# 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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### Finite dimensional Banach algebras whose $K_{0}$ group is a non trivial finite group

Motivated by this question we ask Is there a finite dimensional Banach algebra $A$ such that $K_{0}(A)$ is a nontrivial finite group? I understand from the above link and this post that any ...
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### Examples of a Banach space with an algebra structure having only left continuity

There is a theorem (see for example, Rudin's Functional Analysis, theorem 10.2 ) that if $A$ is a Banach space with an algebra structure, such that both left and right multiplication are continuous, ...
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### trace norm and tensor product

Let $(M_n (\mathbb{C}), n\|.\|)$ , $(M_n (\mathbb{C}), n\|.\|)$ and $(M_{nm} (\mathbb{C}), nm\|.\|)$ be three Banach algebras. where $$\|A\| = \mathrm{tr}\sqrt{(A^* A)}.$$ What is the norm of $\phi$ ...
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### Application of Stone-Weierstrass with a non-unital algebra

Let $X$ be a locally compact Hausdorff space. We say that a function $f\colon X \to \mathbb{R}$ vanishes at $\infty$ if for each $\epsilon >0$ there exists a compact $K_\epsilon \subset X$ such ...
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### Algebra (Not *)-Isomorphisms of von Neumann algebras

Let $A$ and $B$ be any two infinite-dimensional von Neumann algebras, they are operator algebras with operator composition as the multiplication and as infinite dimensional vector spaces they're ...
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### Involutive and C* Banach Algebras.

I want to prove the next theorem: If $\pi: A \rightarrow B$ is a star homomorphism, meaning it's an algebra homomorphism which also satisfies: $\pi(x^*)=(\pi(x))^*$, where $A$ is an involutive Banach ...
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### Let $A$ be a Banach algebra. Suppose that the spectrum of $x\in A$ is not connected. Prove that $A$ contains a nontrivial idempotent $z$.

While trying to solve the exercise below, I came up with a wrong conclusion, but I can't see why it's wrong. Also I'm accepting suggestions to get the right solution. This is the problem 17 from ...
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### Algebra of compact operators on $\ell_p$

Are the algebras of compact operators $K(\ell_p)$ and $K(\ell_q)$ isomorphic as Banach algebras for $1\leq p<q<\infty$?
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I've been into this for days and days and I still can't see why, given the definition of $L^\ast$ as $L^\ast =(L(a^\ast))^\ast$ we get that $(LM)^\ast =L^\ast M^\ast$. Where is my mistake: $$(LM)^\... 1answer 350 views ### Reflexive Banach algebras? I have been reading Gelfand theory for a while and it just occurs to me that the whole theory is an analogy to what we did for Banach spaces. For a Banach space X, we investigate its dual X' ... 1answer 132 views ### Do non-commutative algebras with dense commutative subalgebras exist? Let A be a normed unital algebra. Suppose that C\subseteq A is a commutative subalgebra which is dense in A. I ask myself the following question: Under the above assumptions, is A necessarily ... 1answer 31 views ### Algebra A and its Gelfand spectrum Let A be the set of all function f on \mathbb{R} of the form$$ f(x)=d+\int\limits_{0}^{\infty}e^{ixt}k(t)dt,\qquad\quad x\in\mathbb{R},  where $d\in\mathbb{C}$ and $k\in L_1([0,\infty])$. The ...
As is shown here, the norm of a character in a non-unital Banach algebra with an approximate identity is $1$. I wonder if this result still holds for general non-unital Banach algebras. Let $A$ be ...
The following is a theorem about self-adjoint subalgebra of $C(X)$ where $X$ is compact Hausdorff and the first half of its proof: Here are my questions: Why \$\Gamma:\mathfrak{U}\to C(M_{\...