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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39
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2answers
2k views

The identity cannot be a commutator in a Banach algebra?

The Wikipedia article on Banach algebras claims, without a proof or reference, that there does not exist a (unital) Banach algebra $B$ and elements $x, y \in B$ such that $xy - yx = 1$. This is ...
4
votes
1answer
711 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 ...
12
votes
1answer
1k views

Closure of the invertible operators on a Banach space

Let $E$ be a Banach space, $\mathcal B(E)$ the Banach space of linear bounded operators and $\mathcal I$ the set of all invertible linear bounded operators from $E$ to $E$. We know that $\mathcal I$ ...
10
votes
1answer
572 views

Why is $\ell^1(\mathbb{Z})$ not a $C^{*}$-algebra?

When $\ell^1(\mathbb Z)$ is equipped with the convolution as multiplication and $a^{*}_{n}=\bar{a}_{-n}$, I can prove it satisfies all conditions except $\|a^{*}a\|=\|a\|^2$, which I cannot prove nor ...
12
votes
2answers
981 views

Is there an algebraic homomorphism between two Banach algebras which is not continuous?

According to wikipedia, you need the Axiom of Choice to find a discontinuous map between two Banach spaces. Does this procedure also apply for Banach algebras yielding a discontinuous multiplicative ...
6
votes
1answer
621 views

Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $

Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $. Where $A,\ B$ are bounded operators in Banach space and $\sigma$ denotes spectrum.
5
votes
2answers
332 views

If $(I-T)^{-1}$ exists, can it always be written in a series representation?

If $X$ is a Banach space, and $T:X \to X$ is a bounded linear operator with norm < $1$, then $I-T$ has a bounded inverse defined by $(I-T)^{-1} = \sum_{n=0}^\infty T^n$. Thinking in terms of a ...
4
votes
1answer
371 views

Why is $GL(B)$ a Banach Lie Group?

Banach Lie Groups are what you'd expect: https://www.encyclopediaofmath.org/index.php/Lie_group,_Banach If $B$ is a Banach algebra then why is $GL(B)$, the set of invertible elements of $B$, a ...
7
votes
1answer
958 views

Reference request: Fourier and Fourier-Stieltjes algebras

I would like to learn the basic theory of Fourier algebras and Fourier-Stieltjes algebras. In particular, I want to know how these two objects are defined in the case of not necessarily abelian ...
9
votes
1answer
488 views

Maximal ideals and maximal subspaces of normed algebras

This is a kind of "prove or give a counter-example" question, and I'm having some difficults with it: By a maximal ideal $I$ of an algebra $A$, we mean an ideal $I\neq A$ which is not properly ...
8
votes
1answer
499 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 ...
8
votes
2answers
323 views

Does the Banach algebra $L^1(\mathbb{R})$ have zero divisors?

Assume that the functions $f,g: \mathbb{R}\rightarrow \mathbb{R}$ are integrable and equal to zero on $(-\infty,0)$, (i.e $f,g \in L^+$). Then by Titchmarsh's theorem: $f*g$ is zero almost everywhere ...
5
votes
1answer
295 views

Finding a closed subalgebra generated by functions.

Consider the space of all bounded continuous real-valued functions of $\mathbb{R}$. I am having trouble understanding how to find the closed subalgebra generated by sine and cosine.
8
votes
3answers
346 views

For which $s\in\mathbb R$, is $H^s(\mathbb T)$ a Banach algebra?

According to Theorem 4.39 in Adams & Fournier Sobolev Spaces: If $mp>n$, $m\in\mathbb N$, then $W^{m,p}(\Omega)$ is a Banach algebra, provided that $\Omega\subset\mathbb R^n$ satisfies the ...
6
votes
3answers
1k views

Is $L^2(\mathbb{R})$ with convolution a Banach Algebra?

Is $L^2(\mathbb{R})$ a Banach algebra, with convolution? I am pretty sure the answer is no, because I think that $f,g \in L^2(\mathbb{R})$ does not imply that $f*g \in L^2(\mathbb{R})$. However, I ...
2
votes
2answers
61 views

If $a\in\mathcal {A},$ and $\Vert a\Vert <1$, then $e-a$ is regular element of Banach algebra $\mathcal{A}$ with unit $e$.

I was reading an article yesterday which was silent on the algebra of Banach. In that article was provided this example If $a\in\mathcal {A},$ and $\Vert a\Vert <1$, then $e-a$ is regular element ...
0
votes
1answer
62 views

$C_{c}(X)$ is complete. then implies that $X$ is compact. [closed]

Let $X$ is locally compact Hausdorff space .If $C_{c}(X)$ is complet,then $X$ is compact (this is to be proved). I know that $C_{c}(X)$ is dense in $C_{0}(X)$. As $C_{c}(X)$ is complete implies that $...
1
vote
2answers
242 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 ...
0
votes
1answer
83 views

Is this equality true or it is not necessarily true?

Let $A$ and $B$ are two factor von neumann algebras that act on two infinite dimensional Hilbert spaces H and K respectively. Let $\Phi:A\longrightarrow B$ is an additive bijective map with some other ...
12
votes
1answer
776 views

Liouville's theorem for Banach spaces without the Hahn-Banach theorem?

Let $B$ be a (complex) Banach space. A function $f : \mathbb{C} \to B$ is holomorphic if $\lim_{w \to z} \frac{f(w) - f(z)}{w - z}$ exists for all $z$, just as in the ordinary case where $B = \mathbb{...
8
votes
1answer
286 views

When is the weighted space $\ell^p(\mathbb{Z},\omega)$ a Banach algebra ($p>1$)?

Let $\omega:\mathbb{Z}\to (0,\infty)$ and let $1\leq p<\infty$. Consider the space $\ell^p(\mathbb{Z},\omega)$ of complex valued sequences $f=(a_n)_{n \in \mathbb{Z}}$ such that $$\|f\|=\|f\|_{\...
6
votes
1answer
628 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?
3
votes
3answers
293 views

Spectral radius of an element in a C*-algebra

The following is an proposition of Takesaki's Operator Theory: For any element $x$ of a Banach algebra ${\cal A}$, we have $$||x||_{sp}=\lim_{n\to \infty}||x^n||^{\frac{1}{n}}$$ Proof: My ...
5
votes
1answer
67 views

On the spectrum of a product in a Banach algebra, in specific case

Let $A$ be a Banach algebra, and suppose that $a,b\in A$ have spectra that satisfy: $\sigma(a) \subset U$, and $\sigma(b)\subset U$, where $U$ is the open right half-plane of complex numbers with ...
3
votes
2answers
304 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
1answer
23 views

Showing the sum of a C* subalgebra and ideal is itself a C* subalgebra

In my functional analysis class I was recently met with this in the context of C* algebras: Let A be a C*-Algebra and B is a C*-subalgebra of A and I an ideal of A. We are asked to show that $ B+I ...
2
votes
1answer
140 views

Extensions: Spectrum

Problem Given a C*-algebra $\mathcal{A}_0$ and unital extensions $1\in\mathcal{A}$ and $1'\in\mathcal{A}'$. Regard a common element: $$A_0\in\mathcal{A}_0:\quad A^{(\prime)}:=\iota^{(\prime)}(A_0)$$ ...
5
votes
1answer
202 views

Example of a singular element which is not a topological divisor of zero

We know that every topological divisor of zero in a commutative Banach algebra is singular. I need an example of a singular element which is not a topological divisor of zero.
5
votes
1answer
114 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
2answers
277 views

How far is a Banach algebra from being a multiplicative group?

Given a Banach algebra $\mathcal{A}$, the collection of invertible elements in $\mathcal A$, $G(\mathcal{A})$ is a group. I wonder whether there is a measurement for how far $\mathcal{A}$ is from $G(\...
3
votes
0answers
204 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 ?...
2
votes
1answer
83 views

Does *-operator be automatically continuous

In the C*-algebras, does the * -operator be automatically continuous? I think it is yes, because C*-algebras are semisimple, from the Johnson Theorem, it must be automatically continuous. Am I right? ...
2
votes
1answer
79 views

Character space of $L^{1} (\mathbb Z)$

I have a question about the Gelfand and norm topologies on the character space of $L^{1} (\mathbb Z)$. Are the Gelfand and norm topologies equal, on the character space of $L^{1} (\mathbb Z)$?...
1
vote
1answer
87 views

Norm of a character in a non-unital Banach algebra

Let $\cal A$ be an abelian non-unital Banach algebra and $h:{\cal A}\to {\Bbb C}$ be a homomorphism. If ${\cal A}$ has an approximate identity $\{e_i\}$ such that $||e_i||\leq 1$ for all i, then $||h||...
1
vote
1answer
203 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 - \...
1
vote
1answer
224 views

Closed unit ball of B(H) is not compact in strong operator topology of B(H).

In operator theory we prove that closed unit ball of B(H) is compact in weak operator topology and is closed in strong operator topology. But a book of operator theory states that closed unit ball of ...
0
votes
1answer
56 views

What's difference between spectrum and eigenvectors of an operator

Let $x$ be an operator in $B(H)$. By definition $\sigma(x)=\{\lambda \in \Bbb C ~; \lambda - x \neq inv \}$. Also to find eigenvalue of an operator we should find $\lambda$ such that $x\xi = \lambda \...
7
votes
1answer
406 views

Normed algebra with multiplicative norm, non-isomorphic to $R$, $C$, $H$.

Is there an infinite dimensional real normed algebra $A$ such that $\|xy\|=\|x\|\cdot \|y\|$ for all $x,y \in A$? Thanks.
6
votes
1answer
117 views

K theory of finite dimenional Banach algebras

Is there a reference which studied the K theory of finite dimensional Banach algebras? In particular is there a finite dimensional Banach algebra whose $K_{0}$-group is a non trivial finite group?(I ...
4
votes
0answers
58 views

Submultiplicative Hilbert space norm on $B(H)$

Let $H$ be a complex Hilbert space and let $B(H)$ denote the space of bounded linear operators $H \to H$ equipped with operator norm: $$ \lVert T \rVert = \sup\big\{ \lVert Tx \rVert \: : \: \lVert x \...
4
votes
2answers
196 views

Spectrum of idempotent element

Let $A$ be some unitary algebra over $\mathbb{C}$. If $a^2=a$ and $0\ne a\ne 1$ then $\{0,1\}\subset \sigma_A(a)$ ($\sigma_A(a)$ is the spectrum if $a$). I believe that also $\sigma_A(a)\subset \{0,1\}...
4
votes
2answers
89 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 $...
3
votes
2answers
167 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 ...
3
votes
1answer
31 views

Jordan-homomorphism; equivalent properties

Let $\phi:A\to B$ a linear map between $C^*$-algebras $A$ and $B$. I want to know, why the following properties are equivalent: $(i) \phi(ab+ba)=\phi(a)\phi(b)+\phi(b)\phi(a)$ and $(ii) \phi(a^2)=\phi(...
3
votes
3answers
92 views

Which Hilbert space is isometrically isomorphism with $B(E)$ for some Banach space $E$.

Consider $H$ as a Hilbert space. How can I find a Banach space $E$, for that, $H=B(E)$ where $B(E)$ is the set of bounded linear operator on $E$? (At least under some conditions on $H$) Also if the ...
3
votes
2answers
158 views

For a hermitian element $a$ in a $C^*$-algebra, show that $\|a^{2n}\| = \|a\|^{2n}$

Let $\mathcal{A}$ be a $C^*$-algebra. Suppose that $a \in \mathcal{A}$ with the property that $a^* = a$ (that is, suppose that $a$ is hermitian). I would like to show that $\|a^{2n}\| = \|a\|^{2n}...
2
votes
1answer
55 views

Proof involving invertible elements of Banach algebra

I want to prove for a unital Banach algebra $\mathcal{A}$, it follows that if $\|a-b \| < \frac{1}{\|a^{-1} \|}$ then $b \in \mathcal{A}^{-1}$ (where $\mathcal{A}^{-1}$ is the subset of invertible ...
2
votes
1answer
94 views

Nonhermitian elements whose spectra are real?

Let $\mathcal{A}$ be a complex C* algebra. It is well-known that $\sigma(a)\subset\mathbb{R}$ if $a$ is a hermitian. I wonder whether the converse is true. That is, if $\sigma(a)\subset\mathbb{R}$, ...
2
votes
1answer
178 views

How do we show that prime C* algebras have trivial center

A prime C* algebra is a C* algebra with the property that the product of any two of its non zero ideals is non zero. The claim is that it has trivial center, i.e., the only central elements are ...
2
votes
1answer
296 views

a question about invertibility of Banach Algebra

If $X$ is a Banach algebra with identity, and $0$ is the only element $x \in X$ such that there is a sequence $\{ {x_n}\} \subset X$, $\left\| {{x_n}} \right\| = 1$ and $x{x_n} \to 0$ or ${x_n}x \to ...