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The answer is `No'. Example: Let $A$ be any C*-algebra that is not finitely generated as a C*-algebra. For example, let $A=C(X)$ for a compact, metric space $X$ that has infinite covering dimension, …

Akemann has constructed a C*-algebra that does not contain an approximate unit of commuting elements. See Example 2.1 in Akemann. Approximate units and maximal abelian C*-subalgebras. Pacific J. Math …

Let $A$ and $B$ be C*-algebras, let $I\subseteq A$ and $J\subseteq B$ be closed, two-sided ideals, and let $\pi\colon A\to B$ be a bijective, linear map satisfying $\pi(I)=J$. Assume that both the res …

If $A$ is semiprojective (the C*-analog of an absolute neighborhood retract, ANR), then your question also has a positive answer. The class of semiprojective C*-algebras includes also many non-commuta …

Contrary to what you said, the building block of an AI-algebra, namely $C([0,1],M_n)$, can naturally be embedded into the building block of an AT-algebra, namely $C(\mathbb{T},M_n)$, and not conversel …

Note that $a\precsim b$ if and only if for every $\varepsilon>0$ there exists $r\in A$ such that $\|a-rbr^*\|\leq\varepsilon$. Assume that $a\precsim b\precsim c$. To show that $a\precsim c$, let $\ …

It is known that a C*-algebra is finite-dimensional if (and only if) it is reflexive as a Banach space. What is known about the analog of this question for operator algebras? (Here, an operator algebr …

We may consider each $C^*$-algebra as a Banach space (by forgetting the multiplication and adjoint). I wonder how drastic this step is, i.e., which properties of the $C^*$-algebra are reflected by its …

Let $A$ be a C*-algebra. We identify $A$ with its canonical image in the bidual $A^{**}$. Consider the following conditions: (1) $A$ is a von Neumann algebra. (2) There is a multiplicative condition …

Theorem: Let $J\subseteq A$ be a maximal ideal. Then $J$ is hereditary (if $a\in A_+$ satisfies $a\leq b$ for some $b\in J_+$, then $a\in J_+$), strongly invariant (if $x^*x\in J_+$ then $xx^*\in J_+$ …

Let $\varphi\colon A\to B$ be a bounded, linear map between C*-algebras. Is the bitranspose $\varphi^{**}\colon A^{**}\to B^{**}$ continuous when the von Neumann algebras $A^{**}$ and $B^{**}$ are equ …

The real rank for C*-algebras was defined by Brown-Pedersen in [1] as a noncommutative analog of covering dimension. Given a unital C*-algebra $A$, its real rank $\mathrm{rr}(A)$ is the smallest natur …

If $K$ is compact, extremally disconnected and infinite, then for every compact subset $D\subseteq\mathbf{C}$ there exists $a\in C(K)$ with spectrum $\sigma(a)=D$. [edit: Previously, $K$ was only ass …

Given a countable group $G$ with the AP, and an normal, amenable subgroup $N$ of $G$, does $G/N$ have the AP? In particular, does there exist a group $G$ with the AP and a surjective group homomorphi …

Let $M$ be a von Neumann algebra with predual $M_*$, and let $T\colon M\to M$ be a bounded, linear map. Let us say that $T$ is (sequentially) weak*-norm continuous if for every net (sequence) $(a_j)_j …