# Tagged Questions

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) ...

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### Let $e_0$ be minimal projection, why $E_{e_0}$ is a Hilbert space?

‎A nonzero element $e \in \mathcal{K}(\mathcal{H})$ is called a projection‎, ‎if it is‎ ‎self adjoint and idempotent‎. ‎In addition‎, ‎if $e\mathcal{K}(\mathcal{H})e = \mathbb{C}e$ then‎, ‎it is ...
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### Decomposition of spectrum

We know that if X is a banach space and T be an element in Banach Algebra B(X) then the union of residual spectrum continuous spectrum and point spectrum is spectrum of Banach algebra i.e σ(T) is the ...
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### Let $A$ be a unital Banach algebra and $J$ maximal ideal of $A$. Why is $\overline{J}$ an ideal of $A$?

Let $A$ be a unital Banach algebra with maximal ideal $J$. Why is it true that $\overline{J}$ is also an ideal of $A$, i.e. the closure of a maximal ideal of $A$ is an ideal of $A$?
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### Why does $a(1_A-x)=(1_A-x)a=1_A$

I am working through the following theorem and proof, but I struggle to understand the last part. Theorem: Let $A$ be a Banach algebra with unit element $e$, $x \in A$ and $||x||< 1$. Then ...
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### Projective tensor product continuous?

For $V$, $W$ Banach spaces, is the canonical morphism $B(V) \times_\pi B(W) \to B(V \hat{\otimes}_\pi W)$ continuous, where $B(V):=B(V,V)$ are linear bounded endomorphisms with operator norm? If so, ...
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### A complex unital algebra which is a Banach space is also a Banach algebra

In my functional analysis class I got stuck on this question on Banach algebras: Let $\mathbb{A}$ be a complex unital algebra (it has a unit i.e. a multiplicative identity) and $\mathbb{A}$ is ...
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### Tensor product of algebras of operators on $SQ_p$ spaces

I am aware that there is a canonical tensor product for $L_p$ spaces that allows one to talk about tensor products of algebras of bounded linear operators on $L_p$ spaces. Is there an analogous notion ...
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### Gelfand transform on functions

The Gelfand transformation identifies function spaces $C_0(X)$ for locally compact Haussdorff $X$ with commutative $C^*$ Algebras. Additionally there is a statement that if $f: X \to Y$ is a proper ...
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### Toeplitz operators on $\ell_p$ modulo compact operators

There is the well-known extension of $C^*$-algebras $0\rightarrow K\rightarrow\mathcal{T}\rightarrow C(S^1)\rightarrow 0$ where $\mathcal{T}$ is the Toeplitz algebra (generated by the unilateral shift ...
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### Is the hermitian condition a must?

For a hermitian element $a$ in a $C^*$-algebra, show that $\|a^{2n}\| = \|a\|^{2n}$ In this post, I saw a comment stating that "More generally, if $a$ is normal then $∥a^n∥=∥a∥^n$ for each positive ...
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