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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1answer
25 views
Absolute value of an element in a C*-algebra
Is absolute value of a partial isometry a partial isometry itself?
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1answer
21 views
The spectrum of an invertible element $x$ is $\sigma(x^{-1})=\{\lambda^{-1}: \lambda\in \sigma(x)\}$
Suppose $x$ is invertible in the unital Banach algebra $A$.
How can I prove that $\sigma(x^{-1})=\{\lambda^{-1} : \lambda\in \sigma(x)\}$
4
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1answer
43 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
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1answer
21 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
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1answer
31 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
$$\Gamma: ...
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1answer
34 views
Derivative of norm on Banach algebra
Let $\mathcal A$ be a unital Banach algebra. I want to consider $f(z):= \vert \vert e^{-zA}Be^{zA} \vert \vert, z\in \mathbb C$ and $A,B \in \mathcal A$.
How can I properly define the derivative of ...
4
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0answers
61 views
Maximal ideals in the algebra of continuously differentiable functions on [0,1]
This is an exercise in Rudin's Functional Analysis, in the chapter on commutative Banach algebras. My (uneducated) guess was that every homomorphism on $C^{1}[0,1]$ is an evaluation at some point of ...
2
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1answer
33 views
Norm product inequality
The following is about a proof in Bratteli Robinson vol 1.
Let $\mathcal{A}$ be some C*-algebra. Show that $$\mathcal{B}=\{(A,\alpha)~|~A\in\mathcal{A}, \alpha\in\mathbb{C}\}$$ together with the norm ...
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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 ...
-3
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0answers
47 views
Closure of the set of Fredholm perturbations
fermeture de l'ensemble de perturbation des fredholms
soit A,B deux algébres de banach et T un homomorphisme continue de A dans B
soit a dans A
a est dit de Fredholm si T(a) inversible dans B
on ...
3
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1answer
39 views
Does *-operator be automatically continous
In the C*-algebras, does the * -operator be automatically continous? I think it is yes, because C*-algebras are semisimple, from the Johnson Theorem, it must be automatically continous. Am I right?
...
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left,right,and boundary
let A a banach algebra .
let x is an element of A
let denote σ(x) the spectrum of x ,
σleft(x) the left spectrum of x
σright(x)the right spectrum of x
∂σ(x) the boundary of σ(x)
i need help to ...
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0answers
49 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
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1answer
57 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$ ...
4
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0answers
64 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 ...
6
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1answer
128 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 ...
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1answer
26 views
simply polar elements in a ring
An element $a$ in a ring $A$ with identity is said to be simply polar if there is $b$ for which $a=aba$, with $ab=ba$.
If in addition $b=bab$ then such an element $b$ is unique.
The question is ...
2
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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
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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
...
2
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1answer
54 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 ...
5
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1answer
141 views
strictly positive elements in $C^*$-algebra
Let $A$ be a $C^*\text{-algebra}$ and $A_+$ denote the positive elements. An element $a\in A_+$ is called strictly positive if $\overline{aAa}=A$. Want to find the following:a)What are the strictly ...
3
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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.
...
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1answer
34 views
Prime ideal for the Banach algebra
The maximal ideal and Jacobson radical often appear in the Banach algebra theory, but I do not see the prime and nilradical in it.
We can define a prime for a Banach algebra following the ring ...
5
votes
0answers
75 views
Tensor product of algebra
Can we find define a norm on tensor product $C(X) \otimes C(Y)$ such that the norm completion of $C(X)\otimes C(Y)=C(X\times Y)$
And can we define a norm on tensor product $L^1(X)\otimes L^1(Y)$ such ...
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
75 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
$$
\Exp(A) \stackrel{\text{def}}{=} \{ e^{a_{1}} e^{a_{2}} \cdots e^{a_{n}} ~|~ n \in ...
3
votes
0answers
103 views
Topology of maximal ideal space
We know that strictly positive integers with multiplication form a semigroup $S$. Let $\mathscr A=\ell ^1(S)$ with convolution. What is $\mathscr A$'s maximal ideal space?
It seems enough to find ...
3
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1answer
28 views
Why locally compact in the Gelfand representation?
I'm missing something in the Gelfand representation. Let's just say $\mathfrak{A}$ is a Banach algebra. Then it's a Banach space, and so we have $\mathfrak{A}^\ast$. The multiplicative linear ...
3
votes
1answer
85 views
Polar decomposition of invertible elements in a unital C$ ^{*} $-algebra.
If $ A $ is a unital C$ ^{*} $-algebra and $ a $ is invertible, then
$ a = u|a| $ for a unique unitary element $ u $ of $ A $.
If $ \| a \| = \| a^{-1} \| = 1 $, what can you say about $ |a| $?
I ...
4
votes
1answer
111 views
On the spectrum of the sum of two commuting elements in a Banach algebra
Original: Soit A une algèbre de Banach unitaire et a et b deux éléments tels que a*b=b*a.
Pourquoi σ (a+b) с σ(a)+σ(b)
Et qu’elle est la relation entre σ (a*b) et σ(a) et σ(b)?
Translation: Let ...
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0answers
33 views
Stone-Čech compactification [duplicate]
I'm looking for the proof of Stone-Čech compactification for the following Banach algebra $A=C_b(\Omega)$ where $\Omega$ is a completly regular space and $C_b(\Omega)$ is the space of all bounded ...
6
votes
1answer
99 views
Spectral radius in Banach Algebra
Let $A$ be a unital Banach algebra and $a\in A$ and $\lambda \in \rho(a)$. I want to prove that $$r(R(a,\lambda))=\frac{1}{d(\lambda,\sigma(a))}.$$ where $R(a,\lambda)=(\lambda 1-a)^{-1}$ and $r(.)$ ...
2
votes
1answer
45 views
Spectrum in Hilbert space
Let $H$ be a Hilbert space and $T\in B(H)$ be normal. Want to show that if $\lambda$ is an isoloated point in $\sigma(T)$ then $\lambda$ is an eigenvalue of $T$.I have no idea how to do this one. ...
4
votes
2answers
94 views
Banach-algebra homeomorphism.
Let $ A $ be a commutative unital Banach algebra that is generated by a set $ Y \subseteq A $. I want to show that $ \Phi(A) $ is homeomorphic to a closed subset of the Cartesian product $ ...
3
votes
0answers
35 views
Biduals generated by projections
This question is motivated by a similar question recently posed at MO:
http://mathoverflow.net/questions/122091/masas-in-second-duals-of-banach-algebras
In this setting, let $B$ be a Banach algebra ...
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votes
2answers
141 views
Turning Banach space into Banach algebra
Given a Banach space, how can we determine if we can turn it into a Banach algebra or not?
1
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1answer
67 views
Prove that $L^1$ is a Banach algebra with multiplication defined by convolution
To be more specific, prove that $L^1(\mathbb{R}^n)$ with multiplication defined by convolution:
$$
(f\cdot g)(x)=\int_\mathbb{R^n}f(x-y)g(y)dy
$$
is a Banach algebra. All the properties of Banach ...
4
votes
2answers
77 views
Commutativity in a Unital Banach Algebra
Let $ A $ be a unital Banach algebra and $ S $ a non-empty subset of $ A $. The centralizer of $ S $ is defined as
$$
Z(S) \stackrel{\text{def}}{=} \{ a \in A ~|~ \text{$ as = sa $ for all $ s \in S ...
1
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1answer
128 views
The maximal ideal space of a Banach algebra
Let $G = \mathbb Z^n$, the fi nite cyclic group of order $n$, and let $A = l ^1(G)$, a
Banach algebra over $\mathbb C$ when the product is convolution, de fined for $f, g \in A$ by $f * g(x) =\sum ...
0
votes
1answer
68 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 $$\exp(x)=\sum_{n=0}^{\infty}\frac{x^n}{n!}.$$
0
votes
2answers
41 views
Invertible products in Banach algebras
I found this interesting challenge: give an example of an unital Banach algebra that contains two elements $x$ and $y$ such that $xy$ is invertible but $yx$ is not invertible.
I thought it would be ...
3
votes
1answer
63 views
Double centralizers in the Murphy book
I've been into this for days and days and I still can't see why, given the definition of $L^\ast$ as $L^\ast =(L(a^\ast))^\ast$ we get that $(LM)^\ast =L^\ast M^\ast$. Where is my mistake:
...
1
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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 $ ...
0
votes
1answer
33 views
$\sigma(a)=\sigma(b)$, if $a,b$ are unitarily equivalent
Let $A$ be a *-algebra and $a,b$ are unitarily equivalent in $A$ ( i.e. there exists a unitary $u$ of $A$ s.t $b=uau^{*}$ ).
I want to prove that ...
0
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0answers
39 views
$\widehat{a}: \Omega(A)\rightarrow \mathbb{C}~,~\tau \mapsto \tau(A) $
Suppose that $A$ is abelian Banach algebra for which the space $\Omega(A)$ is non-empty. If $a \in A$, we define the function $\widehat{a}$ by $$\widehat{a}: \Omega(A)\rightarrow ...
1
vote
1answer
33 views
Gelfand representation Theorem
In proof of "Gelfand representation Theorem" (see 1.3.6 Theorem of Murphy's book ), I am understanding that why the map $$ A \rightarrow C_{0}(\Omega(A))~ , ~a\rightarrow ...
0
votes
1answer
120 views
Is the space of bounded functions with the Supremum norm a Banach Algebra
X is an arbitrary , non empty set, B(X) the set of bounded functions $f:X\rightarrow \mathbb{R}$ and $||f||_\infty = \sup_{x\in X }|f(x)|$.
Is $(B(X),||.||_\infty )$ a Banach Algebra?
My attempt ...
1
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1answer
107 views
If $X$ is an infinite-dimensional Banach space and $u\in B(X)$ ,then $\bigcap_{v\in K(X)}\sigma(u+v) =\cdots$
If $X$ is an infinite-dimensional Banach space and $u\in B(X)$,why the following equality is true?
$$\bigcap_{v\in K(X)}\sigma(u+v) =\sigma(u) \setminus \lambda \in\mathbb{C}\mid u - ...
