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) ...

learn more… | top users | synonyms

0
votes
1answer
9 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?
0
votes
1answer
34 views

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

Is there a characterization for all Banach algebras such that $\|a\|\|a^{-1}\|=1$ whenever $a$ is invertible?
0
votes
1answer
32 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
30 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
32 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
17 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
16 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
25 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....
0
votes
1answer
33 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
8 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}...
1
vote
0answers
14 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
35 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
32 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
15 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
28 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 $A$?
3
votes
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
vote
0answers
109 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
163 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 $f$ ...
0
votes
0answers
33 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
543 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
38 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
57 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
28 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 ...
2
votes
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 $...
4
votes
1answer
363 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 ...
0
votes
1answer
37 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 ...
3
votes
1answer
27 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 ...
0
votes
0answers
15 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 ...
1
vote
0answers
34 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 $n\geq 1$. ...
0
votes
0answers
55 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 ...
2
votes
1answer
23 views

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 ...
1
vote
1answer
26 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 ...
0
votes
0answers
8 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 ...
4
votes
1answer
39 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 ...
2
votes
2answers
117 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, ...
0
votes
1answer
39 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
votes
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 ...
3
votes
2answers
43 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
votes
1answer
29 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 ...
2
votes
0answers
17 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 ...
0
votes
1answer
30 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?‎ ‎
0
votes
0answers
17 views

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 ...
1
vote
0answers
27 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
190 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 \{0,1\}...
1
vote
2answers
202 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!