What does it tell us, if we know a unital C*-algebra has approximately inner (half-)flip?

Question

This is a somewhat vague question, but I think it is not too open-ended and should admit well-circumscribed answers by specialists in operator algebras.$\newcommand{\Cst}{{\rm C}^*}$ It arises from some background reading I was doing while working on a different area/problem.

For sake of simplicity I'll restrict to unital separable $\Cst$-algebras, although I think everything can be discussed more generally with appropriate modifications. $\otimes$ will denote the completed minimal tensor product of $\Cst$-algebras.

Definition. A (unital, separable) $\Cst$-algebra $A$ is said to have approximately inner flip if there is a sequence of unitaries $(u_n)\subset A\otimes A$ such that $u_n(x\otimes y)u_n^* \to y\otimes x$ for every $x,y\in A$. It is said to have approximately inner half-flip if $u_n(x\otimes 1_A)u_n^* \to 1_A\otimes x$ for all $x\in A$.

It's known that having a.i. half-flip already imposes some fairly strong restrictions on $A$, such as being simple and nuclear. We also know that ${\mathcal O}_2$ and ${\mathcal O}_\infty$ have a.i. flips.

From a quick look at the 1978 paper of Effros and Rosenberg, I gather that these properties are analogues of von Neumann algebraic properties of the hyperfinite ${\rm II}_1$ factor. What I would like to know is: why is it interesting/important to characterize those $\Cst$-algebras with a.i. (half-)flip? and if we can show that a given simple, nuclear $\Cst$-algebra has a.i. (half-)flip, what further structural consequences does one typically hope to deduce?

I guess part of what I would like to find out is whether we can think of these properties in any of the following ways:

1) "goes unseen by many of the usual invariants"

2) "has some kind of homogeneity not shared by general unital nuclear simple $\Cst$-algebras"

3) "looks like one of several standard examples on a short list".

Of course, if any or all of these three claims are wide of the mark, I'd welcome any clarifications or corrections.

Answer 1

If $A$ is a simple AF algebra, then having an approximately inner flip entails more than that it has unique trace—because $K_0(A \otimes A)$ is order isomorphic to $K_0(A) \otimes_{Z} K_0(A)$, the latter as partially ordered abelian groups—it also says that the infinitesimals are trivial, so in fact, it forces $K_0(A)$ to be a totally ordered subgroup of the rationals, i.e., $A$ is UHF. (The key and perhaps only point is that approximately inner automorphisms induce the identity on taking $K$-groups.) This is drastic, and the result presumably extends to much larger classes of C*-algebras. This was surely in Effros-Rosenberg? (I am too tired to check, and it's fairly late for me.) So maybe (2) and (3) apply?

More generally, if $K_1(A)$ is trivial and UCT applies (whatever that is), then $K_0(A \otimes A)$ is isomorphic to the group tensor product, but I don't know whether anyone has checked that it is a pre-order isomorphism; however, if traces behave well (meaning, are determined by their effects on projections), ai flip should imply at most one trace.

What's an example of an ai half-flip that is not an ai flip?

Edit: It's clear (without using K-theory) that only C*-algebras with zero or one trace can have an ai flip ... .