# Tagged Questions

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### Prove that a norm makes a space Banach

I have to prove that if $A$ is a C*-Algebra then the algebra $A_1$ obtained adjoining the identity is a C*Algebra too (with the usual algebraic operation defined). I have any problem in all the ...
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### The space of all bounded sequences over a Banach Algebra.

If $A$ is a commutative Banach algebra, can we identify $l^\infty(A)$ as the dual of $l_1\hat\otimes A$ (the projective tensor product of $l_1$ with $A$? Also, what do we know about the quotient ...
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### SOT Convergence and Compact Convergence

Let $E$ be a Banach space, and let $A(E)$ denote the closure of the finite rank opertors on $E$. Let $(S_\alpha)$ be a bounded net of operators on $E$ such that $S_\alpha T\to T$ for all $T\in A(E)$. ...
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### Centralizers and containment of $c_0$

Let $X$ be a Banach space over $\mathbb{R}$ or $\mathbb{C}$. By a multiplier on $X$ we mean a bounded linear operator $T$ on $X$ such that every extreme point of $B_{X^*}$ becomes an eigenvector for ...
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### Spectral Radius given by Neumann Series

Good afternoon everybody. So far it is clear that outside the spectral "disk" the resolvent is given by the Neumann Series. But why does it tell us that there is a point on the spectral "circle" where ...
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### Injective norm on tensor algebra of a finite-dimensional Banach space

Suppose that $V$ is a finite n-dimensional Banach space, and suppose that $T(V)$ is the tensor algebra on it. Furthermore, suppose that $T^{(n)}(V)$ is the "abridged" tensor algebra obtained by taking ...
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### what is a “Banach algebra” without the norm condition on a continuous multiplication?

I wish to use the following finite-dimensional Banach spaces. although they do not need to be Banach algebras for my proposed application, a mild curiosity is aroused, because the multiplication ...
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### Turning Banach space into Banach algebra

Given a Banach space, how can we determine if we can turn it into a Banach algebra or not?
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### When is the orbit of a vector a minimal sequence? When does an operator have a minimal orbit vector?

Let $X$ be a banach space. We say a sequence $(x_n)$ is minimal if for each $k$, $x_k\notin [x_n]_{n\neq k}$, where $[x,y,z,\cdots]$ is the closed linear span of the vectors. For ...
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### How to Prove ($\mathbb{C}\langle x, y \rangle$, $\|\cdot\|$) is a Banach Space

Let $\mathbb{C}\langle x,y\rangle$ be the group ring of the complex numbers over the free group in $x,y$. Let $len : \langle x,y \rangle \rightarrow \mathbb{N}$ denote the standard word norm and let ...
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### A subset of $\bar{S}\backslash S$ contains an open ball in $\bar{S}$? (operator theory)

E and S are subsets of a metric space. $E$ is a subset of $\bar{S}\backslash S$. Then $\overline{E}\subset(\overline{S}\backslash S^{o})$, but I wonder whether there is some condition that guarantees ...
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### Why is $GL(B)$ a Banach Lie Group?

Banach Lie Groups are what you'd expect: http://www.encyclopediaofmath.org/index.php/Lie_group,_Banach If $B$ is a Banach algebra then why is $GL(B)$, the set of invertible elements of $B$, a ...
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### Closure of the invertible operators on a Banach space

Let $E$ be a Banach space, $\mathcal B(E)$ the Banach space of linear bounded operators and $\mathcal I$ the set of all invertible linear bounded operators from $E$ to $E$. We know that $\mathcal I$ ...
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### Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\}$

Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\}$. Where $A,\ B$ are bounded operators in Banach space and $\sigma$ denotes spectrum.
Given a non-abelian C*-algebra $A$. I am wondering what are the possible abelian sub-C*-algebras of $A$. Let $K$ be the spectrum of $A$. Does $A$ contain an isomorphic copy (as a Banach space) of the ...
Let $E$ be a Banach Algebra with identity, and $v\in E$, so that $||v|| < 1$. The geometric series $w = \sum_{k=0}^\infty v^k$ converges in the norm. I can show that \$||w|| \le ...