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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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
51 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 ...
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1answer
40 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
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1answer
76 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
45 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
53 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
109 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 ...
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1answer
41 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-* ...
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vote
1answer
89 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
46 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 ...
6
votes
1answer
179 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
84 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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vote
1answer
58 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
votes
1answer
58 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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0answers
74 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 ...
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1answer
132 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 $T$ ...
3
votes
0answers
116 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
votes
1answer
274 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 ...
1
vote
1answer
28 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 ...
3
votes
1answer
84 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
153 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 ...
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votes
2answers
82 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
1answer
77 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
votes
1answer
317 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 ...
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votes
0answers
59 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
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0answers
67 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 ...
1
vote
1answer
73 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 ...
6
votes
1answer
137 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
82 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
178 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
109 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
129 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 ...
5
votes
1answer
61 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
191 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 ...
6
votes
1answer
198 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 ...
0
votes
0answers
35 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 ...
5
votes
1answer
230 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
55 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. ...
3
votes
2answers
128 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 $ ...
4
votes
0answers
108 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
172 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
117 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
93 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
vote
1answer
251 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
152 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 $$\exp(x)=\sum_{n=0}^{\infty}\frac{x^n}{n!}.$$
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votes
2answers
66 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
71 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
vote
2answers
175 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
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1answer
38 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
votes
0answers
43 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 ...