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Let $A$ be a unital Banach algebra. An element $a \in A$ is hermitian if $\|\mathrm{exp}(ita)\|=1$ for every $t \in \mathbb{R}$. An element $a \in A$ is Moore-Penrose invertible if there exists $b \in …

Let $A$ and $B$ be two $C^*$-algebras, and let $A \otimes B$ denote their minimal tensor product. Given positive, linear functionals $\varphi$ on $A$ and $\psi$ on $B$, we obtain a positive, linear fu …

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 …

Let $A$ be a C*-algebra that has no one-dimensional irreducible representations, that is, there is no (closed) two-sided ideal $I\subseteq A$ such that $A/I\cong\mathbb{C}$. Let $J$ denote the (not ne …

A unital C*-algebra has real rank zero if the self-adjoint elements with finite spectrum are norm-dense in the set of all self-adjoints. If a self-adjoint element has finite spectrum, then it is of th …

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 …

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 …

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 $xx^*\in J_+$ …

The answer to question 1 is `yes': Let $A$ and $B$ be any $C^*$-algebras. Let $N$ be a simple $C^*$-algebra of such high cardinality that it does not embed into either $A$ or $B$. Then take $C:=N\oplu …

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 …

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 …

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 …

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 …

(1) If $A$ is $1$-complemented in its bidual, then $A$ is an AW*-algebra. Indeed, assume that $A$ is $1$-complemented in its bidual, via a contractive projection $p\colon A^{**}\to A$. By a theorem o …

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 $\ …