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A unital $C^{*}$ algebra is called a "Path connected algebra" if the spectrum of all its elements is a path connected subset of $\mathbb{C}$. What is an example of a non commutative path connecte …

Edit: According to the Gelfand duality between topological spaces and commutative $C^{*}$algebras, I add some new tags. So the question is that what is the structure of $ Ext (A,A)$ where $A$ is …

The category of $C^{*}$ algebras is denoted by $\mathcal{A}$. Is there a functor $\mathcal{F}$ on $\mathcal{A}$ which send each object $A\in \mathcal{A}$ to its center $Z(A)$. In the other words, ca …

Assume that $\frac{1}{p}+\frac{1}{q}=1$ for two positive real numbers $p,q$. For what kind of $C^*$ algebras $A$ does the following hold: $$\frac{ab+ba}{2}\leq \frac{a^p}{p}+\frac{b^q}{q}$$ $\ …

Under what conditions on a $C^*$ algebra $A$ we have the following inequality: $$x^*a^*ax+a^*x^*xa\leq x^*x+a^*x^*ax+x^*a^*xa\;\;\; \forall x,a\in A$$ The second identity which I am looking for is t …

Is there a non nuclear $C^*$ algebra $A$ for which the minimum and maximum $C^*$ norms on $A\otimes A$ coincide? A somewhat similar question is discussed here.

Edit: According to answer and comments by Prof. Valette we edite the question. The Kadison Kaplansky conjecture says: Kadison-Kaplansky conjecture: If $G$ is a torsion-free discrete group then $C^*_{\ …

Let $A$ be a Banach algebra with the following property: For every two nets $ x_{\alpha}$ and $y_{\alpha}$ in $A$, $x_{\alpha}y_{\alpha}$ converges if and only if $y_{\alpha}x_{\alpha}$ converges. …

Let $A$ be a unital $C^*$-algebra which is equipped with a faithful trace $T$. In particular we may consider $A=C^*_{\text{red}} (G)$ for some discrete group $G$. We consider the following differentia …

Apart from $M_2(\mathbb{C})$. what is an example of a $C^*$ algebra $A$ whose set of non trivial projections form a non empty compact connected set? Is there an example of this situation such that …

Edit: According to the valuable comment of Yemon Choi I revise the question by replacing "faithful" with "irreducible". We say that a $C^*$ algebra $A$ satisfies the invariant subspace p …

Edit: According to comment of Pace Nielsen, I remove question 2 of the previous version: Let $R$ be a unital ring. We define Murray Von Neumann relation $M$ on $R$ as follows: We say $a M b$ iff …

Let x be a positive element in the spatial tensor product of two non unital C* algebras A and B. Is there a single element $a \otimes b \geq x$? How can we noncommutativize the following proof, in …

Assume that $A$ is a $C^{*}$ algebra with self adjoint elements $A_{sa}$. Assume that for all $a,b\in A$ we have $$ab\in A_{sa} \iff ba \in A_{sa}$$ Is $A$ necessarily a commutative algeb …

Let $H$ be the quaternions algebra. An $H^{*}$ algebra is a normed ring $A$ which is simultaneously a unital left $H$ module and has an involution $*$ with the following properties: $\forall \lambda …