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The group you describe should be the infinite symmetric group $S_{\infty}$. The $K$-theory of its $C^*$-algebra has been determined by Kerov and Vershik in The K -functor (Grothendieck group) of the i …

Here are just some trivial observations that came to mind after thinking about this a little longer: You are essentially asking for a canonical class in the $K$-homology group $K^0(B) = KK(B,\mathbb{C …

The unitary exists by the definition of the crossed product. In the case of a $\mathbb{Z}/n\mathbb{Z}$-action you can think of an element of the crossed product as a linear combination $$ \sum_{n = 0} …

This is not necessarily an answer, but it was too long for a comment: Note that for any separable unital $C^*$-algebra $A$ its suspension $SA := C_0(\mathbb{R}) \otimes A$ is quasidiagonal. This can …

This is not really an answer, but an idea. I am not sure if it works, plus you need an extra assumption on the completely positive map. Definition: We call a completely positive map $\psi \colon A \ …

The fact stated in the answer by vap is proven in the paper "Multipliers of C*-algebras" by Akemann, Pedersen and Tomiyama (see Theorem 3.3, I guess). Moreover, they prove in Theorem 3.8 that multipli …

Following Yemon Choi's suggestion I turn my comment into an answer: Lance gave a characterization of amenability in terms of the reduced group $C^*$-algebra: A discrete group $G$ is amenable if and o …

Let $D$ be an infinite UHF algebra, e.g. the infinite tensor product of the matrix algebra $M_k(\mathbb{C})$. The permutation group $\Sigma_n$ acts on the $n$-fold tensor product $D^{\otimes n}$ in a …

As David Handelman already pointed out, by continuity of $K$-theory we obtain that $K_0(C(X)) \cong \bigoplus_{\mathbb{N}} \mathbb{Z}$ and $K_1(C(X)) = 0$. Since $C(X)$ is commutative, it lies in the …

This is by no means a full answer, but Kishimoto has shown in Theorem 4.1 of his paper "Universally weakly inner one-parameter automorphism groups" that for an automorphism $\alpha$ of a separable $C^ …

Let $A$ be a UHF-algebra of type $n^{\infty}$ and denote its unique and faithful trace by $\tau$. Let $L^2(A)$ be the Hilbert space of the GNS-representation associated to $\tau$. We have two commutin …

If you restrict to automorphisms of a $C^*$-algebra $A$ instead of endomorphisms $f \colon A \to B$, then I think what you suspect is true due to a paper by Kadison and Ringrose called "Derivations an …

There is a book by Herbert Schröder called "K-Theory for Real C*-algebras and Applications", which can be found on mathscinet and has a Google Books entry (sadly without preview).

To answer your first question, I think the "usual norm" is the one described on page 63 of Rieffel's paper, i.e. $\lVert z \rVert = \lVert \langle z,z\rangle\rVert^{1/2}$ for $z \in X \otimes Y$. Note …

Let $M_{\mathbb{Q}}$ be the universal UHF-algebra and let $\mathcal{O}_{\infty}$ be the infinite Cuntz algebra. Let $A$ be a Kirchberg algebra that satisfies the UCT with $K_0(A) \cong \mathbb{Q}^n$ a …