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The Cuntz algebra $\mathcal{O}_{\infty}$ is the universal $C^*$-algebra generated by countably infinitely many isometries $s_i$ satisfying the relations $s_i^*s_j = \delta_{ij}$ (there is no condition …

Let $A$ be a $C^*$-algebra, let $p$ and $q$ be Murray-von Neumann equivalent projections in $A$, i.e. there is a partial isometry $v$ such that $v^*v = p$ and $vv^* = q$. Suppose $\alpha \in Aut(A)$ …

Let $X,Y \subset \mathbb{R}^n$ be open subsets. Denote by $C^\infty(X)$ the smooth functions on $X$, let $\mathcal{E}'(Y)$ be its dual space considered as a space of distributions. Let $L(C^\infty(X), …

Recently I learned the definition of the quantum permutation group $A_s(n)$, which starts from the fundamental representation by permutation matrices and exchanges the entries by noncommuting operator …

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 …

By the Atiyah-Jänich theorem the K-group $K^0(X)$ for a compact space $X$ may be represented as $[X, U(Q)]$, where $Q = B(H)/K(H)$ is the Calkin algebra and $H$ is a separable infinite dimensional Hil …

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 $C^*$-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 …

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 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 …

I am looking for functorial ways of replacing a commutative $C^*$-algebra $C$ by a simple one, say $A$ , such that the $K$-theory remains unchanged, i.e. $K_*(C) \cong K_*(A)$. I am particularly int …

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^ …

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 …

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 …

There is a nice overview about algebraic quantum field theory by Halverson and Müger, which covers some of the stuff I mention below and can be found at http://arxiv.org/abs/math-ph/0602036 Concerni …