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).
1,395 questions
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Spectrum of a self-adjoint element and the image of evaluation map on the state space always same?
Let $A$ be unital $C^*$-algebra and $a\in A$ Hermitian. Denote $S_A$ the state space as usual. We know $\hat{a}(S_A):=\{\hat{a}(f): f\in S_A\}\supset \sigma(a)$. But is the inverse always true? Or ...
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Abelian Banach algebra without identity but has compact character space possible?
Let $A$ be an abelian Banach algebra without identity, and $\Omega(A)$ its character space. We know $\Omega(A)\cup \{0\}$ is weak*-closed in the closed dual unit ball, and is therefore compact. This ...
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Proof of a Banach algebra
For a compact Hausdorff space $X$, we denote the Banach algebra of all continuous complex-valued functions on $X$ by $C_R(X)$.
If $A$ be a maximal ideal of $C_R(X)$ with dimeter norm as follows:
$\|f\...
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Borel Functional Calculus for non-unital commutative $C^*$ Algebra
Suppose $A$ be a commutative unital $C^*$ algebra acting on a Hilbert space $H$. We know that $A$ is isometrically isomorphic to $C(X)$ for a compact Hausdorff space $X$, to be more precise $X$ is the ...
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The algebraic and analytic embedding of the disk algebra $A(\mathbb{D})$ in $BH(\mathbb{D})$
**My apologies in advance if this question is elementary, the reason I fear that it may be trivial is that I do not know some concrete examples of elements of $BH(\mathbb{D})\setminus A(\...
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$C(K)$ and closed ideals in $C_0(X)$
Let $X$ be Hausdorff and locally compact. Then $C_0(X)$ is the standard abelian $C^{*}$-algebra.
Question. Let $K\subset X$ be a compact subset. Then $C_K(X):=\left\{f\in C_0(X): {\rm supp}(f)\subset ...
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What are the importance of unitary elements in Banach algebra and beyond?
I have just started studying Banach algebra, *-algebra, and the $C^*$ property. I met with the definition of unitary element. An element $u$ of a unital Banach $^*$-algebra $(A,\|\cdot\|, ^*)$ is ...
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Almost commuting elements and functional calculus
Let $A$ be an unitial $C^*$-algebra, $X \subseteq \mathbb{C}$ a compact set and $f : X \to \mathbb{C}$ a continuous function. Prove that for any $\epsilon > 0$ there is a $\delta > 0$ such that ...
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linear independence of multiplicative functionals
Suppose that $A$ is a complex, commutative, semisimple Banach algebra with maximal ideal space (equiv., set of complex homomorphisms $\phi \rightarrow$ C) $\Delta.$ Is $\Delta$ linearly ...
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Compare a weak topology with the Gelfand topology for the algebra of bounded weakly continuous functions
Let (for simplicity) $H$ be a separable Hilbert space and $A$ be the Banach algebra of bounded weakly continuous functions of $H$. Then
The Gelfand space of $A$, the space of maximal ideals, can be ...
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Generalization of Dirichlet's test and Abel's test to sequences in Banach algebras?
Recently I've been reviewing elementary analysis, and it seems, for a real Banach algebra $A$, we can generalize the classical Dirichlet's test and Abel's test in the following way.
Dirichlet's test. ...
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Confusion with abstract p-operator spaces and Le Merdy's Theorem.
I'm struggling to understand the implications of a theorem, probably due to my lack of background on the subject. It is possible that this is not a very good question.
In "Quantitative K-theory ...
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Spectrum of an element in a non-unital Banach algebra
I have seen questions here on MathStackExchange which look similar to what I need, but in truth aren't, so please read carefully before flagging this as duplicate.
Consider $(A, +, 0, \cdot, 1_A, \...
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functional calculus and isolated points on the spectrum
Consider a Hilbert space $\mathcal{H}$ and a normal operator $T\in B(\mathcal{H})$, i.e. $TT^\ast=T^\ast T$. Let $\lambda \in \sigma(T)$ be an isolated point on the spectrum of $T$ and we denote by $\...
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Existence of a function in L1(G) whose Fourier transform never vanish
I am reading Wiener Tauberian theorem on locally compact abelian group setting and there I have read that:
The closed linear span of the translates of $ f $ is $ L^1(G) $ if and only if $ \hat{f}$ ...
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Deciphering a Proof of Cohen’s Factorization Theorem I
For some personal work, I am trying to understand a proof of Cohen’s Factorization Theorem. Note that we write $\mathscr{A} \cdot E=\{a\cdot x : a\in \mathscr{A}, x\in E\}$ and we use the abbreviation ...
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necessarily continuous maps on Banach algebra [duplicate]
Let $A$ be a Banach algebra such that for all $a\in A$ the implication
$$Aa = 0 \text{ or } aA = 0 \Rightarrow a = 0$$
holds. Let $L,R$ be linear mappings from $A$ to itself such that for all $a,b\in ...
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The Riesz functional calculus
I'm study the Conway's book and I read Cauchy's theorem (below). I don't understand why in the theorem it states that $\sum_{j=1}^{m} n(\gamma_j;a)=0$. I think it is because when point $a$ is not ...
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Spectral theory for Frechet algebra?
In most textbooks discussing spectral theory of operators, they focus on a Banach algebra of operators due to the power that completeness provides. Frechet spaces are complete metric spaces too, so is ...
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An elementary criterion for a normed lineared space to be a Banach space [closed]
Douglas states a criterion in Banach Algebra Techniques in Operator Theory:
1.10 Corollary. A normed linear space $\mathscr{X}$ is a Banach space if and only if for every sequence $\{f_n\}_{n=1}^\...
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Convergence of norm of sequence of invertible elements in a Banach Algebra (another proof)
Let $A$ be a Banach algebra with unit $e$, let $x \in A$ and let $\lambda \in \sigma(x)$. Prove that if there exists a sequence $(\lambda_n) \subset \mathbb{C} \setminus \sigma(x)$ converging to $\...
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Uniqueness of the square root of a positive lineare operator
In their book, Reed-Symon prove that if $A$ is a positive operator from a Hilbert space $H$ to itself, then there exist a positive operator $B$ such that $B^2=A$ and such a $B$ is unique.
For the ...
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Spectral Projections in Conway's Functional Analysis
I'm reading John B. Conway's A Course in Functional Analysis and encountered an expression involving spectral projections that I don't fully understand:
$$
X_\lambda = E(\lambda)X
$$
Context:
$X$ is ...
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Banach ring with respect to two norms, is the max norm also Banach?
We know $(\mathbb{C},\max\{|\cdot|_0,|\cdot|_\infty\})$ is a Banach ring (a concept in Berkovich space, see following definitions), where $|z|_0=1$ for all $z\neq0$ and $=0$ if $z=0$, and $|z|_\infty=...
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Completeness of Banach Algebra and spectral properties [closed]
Central to banach algebra theory are some spectral properties. I was wondering if completeness is a necessary condition to these properties.
The Neumann series are often used to derive that $\sigma(a)\...
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Functional analysis in Banach algebras
"Hi, I'm reading Conway's book, particularly Chapter 7. I'm having difficulty with Proposition 3.8. The proposition states:"
Prop 3.8:
If $\mathcal{A}$ is a Banach algebra with identity and $...
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Why doesn't the Gelfand-Mazur theorem apply to Calkin Algebra?
The Gelfand-Mazur theorem says that, given a complex unital Banach algebra $A$, if every non-zero element in $A$ is invertible, then $A$ is isometrically isomorphic to $\mathbb{C}$.
Given a separable ...
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Is the Spectrum of a Banach Algebra the same as its Ring Spectrum?
Given a commutative unit Banach Algebra $A$, its spectrum is defined as:
$$
\Omega_A= \{\varphi\in A^*| \varphi\text{ is a homomorphism}\}\subset A^*$$
This is a topological space with the subset ...
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Connection Between Invertibility and Norm in the Spectral Radius Formula
I am studying spectral theory in Banach algebras, and I find it surprising how invertibility (an algebraic notion) relates to the norm (a topological one). In the proof of the spectral radius formula, ...
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Alternative Proof of Commutativity in the Gelfand-Mazur Theorem
The Gelfand-Mazur theorem states that if $A$ is a complex Banach algebra with a unit where every nonzero element is invertible, then $A$ is isometrically isomorphic to $\mathbb{C}$. A corollary is ...
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Maximal ideals in subalgebras of the product of Banach algebras
Let $X,Y$ be Banach algebras and $B$ a closed subalgebra of $X\times Y$ with some of the standard equivalent norms, such as $||(x,y)||=||x||+||y||$. Let also $I$ be a maximal modular ideal of $B$, ...
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In $\mathcal{H}^{\infty}(\mathbb{D})$ complex homomorphisms are not the evaluation maps.
For $\Omega$ a locally compact space, $C_0(\Omega)$ is a Banach Algebra and we have that $\psi: C_0(\Omega) \rightarrow \mathbb{C}$ is a complex homomorphism if and only if there exist $t \in \Omega$ ...
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Analog of Mackey-Arens theorem for topological algebras
Is there an analog of the Mackey-Arens theorem for topological algebras? Something to the effect of:
We call a topology on an algebra lcha if it is locally convex Hausdorff and makes its ...
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Does the weak$^*$ topology on a dual Banach algebra make its multiplication continuous?
Let $A$ be a Banach algebra which is a(n isometric) dual space. Let $\tau$ be a weak$^*$-topology on $A.$ That is $\tau=\sigma(A,F)$ for some $F^*=A.$ Let $a\in A.$ Are the maps $(A,\tau)\rightarrow(A,...
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Operators in Banach Spaces satisfying equality in composition norm
It is well-known that for bounded linear operators $T$ and $S$ on a Banach space $X$, the operator norm satisfies $\|T \circ S\| \leq \|T\| \|S\|$. Equality, however, is not generally achieved. A ...
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What is the smallest algebra generated by a self adjoint operator that contains functional calculus?
I have the following question: Assume I have a densely defined Self adjoint operator O. What is the smallest algebra A(say C^* or von Neumann or I don't know what else exists), such that either
a) the ...
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Does Jordan homomorphism preserve spectrum?
I'm studying problem of Kaplansky and it is about Jordan homomorphisms: $\phi:A \to B$ with the property $\phi(a^2)=\phi(a)^2, \ \forall a \in A$, where $A$ and $B$ are for me unital Banach algebras, ...
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A closed ideal of a closed ideal of a unital commutative $C^*-$algebra is also a closed ideal. [closed]
Let $A$ be a unital commutative $C^*$-algebra. If $B$ is a closed ideal of $A$, and $C$ is a closed ideal of $B$, prove that $C$ is a closed ideal of $A$.
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The spectrum of an element in a subalgebra is equal to its spectrum in the algebra
Let $A$ be a Banach algebra with an identity element, and let $B$ be a maximal commutative subalgebra of $A$. Prove that:$Spec_A(b)=Spec_B(b),\forall b\in B$.
It is easy to show that $1in B$ and $...
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Is there an example of a unital homomorphism that breaks involution?
Let $A$ be a unital $C^*$-algebra. It is well-known that characters (nonzero homomorphisms) $h: A\to \mathbb{C}$ preserves the involution $*$. But is this still true for general unital homomorphisms ...
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Are similarity-preserving maps on matrix groups necessarily power series?
Let $n \geq 2$ and $M_n(\mathbb{C})$ denote the set of all $n \times n$ matrices over the complex numbers. Does there exist a continuous map $f: M_n(\mathbb{C}) \to M_n(\mathbb{C})$ such that $$ A f(B)...
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Convexity of domain of unconditional convergence for power series in Banach algebras
Let $X$ be a Banach algebra. Consider the power series $\sum_{k=0}^\infty c_k x^k$. Let $D$ denote the set of elements in $X$ where this power series converges unconditionally (not merely the ...
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Closedness of maximal ideal in unital normed algebras
A classical result in the theory of Banach Algebra states that every maximal ideal in a commutative unital Banach algebra is closed. This generalizes to normed, possibly non-Banach algebras?
In other ...
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Kernel of discrete Fourier transform
Equip $\ell^1(\mathbb Z)$ with the discrete convolution. And define for every $x\in \ell^1(\mathbb Z)$ the element
$$F(x)(\theta) = \sum_{n\in \mathbb Z} x(n)e^{i\theta n}$$
with $\theta \in [0,2\pi]$....
user1446697
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Determining the maximal ideal space
For this exercise I am considering the following space equipped with the operator norm.
$$X = \left\{\begin{pmatrix}\alpha & \beta & 0\\ 0 &\alpha &0\\ 0 &0& \gamma \end{...
user1446697
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$C^n[0,1]\simeq C^m[0,1]$ as Banach algebras if and only if $m=n$
Prove that $C^n[0,1]\simeq C^m[0,1]$ as Banach algebras if and only if $m=n$.
I have proved that $C^n[0,1]$ is a Banach algebra, where the norm is defined as $$\|f\|=\sum_{k=0}^n\frac1{k!}\|f^{(k)}\|_\...
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Can I think of probability convolution as defining a C*-multiplication on a dual space?
This question may not be well-phrased.
A probability measure is a linear functional, and is actually a state on $C_0(X)$ for a locally compact Hausdorff space $X$ with a suitable sigma-algebra. I'm ...
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Bounded Approximate Identity Reference
I'm aware of the following result and its proof:
Let $A,B$ be Banach algebras and $A$ have a bounded approximate identity. If there exists $\theta:A\to B$ a continuous algebra homomorphism with dense ...
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On the norm of the unit element in a Banach algebra
In Serge Lang's Functional Analysis, when defining a Banach algebra, he states that "when there is a unit element, we shall assume that $|e| = 1$." He also notes that this is not always true ...
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Can a Grassmann Algebra turn into a Banach Algebra by some appropriate norm?
As we know a (finite) Grassmann Algebra $\Lambda_q$ is an algebra generated by $\{x_1,...x_q\}$ with the relation $x_ix_j+x_jx_i=0$, and hence a $2^q$-dimension vector space spanned by words of $x_1,.....
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