# Questions tagged [banach-algebras]

A Banach algebra is an algebra over the real or complex numbers which is equipped with a complete norm such that |xy| ≤ |x||y|. The study of Banach algebras is a major topic in functional analysis. If you are about to ask a question on C*-algebras or von Neumann algebras please use (c-star-algebras) or (von-neumann-algebras) instead (or in addition). Further related tags: (operator-algebras), (operator-theory), (banach-spaces), (hilbert-spaces).

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### What is odd element of C*-algebra? [closed]

I cannot nowhere find a definition of an odd/even element of C*-algebra. Can someone write it here?
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### Continuity of product in Banach Algebras

EDIT: The definition literally says that the product is bilinear, as Chris Eagle kindly pointed out. Turns out that my original proof was fine. This question does no longer require an answer and ...
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### Understanding the proof of Fuglede's theorem

I am trying to understand this proof of Fuglede's theorem on wikipedia : Everything was making sense until the last sentence: "Considering the first-order terms in the expansion for small λ, we ...
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### Banach space with continuous multiplication is a Banach algebra

Consider a Banach space $(\mathcal{A},\|\cdot\|_{\mathcal{A}}$) and assume that $\mathcal{A}$ is also an associative algebra with a unit element $I\in \mathcal{A}$ such that $\|I\|_{\mathcal{A}}=1$. ...
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### Predual of disc algebra and $W(\mathbb{T})\neq A(\mathbb{D})$

I want prove that the Wiener algebra, $W(\mathbb{T})$, does not coincide with the disc algebra $A(\mathbb{D})$. I know that there are some ways to do this. For example, one smart way is use Rudin-...
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### If $\Phi$ is injective, $X_1$ normal and Hausdorff and $X_2$ compact, then $F$ is surjective

Let $X_1,X_2$ be topological spaces and $F\in \mathcal{C}(X_2,X_1)$. Let $\Phi:\mathcal{C}_b(X_1,\mathbb{C})\to\mathcal{C}_b(X_2,\mathbb{C}): \Phi(f):=f\circ F$. Then $\Phi$ is a continuous ...
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### ideals, projections and factors in VN algebras

I'm trying to solve some problems on Von Neumann algebra and I got the questions below. $Q1$. By definition we know a factor is a Von Neumann algebra with trivial center, i.e. a center consisting only ...
### Spectrum $\sigma(a)$ taken within the unitisation of a $C^*$-algebra
My question relates to reconciling the definition of the spectrum $\sigma(a)$ of a point $a$ in a $C^*$-algebra $A$ in both the unital case and the more general case using the unitisation, since I ...
### Proving that if $C_0(X)$ is unital, then $X$ is compact
Apologies if this is simple, I'm having one of those days. Let $X$ be a locally compact Hausdorff space and $C_0(X)$ be the commutative $C^*$-algebra of continuous complex-valued functions on $X$ that ...