# Questions tagged [banach-algebras]

The banach-algebras tag has no usage guidance.

268
questions

**6**

votes

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77 views

### 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 ...

**2**

votes

**1**answer

51 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

132 views

### 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

73 views

### 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

116 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

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126 views

### 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**

vote

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109 views

### 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

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124 views

+50

### 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

146 views

### $K$-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,...

**2**

votes

**1**answer

113 views

### 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

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86 views

### 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

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67 views

### 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

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196 views

### 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

117 views

### 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

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22 views

### 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

78 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

137 views

### 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**

votes

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242 views

### 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

313 views

### 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 ...

**5**

votes

**1**answer

152 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 \...

**2**

votes

**0**answers

151 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

134 views

### 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 ...

**5**

votes

**1**answer

132 views

### 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

352 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

148 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

331 views

### 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

130 views

### 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

976 views

### 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

155 views

### 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**

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41 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 ...

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103 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**

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114 views

### 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

75 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 $0$.
...

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147 views

### 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

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78 views

### 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**

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142 views

### 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**

votes

**1**answer

210 views

### 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

98 views

### 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

124 views

### 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

237 views

### 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 ...

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44 views

### 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 ...

**5**

votes

**1**answer

134 views

### 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 $\left|\cdot\right|_X$...

**4**

votes

**1**answer

166 views

### 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 ...

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78 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

132 views

### 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

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104 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 $\...

**0**

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144 views

### Automatic continuity in Banach algebras

I have the following two questions
1: Let $A$ and $B$ be Banach algebras and suppose that $B$ is semisimple. Let $T:A \to B $ be a homomorphism with $\overline {TA}=B.$ Is $T$ automatically ...

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votes

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155 views

### Gelfand spectrum as a measure space

Given a Lebesgue probability measure space $(X,m)$ (say, just the unit interval with the Lebesgue measure on it), let $A$ be a closed subalgebra of the real $L^\infty(X,m)$. Then one can realize the ...

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51 views

### Evaluate $\operatorname{Rad}(A/\operatorname{Rad}(A))$ in a Banach algebra

I've asked this question here
Let $A$ be a Banach algebra with identity $e_A$, I'd like to find
$\operatorname{Rad}(A/\operatorname{Rad}(A)).$
whre we define
$\operatorname{Rad}(A)=\{a\in A:...

**4**

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64 views

### Characterizing the separability of the Gelfand space of a semisimple commutative Banach algebra

Problem. Is the separability of the Gelfand space of a semi-simple commutative Banach algebra $A$ equivalent to the existence of a countable family $\{\varphi_n\}_{n\in\omega}$ of multiplicative ...