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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 \ …

You are basically asking: When is $p \colon U(\mathcal{H}) \to U(\mathcal{H})/H$ a principal $H$-bundle. Equipped with the norm topology $U(\mathcal{H})$ is a Banach-Lie group. There is a theorem for …

Here is a $C^*$-algebraic version of the model described in Andre Henriques' answer (the latter was linked by David Corfield in the accepted answer above): Let $\mathcal{O}_2$ be the Cuntz algebra g …

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 …

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 …

This is not an answer, but too long for a comment: Even for $C^*$-algebras there is more than one method of stabilization. The critical feature of the compact operators is that $K(H) \otimes K(H) \con …

Let $A$ be a $C^*$-algebra. The group $Aut(A)$ of $\ast$-automorphisms of $A$ is usually equipped either with the pointwise norm topology, i.e. the topology generated by the semi-norms $\lVert \varphi …

$B$ does not have to be unital. Think of the case $A = M(I)$. Then $p =1$ is a reasonable projection in $A \backslash I$. In this case $B= I$. Since a unit $1$ in $B$ has to satisfy $1 = p\cdot 1\cdot …

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 $A$ 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 …

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 …

The CAR-algebra is isomorphic to the UHF-algebra $M_{2^{\infty}}$, i.e. the infinite tensor products of $M_2(\mathbb{C})$. This is explained in Example 1.2.6 in the book Classification of Nuclear, Sim …

What can be said about the following crossed product $C^*$-algebra? Let $A$ be a Kirchberg algebra with $K_0(A) = \mathbb{Q}$ and $K_1(A) = 0$. Consider the direct sum of $n$ copies of $A$, i.e. $B = …

I think the situation you describe is impossible: Let $\bar{\phi}$ be the conjugate endomorphism to $\phi$. From the equation $d(\phi)^2 = 2d(\phi)$ we get $d(\phi) = 2$. Denote by $\langle \rho, \sig …