Questions tagged [banach-algebras]

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) or (von-neumann-algebras) instead (or in addition). Further related tags: (operator-algebras), (operator-theory), (banach-spaces), (hilbert-spaces).

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What is odd element of C*-algebra? [closed]

I cannot nowhere find a definition of an odd/even element of C*-algebra. Can someone write it here?
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Continuity of product in Banach Algebras

EDIT: The definition literally says that the product is bilinear, as Chris Eagle kindly pointed out. Turns out that my original proof was fine. This question does no longer require an answer and ...
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If an element in a Banach algebra is anihilated by an analytic function then it must be algebraic.

Let $A$ be a Banach algebra, let $a\in A$ and suppose $f(a)=0$, where $f$ is an analytic function defined on an open set $U$ containing $\sigma(a)$. Prove that $a$ is algebraic in the sense that $p(a)...
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Why is it difficult to define a direct integral of Banach spaces or Banach algebras?

In https://en.wikipedia.org/wiki/Direct_integral I can read about how to define a direct integral on Hilbert spaces and Von-Neumann algebras. Suppose that I want to define a direct integral on either ...
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Banach algebra $L^1(\mathbb{R})$ and its Fourier transform.

We know $L^1(\mathbb{R})$ is a Banach algebra whose product is defined to be convolution: $$f*g(x)=\int f(x-t)g(t)dt.$$ In fact, it is a $*$-Banach algebra (Don't confused with the convolution), with $...
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Understanding the proof of Fuglede's theorem

I am trying to understand this proof of Fuglede's theorem on wikipedia : Everything was making sense until the last sentence: "Considering the first-order terms in the expansion for small λ, we ...
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Banach space with continuous multiplication is a Banach algebra

Consider a Banach space $(\mathcal{A},\|\cdot\|_{\mathcal{A}}$) and assume that $\mathcal{A}$ is also an associative algebra with a unit element $I\in \mathcal{A}$ such that $\|I\|_{\mathcal{A}}=1$. ...
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Mertens theorem in an arbitrary banach algebra

I'm analyzing the Mertens theorem that states that when at least one of two convergent series $\sum_{i=0}^\infty a_i$ and $\sum_{j=0}^\infty b_j$ is absolutely convergent than their Cauchy product $...
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an inequality for completely positive maps

Let $A$ and $B$ be $C$*-algebras and $\phi:A\to B$ a completely positive contractive map. I want to show that, for any $a,b\in A$; $$\Vert \phi(ab)-\phi(a)\phi(b)\Vert\leq\Vert\phi(aa^*)-\phi(a)\phi(a^...
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Solutions of a Linear Differential Equation in a Banach Algebra

Assume $A$ is a real Banach algebra (which need not necessarily be commutative or finite-dimensional) with unit and the function $f: \mathbb{R}\to A$ satisfies the differential equation $$\frac{df\...
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The space of functions vanish at infinity

Why we care about these two Banach spaces $$C_0(\Omega):=\{ f \in C(\Omega) \colon \forall \epsilon>0 \text{ there exists a compact } K_\epsilon \subset \Omega \text{ such that }\\|f(s)|<\...
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Is $r \in \text{Rad}(A)$?

Let $u$ be an element of a Banach algebra $A$ such that $u^2-u \in \text{Rad}(A)$. I am trying to show that there exists a projection in $A$ which is equal to $u$ modulo the radical. We have that $\...
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Show $f\in C(\mathbb{T})$ where $f(z)\neq0$ for all $z$, with summabel Fourier coefficients is invertible.

For $\phi^*:C(\Delta)\rightarrow C(\mathbb{T})$ defined by $(\phi^* f)(\lambda) = f(\omega_{\lambda})$, where $\omega_{\lambda}((a_n)_n) = \sum_{n\in\mathbb{Z}} a_n\lambda^n$, and $\Gamma$ the Gelfand ...
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Example of an algebra that is a Banach space but not a Banach algebra

I'm looking for an example of a space $\mathbb{A} $ such, $\mathbb{A} $ is an algebra; $\mathbb{A}$ is equipped with a norm that makes it a Banach space; $\mathbb{A}$ is not a Banach algebra, i.e., ...
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Analytic Functional Calculus on C*-algebras

Here is one of the practice questions that I have been trying from my C*-algebras exercises. Suppose that $T\in L(X)$ and $f\in H(T)$. Prove that $f(T^*) = f(T)^*$ In this situation $H(T)$ represents ...
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Does a contradiction occurs about a homomorphism on $C(\Omega)$?

In the page 601 of 'HOMOMORPHISMS OF BANACH ALGEBRAS'(Bade and Curtis) https://www.jstor.org/stable/pdf/2372972.pdf, we get an inequality \begin{align*} ||\nu(x_m - x_n)|| \geq \rho_{\mu(C(\Omega))}(\...
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If $\lVert F(z)^{-1}\rVert < M$. Then $z \to F(z)^{-1}$ is holomorphic

I have the following theorem Suppose $B$ a Banach space, $U\subset \mathbb{C}$ is open and $F:U \to B$ is holomorphic such that $\exists~M~\forall z\in U, \lVert F(z)^{-1}\rVert < M$. Then $z \to ...
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Show that $(Tf)(x) = \int_0^x f(t)dt$ is bounded

Let $A = C[0,1]$ be the Banach space of all continuous functions on the interval $[0,1]$. Let $T$ be the linear operator from $A$ to $A$ defined by $$(Tf)(x)=\int_0^x f(t)dt.$$ I know that $T$ is ...
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Find an expression for $(\lambda\textbf{1}-a)^{-1}$.

Let $A$ be a unital Banach algebra, and suppose $a\in A$ has the property that $a^2=1$, but that $a\neq \textbf{1}$ and $a\neq -\textbf{1}$. By the spectral mapping theorem, $\sigma(a^2) = \{\lambda^2 ...
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Weak convergence in $L^{∞}[0,1]$ implies strong convergence in $L^{p}[0,1]$

I'm trying to solve an exercise in Rudin's Functional Analysis. It is the problem 19 in Chapter 11. The problem is: If $f_n\in L^{∞}[0,1]$ and converges to $0$ in the weak topology of $L^{∞}[0,1]$, ...
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Complement of spectrum of a invertible self-adjoint element has no bounded components

Let $A$ be a unital $C^*$-subalgebra of a $C^*$-algebra $B$ and $x$ be an invertible element of $B$. Let $y$ be an element of $B$ defined by $y=x^*x$. Now $x$ is invertible implies $y$ is invertible ...
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$C^\ast$ -algebra : Can we assume image of identity is identity?

In Conway's A Course in Operator Theory, proposition 1.7 (e) is If $\rho:A \to B$ is $\ast$-homomorphism, then $\|\rho(a)\| \leq \|a\|$ for all $a$ in $A$. and the beginning of the proof is First ...
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Regarding invertibility of a certain element in a Banach algebra

Let $A$ be a complex unital Banach algebra. Let $a,b\in A$ such that $\sigma(a)\setminus \{0\}\subset \sigma(b)\setminus \{0\}$. Now let $\lambda\neq 0$ such that $\lambda\notin\sigma(b)$. Then $1+b(\...
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Claim: $\Vert (\lambda_n 1-T)^{-1}\Vert \rightarrow \infty$

Let $\mathcal{A}$ denote a unital Banach algebra, $T \in \mathcal{A}$ and $\sigma(T)$ then spectrum of $T$. Suppose that $(\lambda_n)_n \subseteq \mathbb{C}\setminus\sigma(T), \ \lambda \in \sigma(T)$ ...
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If $T$ commute with all the $\pi(a)$ then $T$ is a scalar multiple of the identity

Let $\pi$ be a representation of the Banach *-algebra $A$ on $H$, prove that every nonzero $h\in H$ is cyclic for $\pi$ if and only if whenever $T\in B(H)$ commute with all of the $\pi(a),\ a\in A$ ...
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The spectrum of the algebra of Fourier multipliers

Let $\mathcal{M}_p(\mathbb{R}^n)$ denote the space of Fourier multipliers on $L^p(\mathbb{R}^n)$, i.e. the set of tempered distributions $m$ such that the operator $T_m:f\mapsto \mathcal{F}^{-1}(m \...
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spectrum of an element of unital banach algebra is closed set

Let $A$ be a unital Banach algebra and $a \in A$. Then the spectrum $\sigma(a)$ and set of invertible elements $\text{Inv}(A)$ are defined as: $$ \begin{align} &\text{Inv}(A):=\{a\in A~|~a \text{ ...
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Reference request for the space $C^k[a,b]$

I am looking for some notes/references that discuss basic properties of the space $C^k[a,b]$ of $k$-times differentiable $\mathbb{C}$-valued functions on an interval $[a,b]$. I am not very familiar ...
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Spectrum of a linear function in the continous functions

I am currently working through my course text and one of the examples for a unital, commutative Banach algebra is the space of continuous functions $C([a,b])$. Further, in the text, there is a theorem ...
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continuity of homomorphisms from a unital Banach algebra to a matrix algebra?

Let $A$ be a unital Banach algebra, and let $\phi:A\rightarrow M_n$ be a homomorphism of complex algebras, $M_n$ denoting the algebra of all $n×n$ matrices over $\mathbb{C}$. Is $\phi$ necessarily ...
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Why is the Wiener Algebra closed under pointwise multiplication of functions?

I want to understand the proof given on the wikipedia page for the Wiener Algebra $A(\mathbb{T})$ (see here) that $A(\mathbb{T})$ is closed under multiplication. We have $$ A(\mathbb{T}) = \{f:\mathbb{...
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The genera linear group in Normed algebras

Let $A$ be a unital Banach algebra with the unit $1$. It is well-known that the elements in $G(A)=\{x\in A: \|1-x\|<1\}$ are all invertible, called the general linear group. Such a property does ...
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Is the set of topological divisors of zero connected?

Let $A$ be a Banach Algebra. An element $a\in A$ is said to be a left topological divisor of zero if there exists a sequence $\{x_n\}$ in $A$ with $\|x_n\|=1$ such that $ax_n\to 0$ as $n\to \infty$. ...
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5 votes
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Prove these two projections are equivalent

$\pmb Problem$: Let $e_1, e_2, f_1$ and $f_2$ be projections of $M$ which is a Von Neumann algebra, such that $e_1e_2=f_1f_2=0$. If $e_1+e_2=f_1+f_2$, $e_1 \sim e_2$ and $f_1 \sim f_2$, prove that $...
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Proving Lipschitz Continuity of Infimum functions

Given a Banach algebra $A$, define $\zeta: A \longrightarrow [0,\infty)$ by $\zeta(a) = \inf_{||c||=1}||ac||$. Prove that $|\zeta(a)-\zeta(b)| \leq ||a-b||$ for all $a,b \in A$. So my first ...
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1 vote
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Equivalence of projections in a Von Neumann algebra factor

$\pmb Problem:$ Let $M$ be a (properly) infinite factor and $e$ and $f$ be projections of $M$. Then $$ e\lor f \sim 1 \iff e\sim1\ \text{ or } f\sim1 .$$ $\pmb Idea:$ As $M$ is a factor, its center is ...
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Element of Norm $1$ in Takesaki Theorem 10.2

Theorem 10.2 (i) of Takesaki's Theory of Operator Algebras I states that if $A$ is a C*-algebra and $S$ is its closed unit ball, then there is an extreme point in $S$ if and only if $A$ is unital. In ...
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1 vote
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Takesaki Proposition 1.5 Epsilon Inequality

In the proof of proposition 1.5 of Takesaki's Theory of Operator Algebras: Part 1, after defining defining a norm on the unitisation $\tilde{A}$ of a non-unital C-algebra $A$ making it a Banach ...
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2 votes
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Square root of positive elements in Banach *-algebras

Let $A$ be a Banach $^*$-algebra, i.e., a complex Banach algebra together with a conjugate linear map $^*:A\to A$ that satisfies $(a^*)^*=a$, $(ab)^*=b^*a^*$ and $||a^*||=||a||$ for all $a,b\in A$. ...
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Can I construct a vector space whose scalar field is a complex algebra?

A definition first: A complex algebra is a vector space $X$ over the complex field $\mathcal{C}$ in which a multiplication is defined that satisfies: $$ x(yz) = (xy)z \\ (x + y)z = xz + yz \\ x(y + z) ...
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Weak * Compactness and Local Compactness

I'm reading through Murphy's C* Algebras, and the following theorem is presented: If $A$ is an abelian Banach algebra, then the set of characters $\Omega(A)$ is a locally compact Hausdorff space. If $...
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Let $A$ be a dense subalgebra of $\mathcal{C}_b(X,\mathbb{K})$. Show that $A$ separates the points of $X$

Let $X$ be a regular Hausdorff space and $A$ a dense subalgebra (with 1) of $\mathcal{C}_b(X,\mathbb{K})$. Show that $A$ separates the points of $X$. My attempt: Let $x\ne y\in X$. We want to show $\...
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Why is $\tau_x(\lambda) - \lambda \in \sigma(-x)$ for every $\lambda \in \mathbb{C}$? Spectral Translation Explanation Needed.

Let $A$ be a Banach algebra and $\phi$ a multiplicative functional on $A$ such that $\phi(x)\in \sigma(x)$ for each $x \in A$. Now for each $x \in A$, define $\tau_x(\lambda):=\phi(\lambda\textbf{1}-x)...
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Predual of disc algebra and $W(\mathbb{T})\neq A(\mathbb{D})$

I want prove that the Wiener algebra, $W(\mathbb{T})$, does not coincide with the disc algebra $A(\mathbb{D})$. I know that there are some ways to do this. For example, one smart way is use Rudin-...
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2 votes
1 answer
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Is Banach algebra $L^1(\mathbb T)$ under convolution a division algebra?

$\mathbf {The \ Problem \ is}:$ If $f$ & $h$ are $L^1(\mathbb T)$ functions with $f\star h=0$ identically on $\mathbb T.$ Then can we say either $f\equiv 0$ or $h\equiv 0 ?$ $\mathbf {My \ ...
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1 vote
2 answers
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Understanding Riesz's functional calculus

I am studying Conway's Functional Analysis, where Riesz's functional calculus is introduced rather suddenly. Let $G \subset \mathbb C$ be open, $\mathcal A$ be a complex Banach algebra with identity $...
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If $\Phi$ is injective, $X_1$ normal and Hausdorff and $X_2$ compact, then $F$ is surjective

Let $X_1,X_2$ be topological spaces and $F\in \mathcal{C}(X_2,X_1)$. Let $\Phi:\mathcal{C}_b(X_1,\mathbb{C})\to\mathcal{C}_b(X_2,\mathbb{C}): \Phi(f):=f\circ F$. Then $\Phi$ is a continuous ...
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2 votes
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ideals, projections and factors in VN algebras

I'm trying to solve some problems on Von Neumann algebra and I got the questions below. $Q1$. By definition we know a factor is a Von Neumann algebra with trivial center, i.e. a center consisting only ...
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6 votes
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Spectrum $\sigma(a)$ taken within the unitisation of a $C^*$-algebra

My question relates to reconciling the definition of the spectrum $\sigma(a)$ of a point $a$ in a $C^*$-algebra $A$ in both the unital case and the more general case using the unitisation, since I ...
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Proving that if $C_0(X)$ is unital, then $X$ is compact

Apologies if this is simple, I'm having one of those days. Let $X$ be a locally compact Hausdorff space and $C_0(X)$ be the commutative $C^*$-algebra of continuous complex-valued functions on $X$ that ...
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