff p \sim_{mvn} q $ and $ 1-p \sim_{mvn} 1-q$.

I'm aware this might be a bit of a weird/unnatural question to ask. It's definitely quite "operator first", which is probably a feature of someone that's been reading a lot of Davidson's book recently! This is just because I'm still at the stage of my Ph.D. of asking questions that aren't necessarily interesting. I would equally be interested in hearing about why or why not there is not so much interest in these types of questions.

I feel thinking about these questions might help deepen my understanding/appreciation of the construction of $K_0$... I suppose it's true that if we had mvn equivalence in some $M_n(A)$, we can blow it up to a larger matrix (would it be respectively $M_{2n}, M_{4n}$?) such that we have unitary/homotopy equivalence. Is this why people don't generally look at what distinguishes these equivalence at the level of $A$?

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    $\begingroup$ For the third question: approximate unitary equivalence of projections is the same unitary equivalence, since close projections are unitarily equivalent. $\endgroup$
    – Jamie Gabe
    Commented Sep 9, 2021 at 10:46
  • $\begingroup$ Aha, that was a silly mistake. Hope you are well Jamie. $\endgroup$
    – Owen Tanner
    Commented Sep 9, 2021 at 12:27
  • $\begingroup$ I wanted to comment that upon some further searching, there is some movement in some areas back in the 90s. This paper by Sang Og Kim is relevant to the second question. $\endgroup$
    – Owen Tanner
    Commented Sep 9, 2021 at 13:50
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    $\begingroup$ The questions feel too general to have a better answer than the description you've given(maybe restricting to classifiable C*-algebras it may be known?). On the other hand, for a large and interesting class of C*-algebras Rieffel has settled the first question. He showed (Corollary 7.8 in Projective Modules over higher dimensional non-commutative tori) that in a (irrational) non commutative torus $A$ if two projections $p,q\in M_n(A)$ have the same $K_0$ class then they are actually unitarily equivalent. So Rieffel's paper may be a fun/good place to start investigating the first question $\endgroup$
    – Caleb Eckhardt
    Commented Sep 9, 2021 at 14:19
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    $\begingroup$ Moreover, homotopy, unitary, and Murray-von Neumann equivalence agree for projections in a stable C*-algebra, i.e.~a C*-algebra $A$ such that $A \cong A \otimes \mathcal K(\ell^2(\mathbb N))$. Here the unitaries can be taken in the minimal unitisation or in the multiplier algebra $M(A)$. This follows since if $p,q\in Proj(A)$ then $1-p, 1-q \in M(A)$ are equivalent (easy exercise using stability), and since the unitary group of $M(A)$ is path-connected by a theorem of Cuntz and Higson (although you can prove homotopy equivalence from unitary equiv. bare hands). $\endgroup$
    – Jamie Gabe
    Commented Sep 10, 2021 at 12:09

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