# Questions tagged [banach-algebras]

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323
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### Solvability and nilpotency for Banach algebras

Do we have topological counterparts of solvability and nilpotency, which are central concepts for (finite-dimensional) Lie algebras, for infinite dimensional Banach algebras with the commutator ...

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### Contractive projections on operator algebras

This is a follow up on an earlier question.
In [Lau&Loy, 2008] a Banach algebra $\mathcal{U}$ was called to have the Tomiyama property if any contractive projection $P:\mathcal{U}\to \mathcal{U}$, ...

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### A closed ideal in $L^1(T)$

Let $\mathbb{T}$ be the unit circle and consider the convolution group algebra $L^1(\mathbb{T})$. Let $I_n$ be the closed ideal generated by the polynomial $p_n(z)=z^n-1$ in $L^1(\mathbb{T})$.
Let $I=...

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### Banach algebras for which left invertible implies invertible

Are there noncommutative Banach algebras in which left invertibility implies invertibility? If so, what are they called?

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92
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### Regarding variation of spectra

I have been reading the article The variation of spectra by J.D Newburgh. in this article and all related reference/ articles, the term 'variation of spectra' keeps coming in, but I nowhere find a ...

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### The functional calculus on continuous functions

Let us consider $C[0,1]$ the space of continuous functions on the closed unit interval. For a given $x$ in $C[0,1]$, let us consider $A(sp(x))$, all analytic functions on a neighborhood of $sp(x)$.
...

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### Normalization of unital Banach algebras whose units are not norm one

Let $A$ be a unital Banach algebra whose unit is not necessarily norm one.
By a normalization of $A$ we mean a pair $(A^e,\Gamma)$ where
1- $A^e$ is a unital Banach algebra whose unit is norm one.
2- ...

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297
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### Permanent invertible elements

Let $A$ be a unital complex algebra with the unit $\bf1$. Let $\mathcal{N}$ be the family of all norms on $A$ making it a unital normed algebra with the same unit $\bf1$. Let us put
$B_{\|\cdot\|}...

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### Two completely different norms on a unital algebra!

Does there exist any unital normed algebra $(A,\|\cdot\|)$ enjoying another norm $\|\cdot\|_1$ such that
$(A,\|\cdot\|_1)$ forms a unital normed algebra with the same unit.
Any element contained ...

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### Are fibers in the corona of $H^\infty$ separable?

Let $\mathcal{M}(H^\infty(\mathbb{D}))$ denote the spectrum of the Banach algebra $H^\infty$ and $\mathcal{M}_z(H^\infty(\mathbb{D}))$ the fiber over $z\in \mathbb{D}$, i.e. $\{\varphi\in \mathcal{M}:...

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### $ f,g\in \mathrm{VMO} $ but $ f\cdot g\notin \mathrm{VMO} $

We say a function $ f\in L^1_{\mathrm{loc}}(\mathbb{R}) $ is in $\mathrm{BMO}(\mathbb{R})$ if
$$\|f\|_{\mathrm{BMO}}=\sup_{I}\frac{1}{|I|}\int\limits_I |f(y)-f_I|\, dy<\infty$$ for all intervals $I\...

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### Is a certain property of a continuous map preserved under "surjectification"?

Let $X$ and $Y$ be compact Hausdorff spaces and let $\varphi:X\to Y$ be continuous with a property that if $A$ is a nowhere dense zero-set in $Y$, then $\varphi^{-1}(A)$ is nowhere dense in $X$. Let $...

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### A list of unital Banach *-algebras whose left annihilators are principal

Let $A$ be a unital Banach *-algebra. For a given subset $S\subseteq A$, the left annihilator of $S$, denoted by Ann($S$), is given by Ann$(S)=\{x\in A: xS=0\}$. Let us say Ann($S$) is principal if ...

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### Unital commutative dual Banach *-algebras whose $w^*$-closed ideals are principal

Let $A$ be a commutative Banach *-algebra. For a given ideal $I$ of $A$, we say that, it is principal if there is a projection $p$ (i.e. $p^2=p=p^*$) in $A$ with $I=Ap$.
Q. Any characterization ...

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### Are nearby subalgebras of matrix algebras conjugate?

Let $k=\mathbb{R}$ or $\mathbb{C}$ and let $A$ be a finite-dimensional $k$-algebra. If $A$ is simple, then the Skolem-Noether theorem says that any two algebra homomorphisms $f, g: A \to M_n(k)$ are ...

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### Does every infinite-dimensional Banach algebra contain an infinite-dimensional subalgebra with second-countable primitive ideal space?

Let $A$ be an infinite dimensional Banach algebra. Even if separable the primitive ideal space of $A$ need not be second-countable when endowed with the hull-kernel topology. Can we at least find an ...

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### Non-emptiness of spectrum $\sigma(a)$ in non-Archimedean Banach algebras

I'm trying to determine whether or not the standard proof that the spectrum of a point in a unital Banach algebra is non-empty can be adapted to prove the same thing over certain non-Archimedean ...

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### A generalisation of closed and bounded subsets of non-Archimedean fields to topological spaces

The definition of compactness in topological spaces generalises the notion of a subset of $\mathbb{R}^n$ being closed and bounded, as expressed by the Heine-Borel Theorem.
In finite-dimensional vector ...

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### Left ideals of $\ell^{\infty}(A)$ containing all weakly null sequences in a Banach algebra $A$

Let $A$ be a Banach algebra. $\ell^{\infty}(A)$, the space of all bounded sequences in $A$, is a Banach algebra with pointwise operations.
Let $w_0(A)$ be the subspace of all weakly null sequences in $...

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### If $A\hat\otimes B$ has identity then so are $A$ and $B$

Let $A$ and $B$ be commutative Banach algebra. I have proven that if $A$ and $B$ have identity $e_A$ and $e_B$ respectivly , then $e_A\hat\otimes e_B$ is identity for $A\hat\otimes B$ (the ...

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### $B(X)$ and $B(X)^{**}$ for superreflexive Banach spaces $X$

Let $A$ be a Banach algebra. It is well-known that $A^{**}$ with either Arens product has separately weak$^{*}$ continuous multiplication if and only if $A$ is Arens regular [Theorem 1, ...

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### Projectional skeletons in dual Banach algebras

A Banach algebra $A$ is a dual Banach algebra if it is a dual Banach space with a (not necessarily unique) predual $A_{\ast}$, and the multiplication on $A$ is separately weak*-continuous. Dual Banach ...

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### Separable subalgebras of non-separable reflexive Banach algebras

Let $A$ be a non-separable reflexive Banach algebra. Every separable subspace of $A$ is contained in a separable 1-complemented subspace [Lindenstrauss,1966]. It is straightforward to show that every ...

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### What are the maximal ideals of $C_0 (X),$ where $X$ is a locally compact Hausdorff space?

Crossposted from MSE
How do the maximal ideals of $C_0(X)$ look like where $X$ is a locally compact Hausdorff space?
I know that if $X$ is a compact Hausdorff space then the maximal ideals of $C(X)$ ...

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### When is the annihilator of the commutator subspace a complemented subspace?

Let $A$ be a unital Banach algebra and $C$ be its commutator subspace, i.e., $C$ is the norm-closure of the subspace spanned by the elements of the form $xy-yx$ in $A$.
Notation: Let $C^{\perp}=\{f\in ...

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### Second dual $X^{**}$ of ternary $C^*$-ring $X$ is again ternary $C^*$-ring?

Recall that a ternary $C^*$-ring is a complex Banach space $X$, equipped with a associative ternary product $[.,.,.]:X^3 \to X$ which is linear in outer variables and conjugate linear in middle ...

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### Trying to understand construction of $C^*$-algebra corresponding to a ternary $C^*$-ring from a paper

Recall that a ternary $C^*$-ring is a complex Banach space $X$, equipped with a associative ternary product $[.,.,.]:X^3 \to X$ which is linear in outer variables and conjugate linear in middle ...

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### Connectedness of Invertible elements in a non- commutative C*- algebra

The Gelfand Naimark Segal theorem says that any complex C* algebra $A$ is isometrically isomorphic to a C* sub-algebra of bounded operators on a Hilbert space.
Here we see that the set of all ...

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140
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### Regarding socle of a C* algebra

I wanted to know if the socle of a complex C*-algebra is essential?
Can anyone suggest a text where the socle is studied in detail. I tried reading it from the book by Bernard Aupetit, A Primer in ...

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### Operator algebra on an invariant subset

In Rickart, page 50 Theorem 2.2.1, the statement is made: A linear subspace $\mathfrak{M}$ of the algebra $\mathfrak{A}-\mathfrak{L}$ is invariant with respect to the representation $a{\rightarrow}A_a^...

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### Trace norm of operators obtained by restricting the matrix of a trace class operator

Suppose $H$ is a Hilbert space with orthonormal basis $\{e_i\}_{i\in \mathbb N}$. To every operator $T$, we associate a infinite matrix $[T_{ij}]$, where $T_{ij}=\left<Te_j,e_i\right>$. We know ...

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### Does there exists a $C^*$-algebra corresponding to every Banach ternary algebra?

Let $V$ be a TRO (closed subspace of $B(H,K)$, closed under the product $xyz\to xy^*z$). Let $C(V)$, $D(V)$ denote the $C^{\ast}$-algebra generated by $VV^{\ast}$ and $V^*V$ respectively. We define $A(...

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### Weak sequential continuity of certain bilinear forms on Banach algebras

Let $A$ be a Banach algebra and $Bil(A)$ denote the space of bounded bilinear forms on $A$.
$Bil(A)$ is a Banach $A$-bimodule with the module operations
\begin{eqnarray*}
\beta a(x,y) &:=& \...

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### Non commutative Teichmuller theory

Perhaps the first example in Teichmuller theory is the following proposition:
Proposition: Let $1<r<R$. Then two annular region $U_r=\{z\in \mathbb{C}\bigm|1<|z|<r\}$ and $U_R=\{z\in \...

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### Classical fixed-point argument and invertible function

Let $n\in\mathbb{N}$ and $W^{1,\infty}(\mathbb{R}^n)=\lbrace f:\mathbb{R}^n\rightarrow \mathbb{R}^n : \text{ f is bounded and Lipschitz continuous } \rbrace$. Suppose $f\in W^{1,\infty}(\mathbb{R}^n)$ ...

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### When does a positive operator preserve invertibility

Let $\Omega_1,\Omega_2$ be compact Hausdorff spaces and let $P:C(\Omega_1)\longrightarrow C(\Omega_2)$ be a unital positive operator. I wanted to know if there is a necessary and sufficient condition ...

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### Regarding an element being self adjoint

Let $A$ be a unital C*-algebra. Let $x,y\in A$ be self adjoint elements in $A$, with $x$ being invertible. Can we say that the spectrum of $x^{-1}y$ is a subset of the real line? I know this is true ...

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### Density of normal elements in a C*- algebra [closed]

Let $A$ be a unital C*-algebra.
I wanted to know if there is a necessary and sufficient condition for normal elements to be dense in $A$?

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### Gelfand ring in Bourbaki's exercises

In Bourbaki's General Topology, Chapitre III §6 Exercise 11, they define a Gelfand Ring as a topological ring $A$ such that
The set $A^*$ ($=A^{-1}$) of invertibles is open.
The uniform structure ...

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### Reflexive norm-closed subalgebras of $B(X)$

Let $X$ be a reflexive Banach space, and let $B(X)$ denote the set of all bounded linear operators $X\to X$.
Does there exist a norm-closed subalgebra $A\subseteq B(X)$ with the following properties?
...

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### Universality in the class of separable Banach algebras

Let us consider the class of Banach algebras with homomorphisms that are bounded below but not necessarily isometric.
Is there a separable Banach algebra that contains isomorphic images of all ...

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### Understanding vector-valued analytic functions vs holomorphic functional calculus

Let $A$ be a unital Banach algebra over complex numbers and call elements of $A$ "vectors". Let $\Omega$ be an open set in $\mathbb{C}$ and $H(\Omega)$ the space of analytic functions on $\...

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### Infinite ordered products (reference request)

While writing arXiv:1510.05757v2, I found myself proving some basic facts about products of Banach algebra elements over an infinite totally ordered set. (Statements and proofs are in Appendix C.) The ...

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### Has this Banach algebra been studied?

Given $\Omega$ as $[0,1]^n$ or the closed unit ball in $\mathbb{R}^n$, we can consider the algebra of complex valued polynomials with pointwise multiplication and its closure with respect to the norm
...

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### Is the union of good equivalence relations on a compact space good?

Let $X$, $Y_1$ and $Y_2$ be a compact Hausdorff spaces and let $\varphi_i:X\to Y_i$ be a continuous surjection (and so a quotient map).
Let $\sim$ be the minimal closed equivalence relation on $X$ ...

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### Two inequalities in $C^*$ algebras

Under what conditions on a $C^*$ algebra $A$ we have the following inequality:
$$x^*a^*ax+a^*x^*xa\leq x^*x+a^*x^*ax+x^*a^*xa\;\;\; \forall x,a\in A$$
The second identity which I am looking for is ...

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### Irreducible sub-modules of $\ell^2(\mathbb{Z})$

It is known that $\ell^2(\mathbb{Z})$ is $\ell^1(\mathbb{Z})$-module (the module operation is the convolution).
What about the irreducible submodules? Can we characterize them?

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### Is the product of two Banach algebras given by the injective cross-norm itself a Banach algebra?

I understand that you can take the tensor product of Banach spaces in many different ways by specifying different norms; of particular interest to me are the cross-norms. The projective and injective ...

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### A noncontinous algebra map between Banach algebras

What is an example of two Banach algebras $A$ and $B$, and an algebra map $\phi:A \to B$ which is not continuous?

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### Good source for Jordan Fréchet algebras

Is there any good source for Jordan Fréchet (or more generally, Jordan locally convex) algebras?
I'm looking for something on the level similar to the level of the book "Banach and Locally Convex ...