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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Invertible elements in a Banach algebra connected to identity

I'm currently working on a problem for homework in my Banach algebras course and I've run into a bit of an issue with terminology. Let $\mathcal{A}$ be a Banach algebra, then $\mathcal{A}^{-1}_0$ ...
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302 views

Converse of Wiener's lemma

Let $A$ be a commutative Banach algebra with unit. It is well known that if the Gelfand transform $\hat{x}$ of $x\in A$ is non-zero, then $x$ is invertible in $A$ (the so called Wiener Lemma in the ...
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47 views

$\int_{\mathbb R^{2}} |\int_{\mathbb R} (f(t-y)- f(t)) g(t-x) e^{-2\pi i w\cdot t} dt|dx dw \to 0 $ as $y\to 0$?

Fact: It is well-known that translation is continuous in the $L^{1}$ norm, that is, if $f\in L^{1}(\mathbb R)$ then $\lim_{y\to 0} \|f_{y}-f\|_{L^{1}(\mathbb R)}=0;$ (where, $f_{y}(x)= f(x-y)$, ...
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81 views

How to prove $F$ is a contraction mapping?

Given a $n\times n$ matrix $\mathbf{D}$ in which each entry $\mathbf{D}_{ij} \in [0,1]$. Let $i$-th row vector of $\mathbf{D}$ denote by $\mathbf{D}_{i\ast}$. The mapping $F:[0,1]^n \mapsto [0,1]^n$, ...
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116 views

Bergman-Shilov Boundary and Peak Points

Let $\Omega$ be a domain in $\mathbb{C}^n.$ Consider the Banach algebra $A(\Omega):=\mathcal{C}({\overline{\Omega}})\cap\mathcal{O}(\Omega).$ Denote the Bergman-Shilov boundary of $A(\Omega)$ by ...
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101 views

Unitary Equivalence of Two Irreducible $ * $-Representations of a GCR $ C^{*} $-Algebra that Have the Same Kernel.

In general, if two irreducible $ * $-representations of a $ C^{*} $-algebra $ A $ have the same kernel, then we can say that they are approximately unitarily equivalent. When $ A $ is GCR, how can we ...
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89 views

What is the generator of the disc algebra

The disc algebra, as a set, consists of the functions on the unit disc $D$, which are analytic on the interior of the disc and continuous on its boundary. Its addition and multiplication is obvious. ...
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Using Stone–Weierstrass theorem for completely regular space

Let $X$ be completely regular. If $K$ is a compact subset of $X$, define $$p_K(f)=\sup\{|f(x)|;x\in X\}$$ then $\{p_k; \text{K is a compact }\}$ is a family of seminorms that makes $C(X)$ into a ...
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88 views

Positive Operators: Definition?

Let $A$ be a self adjoint element of a C*-algebra $\mathcal{A}$ resp. a self adjoint operator of the operator algebra $\mathcal{B}(\mathcal{H})$ of bounded operators over a Hilbert space ...
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25 views

On maximal ideal spaces of a banach algebra

I am reading this article on maximal ideal spaces and there is this part that I don't quite understand very well, hope you guys can help me out. "Let $M(A)$ denote the maximal ideal space of a ...
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117 views

application of c*algebras

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. ...
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Is the range of the Fourier transform $L^1(\mathbb{R}) \to C_0(\mathbb{R})$ closed under “quasi-inversion”?

Definitions: A function $f \in C_0(\mathbb{R})$ is quasi-invertible if $1 \notin \operatorname{ran}(f)$. The quasi-inverse of such an $f$ is $\frac{f}{f-1}$. Some discussion: Let $C_1(\mathbb{R})$ ...
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Do inclusions of Banach algebras preserve spectral radius?

Let $f : A_1 \to A_2$ be an injective homomorphism of unital Banach algebras. It's a standard fact that if $f$ is has closed range, i.e. $A_1$ is embedded as a closed subalgebra of $A_2$, then for ...
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132 views

C* algebra of bounded Borel functions

Let $T\in B(H)$ is normal, and $B(\sigma(T))$ denote the $C^*$ algebra of all bounded Borel functions on $\sigma(T)$. Then is it true that $B(\sigma(T))$ is a closed $C^*$ algebra under the sup. norm ...
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68 views

In relation with the set of Fredholm perturbation elements

Let $A$ and $B$ be two unital Banach algebras and $T\colon A\to B$ an homomorphism of Banach algebras. Let denote the set of Fredholm perturbation elements in $A$, i.e. $\operatorname{Ft}:=\{r\in ...
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131 views

Topology of maximal ideal space

We know that strictly positive integers with multiplication form a semigroup $S$. Let $\mathscr A=\ell ^1(S)$ with convolution. What is $\mathscr A$'s maximal ideal space? It seems enough to find ...
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63 views

Let $K$ be a circle. Describe the spectra of two subalgebras of $C(K)$

Suppose $K=\{\lambda\in\mathbb{C}: 1<\vert\lambda\vert<2\};$ put $f(\lambda)=\lambda$. Let $A$ be the smallest closed subalgebra of $C(K$) that contains $1$ and $f$. Let $B$ be the smallest ...
3
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103 views

Density of operators

I am interested in operators on non-reflexive Banach space. Let $X$ be a Banach space and let $L(X)$ be the algebra of operators acting on $X$. We may embed $L(X)$ into $L(X^{**})$ by ...
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110 views

Does the non-emptiness of the spectrum of an element of a Banach algebra depend on the Axiom of Choice?

One of the most basic results in functional analysis states that the spectrum of any element of a Banach algebra is non-empty. The proof, as most people might have seen, makes use of Liouville's ...
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92 views

What does $(B+I)/I\sim B/(B\cap I)$ tell us?

Let $A$ be a $C^*$-algebra in which $B$ is a $C^*$-subalgebra and $I$ is a closed ideal. In several books on $C^*$-algebras I have encountered the following: $(B+I)/I$ is $*$-isomorphic to ...
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271 views

a question about invertibility of Banach Algebra

If $X$ is a Banach algebra with identity, and $0$ is the only element $x \in X$ such that there is a sequence $\{ {x_n}\} \subset X$, $\left\| {{x_n}} \right\| = 1$ and $x{x_n} \to 0$ or ${x_n}x \to ...
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43 views

C*-algebra representations

Let A be a C*-algebra and $\left\{\phi_n\right\}$ a weak* dense sequence in the state space. Putting $\phi=\sum_n 2^{-n} \phi_n$, can you show that $\phi$ is a state and the representation $\pi_\phi$ ...
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112 views

Norm on unitisation of a $C^\ast$ algebra

In the theory of $C^\ast$ algebras there exists the following theorem: If $A$ is a $C^\ast$ algebra and $\widetilde{A}$ denotes its unitisation then there exists exactly one norm that extends the ...
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1answer
70 views

Identifying the dual group of a locally compact abelian group with the spectrum of $ {L^{1}}(G) $.

Folland stated the following theorem in his book A Course in Abstract Harmonic Analysis on Page 88. The dual group of a locally compact abelian group can be identified with the spectrum of $ ...
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1answer
27 views

Show that there is a unique continuous function

I have no idea where to even start, i have never dealt with question like this before, any direction you can give me would be greatly appreciated.
2
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1answer
131 views

Constructing a functions with Gelfand Naimark

If $X$ and $Y$ are compact Hausdorff spaces, show that for any algebra homomorphism $$ F:C(Y) \to C(X) $$ there exists a continuous function $f:X\to Y$ such that $$ F(\phi)=\phi \circ f, \forall \phi ...
2
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1answer
58 views

Subalgebras of certain C*-algebras

Let $A$ be a C*-subalgebra of $C(X, M_{n}(\mathbb{C}))$ where $X$ is a compact Hausdorff space, does it follow that $A$ is isomorphic to $C(Y, M_{m}(\mathbb{C}))$ for some $Y\subseteqq X$ and ...
2
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‎‎If $A$ contains ‎an ‎idempotent $e‎$ (‎‎$‎e‎\neq ‎‎0,1‎‎$‎) , then $‎\Omega(A)‎$ ‎is ‎disconnected

If $A$‎ ‎be a‎ ‎unital ‎abelian ‎Banach ‎algebra ‎and ‎contains ‎an ‎idempotent $e$‎ ‎(that ‎is ‎‎$‎e=‎e‎^{‎2‎}‎‎$‎) ‎other ‎than $0$‎ ‎and $1$‎ ,‎ ‎then help me to show that ‎‎$‎\Omega(A)‎$ ‎is ...
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2answers
34 views

Why is the Gelfand transform injective?

There is a theorem that proves that if $A$ is a commutative C*-Algebra, the Gelfand map is an isometric *-isomorphism of $A$ onto $\hat{A}$ i.e. the spectrum of $A$. (Theorem 1.1 in Averson's "An ...
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1answer
19 views

projection generated by intersection of two projection

let $H$ be a Hilbert space and $P,Q$ be projections on $H$. suppose $P,Q$ do not commute. $P\wedge Q$ is a projection on $PH\cap QH$. I want to calculate $P\wedge Q$ but I can not. Please help me. ...
2
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1answer
18 views

Commutative subspace lattice

I have seen an article in which there is an algebra which was named CSL-algebra (Commutative Subspace Lattice). This algebra is about projection on Banach algebra? I couldn't find any good source to ...
2
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1answer
90 views

Example of a Banach algebra $ A $ whose only closed ideals are $ \{ 0 \} $ and $ A $.

I'm trying to come up with an example of a Banach algebra $A$ that is not commutative, unital and such that the only closed ideals are $\{0\}$ and $A$. I already struggled to even come up with a non ...
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1answer
47 views

Module algebras

Spectrum: For Banach algebra $A$ spectrum is denoted by $\sigma(A)$ and defined as the set of all non-zero bounded linear multiplicative function from $A$ to $\Bbb C$.(Function $\psi:A\to\Bbb C$ is ...
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1answer
65 views

Uniform boundedness

I was thinking about the following problem: Let $(A_t)_{t \geq 0}$ be a family of bounded operators on a Banach space $X$ which is uniformly bounded and let $(B_{t,\alpha})$ be a net in the Banach ...
2
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1answer
186 views

Application of Stone-Weierstrass with a non-unital algebra

Let $X$ be a locally compact Hausdorff space. We say that a function $f\colon X \to \mathbb{R}$ vanishes at $\infty$ if for each $\epsilon >0$ there exists a compact $K_\epsilon \subset X$ such ...
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votes
1answer
79 views

A question about positive elements in $C^*$ algebras

Let $A$ be a $C^*$-algebra If $a\in A$ is positive, is it true that for any $0<\alpha<\frac{1}{2}$ we have $$\left(a+\frac{1}{n}1\right)^{\frac{-1}{2}}a^{\frac{1}{2}-\alpha}$$is self adjoint?A ...
2
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1answer
31 views

Extension theory and automorphism extension

My question is motivated by the following two posts On finite 2-groups that whose center is not cyclic and Automorphisms of group extensions Question: Assume that $A,B,C$ are there algebraic ...
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1answer
92 views

Why characters are continuous

According to Wikipedia: ''Every character is automatically continuous from $A$  to $\mathbb C$, since the kernel of a character is a maximal ideal, which is closed. '' where $A$ is a Banach algebra. ...
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1answer
106 views

Prove that the Fourier series of $\dfrac{1}{f}$ is absolutely convergent.

I have a problem: Let $f$ be a continuous function on the unit circle $(\Gamma)$: $$\Gamma=\{e^{i\theta}: \theta\in [0, 2 \pi]\}$$ Assume that $f \ne 0$ on $\Gamma$, and the Fourier ...
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112 views

Prove the approximate identity from the unitization

Suppose $A$ is a $C^*$-algebra without unit, $A^+$ is a unitization of $A$, treat $A$ in the $A^+$, if $\{x_n\}$ in $A$ converge (or monotonous converge) to $1$ in $A^+$, does $\{x_n\}$ must be the ...
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1answer
93 views

Why is there an element with infinite spectrum in a commutative Banach algebra with infinitely many characters?

Let $A$ be a commutative Banach algebra such that set of all characters is infinite. I want to prove that there exists an element in $A$ such that its spectrum is infinite.
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1answer
219 views

Are nilpotent Lie groups unimodular?

The continuous homomorphism $\Delta:G \rightarrow \mathbb{R}^{\times}_+$ is defined by \begin{equation*} \int_G f(xy)dx = \Delta(y)\int_Gf(x)dx \end{equation*} where $dx$ is a left Haar measure on ...
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1answer
60 views

Spectrum in Hilbert space

Let $H$ be a Hilbert space and $T\in B(H)$ be normal. Want to show that if $\lambda$ is an isoloated point in $\sigma(T)$ then $\lambda$ is an eigenvalue of $T$.I have no idea how to do this one. ...
2
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2answers
91 views

Multiplication operators

Consider a commutative Banach algebra $A$ and the Banach algebra of bounded operators $B(A)$ on $A$. Associate to each $a\in A$ the multiplication operator $T_ax =ax$ ($x\in A$). Is always the mapping ...
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1answer
28 views

Condition in the definition of Banach star algebra

Here the definition of Banach star algebra is given as Banach algebra with an involution. In the book by Murphy for example, it is given as Banach algebra with an involution plus the condition that ...
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1answer
28 views

Approximating a mean on $L^{\infty}(G)$ by a norm $1$ non-negative $f\in L^{1}(G)$

Let $G$ be a locally compact group. A mean $M$ on $L^{\infty}(G)$ is a continuous linear functional on $L^{\infty}(G)$ such that $1 = \langle 1 , M\rangle = \|M\|$. My Exercise: Show that the set ...
2
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1answer
68 views

Convergence Radius => Nonanalytic

Why is a function certainly nonanalytic at some point on the radius of convergence? I mean considering a power series around somewhere and if theres a power series expansion at every point on circle ...
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1answer
33 views

Why is $F_\phi$ defined on the whole disk

This is a question about a proof on page 97 in these lecture notes. In exercise 13, I don't understand On the hand, $F_\phi$ is defined on the whole open disk $D$ Why is $F_\phi$ defined on ...
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1answer
90 views

Ideals and exactness of projective tensor product of Banach spaces / algebras

Thanks for suggesting this question: Image of the tensor product of strict maps of Banach spaces I read the reference and realize that for a short exact sequence of Banach algebra: $0 \to J \to A \to ...
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1answer
54 views

How do we show that prime C* algebras have trivial center

A prime C* algebra is a C* algebra with the property that the product of any two of its non zero ideals is non zero. The claim is that it has trivial center, i.e., the only central elements are ...