# Tagged Questions

1answer
39 views

### A problem on bounded invertible linear operator in Banach space

Let $X$ be a Banach space. Let $T : X \to X$ be a invertible linear operator and $M > 0$ be such that $\|T^{-k}\| \le M$ for all $k \ge 1$. Prove that $\inf_ {n\ge1} \|T^n(x)\| > 0$ for all \$x ...
1answer
20 views

### Multiplicative functionals on Banach algebra closed in weak-* topology

Let $\mathcal A$ be commutative unital Banach algebra denote by $M(A)$ the space of all non-zero multiplicative functionals on $\mathcal A$. I want to show that $M(A)$ is closed in the weak-* ...
1answer
30 views

### Gelfand transform and spectrum

Let $\mathcal A$ commutative, unital Banach algebra and denote by $\mathcal M(\mathcal A)$ the space of multiplicative functionals on $\mathcal A$. The Gelfand transform is defined by \Gamma: ...
1answer
32 views

### Derivative of norm on Banach algebra

Let $\mathcal A$ be a unital Banach algebra. I want to consider $f(z):= \vert \vert e^{-zA}Be^{zA} \vert \vert, z\in \mathbb C$ and $A,B \in \mathcal A$. How can I properly define the derivative of ...
0answers
56 views

### Maximal ideals in the algebra of continuously differentiable functions on [0,1]

This is an exercise in Rudin's Functional Analysis, in the chapter on commutative Banach algebras. My (uneducated) guess was that every homomorphism on $C^{1}[0,1]$ is an evaluation at some point of ...
1answer
32 views

### Norm product inequality

The following is about a proof in Bratteli Robinson vol 1. Let $\mathcal{A}$ be some C*-algebra. Show that $$\mathcal{B}=\{(A,\alpha)~|~A\in\mathcal{A}, \alpha\in\mathbb{C}\}$$ together with the norm ...
1answer
27 views

### Banach algebra: norm distance of non-invertible elements to unit element

Let $\mathcal A$ be a commutative, unital Banach algebra. Take $A \in \mathcal A$ such that $A$ is non-scalar, i.e. $A\neq \alpha \mathbb I$, where $\mathbb I$ is the unit element. Denote the ...
0answers
48 views

### Unitary equivalent

In general, if two irreducible representations of a $C^*$-algebra have the same kernel we can say this two representations are approximately unitarily equivalent. When our $C^*$-algebra is GCR, how to ...
1answer
47 views

### Continuous functional calculus

Let $\mathscr H$ be a Hilbert space, and $\mathscr B(\mathscr H)$ is a $C^*$-algebra, $T\in \mathscr B(\mathscr H)$ is a normal operator. Let $C^*(T)$ denote the $C^*$- subalgebra generated by $T$ ...
1answer
127 views

### Maximal abelian subalgebra of Banach algebra is closed and contains the unity

I'm studying Murphy's book: C*-Algebras and Operator Theory, and got stuck in exercise 8 from chapter 1: "Show that if $B$ is a maximal abelian subalgebra of a unital Banach algebra $A$, then $B$ is ...
1answer
57 views

### Algebra (Not *)-Isomorphisms of von Neumann algebras

Let $A$ and $B$ be any two infinite-dimensional von Neumann algebras, they are operator algebras with operator composition as the multiplication and as infinite dimensional vector spaces they're ...
0answers
55 views

### Proving properties of exponential map on a Banach algebra

$$\exp(a) := \sum\frac {a^k}{k!}$$ Can you help me prove that: $\exp$ is well defined (i.e. converges for all $a$ in $A$) $\exp$ is continuous $\exp(A)$ is a subset of $A_0$ (where $A_0$ is the ...
2answers
59 views

### Banach algebra problem?

Let $A$ be a Banach algebra and let $$A_1=\{(x,\alpha)\;;\;:x∈A, \alpha\in\mathbb{C}\}$$ with the following operations:  (x_1,\alpha_1 )+(x_2,\alpha_2 )=(x_1+x_2 ,\alpha_1+\alpha_2 )\qquad ...
0answers
59 views

### In relation with the set of polynomially 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 polynomially Fredholm perturbation elements in $A$, i.e. ...
0answers
62 views

### Open map in Banach algebra

I'm having trouble showing a certian function is open and can be extended. Let $\Omega$ be a completely regular topological space and $A=C_b(\Omega)$ the space of all complex-values bounded ...
1answer
74 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 ...
1answer
67 views

### Properties of $\text{Exp}(A)$, where $A$ is a Banach algebra.

$\newcommand{\Exp}{\operatorname{Exp}}$ Let $A$ be a unital Banach algebra. For $a \in A$, consider  \Exp(A) \stackrel{\text{def}}{=} \{ e^{a_{1}} e^{a_{2}} \cdots e^{a_{n}} ~|~ n \in ...
1answer
26 views

### Why locally compact in the Gelfand representation?

I'm missing something in the Gelfand representation. Let's just say $\mathfrak{A}$ is a Banach algebra. Then it's a Banach space, and so we have $\mathfrak{A}^\ast$. The multiplicative linear ...
1answer
85 views

### Polar decomposition of invertible elements in a unital C$^{*}$-algebra.

If $A$ is a unital C$^{*}$-algebra and $a$ is invertible, then $a = u|a|$ for a unique unitary element $u$ of $A$. If $\| a \| = \| a^{-1} \| = 1$, what can you say about $|a|$? I ...
1answer
111 views

### On the spectrum of the sum of two commuting elements in a Banach algebra

Original: Soit A une algèbre de Banach unitaire et a et b deux éléments tels que a*b=b*a. Pourquoi σ (a+b) с σ(a)+σ(b) Et qu’elle est la relation entre σ (a*b) et σ(a) et σ(b)? Translation: Let ...
1answer
99 views

### Spectral radius in Banach Algebra

Let $A$ be a unital Banach algebra and $a\in A$ and $\lambda \in \rho(a)$. I want to prove that $$r(R(a,\lambda))=\frac{1}{d(\lambda,\sigma(a))}.$$ where $R(a,\lambda)=(\lambda 1-a)^{-1}$ and $r(.)$ ...
2answers
94 views

### Banach-algebra homeomorphism.

Let $A$ be a commutative unital Banach algebra that is generated by a set $Y \subseteq A$. I want to show that $\Phi(A)$ is homeomorphic to a closed subset of the Cartesian product \$ ...
0answers
35 views

### Biduals generated by projections

This question is motivated by a similar question recently posed at MO: http://mathoverflow.net/questions/122091/masas-in-second-duals-of-banach-algebras In this setting, let $B$ be a Banach algebra ...
2answers
141 views

### Turning Banach space into Banach algebra

Given a Banach space, how can we determine if we can turn it into a Banach algebra or not?
1answer
67 views

### Prove that $L^1$ is a Banach algebra with multiplication defined by convolution

To be more specific, prove that $L^1(\mathbb{R}^n)$ with multiplication defined by convolution: $$(f\cdot g)(x)=\int_\mathbb{R^n}f(x-y)g(y)dy$$ is a Banach algebra. All the properties of Banach ...
1answer
67 views

### The exponential function of Banach algebra

I am wondering how to prove the following question: In any unital Banach algebra, we have $\exp(x+y)=\exp(x)\exp(y)$, if $xy=yx$, where $$\exp(x)=\sum_{n=0}^{\infty}\frac{x^n}{n!}.$$
2answers
115 views

### Left topological zero-divisors in Banach algebras.

Let $A$ be a unital Banach algebra. Define $\zeta: A \longrightarrow [0,\infty)$ by $$\forall a \in A: \quad \zeta(a) \stackrel{\text{def}}{=} \inf_{b \in \mathbb{S}(A)} \| ab \|,$$ where \$ ...
2answers
69 views

### ‎‎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 ...
0answers
84 views

### invariant subspace of a Hardy space

Let $T$ be the unit circle and $H^1=\{f\in L^1(T): \int_0^{2\pi} f(e^{it})\chi_n(e^{it})dt=0 \text{ for } n>0\}$ where $\chi_n(e^{it})=e^{int}$. Let $M$ be a closed subspace of $H^1$. Then ...
1answer
128 views

### Fourier transform as a Gelfand transform

One question came to my mind while looking at the proof of Gelfand-Naimark theorem. Is Fourier transform a kind of Gelfand transform? Are there any other well-known transforms which are so?
2answers
120 views

### Stone-Čech via $C_b(X)\cong C(\beta X)$

I am having some trouble constructing the Stone-Čech compactification of a locally compact Hausdorff space $X$ using theory of $C^*$-algebras. I did some search but could not find a good answer on ...
2answers
63 views

### Characterization of small Banach subalgebras

Let $A$ be a unital Banach algebra and $x \in A$ nonzero. We can consider the subalgebra $B$ of $A$ generated by $\{1,x\}$. This is the norm closure of the subspace of polynomials in $x$. So for any ...
2answers
95 views

### $(\lambda-a)^{-1}$ as limits of 'polynomials'

For a unital $C^*$-algebra $\mathcal{A}$ the spectral permanence gives \begin{equation} \sigma_{\mathcal{B}}(a)=\sigma_{\mathcal{A}}(a) \end{equation} for any unital $C^*$-subalgebra $\mathcal{B}$. ...
1answer
73 views

### Spectrum of elements in $C^*$-subalgebras

Assume $\mathcal{A}$ is a $C^*$-algebra with unit $1$ and $\mathcal{B}\subset\mathcal{A}$ is a $C^*$-subalgebra (i.e. a closed $*$-subalgebra) such that $1\in\mathcal{B}$. It is said that under these ...
1answer
148 views

### $\sigma(x)$ has no hole in the algebra of polynomials

Let $A$ be a unital banach algebra generated by two elements $1$ and $x$. Then it seems $\sigma(x)$ cannot have holes. At least this is true in the case for disk algebras. This amounts to prove that ...
2answers
174 views

### $Conv(Ex((C(X))_1))$ is dense in $(C(X))_1$?

Let $X$ be compact Hausdorff, and $C(X)$ the space of continuous functions over $X$. Denote the closed unit ball in $C(X)$ by $(C(X))_1$, then it can be shown $f$ is an extreme point of $(C(X))_1$ if ...
0answers
89 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 ...
2answers
55 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 ...
0answers
91 views

### Decomposing $\mathcal{B}(H)$

Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I ...
1answer
169 views

### spectral radius.

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 ...
1answer
344 views

### Some examples in C* algebras and Banach * algebras

I would like an example of the following things. A Banach * algebra that is not a C* algebra for which there exists a positive linear functional (it takes $x^*x$ to numbers $\geq 0$) that is not ...
1answer
93 views

### Universal separable Banach algebras

The well-known Banach-Mazur theorem says that $C([0, 1])$ is a universal separable Banach space, in the sense that if $X$ is any separable Banach space then there is a map $f : X \to C([0, 1])$ which ...
0answers
97 views

### When is a Banach Algebra stellar?

I know that if there are enough Hermitian elements in a Banach algebra, then the Banach algebra is stellar. In particular, I'm interested in the two spaces $B(L^1(S^1,\Sigma,\mu))$ the space of ...
1answer
112 views

### Schwarz inequality for unital completely positive maps

I came across the following form of Schwarz inequality for completely positive maps in Arveson's paper: Let $\delta:\mathcal{A}\to\mathcal{B}$ be a unital completely positive linear map between ...
2answers
137 views

### Why are compact operators 'small'?

I have been hearing different people saying this in different contexts for quite some time but I still don't quite get it. I know that compact operators map bounded sets to totally bounded ones, that ...
0answers
45 views

### Topology of $(\mathcal{A},*)$ determined by $\mathcal{A}_{sa}$?

Let $(\mathcal{A},*)$ be a $*$-algebra, we have the following observation: Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two norms on $\mathcal{A}$ such that the involution is an isometry with respect to ...
2answers
80 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 ...
1answer
79 views

### Every ideal has an approximate identity?

Averson's 1970 paper on extensions of $C^*$-algebras seems to assume that every ideal has an approximate identity. However, I am a little bit suspicious here, since he does not assume the closeness ...
2answers
127 views

### Linear functionals can be decomposed as linear combinations of positive ones?

I am reading Arveson's Notes on Extensions of $C^*$-algebras. In proving theorem 1, he needs to establish some results concerning bounded linear functionals. However, he said it suffices to prove for ...
2answers
85 views

### If $a\ge 0$ and $b\ge 0$, then $\sigma(ab)\subset\mathbb{R}^+$.

This is an exercise in Murphy's book: Let $A$ be a unital $C^*$-algebra and $a,b$ are positive elements in $A$. Then $\sigma(ab)\subset\mathbb{R}^+$. The problem would be trivial if the algebra ...
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