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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5
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386 views

application of c*algebras to PDEs

I am preparing an introductory talk about c* algebras and I'd like to motivate C*algebras or show an application of them and I'd prefer an application in the field of partial differential equations. ...
4
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1answer
32 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 ...
4
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0answers
25 views

Commutative Banach algebra and its Gelfand spectrum

Let $A$ be the set of all functions on $\mathbb{R}^2$ of the form $$ f(t,s):=\sum_{m=-\infty}^{\infty}\sum_{n=-\infty}^{\infty}{a_{mn}e^{i(mt+ns)}}, $$ with the following norm: $$ \|f\|:=\sum_{m=-\...
1
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0answers
19 views

noncommutative algebra

If we let $B$ be a noncommutative Banach algebra with unit $e$, then, obviously, $xy\not=yx$, but are they related? I suspect that the spectral radii of $xy$ and $yx$ are the same, but I couldn't ...
2
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1answer
14 views

Commutative Banach algebra and its maximal ideal space

Let $A:=C^{(n)}([0,1])$ be the set consisting of the n-times continuously differentiable complex-valued functions. Consider $A$ with the norm $$ \|f\|:=\max\limits_{0 \leq t \leq 1} \sum_{k=0}^{n}{\...
0
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1answer
14 views

Show that the Gelfand transform is a morphism?

Let $A$ be a commutative Banach algebra and let $x \in A$. We define the Gelfan transform of $x$ by $$\hat{x} (\chi)= \chi (x)$$ where $\chi$ is a nonzero multiplicative linear functional on $A$. I ...
2
votes
0answers
27 views

About the Banach algebra $\ell^{\infty} (K( \ell^{1}))$

If $P_{n}: \ell^{1} \rightarrow \ell^{1}$ is the projection onto the first n coordinates, then it's well known that $ P_{n} K(\ell^{1}) P_{n}$ is isomorphic to $B(\ell^{1}_{n})$, the Banach space of ...
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0answers
18 views

R.Douglas “Banach Algebra Technique Operator Theory” - Chapter 2 issue

Just before 2.37 Corollary (Spectral Mapping Theorem) Douglas says: If $\varphi (z)= \sum_{n=0}^\infty a_nz^n$ is an entire function with complex coefficients and $f$ is an element of the Banach ...
0
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0answers
14 views

If $L(E)$ is a prime C$^*$-algebra?

We say that a C$^*$-algebra is prime if and only if whenever $I$ and $J$ are closed two-sided ideals in $A$ and $I ∩ J = {0}$, then either $I = {0}~ or~ J = {0}$. Let $L(E)$ be the set of all ...
6
votes
2answers
62 views

Correspondence between maximal ideals and multiplicative functionals of a non unital, commutative Banach algebra.

Let $\mathcal{A}$ be a non (necessarily) unital commutative Banach algebra, and let $$ M_{\mathcal{A}} = \{ \phi:\mathcal{A} \to \mathbb{C} : \phi \mbox{ is multiplicative and not trivial}\} $$ and $$...
2
votes
1answer
49 views

Why are two characters of a commutative Banach algebra with the same kernel equal?

Let $A$ be a commutative Banach algebra. Let $\chi_1$ and $\chi_2$ be characters of $A$. I am having some difficulty seeing why the following statement is true: If $\ker \chi_1 = \ker\chi_2$, then ...
1
vote
1answer
14 views

Spectral radius of an invertible element in a Banach algebra.

Let $A$ be a commutative Banach Algebra. Suppose $a\in A$ is invertible and $r(a)=\|a\|$, then is $r(a^{-1})=\|a^{-1}\|$, where $r$ denotes the spectral radius?
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1answer
18 views

Is the range of the Gelfand transform closed?

Let $A$ be a commutative unital Banach algebra. Consider the Gelfand map $\Gamma:A\longrightarrow C(M_A)$, $\Gamma(a)=\hat{a}$, where $M_A$ is the Character space of $A$. Is the image of the Gelfand ...
1
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1answer
16 views

Uses of spectral radius equal to norm.

Let $A$ be a unital commutative Banach algebra.What can be the consequences of the spectral radius of an element $a$ equaling its norm $\|a\|$.
6
votes
1answer
53 views

If B(X) is isomorphic to B(Y), does that mean X is isomorphic to Y (for X and Y Banach spaces)?

Let $X$ and $Y$ be Banach spaces such that $\mathcal{B}(X)$ is linearly isomorphic to $\mathcal{B}(Y)$ (where $\mathcal{B}(\cdot)$ denotes the algebra of bounded linear operators). Must it always be ...
0
votes
2answers
43 views

What is the maximal ideal space of $H^\infty$?

What is the spectrum of $H^\infty$, the Banach algebra of all bounded holomorphic functions in the open unit disk $D=\{z\in \mathbb{C}\mid |z| <1 \} $?
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12 views

Does tensor product with $L_p$ operator algebra preserve exact sequences?

By $L_p$ operator algebra I mean a closed subalgebra of the algebra of bounded linear operators on some $L_p$ space where $p\in(1,\infty)$. There is a notion of tensor product of $L_p$ spaces (as ...
0
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2answers
48 views

Can you give an examples of non commutative non C*algebras?

Are there examples apart from $B(X)$ where $X$ is not a Hilbert space and not finite dimensional. Do they have a characterization or representation?
1
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1answer
89 views

Understanding a proof from Conway: showing existence of idempotents using functional calculus

I am studying the spectral theory of operators on Banach and Hilbert spaces, making use of Conway's A Course in Functional Analysis. In section VII.4, Conway states Proposition 4.11 as a consequence ...
0
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1answer
42 views

Introduction to subfactor theory

I have almost no knowledge about subfactor theory but I would like to understand what it is. As a self-learner, I do not know where to start. Could you suggest introductory text/paper/book to study ...
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votes
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28 views

Stone-Čech compactification of a completely regular topological space

Let $X$ be a completely regular space. show that $X$ is homeomorphic to a dense subset of the commutative $C^*$-algebra $C^b(X)$, and that every function in $C^b(X)$ extends to a continuous function ...
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0answers
13 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 ...
0
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1answer
27 views

Motivation for condition of Normed Algebra

Is there any particular motivation for the defining Normed Algebra with the condition $\|xy\| \leq \|x\|\|y\|$. Is there any Geometrical view of this condition?
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1answer
44 views

What are all the Banach algebras where $\|a\|\|a^{-1}\|=1$? [closed]

Is there a characterization for all Banach algebras such that $\|a\|\|a^{-1}\|=1$ whenever $a$ is invertible?
0
votes
1answer
33 views

Dirac functional embedding

I got the following statements to show. Let $S \neq \emptyset$ equiped with the discrete topology and let $\ell_\infty(S) = \{f: S \to \mathbb C \mid f \text{ bounded}\}$. Not $\ell_\infty(S)$ with ...
1
vote
1answer
41 views

Norm of a unital homomorphism

Let $\mathcal{A}_1$ and $\mathcal{A}_2$ be two unital $C^*$-algebras and $\varphi : \mathcal{A}_1 \to \mathcal{A}_2$ a unital $*$-homomorphism, i.e. a linear map such that $\varphi(xy)=\varphi(x)\...
2
votes
1answer
33 views

A certain limit of a state is zero

Let A be a C$^*$-algebra, $a\in A$ strictly positive (this means: for every state $\varphi$ of A is $\varphi(a)>0$). Let $u_n=(\frac{1}{n}+a)^{-1}$. Then for all $b\in A$ and all states $\varphi$ ...
0
votes
0answers
25 views

Ideal generated by a set of singular elements in a Banach algebra.

Let $A$ be a commutative unital Banach Algebra. Suppose for every $a\in A$, $\|a\|=1$, I get a singular element $b_a$. I know that each such $b_a$ is contained in a proper maximal ideal of $A$. Is it ...
0
votes
1answer
17 views

Relationship between spectral rays commuting

Show sing binomial Newton that if $S$ and $T$ are continuous operators on Banach space $B$ such that $ST = TS$ then $$r_\sigma(T + S) \leq r_\sigma(T) + r_\sigma(S)$$ where $r_\sigma$ is spectral ...
4
votes
1answer
26 views

Proof that group of invertible elements in a Banach algebra have 1 or infinite connected components?

I'm trying to reconcile this proof that I've read that a group of invertible elements in a commutative (complex) Banach algebra have 1 or infinite connected components with this example I'm looking at....
1
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1answer
40 views

Exercise 16 - chapter 11 - From Rudin's Functional Analysis

I'm trying to solve this problem, which comes from the book mentioned in the title. Suppose $A$ is a Banach algebra, $m$ is an integer, $m\geq2$, $K<\infty$, and $$\|x\|^m \leq K\|x^m\| $$ ...
0
votes
0answers
9 views

Approximate unit for an ideal of a C* algebra.

Suppose I have a C* algebra $A$ and an ideal $J$ with approximate unit $\{e_{\lambda}\}$. Let $x\rightarrow{}q(x)$ denote the projection onto $A/J$. I want to show that $\lim_{\lambda}||x-xe_{\lambda}...
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0answers
15 views

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 ...
2
votes
1answer
36 views

Definition of C$^\ast$-algebra: which conditions can be deduced from the others?

A C$^\ast$-algebra is a Banach algebra $A$ with an involution, i.e. a map $\ast$ such that: $(x^\ast)^\ast=x$ for all $x\in A$; $(x+y)^\ast=x^\ast+y^\ast$ for all $x,y\in A$; $(ax)^\ast=\overline ax^...
0
votes
2answers
19 views

Are compact operators trace class operators?

We say that $A\in B(\mathcal{H})$ is a trace class operator, if $\sum_{i\in I}\langle|A|e_i,e_i\rangle<\infty$,$\hspace{0.1cm}$ such that {$e_i; i\in I$} is a orthonormal bass for Hilbert space $\...
0
votes
0answers
34 views

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 ...
2
votes
0answers
17 views

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 ...
2
votes
1answer
29 views

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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0answers
34 views

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, ...
1
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0answers
111 views

Are there bounded nonconvergent sequences satisfying this recurrence relation?

In this question, we ask about convergent sequences $(p_n)$ satisfying $p_{i} = 2p_{i+6} - {1 \over 2}p_{i+1} - {1 \over 4}p_{i+3} - {1 \over 8}p_{i+7} - {1 \over 16}p_{i+15} - {1 \over 32}p_{i+31} - ...
0
votes
1answer
40 views

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 ...
2
votes
1answer
164 views

Two-dimensional Banach algebras

Let $A_1$ be the matrix algebra consisting of matrices of the form $$ \pmatrix{ \alpha & 0 \\ 0 & \beta\\ }$$ and let $A_2$ be the matrix algebra consisting of matrices of the form $$ \pmatrix{...
2
votes
1answer
58 views

Injectivity of the evaluation map from holomorphic functions to a Banach algebra

In Functional Analysis, we have covered Functional Calculus, that is, a way to associate, once having fixed a Banach algebra $A$ and an element $a\in A$, an element $\tilde f(a)\in A$ to every $f$ ...
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0answers
34 views

What are the Hermitian idempotent matrices with respect to $l_{1}$ norm

Let $A=(M^{n}(\mathbb{C}), \|\cdot\|_{1})$. The numerical range of $a\in A$ is defined as $V(a)=\{f(a):f\in A',\|f\|=1=f(1)\}$. $a\in A$ is said to be Hermitian if $V(a)\subseteq \mathbb{R}$. $a\in A$ ...
1
vote
1answer
29 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 ...
4
votes
1answer
545 views

Show that the spectral radius is an upper semi-continuous function

I am stuck in a problem of Conway's A course in a Functional Analysis. Can anyone give me a hint to solve the problem? The question is "If $A$ is a Banach Algebra, then show that the function $r:A\to ...
4
votes
1answer
39 views

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 ...
5
votes
0answers
58 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 $\{e_\alpha\}\...
0
votes
0answers
29 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 ...
2
votes
1answer
38 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 ...