# Questions tagged [banach-algebras]

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### Solvability and nilpotency for Banach algebras

Do we have topological counterparts of solvability and nilpotency, which are central concepts for (finite-dimensional) Lie algebras, for infinite dimensional Banach algebras with the commutator ...
1 vote
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### Contractive projections on operator algebras

This is a follow up on an earlier question. In [Lau&Loy, 2008] a Banach algebra $\mathcal{U}$ was called to have the Tomiyama property if any contractive projection $P:\mathcal{U}\to \mathcal{U}$, ...
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### Two completely different norms on a unital algebra!

Does there exist any unital normed algebra $(A,\|\cdot\|)$ enjoying another norm $\|\cdot\|_1$ such that $(A,\|\cdot\|_1)$ forms a unital normed algebra with the same unit. Any element contained ...
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### If $A\hat\otimes B$ has identity then so are $A$ and $B$

Let $A$ and $B$ be commutative Banach algebra. I have proven that if $A$ and $B$ have identity $e_A$ and $e_B$ respectivly , then $e_A\hat\otimes e_B$ is identity for $A\hat\otimes B$ (the ...
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### $B(X)$ and $B(X)^{**}$ for superreflexive Banach spaces $X$

Let $A$ be a Banach algebra. It is well-known that $A^{**}$ with either Arens product has separately weak$^{*}$ continuous multiplication if and only if $A$ is Arens regular [Theorem 1, ...
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### Projectional skeletons in dual Banach algebras

A Banach algebra $A$ is a dual Banach algebra if it is a dual Banach space with a (not necessarily unique) predual $A_{\ast}$, and the multiplication on $A$ is separately weak*-continuous. Dual Banach ...
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### Separable subalgebras of non-separable reflexive Banach algebras

Let $A$ be a non-separable reflexive Banach algebra. Every separable subspace of $A$ is contained in a separable 1-complemented subspace [Lindenstrauss,1966]. It is straightforward to show that every ...
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### What are the maximal ideals of $C_0 (X),$ where $X$ is a locally compact Hausdorff space?

Crossposted from MSE How do the maximal ideals of $C_0(X)$ look like where $X$ is a locally compact Hausdorff space? I know that if $X$ is a compact Hausdorff space then the maximal ideals of $C(X)$ ...
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### Trace norm of operators obtained by restricting the matrix of a trace class operator

Suppose $H$ is a Hilbert space with orthonormal basis $\{e_i\}_{i\in \mathbb N}$. To every operator $T$, we associate a infinite matrix $[T_{ij}]$, where $T_{ij}=\left<Te_j,e_i\right>$. We know ...
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1 vote
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### Classical fixed-point argument and invertible function

Let $n\in\mathbb{N}$ and $W^{1,\infty}(\mathbb{R}^n)=\lbrace f:\mathbb{R}^n\rightarrow \mathbb{R}^n : \text{ f is bounded and Lipschitz continuous } \rbrace$. Suppose $f\in W^{1,\infty}(\mathbb{R}^n)$ ...
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### When does a positive operator preserve invertibility

Let $\Omega_1,\Omega_2$ be compact Hausdorff spaces and let $P:C(\Omega_1)\longrightarrow C(\Omega_2)$ be a unital positive operator. I wanted to know if there is a necessary and sufficient condition ...
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### Regarding an element being self adjoint

Let $A$ be a unital C*-algebra. Let $x,y\in A$ be self adjoint elements in $A$, with $x$ being invertible. Can we say that the spectrum of $x^{-1}y$ is a subset of the real line? I know this is true ...
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### Density of normal elements in a C*- algebra [closed]

Let $A$ be a unital C*-algebra. I wanted to know if there is a necessary and sufficient condition for normal elements to be dense in $A$?
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### Gelfand ring in Bourbaki's exercises

In Bourbaki's General Topology, Chapitre III §6 Exercise 11, they define a Gelfand Ring as a topological ring $A$ such that The set $A^*$ ($=A^{-1}$) of invertibles is open. The uniform structure ...
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### Reflexive norm-closed subalgebras of $B(X)$

Let $X$ be a reflexive Banach space, and let $B(X)$ denote the set of all bounded linear operators $X\to X$. Does there exist a norm-closed subalgebra $A\subseteq B(X)$ with the following properties? ...
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### Universality in the class of separable Banach algebras

Let us consider the class of Banach algebras with homomorphisms that are bounded below but not necessarily isometric. Is there a separable Banach algebra that contains isomorphic images of all ...