Tagged Questions
4
votes
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 ...
1
vote
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-* ...
1
vote
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
$$\Gamma: ...
0
votes
1answer
32 views
Derivative of norm on Banach algebra
Let $\mathcal A$ be a unital Banach algebra. I want to consider $f(z):= \vert \vert e^{-zA}Be^{zA} \vert \vert, z\in \mathbb C$ and $A,B \in \mathcal A$.
How can I properly define the derivative of ...
4
votes
0answers
56 views
Maximal ideals in the algebra of continuously differentiable functions on [0,1]
This is an exercise in Rudin's Functional Analysis, in the chapter on commutative Banach algebras. My (uneducated) guess was that every homomorphism on $C^{1}[0,1]$ is an evaluation at some point of ...
2
votes
1answer
32 views
Norm product inequality
The following is about a proof in Bratteli Robinson vol 1.
Let $\mathcal{A}$ be some C*-algebra. Show that $$\mathcal{B}=\{(A,\alpha)~|~A\in\mathcal{A}, \alpha\in\mathbb{C}\}$$ together with the norm ...
1
vote
1answer
27 views
Banach algebra: norm distance of non-invertible elements to unit element
Let $\mathcal A$ be a commutative, unital Banach algebra. Take $A \in \mathcal A$ such that $A$ is non-scalar, i.e. $A\neq \alpha \mathbb I $, where $\mathbb I$ is the unit element. Denote the ...
1
vote
0answers
48 views
Unitary equivalent
In general, if two irreducible representations of a $C^*$-algebra have the same kernel we can say this two representations are approximately unitarily equivalent. When our $C^*$-algebra is GCR, how to ...
1
vote
1answer
47 views
Continuous functional calculus
Let $\mathscr H$ be a Hilbert space, and $\mathscr B(\mathscr H)$ is a $C^*$-algebra, $T\in \mathscr B(\mathscr H)$ is a normal operator. Let $C^*(T)$ denote the $C^*$- subalgebra generated by $T$ ...
6
votes
1answer
127 views
Maximal abelian subalgebra of Banach algebra is closed and contains the unity
I'm studying Murphy's book: C*-Algebras and Operator Theory, and got stuck in exercise 8 from chapter 1:
"Show that if $B$ is a maximal abelian subalgebra of a unital Banach algebra $A$, then $B$ is ...
2
votes
1answer
57 views
Algebra (Not *)-Isomorphisms of von Neumann algebras
Let $A$ and $B$ be any two infinite-dimensional von Neumann algebras, they are operator algebras with operator composition as the multiplication and as infinite dimensional vector spaces they're ...
1
vote
0answers
55 views
Proving properties of exponential map on a Banach algebra
$$\exp(a) := \sum\frac {a^k}{k!}$$
Can you help me prove that:
$\exp$ is well defined (i.e. converges for all $a$ in $A$)
$\exp$ is continuous
$\exp(A)$ is a subset of $A_0$ (where $A_0$ is the ...
0
votes
2answers
59 views
Banach algebra problem?
Let $A$ be a Banach algebra and let
$$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
...
3
votes
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.
...
4
votes
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 ...
2
votes
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 ...
3
votes
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
$$
\Exp(A) \stackrel{\text{def}}{=} \{ e^{a_{1}} e^{a_{2}} \cdots e^{a_{n}} ~|~ n \in ...
3
votes
1answer
26 views
Why locally compact in the Gelfand representation?
I'm missing something in the Gelfand representation. Let's just say $\mathfrak{A}$ is a Banach algebra. Then it's a Banach space, and so we have $\mathfrak{A}^\ast$. The multiplicative linear ...
3
votes
1answer
85 views
Polar decomposition of invertible elements in a unital C$ ^{*} $-algebra.
If $ A $ is a unital C$ ^{*} $-algebra and $ a $ is invertible, then
$ a = u|a| $ for a unique unitary element $ u $ of $ A $.
If $ \| a \| = \| a^{-1} \| = 1 $, what can you say about $ |a| $?
I ...
4
votes
1answer
111 views
On the spectrum of the sum of two commuting elements in a Banach algebra
Original: Soit A une algèbre de Banach unitaire et a et b deux éléments tels que a*b=b*a.
Pourquoi σ (a+b) с σ(a)+σ(b)
Et qu’elle est la relation entre σ (a*b) et σ(a) et σ(b)?
Translation: Let ...
6
votes
1answer
99 views
Spectral radius in Banach Algebra
Let $A$ be a unital Banach algebra and $a\in A$ and $\lambda \in \rho(a)$. I want to prove that $$r(R(a,\lambda))=\frac{1}{d(\lambda,\sigma(a))}.$$ where $R(a,\lambda)=(\lambda 1-a)^{-1}$ and $r(.)$ ...
4
votes
2answers
94 views
Banach-algebra homeomorphism.
Let $ A $ be a commutative unital Banach algebra that is generated by a set $ Y \subseteq A $. I want to show that $ \Phi(A) $ is homeomorphic to a closed subset of the Cartesian product $ ...
3
votes
0answers
35 views
Biduals generated by projections
This question is motivated by a similar question recently posed at MO:
http://mathoverflow.net/questions/122091/masas-in-second-duals-of-banach-algebras
In this setting, let $B$ be a Banach algebra ...
0
votes
2answers
141 views
Turning Banach space into Banach algebra
Given a Banach space, how can we determine if we can turn it into a Banach algebra or not?
1
vote
1answer
67 views
Prove that $L^1$ is a Banach algebra with multiplication defined by convolution
To be more specific, prove that $L^1(\mathbb{R}^n)$ with multiplication defined by convolution:
$$
(f\cdot g)(x)=\int_\mathbb{R^n}f(x-y)g(y)dy
$$
is a Banach algebra. All the properties of Banach ...
0
votes
1answer
67 views
The exponential function of Banach algebra
I am wondering how to prove the following question:
In any unital Banach algebra, we have $\exp(x+y)=\exp(x)\exp(y)$, if $xy=yx$, where $$\exp(x)=\sum_{n=0}^{\infty}\frac{x^n}{n!}.$$
1
vote
2answers
115 views
Left topological zero-divisors in Banach algebras.
Let $ A $ be a unital Banach algebra. Define $ \zeta: A \longrightarrow [0,\infty) $ by
$$
\forall a \in A: \quad \zeta(a) \stackrel{\text{def}}{=} \inf_{b \in \mathbb{S}(A)} \| ab \|,
$$
where $ ...
2
votes
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 ...
4
votes
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 ...
4
votes
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?
6
votes
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 ...
4
votes
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 ...
4
votes
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 $\mathcal{B}$.
...
3
votes
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 ...
4
votes
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 ...
1
vote
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 ...
3
votes
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 ...
2
votes
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 ...
6
votes
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 ...
2
votes
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 ...
6
votes
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 ...
4
votes
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 ...
0
votes
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 ...
4
votes
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 ...
5
votes
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 ...
2
votes
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 ...
2
votes
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 ...
1
vote
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 ...
0
votes
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 ...
3
votes
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 ...
