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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The second isomorphism theorem for C*-Algebras

in my functional analysis class right now we are studying the basics of C* Algebras and I was recently asked this question about the second isomorphism theorem for C* Algebras, but first let me cite ...
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51 views

Compactum of Banach algebra

I need an example of Banach algebra $A$ and a left non-trivial closed ideal $I$ with all of following properties: There exists a bounded approximate identity in $I$ for $I$ i.e., a net ...
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15 views

Spectral radius of an element in a Banach algebra

I want to solve this problem: Let $A$ be an unital Banach algebra and $a\in A$ then \begin{equation} r(a)<1 \Longrightarrow \lim_{n\to \infty} a^n=0 \Longrightarrow (1-a)^{-1}=\sum_{n=0}^\infty ...
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37 views

example of the arens multiplication; I want to understand the construction

I want to understand the double dual as a $C^*$-algebra of a given $C^*-$algebra $A$ but my first problem is to understand the Arens multiplication on the double dual $A^{**}$ (considered as Banach ...
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1answer
60 views

When does the Fourier algebra coincide with the Fourier-Stieltjes algebra?

For a given locally compact group $G$ the Fourier-Stieltjes algebra $B(G)$ is defined as the algebra of matrix coefficients of unitary representations $\pi:G\to B(H)$. Similarly, the Fourier algebra ...
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349 views

Why is $GL(B)$ a Banach Lie Group?

Banach Lie Groups are what you'd expect: https://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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34 views

Two normal operators are similar if and only if they are unitarily similar

I need to prove that in a $C^*$-Algebra two normal operators are similar if and only if they are unitarily similar. Can anybody help, please? One side is obvious, so our concern is the other side. I ...
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23 views

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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11 views

Smallest closed unital subalgebra containing an element has connected resolvent.

Suppose I have $A$ a commutative, unital Banach algebra. Let $C$ be the smallest closed unital subalgebra containing $a\in{}A$ (i.e. the closure of polynomials in $a$). I want to show that the ...
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26 views

Proving that the point spectrum of $T$ is not empty

This problem comes from Rudin's Functional Analysis, Chapter 10. It's the problem 15. Suppose $X$ is a Banach space, $T\in\mathcal{B}(X)$ is compact, and $\|T^n\|\geq 1$ for all $n\geq 1$. ...
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45 views

Rudin functional Analysis chapter $10$, exercise $13$

This is Rudin's functional Analysis chapter $10$, exercise $13$. I am confused about the notation $\sigma_A(f)$, what does that mean?(What role does the subscript $A$ play here). And can someone ...
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Showing the sum of a C* subalgebra and ideal is itself a C* subalgebra

In my functional analysis class I was recently met with this in the context of C* algebras: Let A be a C*-Algebra and B is a C*-subalgebra of A and I an ideal of A. We are asked to show that $ B+I ...
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1answer
14 views

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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7 views

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 ...
3
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1answer
29 views

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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6 views

$L_p$ version of Toeplitz extension

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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2answers
99 views

How does the general spectral theorem generalize the simpler versions?

I read the following version of the spectral theorem in Banach Algebra Techniques in Operator Theory by Douglas: I'm trying to understand why this is a generalization of the following version, ...
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1answer
36 views

Every positive functional achieves its norm on the identity - converse?

Let $X$ be a unital $C^*$ algebra. I can show that if $f$ is a positive linear functional, i.e. $f(xx^*)\geq0$, then $f$ is continuous and achieves its norm at the identity. Is the converse also true ...
3
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1answer
34 views

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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2answers
40 views

Are there finite dimensional $\mathbb{R}$-algebras which are not Banach algebras?

It is known that not every algebra (over a ground field $\mathbb{R}$ or $\mathbb{C}$) is a Banach algebra. It might be a silly question, but are there examples of finite dimensional ($\mathbb{R}$- or ...
3
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1answer
28 views

Every Jordan function $\phi$ on $A$ is multiplicative.

I am reading the following proposition by : FF. Bonsall and J. Duncan, Complete Normed Algebras, pg. 79 Definition: A Jordan function on $A$ is a nonzero linear functional $\phi$ on $A$ such that ...
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16 views

Question on maximal left ideals.

Suppose I have a unital Banach algebra $A$ and a maximal left ideal $L$ and an element $a$ such that $La\subset{}L$. I want to show that there exists (uniqueness is easy) some complex $\lambda$ such ...
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21 views

Differentiability of the norm in connexion with duality map

Let $(X,\|\cdot\|)$ be a Banach space and let $J$ be the duality mapping defined for all $x\in X$ by: $J(x)=\{x^∗∈X^∗\mid ⟨x^∗,x⟩=\|x\|^2=\|x^∗\|^2\}$, where $X^∗$ is the dual space of $X$. I'm ...
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1answer
29 views

projecton and positive element in $C^*$algebras

Let ‎$‎‎A$ ‎be ‎a‎ ‎$‎‎C^*$-algebra.‎$‎‎p\in A$ ‎is a ‎‎projections. ‎‎‎ Assume ‎that ‎‎$‎‎a$ ‎is a element in‎$‎‎ Ball(A_+)$ ‎such ‎that ‎‎$‎‎a‎\leq p‎$‎ Q:May I‎ ‎say ‎‎$‎‎ap=pa$?why?‎ ‎
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calculating the abstract index of $C(T)$

Consider the following definition in operator theory: I'm reading an example of the abstract index of $\mathcal{A}$ in Zhu's An Introduction to Operator Albebras: Here $G_0(\mathcal{A})$ is the ...
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24 views

When is the set $\{ x^* x : x \in A\} $ a cone in a *-algebra?

So in a $C^*$-algebra, every positive element can be written as $x^*x$, and the set of positive elements form a cone. What if we remove all information about the norm? Say $A$ is an algebra, with ...
4
votes
2answers
178 views

Spectrum of idempotent element

Let $A$ be some unitary algebra over $\mathbb{C}$. If $a^2=a$ and $0\ne a\ne 1$ then $\{0,1\}\subset \sigma_A(a)$ ($\sigma_A(a)$ is the spectrum if $a$). I believe that also $\sigma_A(a)\subset ...
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194 views

An abelian Banach algebra without characters

Can one give an example of an abelian Banach algebra with empty character space? Such algebra must be necessarily non-unital. I couldn't find any examples of such algebras. Thanks!
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46 views

Submultiplicative Hilbert space norm on $B(H)$

Let $H$ be a complex Hilbert space and let $B(H)$ denote the space of bounded linear operators $H \to H$ equipped with operator norm: $$ \lVert T \rVert = \sup\big\{ \lVert Tx \rVert \: : \: \lVert x ...
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2answers
52 views

If $X$ is maximal ideal then it consists of non-invertible elements?

I'm reading through a paper where I came across the following theorem Let $A$ be a commutative complex Banach algebra with unit element $e$. Theorem: A subspace $X\subset A$ of codimension $1$ is ...
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2answers
62 views

Show that $\exp(-\lambda x) \cdot\exp(\lambda x)=1$ using the power series

Let $A$ be a commutative Banach algebra. Consider the exponential function $$\exp(\lambda x) = \sum_{n=1}^\infty\frac{(\lambda x)^n}{n!},$$ where $x \in A$ and $\lambda \in \mathbb C$. We can easily ...
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votes
1answer
39 views

Why is the canonical approximate identity positive?

Let $\{u_n\}$ be an approximative identity for C*-algebra A. Why the element $1-u_n$ is also positive? I'm asking about it because in some theorem we use something like this: $\mid \mid a-u_n a \mid ...
1
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1answer
21 views

Why if $X \subset A$ has codimension $1$, the only subspace properly containing $X$ is $A$?

I saw the following question a while ago on Math.SE This question. The answer provided seems to give a satisfactory result, but one thing in the answer I can not quite see. The author of the answer ...
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0answers
12 views

Matrices over tensor product of Banach algebras

Suppose that $A$ is a Banach algebra, or even a $C^*$-algebra, and $B$ is a closed subalgebra of $B(L^p)$ for some $L^p$ space. In particular, $M_n(B)$ has a canonical matrix norm. Is there some ...
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1answer
28 views

Explicit example of Gel'fand transform

I would like to determine explicitly the Gel'fand transform for the (commutative unital) Banach algebra of $2 \times 2$ matrices of the form $$\pmatrix{ a && b \\ 0 && a}, \quad a,b ...
3
votes
1answer
43 views

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 $1\leq p<q<\infty$?
2
votes
2answers
48 views

Functional calculus for unitization of an algebra?

I have been stuck on this problem for a week now: "Let $A$ be a Banach algebra without identity, let $a\in A$, and let $f$ be holomorphic on a neighborhood of $\sigma(a)$, so that $f(a)\in A^\# $ is ...
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1answer
53 views

Gelfand transform of $a \in A$ vanishes at infinity?

Let $A$ be a non-unital commutative Banach algebra defined over the field of complex numbers. Given $a \in A$, one defines the function $\widehat{a}:\Phi_A\to{\mathbb C}$ by ...
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votes
1answer
74 views

Is the algebra of adjointable operators on a Hilbert module prime?

In abstract algebra, a nonzero ring $R$ is a prime ring if for any two elements $a$ and $b$ of R, $arb = 0$ for all $r$ in $R$ implies that either $a = 0$ or $b = 0.$ Or for any two ideals $A$ and ...
2
votes
0answers
26 views

Operator norm of matrix of scalars regarded as matrix with entries in the unitization of a Banach algebra

Let $A$ be a non-unital Banach algebra, and let $A^+=\{(a,z):a\in A,z\in\mathbb{C}\}$ with product $(a,z)(b,w)=(ab+zb+wa,za)$ and norm $||(a,z)||=||a||+|z|$. Equip $M_n(A^+)$ with the operator norm by ...
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Why the set of all bounded operators on Hilbert space is a prime?

In abstract algebra, a nonzero ring $R$ is a prime ring if for any two elements $a$ and $b$ of $R$, $arb=0$ for all $r$ in $R$ implies that either $a=0$ or $b=0$. Or for any two ideals $A$ and $B$ of ...
3
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1answer
37 views

Norm of a character in a non-unital Banach algebra without approximate identity

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 ...
3
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30 views

Tensor product of $L^1([0,1],\omega)$ with some Banach space

We know that for any Banach space $E$ if $\hat{\otimes}$ denotes projective tensor product then $L^1([0,1])\hat{\otimes} E$ is isomorphism isometric with $$L^1([0,1],E)=\{f:[0,1]\to E:\ \ ...
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1answer
30 views

Why are matrix norms defined the way they are?

Given $A$ a square matrix Define: $\|A\|_1$ as the max absolute column sum $\|A\|_2$ as the sum of the squares of each element $\|A\|_\infty$ as the max absolute row sum Pray tell, why are ...
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1answer
27 views

Applying Complex Stone-Weierstrass Theorem for Compact Subsets

The complex Stone-Weierstrauss theorem can be applied to compact spaces. However, say I have an open set $D\subset \mathbb{C}$, and I want to approximate a holomorphic function $f$ on $D$ via ...
3
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1answer
29 views

One-sided identities in Banach algebras

What is an example of a Banach algebra with a left identity but with no right identities? Is there an example of an operator algebra with this property?
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Norms on matrices over Banach algebras

Let $A$ be a unital Banach algebra. Is there a sequence of norms $||\cdot||_n$ such that $(M_n(A),||\cdot||_n)$ is a Banach algebra for each $n$, identity matrices of all sizes have norm 1, the ...
2
votes
1answer
53 views

Why is $f(e)=1$?

Let $A$ be a commutative complex Banach Algebra with unit element $e$. Now, let $f \in A^*$ be a non-zero multiplicative linear functional. Why does it follow, directly from the above, that ...
2
votes
2answers
65 views

Why can entire function be written as exponential, and why is it bounded in this way?

Let $A$ be a commutative complex Banach algebra with unit element $e$. Suppose now that $f(x) \in \sigma(x)$ for every $x \in A$ where $\sigma(x)$ denotes the spectrum of $x$. Now, let $x\in A$ and ...
0
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1answer
18 views

Idempotents with arbitrarily large norms in Banach algebras

While nonzero projections in $C^*$-algebras have norm 1, there is no such restriction for idempotents in Banach algebras. What is an example of a Banach algebra that has idempotents of arbitrarily ...