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What can be said about the tensor product $T\otimes S$ of two Fredholm operators $T:X_1\to Y_1$ and $S:X_2 \to Y_2$ where $X_1,X_2,Y_1, Y_2$ are Banach spaces and tensor product of operat …

In this question the disc algebra $\mathcal{A}(\mathbb{D})$ is the Banach algebra of all holomorphic functions on the unit open disc $\mathbb{D} \subset \mathbb{C}$ which have a continuou …

Let $A$ be a $C^*$ algebra. Assume that the spectrum $Sp(a_1a_2\ldots a_{n-1}a_n)$ is unchanged as a set after a permutation of $a_i$'s. (unless possible emerge or removing 0 from the spectrum) Does …

Inspired by this question About noncommutative sphere and inspired by the fact that the classical sphere is the one point compactificatiin of $\mathbb{R}^2$ we ask the question below: Is the non com …

Is there a commutative subalgebra $A\subset B(H)$ containing the 1 dimensional scalars with the following property: The algebra $A$ has trivial intersection with the set of commutator element …

Inspired by Pitt's theorem and this post we ask the following question: First we remind the Pitt's theorem and also introduce a particular Banach space as averaging of $\ell^p$ spaces for $1\leq p …

Let $H$ be a seperable complex Hilbert space. What is the closure of the set of all self adjoint operators in $B(H)$ whose spectrum is a subset of the rational number $\mathbb{Q}$. Apart from finite d …

I have already asked this question on MSE; now I repeat it on MO. https://math.stackexchange.com/questions/4132346/on-which-subspace-w-subset-c-infty0-1-is-df-xfx-a-bounded-operator First we introd …

Let $H$ be a Hilbert space. A vector subspace $W\subset B(H)$ is called a Fredholm subspace if there is an upper bound for the absolute value of Fredholm index of all Fredholm operators $T$ in $W$. …

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 terminology for the following property of $C^*$ algebra $A$: For every $C^*$ algebra $B$ and surjective $C^*$ morphism $\phi: B\to A$, every normal element $y\in A$ admits a normal ele …

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 …

An element $a$ of a Lie algebra $L$ is called a Fredholm element if the adjoint operator $\mathrm{ad}_a:L \to L$ is a Fredholm linear map. That is: its kernel is a finite-dimensional space and its ran …

Let $F\subset B(H)$ be a finite dimensional subvector space of the space of all bounded operators on a Hilbert space. Question: Is there an upper bound for $$\{|tr(T)| \text{where} \quad T\in …

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 …