Questions tagged [banach-algebras]
The banach-algebras tag has no usage guidance.
372
questions
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Examples of amenable Banach algebras which have non-amenable subalgebra
I am looking for examples of amenable Banach algebras which have non-amenable subalgebra
I know
1: Each amenable Banach algebra has a bounded approximate identity
2: If $I$ be a closed ideal in an ...
2
votes
0
answers
107
views
Finding non-zero elements with $x^*x\leq\frac{1}{n}$
For a given unital Banach *-algebra $A$, let us put $A_+=\{\sum_1^n x^*_ix_i : x_i\in A, n\in \mathbb{N}\}$. We write $x\geq0$ if $x\in A_+$.
What types of (non semi-simple) unital Banach *-algebras ...
4
votes
1
answer
176
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Continuity of the involution in Banach *-algebras
My question is concerned with the involution in Banach *-algebras.
1- Should the involution be assumed continuous in every Banach *-algebra?
If the answer is negative,
2- Does there exist any ...
1
vote
0
answers
126
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Maximal ideal space of $\ell^\infty(A)$, $A$ a commutative unital Banach algbera
Let $A$ be a commutative unital complex Banach algebra with norm $\|\cdot\|_A$, and let $\ell^\infty(A)$ denote all bounded sequences $(a_n)_{n\in \mathbb{N}}$ with $a_n\in A$, $n\in \mathbb{N}$, with ...
6
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0
answers
104
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Real-world example of a Banach *-algebra with a nonzero *-radical
Is there a real-world example of a Banach *-algebra with a nonzero *-radical (intersection of kernels of all *-representations)? Textbooks give examples of finite-dimensional algebras with degenerate ...
3
votes
1
answer
109
views
Classes of Banach algebras (that aren't operator algebras) whose bidual comes from a "universal representation"
Are there any classes of (Arens regular) Banach algebras that are not operator algebras whose bidual comes from a “universal representation”, as in the case of C*-algebras?
4
votes
2
answers
166
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Measure algebra on the Bohr compactification vs the bidual algebras
The following question probably reduces to some standard abstract harmonic analysis Twister play, but I'd still welcome some comments on it.
Let $G$ be a locally compact Abelian group and let $bG$ ...
4
votes
0
answers
123
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Closed subgroup (Cartan) theorem without transversality nor Lipschitz condition within Banach algebras
Yesterday, I came across the following preliminary theorem.
Theorem Let $\mathcal{B}$ be a Banach algebra (with unit $e$) and $G$ be a closed subgroup
of $\mathcal{B}^{-1}$ (the group of ...
0
votes
1
answer
418
views
Separability of an algebra is equivalent to separability of its spectrum
Let $A$ be a commutative C*-algebra.
I would like to show that $A$ is separable (i.e. has a countable dense subset) if and only if the spectrum of $A$ (denoted by $\Omega(A)$) is separable.
Notes ...
0
votes
0
answers
153
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Representations of Banach algebras
If $A$ is a Banach algebra and $L$ a left ideal of $A$, consider the representation $T_{L}$ of $A$ into the algebra $B(A/L)$ of bounded linear operators on the quotient space $A/L$ defined by $T_{L}(a)...
1
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0
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150
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When a finite codimensional subalgebra contains a finite codimension ideal?
What is a classification of all algebras $A$ (purely algebraic algebras, Banach or $C^*$ algebras or Lie algebras) with the following property:
Every finite codimensional subalgebra $B$ of $A$ ...
3
votes
0
answers
153
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Left and right topological K-theory of Banach algebras
Let us consider the topological $K$-functor on the category of Banach algebras as described in page $18$ of "Introduction to the Baum–Connes conjecture" by Alain Valette.
The definition is based on ...
4
votes
2
answers
245
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$K$-theory and surjective norm-decreasing $*$-homomorphisms between $C^*$-algebras
Let $A$ and $B$ be two $C^*$-algebras, and let $p:A \to B$ be a surjective norm-decreasing $*$-homomorphism which is injective on a dense $*$-sub-algebra of $A$. Can such a map have non-trivial kernel,...
3
votes
1
answer
163
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Can C*-algebras be characterized among Banach *-algebras by the spectral radius?
Let $(A,\,^\ast,\lVert\cdot\rVert)$ be a Banach $\ast$-algebra with the property that $\lVert a \rVert^2=\rho(a^\ast a)$ holds for all $a\in A$,
where $\rho(x)$ denotes the spectral radius of an ...
4
votes
0
answers
99
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Generating $H^{\infty}(X)$
Let $X$ be a domain in $\mathbb{C}^d$ and let $\mathbb{D}$ be the open unit disk in $\mathbb{C}$. Consider the Banach algebra $H^{\infty}(X)$ consisting of bounded holomorphic functions on $X$ with ...
3
votes
0
answers
84
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Norm-controlled inverses vs uniform openness of multiplication
Let $A$ be a unital commutative Banach algebra and let $\hat{a}\in C(\Phi_A)$ be the Gelfand transform of an element $a\in A$. The algebra $A$ has norm-controlled inverses, whenever there exists a ...
4
votes
0
answers
205
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Simultaneous Hahn-Banach theorem
Let $C(\mathbb{T})$ be the Banach algebra of continuous functions on the unit circle. Let $n \in \mathbb{N}$ and let $P_n(\mathbb{T})$ be the subspace of trigonometric polynomials of degree at most $n$...
0
votes
1
answer
126
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Tauberian operators
Let $X$ be a Banach space non reflexive and $T$ from $l_2(X)$ to $l_2(X)$ a bounded operator defined by:
$$T(x_n )=\frac{x_n }{n}.$$
We know that :
$$T^{**-1}(l_2(X))=\{x_n^{**} \in l_2(X^{**}) : \...
1
vote
0
answers
27
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Approximation of multipliers by multipliers of a smaller set 2
This question is a refinement of my previous question.
Let $X$ be a compact metric space, and let $B$ be a bounded Banach Disk in $C(X)$ such that for every $x\in X$ there is $f\in B$ with $f(x)\ne 0$...
4
votes
1
answer
86
views
Approximation of multipliers by multipliers of a smaller set
Let $X$ be a compact metric space, and let $B$ be a convex balanced bounded set in $C(X)$ such that for every $x\in X$ there is $f\in B$ with $f(x)\ne 0$.
Let $M=\{u\in C(X),~ uf\in B,~\forall f\in B\...
2
votes
1
answer
151
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Automorphism of algebras with certain initial conditions on given idempotents
The First question
Let $A$ be a Banach or a $C^*$ algebra. Assume that $e,f$ are two idempotents or prjections in $A$ which satisfy $ef=fe=0$. Assume that there are two automorphisms $\phi, \psi: A \...
4
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0
answers
326
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Other kinds of equivalence relations on the set of idempotents of a Banach or $C^*$-algebra or a ring (Can we get a new kind of K-theory?)
The standard equivalent relations on idempotents of a $C^*$ algebra or a Banach algebra are Murray von Neumann, similarity and homotopy equivalent. In this post we consider two other kinds of ...
4
votes
1
answer
336
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Removing the interior of spectrums
Let $A$ be a Banach algebra. Is there a Banach algebra $B$ which contains $A$ but the spectrum of each elements of $B$ has empty interior(as a subset of $\mathbb{C}$)?
The motivation comes from the ...
3
votes
1
answer
201
views
Relaxed/Truncated Version of Wiener's Tauberian Theorem
Background
Let $(U_t)_{t \in \mathbb{R}}$ be the (translation) $C_0$-group on $L^1(\mathbb{R})$ defined by
$$
U_t(f)(x) = f(x-t) \quad \text{for almost every } x \in \mathbb{R}
$$
(for $t \in \...
3
votes
0
answers
262
views
A complete Tate Huber ring is Banachizable (maybe not)?
I have questions of technical nature.
A complete Tate Huber ring is a complete topological (commutative) ring $A$ admitting an open subring $A_0$ whose topology is the $\varpi A_0$-adic topology, for ...
1
vote
1
answer
160
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A question about Johnson's theorem on the first and second cohomology of commutative amenable algebras
Johnson in cohomology of Banach algebra proved the following proposition.
I need to some guidance for the bold part of the following proof. Do you know any papers or book with more details for this ...
6
votes
1
answer
239
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Arens regularity of Banach algebras
I was trying to learn the concept of Arens regularity of Banach algebras from T.W Palmers book -"Banach algebras and the general theory of $*$-algebras". There he have discussed the Arens regularity ...
5
votes
1
answer
683
views
When are homomorphisms between Banach algebras contractions?
When are homomorphisms between Banach algebras contractions?
I recall from my student days that there are results which show that a positive answer to the above question holds under very general ...
1
vote
1
answer
265
views
Is it possible to extend this homomorphism?
Let $G$ be a torsion free group and $\alpha$ be a non-zero element in its complex group algebra. Assume that $\mathfrak A$ is the Banach sub-algebra of $\ell^1(G)$ generated by $\alpha$. Is it ...
4
votes
1
answer
383
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First and second cohomology groups of Banach algebras
Johnson in the introduction section (page 1) in "Cohomology in Banach algebras" ZBL0256.18014, wrote that Guichardet in [14,15] obtained for a Banach algebra $A$,
one has $H^1(A,X)=H^2(A,X)=0$, ...
1
vote
1
answer
251
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Bounding the norm of the inverse in a commutative Banach algebra from above
Let $B$ be a commutative, Jacobson semi-simple unital Banach algebra and take an invertible element $x$ in $B$. We may then compute the infimum of the Gelfand transform:
$\delta = \inf |f(x)|$
where ...
22
votes
5
answers
1k
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Rigorous justification for this formal solution to $f(x+1)+f(x)=g(x)$
Let $g\in C(\Bbb R)$ be given, we want to find a solution $f\in C(\Bbb R)$ of the equation
$$
f(x+1) + f(x) = g(x).
$$
We may rewrite the equation using the right-shift operator $(Tf)(x) = f(x+1)$...
4
votes
2
answers
181
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Largest ideal in bounded linear maps on Schatten-$p$ class
Let $1\leq p<\infty.$ Denote $S_p(\ell_2)$ be the set of all compact operator $x$ on $\ell_2$ such that $Tr(|x|^p)<\infty.$ Define $\|x\|_{S_p(\ell_2)}:=Tr(|x|^p)^{\frac{1}{p}}.$ This makes $S_p(...
2
votes
0
answers
47
views
Weak amenability hereditary properties
Let $\mathcal{A}$ be a commutative weakly amenable Banach algebra and $\mathcal{B}$ be a Banach algebra, let $\theta:\mathcal{A} \to \mathcal{B}$ be a continuous homomorphism with dense range; then it ...
1
vote
0
answers
105
views
The tower of path algebras associated to a tower of finite dimensional $C^*$-algebras is isomorphic to the original tower
Let $A_0\subseteq A_1\subseteq...$ be an infinite tower of unital inclusions of finite dimensional $C^*$-algebras and $B_0\subseteq B_1\subseteq ...$ be its associated infinite tower of path algebras. ...
4
votes
0
answers
132
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Contractible Banach algebras
A Banach algebra $A$ is contractible if $H^1(A, X)=0$ for all Banach $A$-bimodules $X$. Now to my question
Let $A$ be Banach algebra and $I$ be closed ideal of $A$. If $I$ and $A/I$ are both ...
3
votes
1
answer
159
views
How does $E$ closed follow from the upper semicontinuity of the spectrum?
Let $f$ be an analytic function for a domain $D$ of $\mathbb{C}$ into a Banach algebra $A$. Suppose that, for all $\lambda \in D$, $\text{Sp}f(\lambda)$ is finite or a sequence converging to $0$.
...
2
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0
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150
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A Banach or $C^*$ algebraic analogy of a classical fact in real analysis
Let $A$ be a commutative unital Banach algebra.The maximal ideal space of $A$ is denoted by $\hat A$.
Assume that $D:A \to A$ is a derivation. Fix an element $a\in A$.
Assume that for every $\phi\in \...
2
votes
0
answers
92
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Sub Banach spaces (Banach algebras) of the disc algebra which are invariant under the differentiation operator
In this question the disc algebra $\mathcal{A}(\mathbb{D})$ is the Banach algebra of all holomorphic functions on the unit open disc $\mathbb{D} \subset \mathbb{C}$ which have a ...
4
votes
0
answers
147
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A characterization of nuclear functionals in terms of continuity with respect to some special topologies on $B(X)$?
I think, nuclear functionals on the space of operators $B(X)$ (on a Banach space $X$) must have a characterization in terms of some special continuity. I would be grateful if somebody could help me ...
8
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1
answer
621
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When does the dual to the space $K(X)$ of compact operators consist of nuclear functionals?
Let $X$ be a Banach space and $B(X)$ be its space of all (bounded) operators. A nuclear functional on $B(X)$ is a linear functional $u:B(X)\to{\mathbb C}$ that can be represented in the form
$$
u(A)=\...
2
votes
1
answer
137
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Continuity of the derivations from semisimple Banach algebras
Let $A$ be a Banach algebra and $X$ a Banach $A$-bimodule. It is known that if $A$ is a $C^*$-algebra, then by Ringrose theorem every derivation $D:A\rightarrow X$ is continuous. Also, a famous ...
0
votes
1
answer
240
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Banach algebra $A$ without an approximate identity but $A^2=A$
Please help me with the following question.
What are some examples of Banach algebra $A$ satisfying the following two conditions?
$1$.$ A $ does not have an approximate identity.
$2$. $A^2=A$. ...
4
votes
1
answer
447
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Bicommutant theorem for commutative operator algebras
Let $\mathcal{B}(H)$ denote the space of bounded linear operators on a complex Hilbert space $H$. The von Neumann bicommutant theorem says:
Theorem. Suppose that $\mathcal{A}$ is a $C^*$-subalgebra ...
1
vote
0
answers
46
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Show that $\Phi_{P(X)}=\hat{X}$
Let $X$ be a compact subset of $\mathbb {C^n}$. The polynomial convex hall of $X$ is the set
$\hat{X}=\{z\in \mathbb {C^n}: \left|P(z) \right|\leq \left||P |\right|_\infty , \text{for all polynomial ...
5
votes
1
answer
147
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Whether every algebra norm $\left|\cdot\right|$ on $C(X)$ is equivalent to uniform norm $\left|\cdot\right|_X$
Suppose that $X$ be a compact space and $\left|\cdot\right|$ be an algebra norm on $C(X)$
Is every algebra norm $\left|\cdot\right|$ on $C(X)$ equivalent to uniform norm $\left|\cdot\right|_X$...
6
votes
1
answer
428
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Projections in the tensor product of von Neumann algebras
This question seems elementary, but I have already asked an expert who does not know the answer, so I would like to post here.
Let $M$ and $N$ be von Neumann algebras, and let $M\bar{\otimes}N$ be ...
1
vote
0
answers
87
views
An example of a Banach algebra with a specified property
I asked this question
(https://math.stackexchange.com/questions/3076735/an-example-of-a-banach-algebra-satisfying-given-conditions)
but unfortunately no one answered it. Please help me to find an ...
0
votes
1
answer
233
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The definiton of a multiplier on a Banach algebra
Let $A$ be a Banach algebra. Some textbooks define a (left ) multiplier as a map $T:A\rightarrow A$ satisfying $T(ab)=T(a)b$ for all $a,b\in A$ and assume that $A$ needs to be a without order Banach ...
6
votes
0
answers
113
views
Homomorphisms from BV
Denote by $\mathsf{BV}(\mathbb T)$ the Banach space of functions on the circle with bounded variation which is a Banach algebra under the pointwise product. Is there a surjective homomorphism from $\...