# Questions tagged [banach-algebras]

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### Does a Banach algebra version of "the sum of a closed subspace and a finite dimensional subspace is always closed" exist?

In the setting of Banach spaces, it is well known that if $M$ is a closed subspace of a Banach space $X$ and $F$ is a finite dimensional subspace of $X$, then $M+F$ is closed. Does a Banach algebra ...
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### Can the Banach algebra structure on $B(E)$ be (almost) retrieved from its Banach space structure?

This is basically just out of curiosity. Also, since my research area is in von Neumann algebras and my knowledge of general Banach algebras as well as general Banach spaces is somewhat limited, I ...
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### When is the "Gelfand Remainder" compact?

Suppose we have a noncompact Hausdorff space $S$ and a Banach algebra $A \subset C^*(S,\mathbb R)$ of the space of real-valued bounded functions on $S$. For niceness let's assume $A$ separates the ...
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### Spectrum of ring in algebraic geometry vs spectrum of Banach algebra

For a commutative unital Banach algebra $A,$ and $x\in A,$ we have $\lambda \in \sigma_A(x)$ if and only if $\phi(x) = \lambda$ for some algebra homomorphism $\phi:A \to \mathbb C.$ The set of all ...
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### Does integration by parts formula hold in $H^1(0,T,L^2(\Omega))$?

Let $\Omega$ be an open set from $\mathbb{R}^N$. How can we prove that if $u,v\in H^1(0,T,L^2(\Omega))$ (in Bochner sense) then $(u\cdot v)'\in L^2(0,T,L^1(\Omega))$ with $(uv)'=u'v+v'u$ and the ...
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### Amenability of $\textrm{w}_0(A)$ for a $C^*$-algebra $A$

Let $A$ be a $C^*$-algebra with only finite dimensional irreducible representations. As in a previous question, let $\textrm{w}_0(A)$ denote the subspace of $\ell^{\infty}(A)$ consisting of all weakly ...
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### Complemented C*-algebras

Let $A$ and $B$ be unital separable commutative $C^*$ algebras, with $A\subset B$. Is it true that $A$ is complemented in $B$?
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### Non-degenerate representation of a Banach algebra

Let $\mathcal{A}$ be a non-reflexive Banach algebra. For the definition of Arens product, please refer to this link. Here we let $\square$ denote the first Arens product and $\diamond$ denote the ...
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### What is the socle of the $2\times 2$ matrix algebra over a Banach algebra?

$\DeclareMathOperator\soc{soc}$Let $\mathcal{A}$ be a unital semisimple Banach algebra. The socle of $\mathcal{A}$, $\soc(\mathcal{A})$ is defined as the sum of the minimal right ideals (which equals ...
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### Why do von Neumann algebras possess identity?

My starting point is that a von Neumann algebra is a $C^*$-algebra with a predual. The usual approaches to showing the existence of identity involve spectral theory (for approximate identity), but ...
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### The group of quasi unitary elements of a (simple) Banach algebra

For a Banach algebra $A$ with invertible group $G(A)$ we define the following group: $$QG(A)=\{u\in G(A)\mid \;\text{the mapping}\; a\mapsto u^{-1} a u \;\text{is an isometry}\}$$ What is an ...
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Let $A$ and $B$ be two semisimple commutative Banach algebras. Assume that $A\mathbin{\tilde\otimes} B$ is a Banach algebra obtained by completing $A\otimes B$ with respect to a cross norm not ...