Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [banach-algebras]

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) or (von-neumann-algebras) instead (or in addition). Further related tags: (operator-algebras), (operator-theory), (banach-spaces), (hilbert-spaces).

2
votes
1answer
30 views

Spectrum invariance under the passage to a sub Banach Algebra

Let $\mathcal{B}$ be a unital Banach Algebra, fix $A \in \mathcal{B}$. $\sigma_\mathcal{B}(A) = \{ \lambda \vert \lambda I - A \, not \,invertible \, in \, \mathcal{B}\}$ the specturm of $A$ in $\...
1
vote
1answer
13 views

exponential null operator

In $A$ banach algebra with unit, and $X\in A$. if i define $e^X=\sum_{n=0}^{\infty} \frac{1}{n!}X^n$ why $e^0=Id$ , i am aassuming $O^0=Id $ with $0$ null operator thanks
0
votes
0answers
21 views

Qutient Topologies in Banach Spaces

I hope the title is not misleading. I am currently reading a paper, where the author uses the following argument: First, let $ f:A\rightarrow B $ be a linear and continuous map between Banach spaces $...
1
vote
0answers
15 views

The set of all inner derivations $\mathcal{B}^1(A, X)$ is not closed in the set of continuous derivations $\mathcal{Z}^1(A, X)$,

Let $A$ be a complex Banach algebra and $X$ be a Banach $A$-module The set of all continuous $X$-derivations on $A$ is a complex linear space, denoted by $\mathcal{Z}^1(A, X)$. and The set of all ...
5
votes
2answers
73 views

Is the limit of the spectral radius the spectral radius of the limit?

Let $A$ be an unital Banach algebra, $x \in A$ and $(x_n)$ a sequence in $A$ converging to $x$. I want to show that $$ \lim\limits_n \rho (x_n) = \rho (x).$$ I can show that $$\limsup \rho(x_n) \leq \...
3
votes
0answers
107 views

Does $e^{a+b}=e^{a}e^{b}$ implies that $ab=ba$?

Let $A$ be a Banach algebra we khow that If $ab=ba$ then $e^{a+b}=e^{a}e^{b}$ my question is Does $e^{a+b}=e^{a}e^{b}$ implies that $ab=ba$? Any comment or response is appreciated.
1
vote
1answer
36 views

Explaining the group structure in homotopy classes of map $[X,K(H)]$

It is Higon's notes on Index Theory, Ex 3.16, pg 43 where he describes a group structure (where I suppose works in more general context) of the $*$-algebra homotopy classes of map $$ [S,K(H)]$$ ...
2
votes
1answer
25 views

Simple proof that $x^*x$ is positive in a commutative $C^*$-Algebra

Is there any simple proof (one that does not use continuous functional calculus) for the statement that $\sigma(x^*x) \subseteq [0,\infty)$ for any $x \in \mathcal{A}$ where $\mathcal{A}$ is a ...
2
votes
0answers
31 views

some examples of a Banach algebra with given conditions

What are some examples of Banach algerba $A$ satsifying the following two conditions? $1$. $A$ does not have an approximate idebtity. $2$. $A^2=A$. That is, for any $a\in A$, there exist some $b,c\...
-1
votes
2answers
24 views

Example of a Banach algebra with given property

Please let me know what are some examples of Banach algerba $A$ for which $A^2=A$ ( that is, $a^2=a$ for each $a\in A$). A direct application of the cohen factorization theorem shows that if $A$ has ...
0
votes
1answer
65 views

Modern introduction to $C^*$-algebras

Well, I am asking for the references on the subject for those who can't stand the Murphy's book. I have a background in functional analysis (including Banach algebras, functional calculus, Gelfand ...
0
votes
0answers
7 views

enlarging modules to get innerness of a given derivation

Let $A$ be a Banach algebra, let $E$ be an $A$-bimodule and let $D\colon A\to E\,$ be a continuous derivation. Is it true that by enlarging a bimodule one can make $D$ inner?
0
votes
0answers
19 views

spectrum on finite dimensional Banach algebra

How can I prove that in a finite dimensional complex Banach algebra $A$ we have $\sigma(ab)=\sigma(ba),~\forall a,b\in A$, which $\sigma(a) $ denotes the spectrum of $a$? Every hint is appreciated
0
votes
1answer
36 views

Prove that $(\ell^2, \left||.\right\|_2)$ is a commutative and semisimple Banach algebra

Let $(\ell^2, \left||.\right\|_2)$ with the coordinatewise product.Prove that $(\ell^2, \left||.\right\|_2)$ is a commutative and semisimple Banach algebra this is my attempt: Since $(\ell^2, \...
5
votes
1answer
50 views

Invertible elements in $l^1(\mathbb Z)$

The vector space $l^1(\mathbb Z)$ with $||x|| = \sum_{n \in \mathbb Z} |x_n|$ and $x * y(t) = \sum_{k \in \mathbb Z} x(k)y(t-k)$ forms a unital complex Banach algebra, with the unit being $\mathbf 1(0)...
2
votes
1answer
64 views

Prove that $|f|_X\leq \|f\|$ for every $f\in C(X)$

Let $X$ be a compact space and let $\|.\|$ be an algebra norm on $ C(X)$ Show that $|f|_X \leq \|f\|$ for every $f\in C(X)$ Could anyone please suggest me how to deal with these questions
0
votes
1answer
30 views

In Banach algebra $A$ find an example such hat $e^{a+b} \not =e^ae^b$

Let $A$ be a Banach algebra if $ab=ba$ then we have $e^{a+b}=e^ae^b$ without $ab=ba$, I want to find an example such hat $e^{a+b} \not =e^ae^b$ Any help will be greatly appreciated.
2
votes
1answer
43 views

Bounding || zx - yz || given that || x - y || < M in a Banach algebra.

Let $ X $ be an Banach algebra (not necessarily commutative), and let $ x, y, z \in X $. Suppose that $ \| x - y \| < M $. I want to bound $ \| zx - yz \| $ in terms of $ M $ by writing $ zx - yz ...
1
vote
1answer
22 views

A natural Banach algebra where adjoint doesn't preserve the norm?

Are there any natural Banach algebras with some natural operator $A$ where $\Vert A \Vert \ne \Vert A^* \Vert$?
2
votes
0answers
62 views

An example of a Banach algebra satisfying given conditions

Is there a non-commutative non-unital Banach algebra $A$ for which $aa_0 -a_{0}a$ lies in the annihilator of $A$ for any $a\in A$? Here $a_0$ is an element of $A$ not belonging to its centre $Z(A)$. ...
1
vote
1answer
54 views

Verapoulous Algebra $C(K) \mathbin{\hat\otimes} C(L)$ is a subalgebra of $C(K\times L)$?

Let $K$ and $L$ be compact spaces. Consider the Banach algebra $V(K,L)=C(K)\mathbin{\hat\otimes} C(L)$ , which is the completion of the $C(K)\otimes C(L)$ with respect to the projective tensor norm. ...
0
votes
1answer
18 views

Regarding continuous function not in the Disc algebra

Let $D=\{z\in\mathbb{C}: |z|<1\}$. $C(\bar{D})=\{f:\bar D\longrightarrow \mathbb{C}: f \;\text{is continuous on}\; \bar{D}\}$ $A(D)=\{f\in C(\bar{D}): f \;\text{is analytic in} \;D\}$ Can you ...
2
votes
0answers
13 views

$C_0(\Delta(l^p))$ description [duplicate]

If $X=l^p$ ,$p \in [1,\infty)$ is nonunitial Banach algebra with coordinate multuplication and $\Delta(l^p)=\{e_n:n \in \mathbb{N}\}$, where $e_n(x)=x_n$ for $x \in l^p$, what is $C_0(\Delta(l^p))$ ?
2
votes
0answers
40 views

Banach algebra $l^p$ is not isomorphic to $C^{*}$ algebra

Consider commutative Banach algebra $l^p$, $p \in [1,\infty)$ with multiplication by coordinates. I know, that $\Delta (l^{p})=\{e_n : n \in \mathbb{N}\}$ - set of canonical functionals. We know that $...
0
votes
0answers
12 views

Approximate identity for induced algebra

Let $H$ be a closed subgroup of a locally compact group $G$, and let $A$ be a Banach algebra with an $H$-action $\alpha$. Then we can define an induced algebra $\mathrm{Ind}A$ with a $G$-action by ...
0
votes
1answer
38 views

Banach Algebra Isomorphism

Let $X=l^1$ with coordinate multiplication - it is commutative Banach algebra without unit. The Gelfand transformation is defined as $\widehat{x}(e_n)=x_n$ for $x \in l^1$. I would like to prove, that ...
4
votes
1answer
53 views

Commutative Banach algebras and maximum ideal space

Let $A, B$ be commutative unital Banach algebras and let $\varphi: A \rightarrow B$ be a continuous unital map such that $$\overline{\varphi(A)} = B$$ Let $$\varphi^{*}: \text{Max}(B) \rightarrow \...
2
votes
1answer
36 views

When does the spectrum of an element in a Banach algebra with involution lie in the open right half-plane?

Let $A$ be a Banach algebra with involution, $x\in A$ and $t\in {\mathbb R}$ such that $t>\rho(xx^*)$. Show that $\sigma(te-xx^*)$ lies in the open right half-plane. I have no idea! It's obvious ...
3
votes
1answer
57 views

Gelfand map is topologically injective

Let $A$ be a commutative Banach algebra and let $$G: A \rightarrow C_{0}(\text{Spec}(A))$$ be a topologically injective map. Recall that the map $T: X \rightarrow Y$ is called topologically ...
1
vote
0answers
17 views

Lower bound for the speed convergence of the spectrum

Let $A$ be a (complex unital) Banach algebra and $a \in A$ with spectrum $\sigma(a) = \{ 0 \}$ ($a$ is quasinilpotent). For $b \in A$ and $\varepsilon > 0$ consider the linear perturbation $a + \...
2
votes
1answer
67 views

Gelfand transformation of $l^p$

I would like to describe Gelfand transofrmation of commutative Banach algebra $l^p(\mathbb{N}),p \in [1,\infty)$ with multiplication define by $(a_n)_n(b_n)_n=(a_n b_n)_n$, but I have no idea, how to ...
2
votes
1answer
70 views

Spectrum of Banach algebra with coordinate multiplication

Consider $X=l^p$ , $p \in [1, \infty )$. I proved that $X$ with coordinate multiplication is commutative Banach algebra without unit. I have got a problem to find the spectrum of general element of ...
1
vote
3answers
66 views

Rudin Functional Analysis, Chater $10$, exercise $12$

Here is the exercise: $12$. Find the spectrum of the operator $T\in {\mathcal B}(\ell^2)$ given by $$T(x_1, x_2, x_3, x_4,...)=(-x_2, x_1, -x_4, x_3,...).$$ I am confused by ${\mathcal B}(\ell^2)$. ...
0
votes
0answers
38 views

Lemma 5.1.5 from Garth Dales, Introduction to Banach algebra

The problem is from Garth Dales, Introduction to Banach algebra, chapter 5 and Lemma 5.1.5 Lemma: Let $(A, \|.\|)$ be a unital Banach algebra, let $a\in A$ and let $\epsilon>0.$ then there is a ...
0
votes
1answer
30 views

$A$ self-adjoint, $f \in C(\sigma(A), \mathbb{C})$, then $\sigma(f(A)) = f(\sigma(A))$

Consider $\mathcal{A}, \mathcal{B}$ two unital $C^*$-algebras and $\varphi: \mathcal{A} \rightarrow \mathcal{B}$. Let $A \in \mathcal{A}$ be a selfadjoint element and consider $f \in C(\sigma(A), \...
3
votes
1answer
69 views

$\mathcal{E}(C(K, \mathbb{C})) = \{\omega_a : a \in K\}$

We know that $C(K, \mathbb{C})$ is a $C^*$-algebra of complex valued functions on a compact $K$. Now, for all $a \in K$, define the linear functional $\omega_a \colon C(K, \mathbb{C}) \rightarrow \...
4
votes
1answer
27 views

If $A$ is invertible and $r^{\text{Gelf}}(BA^{-1}) < 1$, then $(A - B)$ is invertible.

Let $\mathcal{A}$ be a unital Banach algebra and define $r^{\text{Gelf}}(A) = \lim_{n \rightarrow +\infty} \| A^n \|^{1/n}$. It is possible to show that $r^{\text{Gelf}}(A) = \lim_{n \rightarrow +\...
0
votes
0answers
27 views

In Banach algebra $A$ if $a_n\to a$ then $e^{a_n}\to e^a$

My question is as follows let $(a_n)$ be a sequence in banach algebra $A$ and $a_n\to a$ then prove or disprove $e^{a_n}\to e^a$ Any help will be greatly appreciated
0
votes
1answer
43 views

In Banach algebra $A$ if $ab=ba$ prove that $e^{a+b}=e^ae^b$

Let $A$ be a Banach algebra if $ab=ba$ then prove that $e^{a+b}=e^ae^b$ I've started by $e^a=\sum _{n=0}^{\infty }\frac{a^n}{n!}$, I want to know if this is correct way? Any help will be ...
0
votes
1answer
23 views

$C^*$ algebra, existence of particular state

If we have $a\in A$ be arbitrary element of $C^*$ algebra $A$. Can we find a faithful state $\phi$ such that $\phi(a) = k$ for $k$ in $spec(a)$?
0
votes
0answers
23 views

If ${A}/{I}$ and $I$ , where $I$ is a closed ideal, have approximate identities then so does $A$.

Let $A$ be a Banach algebra and $I$ a closed ideal in $A$. I want to prove that If ${A}/{I}$ and $I$ have left approximate identities then $A$ has a left approximate identity Any help will be ...
0
votes
1answer
23 views

Approximate identity in $\ell _p$

Show that for $1\leq p<\infty$, $\ell _p$ with multiplication defined by $(a_n)_n(b_n)_n=(a_nb_n)_n$has an unbounded approximate identity but no bounded approximate identity, I don't know how to ...
1
vote
0answers
28 views

Can C*-envelope introduce the unity?

Let $A$ be a non-unital Banach $*$-algebra with isometric involution. Is it possible that the enveloping $C^*$-algebra of $A$ is unital? I guess not at least when $A$ admits an approximate identity, ...
0
votes
0answers
21 views

In a unital Banach algebra $A$ with $ab=ba$ prove that $\sigma(a+b)\subset \sigma(a)+\sigma(b) $ [duplicate]

Let $A$ be a unital Banach algebraIn with $ab=ba$ prove that $\sigma(a+b)\subset \sigma(a)+\sigma(b) $ and $\sigma(ab)\subset \sigma(a)\sigma(b)$
3
votes
1answer
37 views

Showing uniqueness of holomorphic functions on Banach spaces using uniquess of scalar holomorphic functions?

Theorem Let $D ⊆ C$ be a connected open set, and let $z_n ∈ D$, $n ∈ \mathbb{N}$, be a sequence which converges to a point $z_0 ∈ D$ such that $zn \neq z_0$ for all $n ∈ \mathbb{N}$. Further let $E$ ...
0
votes
0answers
31 views

Existence of a holomorphic function on the unit disk

Let $z_1$ and $z_2$ be two distinct points in the unit sphere of $\mathbb{C}$, i.e., $z_1 \neq z_2$ and $|z_1|=|z_2|=1$. I'd like to construct the bounded uniformly continuous holomorphic function $...
1
vote
0answers
34 views

Calculate $\operatorname{Rad}(A/\operatorname{Rad}(A))$ in a Banach algebra

Let $A$ be a Banach algebra with identity $e_A$, I'd like to find $\operatorname{Rad}(A/\operatorname{Rad}(A)).$ whre we define $\operatorname{Rad}(A)=\{a\in A:e_A-ba \in \text{InvA},b\in A\}...
0
votes
2answers
34 views

In Banach algebra $A$ if $a^2=a$ show that $\sigma_A(a)\subseteq\{0,1\}$

Let $A$ be a Banach algebra with identity $e$ and let $x\in A $ be such that $a^2=a$ show that $\sigma_A(a)\subseteq\{0,1\}$ and compute the resolvet function $$R(a, \lambda)=(\lambda e-a)^{-1}$$ ...
0
votes
0answers
27 views

Relation between spectrum of two isomorphic Banach algebra

Suppose that $A$ and $B$ are two Banach algebra with identity and $T:A \to B$ be an isomorphism prove that $\sigma _{A}(a)=\sigma _{B}(T(a)) $ for all $a\in A$ My atempp: Let $\lambda \not\...
1
vote
0answers
24 views

Reference for Banach Lattices

I am looking for a survey or a basic book on Banach Lattices (I know next to nothing about the subject, so a basic survey is what I believe will be helpful). Can somebody please suggest any good ...