Stack Exchange Network

Stack Exchange network consists of 174 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).

0
votes
0answers
4 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
39 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
39 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
15 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
9 views

$C_0(\Delta(l^p))$ description

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
31 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
6 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
34 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
40 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
28 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
51 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
15 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
53 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
58 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
47 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
37 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 ...
4
votes
0answers
105 views

Rudin functional Analysis chapter 10, exercise 20

This is the exercise: Suppose $x\in A$, $x_n\in A$, and $\lim x_n=x$. Suppose $\Omega$ is an open set in $\mathbb C$ that contains a component of $\sigma(x)$. Prove that $\sigma(x_n)$ intersects $\...
0
votes
1answer
25 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
68 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
25 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
23 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
34 views

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

Let $A$ be aBanach algebra if $ab=ba$ then prove that $e^{a+b}=e^ae^b$ I've started by $e^a=\sum _{n=0}^{\infty }\frac{a^n}{n!}$, Iwant to khow this is correct way? Any help will be greatly ...
1
vote
1answer
19 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
22 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
21 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
26 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
14 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
26 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
26 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
33 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
32 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
23 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
22 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 ...
1
vote
0answers
35 views

Does Grothendieck's theorem hold for Bounded borel functions?

In the case of continuous functions on a compact Hausdorff space, we have that any bounded set is pointwise compact if and only if it is weakly compact and the two topologies coincide with this set. ...
1
vote
1answer
41 views

Compute the norm of a bounded linear operator

Let $T$ be a nonzero bounded linear operator in $B(H)$, where $H$ is an infinite dimensional Hilbert space. If the norm of $T$ is known, how to compute the norm $\|I–T\|$,where $I$ is the identity ...
4
votes
2answers
227 views

What is the limit of $\mathrm{Tr}(G^kM{G^*}^k)^{1/2k}$ when $k$ goes to infinity?

If $G\in \mathscr M_n(\mathbf C)$ then it's well known that $\lim_{k\to \infty}\|G^k\|^{1/k}=\rho(G)$ where $\rho(G)$ is the spectral radius of $G$, the value of the limit does not depend on the ...
4
votes
1answer
40 views

$GL_n^+(A)$ is open but $U_n^+(A)$ is not

Let $A$ be a C*-algebra. $\tilde{A}$ be the unitization of $A$. I checked the following lemma: If $x$ and $y$ are elements of $M_n(\tilde{A})$ such that $x$ is invertible and $\|x-y\| \leq \frac{1}{...
2
votes
2answers
63 views

Prove that Banach algebra $B(X)$ has a unique (up to equivalence) complete algebra norm.

Let $X$ be a Banach Space, and denote by $∥ · ∥$ the standard norm on $B(X)$, the space of bounded linear functions $T:X\to X$. (a) Suppose that $||| · |||$ is another algebra norm on $B(X)$. ...
1
vote
1answer
44 views

How can we use Hahn-Banach Theorem to prove a functional is linear and bounded?

Let $V$ and $W$ be Banach spaces, bounded linear operators $T: V \to W$ and $S: W' \to V'$ such that \begin{align*} f(T(v)) = S(f)(v), \qquad \forall v \in V, \quad \forall f \in W' \end{align*} ...
0
votes
1answer
19 views

To show an element in the sprectum of a matrix $M$ is negative in the spectrum of $-M$

I am trying to solve the following problem: Let $V$ be a Banach space , $\lambda \in \mathbb{C}$, an operator $H \in \mathbb{B}(V \oplus V)$ in a form of \begin{align} H = \begin{pmatrix} 0 &...
0
votes
1answer
14 views

Why is $E$ closed?

I am reading through the following statement and proof in Aupetit’s A Primer on Spectral Theory He provides the following proof: Towards the end of the proof, Aupetit says that the set $E$ as ...
1
vote
1answer
36 views

Some concepts of $C^*$-algebra generalized from linear algebra. Can anyone help me to check if they are correct, and give some examples?

A Banach algebra is just a Banach space equipped with an operation of multiplication defined such that $\|a b\| \le \|a\|\|b\|$ for all $a,b$ in it. If, in addition, there exists an identity, then it ...
0
votes
0answers
54 views

The spectrum of a matrix operator is the set of its eigenvalues. But how to prove?

I am studying spectral theory of functional analysis. I understand there are deep connections between the spectrum of operators and the eigenvalues of matrices. But I am unable to solve the following ...
0
votes
0answers
62 views

Prove a mapping $C(X,Y)\to \operatorname{Hom}(C(Y), C(X))$ is surjective [duplicate]

Let $X$ and $Y$ be compact Hausdorff spaces, and let $F$ be a continuous function from $X$ to $Y$. Define a function $\Phi_F$ from $C(Y)$ to $C(X)$ by $\Phi_F(f)=f\circ F.$ I have shown $\Phi_F$ is ...
0
votes
0answers
26 views

Exercise books in Banach algebra

I'm studying Banach algebra and I was wondering if there are some exercise books (that is, books with solved problems and exercises) The books I'm searching for should be: full of hard, non-obvious,...
1
vote
1answer
38 views

Identification of Gelfand spectrum, $\Sigma(\mathcal{A})$, with $\sigma(A)$, spectrum of Banach Algebra element

I am reading Mathematical Structure of Quantum Mechanics A short Course for Mathematicians by F. Strocchi. My question is about the text after Proposition 1.5.3. The proposition is that For a ...
0
votes
0answers
18 views

Existence of Bounded Approximate Identities for Modules of a Normed Algebra

In Chapter 11 of "Complete Normed Algebras" by Bonsall and Duncan, the following definitions are given: Definition Let $ A $ be a normed algebra. A Bounded Approximate Identity for $ A $ is a ...
2
votes
2answers
57 views

Show a relation for the state on $C^*$ algebra

Let $\varphi$ be a state on $C^*$-algebra $A$. Assume $\varphi(a^2)=\varphi(a)^2$ for some self-adjoint elements $a\in A$. Show that $\varphi(ab)=\varphi(ba)=\varphi(a)\varphi(b)$ for any element $b\...
1
vote
1answer
70 views

Show that the space of continuous functions on the compact Hausdorff space with matrix-value is a $C^*$-algebra

Given $X=0 \cup \{1/n\}$. We can show that it is a compact Hausdorff space. Now take $M_n$ to be the matrix algebra. I am confusing on showing that both $C(X, M_2)$ and $B=\{f \in C(X, M_2)$ where $f(...
2
votes
2answers
61 views

When the quotient map is closed?

Let $A$ be a Banach algebra and $I$ be a closed two sided ideal of it. Is the quotient map $\pi:A\to A/I$, a closed map?