Tagged Questions
3
votes
2answers
107 views
Banach Algebra counterexample
Can someone give me an exemple of a Banach Algebra $\mathbb{A}$, for which there is no isometric representation in a Hilbert Space ? (with proof or references to proof)
Thank you very much :)
1
vote
1answer
69 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
votes
1answer
33 views
limit of evaluated automorphisms in a Banach algebra
Let $\mathcal{A}=\operatorname{M}_k(\mathbb{R})$ be the Banach algebra of $k\times k$ real matrices and let $(U_n)_{n\in\mathbb{N}}\subset\operatorname{GL}_k(\mathbb{R})$ be a sequence of invertible ...
1
vote
0answers
63 views
Approximation of certain continuous functions by analytic functions
Let $f\in C(S^{1},M_{n}(\mathbb{C}))$ be a unitary. Does there exist an analytic unitary function $g$ from $S^{1}$ to $M_{n}(\mathbb{C})$ that approximates $f$?
0
votes
2answers
53 views
The maximal ideal space of $A$ is contained in the unit ball of $A^\ast$
On page 281 of Rudin's book Functional Analysis, let $\Delta$ be the maximal ideal space of a commutative Banach algebra $A$, $K$ be the norm-closed unit ball of $A^*$, then $\Delta\subset K$ (by ...
1
vote
0answers
39 views
$\ast$-homomorphism
Let $\phi: C(X,M_{4}(\mathbb{C})) \rightarrow C(Y,M_{8}(\mathbb{C})) $ be a $\ast$-homomorphism where $X$ and $Y$ are compact Hausdorff spaces. Let $M_{2}(\mathbb{C})$ be the C*-subalgebra of ...
2
votes
0answers
50 views
A semisimple commutative Banach algebra with a non-semisimple quotient
I am looking for an example of a semisimple commutative Banach algebra which admits a non-semisimple quotient.
Attempt from the comments:
"I take $A$ to be the algebra of all continuously ...
2
votes
1answer
76 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.
2
votes
1answer
39 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 ...
0
votes
1answer
36 views
Real commutative Banach algebra with identity
I am looking for example of real commutative Banach algebra with identity which does not admit a nonzero real multiplicative linear functional
-1
votes
1answer
57 views
Spectrum of a unitary
I have a unitary element $v$ in $C(S^{1}, \mathbb{C})$ with full spectrum (the whole circle). Is it possible to construct another unitary $u$ in $C(S^{1}, \mathbb{C})$ out of $v$ such that the ...
1
vote
1answer
30 views
Absolute value of an element in a C*-algebra
Is absolute value of a partial isometry a partial isometry itself?
4
votes
1answer
52 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
26 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
36 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
35 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
73 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
36 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
32 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
53 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
62 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
151 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
61 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
58 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
63 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:
$$
(x_1,\alpha_1 )+(x_2,\alpha_2 )=(x_1+x_2 ,\alpha_1+\alpha_2 )\qquad
...
2
votes
0answers
53 views
The group algebra is separable
Let $G$ be a locally compact Hausdorff group. Is the group algebra $L^1(G)$ separable?
Would you please help me or introduce references that can help me.
Thanks
3
votes
0answers
59 views
In relation with the set of polynomially Fredholm perturbation elements
Let $A$ and $B$ be two unital Banach algebras and $T\colon A\to B$ an homomorphism of Banach algebras. Let denote the set of polynomially Fredholm perturbation elements in $A$, i.e.
...
4
votes
0answers
64 views
Open map in Banach algebra
I'm having trouble showing a certian function is open and can be extended.
Let $\Omega$ be a completely regular topological space and $A=C_b(\Omega)$ the space of all complex-values bounded ...
2
votes
1answer
81 views
Are nilpotent Lie groups unimodular?
The continuous homomorphism $\Delta:G \rightarrow \mathbb{R}^{\times}_+$ is defined by
\begin{equation*}
\int_G f(xy)dx = \Delta(y)\int_Gf(x)dx
\end{equation*}
where $dx$ is a left Haar measure on ...
1
vote
1answer
72 views
Properties of $ \text{Exp}(A) $, where $ A $ is a Banach algebra.
$ \newcommand{\Exp}{\operatorname{Exp}} $ Let $ A $ be a unital Banach algebra. For $ a \in A $, consider
$$
\Exp(A) \stackrel{\text{def}}{=} \{ e^{a_{1}} e^{a_{2}} \cdots e^{a_{n}} ~|~ n \in ...
3
votes
1answer
34 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 ...
2
votes
1answer
94 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
118 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 ...
5
votes
1answer
103 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 $r(.)$ ...
3
votes
2answers
102 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
36 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
144 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
71 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
75 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
120 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
70 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
87 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 ...
5
votes
1answer
138 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
126 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
66 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
78 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
152 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
177 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
90 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 ...
