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

1
vote
0answers
26 views

Riemann Lebesgue Lemma for locally compact ableian groups

I'm looking for a reference (or proof here) of the generalized RL Lemma for LCAGs. One result is that if $G$ is a LCAG then $$\{ \hat{f} : f \in L^1(G)\} \subset C_0(\Gamma)$$ where $\Gamma$ is the ...
0
votes
0answers
31 views

How to generate weak topology by the family of semi-norm?

How to generate a weak topology by the family of semi-norm $$ \{P_i \colon A \rightarrow B, i \in I\} $$ in which $$ P_i(a)=\lVert ia\rVert+\lVert ai\rVert$$ when $A$ is a Banach algebra and $I$ is ...
0
votes
0answers
28 views

Functoriality in $K$-theory for $C^*$-algebras or Banach algebras

I'm trying to clear up some confusion I'm having over how one establishes functoriality in $K$-theory for $C^*$-algebras or Banach algebras. Let me stick to $K_0$. Given a *-homomorphism (or bounded ...
2
votes
1answer
29 views

Norms on unitization of a Banach algebra

Let $A$ be a non-unital Banach algebra, and let $A^+$ be its unitization. Then $||(a,z)||_1=||a||+|z|$ is a Banach algebra norm on $A^+$. Can we also make $A^+$ a Banach algebra by giving it the norm ...
1
vote
1answer
39 views

Inverse Tensor map

Let $$\phi: M_n (\mathbb{C})\otimes M_m (\mathbb{C})\to M_{nm} (\mathbb{C})$$ $$ \phi({A}\otimes{B}) = \begin{bmatrix} a_{11} {B} & \cdots & a_{1n}{B} \\ \vdots & \ddots & \vdots \\ a_{...
0
votes
1answer
61 views

$C_{c}(X)$ is complete. then implies that $X$ is compact. [closed]

Let $X$ is locally compact Hausdorff space .If $C_{c}(X)$ is complet,then $X$ is compact (this is to be proved). I know that $C_{c}(X)$ is dense in $C_{0}(X)$. As $C_{c}(X)$ is complete implies that $...
1
vote
0answers
20 views

$C_{c}(X)$ is complete implies $X$ is compact. [duplicate]

If $X$ locally compact Hausdorff space. Then $C_{c}(X)$ is complete implies $X$ is compact. I know that $C_{c}(X)$ dense in $C_{0}(X)$. So in that case $C_{c}(X)=C_{0}(X)$. I know only Tiez ...
1
vote
1answer
43 views

invertibility in C*-algebra

I have a question about a passage in the book C*-algebras and Operator Theory by Murphy, in the proof of Theorem 2.1.8. Let $a$ be a hermitian element of a unital C*-algebra and let $\lambda\in\sigma(...
0
votes
0answers
23 views

Norm-closed subalgebras of $B(L^p(X,\mu))$

If $A$ is a Banach algebra that is isometrically isomorphic to a norm-closed subalgebra of $B(L^p(X,\mu))$ for some $p\in[1,\infty)$ and some measure space $(X,\mu)$, and is also isometrically ...
2
votes
0answers
14 views

Approximation by elements in intersection of two Banach subalgebras

Let $A$ be a Banach algebra, and let $A_1,A_2$ be Banach subalgebras of $A$. Suppose that there exists $c>0$ such that whenever $a_i\in A_i$ ($i=1,2$) and $||a_1-a_2||<\varepsilon$, then there ...
4
votes
0answers
172 views

trace norm and tensor product

Let $(M_n (\mathbb{C}), n\|.\|)$ , $(M_n (\mathbb{C}), n\|.\|)$ and $(M_{nm} (\mathbb{C}), nm\|.\|)$ be three Banach algebras. where $$\|A\| = \mathrm{tr}\sqrt{(A^* A)}. $$ What is the norm of $\phi$ ...
1
vote
0answers
23 views

Banach Algebras in the context of commutative algebra

Banach algebras are primarily studied within the context of functional analysis. Is there any theory attempting to study commutative Banach algebras from the viewpoint of commutative algebra?
2
votes
1answer
64 views

character space of C*-algebra

I have a question about a passage in the book C*-algebras and Operator Theory by Murphy. On page 41, between theorems 2.1.9 and 2.1.10 he proves that the character space of a non-unital, non-zero, ...
3
votes
1answer
31 views

Norm on reduced crossed product - $C^*$ version v.s. $L^p$ version

Let $(G,A,\alpha)$ be a $C^*$-dynamical system where $G$ is a countable discrete group. When defining the reduced crossed product, one can proceed as follows: Let $\pi$ be a faithful representation ...
1
vote
1answer
39 views

Is there any examples of a Banach algebra which every ideal of it, is a maximal ideal?

Is there any examples of a Banach algebra which every ideal of it, is maximal ideal? Or, Is there any conditions which turn all of the ideals of a Banach algebra to maximal ideals?
1
vote
0answers
55 views

Integration of $A$-valued functions (Functional Analysis)

Premise 1: my source is the Rudin - Functional Analysis. Premise 2: i'm not a mathmo so forgive for the mistakes A couple of question on the subject... An example of Banach Algebra is the set of ...
2
votes
1answer
118 views

Why the disk algebra is not a C* algebra.

I'm trying to figure out why the set of bounded analytic functions on the unit disk, A(D), is not a C* algebra. The norm is the sup norm and the involution is $f(z) \to \overline{f(\bar z)}$. I want ...
0
votes
1answer
23 views

Norm of the projection onto a maximal ideal

Let $A$ be a complex Banach algebra and $\chi \ne 0$ be a complex character. Consider the quotient space $\hat A = \dfrac A {\ker \chi} \simeq \Bbb C$. If $\hat x \in \hat A$, how can one quickly ...
0
votes
0answers
35 views

Necessary and sufficient condition on analytic function $f$ such that $f(A)=0$

Given a matrix $A\in M_n(\mathbb C)$, I want to find a necessary and sufficient condition on analytic function $f:G\to M_n(\mathbb C)$ such that $f(A)=0$. I've been trying to use the spectral mapping ...
1
vote
1answer
25 views

strictly positive element in a $C^*$-algebra. Where is the mistake in the proof?

Let $a\in A$ a stricly positive element (this means: for all states $\varphi$ of $A$ is $\varphi(a)>0$), let $u_n=a(\frac{1}{n}+a)^{-1}$, $n\in\mathbb{N}$ . Claim: for all $b\in A$, for all states $...
0
votes
0answers
14 views

$I\subseteq C_0(X)$ closed Ideal. Does for all $x\in X$ exist $f\in I$ such that $f(x)\neq 0$?

$I$ a closed ideal in the Banach algebra $C_0(X)$, $X$ locally compact Hausdorff space. Is the claim correct: For all $x\in X$ exists $f\in I$ such that $f(x)\neq 0$? I need this for a proof. But I ...
3
votes
0answers
30 views

Unital amenable Banach algebras which is a proper two sided ideal in its second dual

I need some examples of "unital amenable Banach algebras which is a proper two sided ideal in its second dual".
1
vote
1answer
39 views

How to show that the set of square matrices $R^{n \times n}$ is complete under the operator norm $\|A\| = \sup\limits_{\|x\|\leq 1} \|Ax\|$

I want to show that the set of square matrices $R^{n \times n}$ is a Banach algebra with property $\|AB\| \leq \|A\|\|B\|$. I have already showed that $R^{n \times n}$ is a linear space and it is a ...
0
votes
2answers
89 views

How do you show that the norm for square matrices is submultiplicative?

In these notes: http://www.math.usm.edu/lambers/mat610/sum10/lecture2.pdf It says that square matrices satisfy so called submultiplicative norm $\|AB\| \leq \|A\|\|B\|$. Is it by definition or is ...
1
vote
1answer
75 views

Inductive limit of separable Banach spaces

Let $\mathcal{C} = \{A_\alpha\}_\alpha$ be a chain of nesting separable Banach subspaces of $C_b(X)$, the space of all bounded continuous functions, on a locally compact space $X$. I am wondering if ...
2
votes
0answers
29 views

Norm of homomorphism embedding Banach algebra into matrix algebra

Let $A$ be a nonunital Banach algebra, and let $\tilde{A}$ be the unitization of $A$, i.e. $\tilde{A}=\{(a,\lambda):a\in A,\lambda\in\mathbb{C}\}$ with multiplication given by $(a,\lambda)(b,\mu)=(ab+\...
3
votes
1answer
80 views

Using Fubini's Theorem in Contour Integrals proof

I have a few questions regarding the following proof: Suppose that $\mathcal{A}$ is a unital Banach algebra, and that $g$ is a complex-valued function which is analytic on $\sigma(a)$ while $f$ is a ...
2
votes
1answer
54 views

Proof involving invertible elements of Banach algebra

I want to prove for a unital Banach algebra $\mathcal{A}$, it follows that if $\|a-b \| < \frac{1}{\|a^{-1} \|}$ then $b \in \mathcal{A}^{-1}$ (where $\mathcal{A}^{-1}$ is the subset of invertible ...
1
vote
0answers
34 views

Proposed proof of operator theory result

Hi I am interested in checking my proposed solution to the following problem in Operator Theory: Please give me hints as to how to improve the proposed proof rather than the full correct solution. ...
1
vote
1answer
42 views

Partial isometries

Given a unital $C^*$-algebra A and partial isometries $w_1, \cdots, w_n$ such that $\sum_{i = 1}^{n}w_iw_i^* = 1$ and $w_i^*w_j = 0$ if $i \neq j$ then is it true that $w_i$ is an isometry?
2
votes
1answer
30 views

How to handle direct sums and unitizations of $L^p$ operator algebras?

Let $p\in[1,\infty)$. An $L^p$ operator algebra refers to a Banach algebra that is isometrically isomorphic to a closed subalgebra of $B(L^p(X,\mu))$ for some ($\sigma$-finite) measure space $(X,\mu)$....
1
vote
1answer
27 views

Linear Operators and Convergence

I am struggling with this question from my Differential Equations course: If $T$ is a linear transformation on $\mathbb{R}^n$ with $||T-I||<1$, prove that $T$ is invertible and that the ...
5
votes
1answer
51 views

Radius of convergence of Taylor expansion of $z \mapsto (1 - z \cdot a)^{-1}$

Let $A$ be a Banach $\mathbb{C}$-algebra with norm $\text{N}(-)$ and let $a \in A$. Where can I find a reference to/can somebody supply a proof of the following posited equality?$$\max_{z \in \text{...
2
votes
0answers
32 views

For which $F$ we have $F(f\ast g)= f\ast F(g)$ for all $f,g \in L^{1}$?

Young's inequality tells us that: $L^{1}\ast L^{p} \subset L^{p}, (1\leq p \leq \infty)$ My Question: What are examples of functions $F:L^{1}(\mathbb R)\to L^{p}(\mathbb R)$ with the property $...
0
votes
1answer
33 views

An isomorphism between two Banach algebras

Consider the compact set $[-1,1]$ and $C([-1,1])$ the set of all continuous functions $\phi: [-1,1] \rightarrow \mathbb{C}$. I want to show that the quotient of $C([-1,1])$ by $\mathbb{C}$ is the ...
3
votes
1answer
36 views

Does $(\ell^{1}(\mathbb Z), \cdot)$ have a bounded approximate identity?

Put $\ell^{1}(\mathbb Z)=\{f:\mathbb Z \to \mathbb C: \|f\|_{\ell^{1}}:=\sum_{n\in \mathbb Z}|f(n)|< \infty \}$ and we note that $\ell^{1}(\mathbb Z)$ is an algebra under pointwise multiplication. ...
2
votes
2answers
88 views

Examples of algebras that have a bounded approximate identity

We note that $L^{1}(\mathbb R)$ is an algebra with respect to convolution without identity element. However, regarded as a Banach algebra, $(L^{1}(\mathbb R), \ast) $ has a bounded approximate ...
3
votes
2answers
55 views

Does there exists an approximate identity in Fréchet algebra $\mathcal{S}(\mathbb R)$?

We put $\|f\|_{(N, \alpha)}:= \sup_{x\in \mathbb R} (1+|x|)^{\alpha} | D^{\beta}f(x)|; $ and he Schwartz space, $S(\mathbb R): = \{f\in C^{\infty}(\mathbb R): \|f\|_{(N, \alpha)}< \infty , \...
3
votes
0answers
17 views

Definition of $K_1(A)$ for a Banach algebra $A$

When defining $K_1(A)$ for a Banach algebra $A$, one may consider $\bigcup_{n\in\mathbb{N}}\{x\in GL_n(A^+):x\equiv I_n\mod M_n(A)\}$ and take the quotient by the component containing the identity, or ...
2
votes
0answers
46 views

Clarification on Wolfram Mathworld's explanation of the connection between Gelfand Transform and Fourier Transform

http://mathworld.wolfram.com/GelfandTransform.html In the definition, what does $x$, $\hat x(\phi)$, and $\phi$ represent exactly if we were to consider definition of the Fourier transform? Can ...
3
votes
0answers
32 views

Derivation into dense ideal of Banach algebras

Let $A$ be a Banach algebra and $I$ be an ideal of $A$. A derivation $D:A\to I$ is a linear bounded map, with the following property: $$D(ab)=aD(b)+D(a)b,\qquad a,b\in A.$$ Suppose that $I$ is dense ...
0
votes
0answers
20 views

What are the appropriate morphisms for forming inductive limits of Banach algebras?

For Banach algebras, if we take the morphisms to be bounded homomorphisms, the inductive limit construction may not result in a Banach algebra. (I remember seeing this fact but I don't know a specific ...
2
votes
2answers
31 views

How does the product of sets of complex numbers give a character?

I'm working through this "Introduction to Banach Algebras" and just after proposition 8.2 they say: If $A$ is a commutative Banach algebra, $a\in A$ and $\phi\in M(A)$, then $\phi(a)\in sp(a)$. ...
4
votes
1answer
56 views

Distributivity of projective tensor product over direct sum

Let $I$ is a non-empty set and $\{A_i\}_{i\in I}$ is a family of Banach algebras and $B$ is a Banach algebra. Define $$\ell^1-\oplus_{i\in I}A_i=\{a=\{a_i\}_{i\in I}: \|a\|_1=\sum_{i\in I}\|a_i\|<\...
2
votes
1answer
39 views

How are $M_n(A)$ and $M_n(\mathbb{C})\otimes A$ identified as Banach algebras?

When $A$ is a Banach algebra, one can make $M_n(A)$ into a Banach algebra by equipping it with various norms. One can also make the algebraic tensor product $M_n(\mathbb{C})\otimes A$ into a Banach ...
3
votes
3answers
92 views

Which Hilbert space is isometrically isomorphism with $B(E)$ for some Banach space $E$.

Consider $H$ as a Hilbert space. How can I find a Banach space $E$, for that, $H=B(E)$ where $B(E)$ is the set of bounded linear operator on $E$? (At least under some conditions on $H$) Also if the ...
7
votes
2answers
148 views

Is every Hilbert space a Banach algebra?

Let $H$ be a Hilbert space. Could we say that, always there is a multiplication on $H$, that makes it into a Banach algebra? If not, under which conditions does it exist?
4
votes
1answer
47 views

The same topologies

Let $L^1 (\mathbb{Z})$ be the space of all functions $f:\mathbb{Z}\rightarrow \mathbb{C}$ such that $\left\{\|f\|=\sum_{k\in \mathbb{Z}}|f(k)|<\infty\right\}$. Clearly, $L^1 (\mathbb{Z})$ is a ...
3
votes
0answers
28 views

Semiprimitivity of second dual of semiprime Banach algebras

Let $A$ be a Banach algebra. Then $A^*$ is right Banach $A$-module with product $\langle b,f.a\rangle=\langle ab,f\rangle$ for every $a,b\in A, f\in A^*$. Define $\langle a,F*f\rangle=\langle f.a,F\...
3
votes
1answer
32 views

Weakly compact operator with different domains

Let $A$ be a Banach algebra. Suppose that $e\in A$ such that $e^2=e$ and $eAe$ is division algebra(i.e., $eAe$ is unital and every element of $eAe$ has inverse in $eAe$). Define $T_e:A\to A$ with $T_e(...