The banach-algebras tag has no usage guidance.

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### Unitizations of Banach algebras and matrix norms

Consider the short exact sequence of Banach algebras $0\rightarrow A\rightarrow A^+\rightarrow\mathbb{C}\rightarrow 0$ where $A$ is a Banach algebra without unit and $A^+$ denotes the unitization of ...

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129 views

### $K$-Theory of finite dimensional Banach algebras

Is there a finite dimensional Banach algebra $A$ for which $K_{0}(A)$ is a finite group?
I asked this question in MSE but I received no answer
...

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**1**answer

77 views

### The image of a derivation on a Banach algebra is contained in the kernel of a character

It is known that if $D$ is a continuous derivation on a commutative Banach algebra $\mathcal{A}$, then for any nonzero character $\theta$ on $\mathcal{A}$ we have $D(\mathcal{A})⊆ker\theta $.
...

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115 views

### Spectral mapping theorem for polynomials in $z,\overline{z}$ and direct construction of the function calculus for a normal operator

Suppose that $A$ is an element in Banach algebra and $p$ is a polynomial. Then we have an equality $p(\sigma(A))=\sigma(p(A))$ where $p(A)$ has an elementary meaning. This theorem (the spectral ...

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120 views

### Spectrum of unitary elements of a Banach algebra

Unitary elements of a Banach space have been defined in this paper as follows:
Let $A$ be a Banach space and $a\in A, \|a\|=1$. Let $S_{a}=\{f\in A':\|f\|=1=f(a)\}$. Then $a$ is said to be ...

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votes

**1**answer

169 views

### Is the Fourier-Stieltjes algebra of a locally compact group semi-simple?

Let $G$ be a locally compact group. Is the Fourier-Stieltjes algebra $B(G)$ semi-simple?

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### Every norm-decreasing algebra morphism $L_1(G)\to\mathcal{B}(E)$ comes from a group representation

In section 8 of this paper http://arxiv.org/abs/math/0611833v3 the author proves the following:
If $E$ is a reflexive Banach space, $G$ a locally compact group and $\pi:L_1(G)\to\mathcal{B}(E)$ a ...

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203 views

### Definition of a normed ring

A normed ring "should" be a monoid object in the monoidal category of normed abelian groups. There are (at least) two choices of morphisms of normed groups, namely bounded or short homomorphisms, ...

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158 views

### Infinite topological direct sum of amenable Banach algebra

Is an infinite topological direct sum of amenable Banach algebras amenable again?
Can you give me a good reference about this notion?
Thanks

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**1**answer

134 views

### comparing norms of tensor product of two Hilbert spaces

Suppose $H_1$ and $H_2$ are two Hilbert spaces with dimension $n$ and $m$, for $ x \in H_1 \otimes H_2$
consider $$\|x\|_\pi = \inf \left\{ \sum_{i=1}^n \|a_i\| \|b_i\| : x = \sum_{i} a_i \otimes b_i ...

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votes

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420 views

### Banach algebraic proof of the Borsuk Ulam theorem

I am wondering whether there exists a proof of the Borsuk Ulam theorem
for the Euclidean n-sphere that is based only on the theory
of Banach algebras. I checked on MR but had no success.

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152 views

### Connected component of the identity in graded Banach algebras

I search for a noncommutative idempotent-less Banach algebra $A$ which is graded by a finite abelian group $G$ such that a nontrivial homogenous element lies in the same connected component as ...

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215 views

### Is there support for the term “Gelfand algebra”?

In this question Yemon Choi asked whether there is a standard term for Banach algebras for which the submultiplicative law
($\|ab\| \leq \|a\| \|b\|$) is weakened to merely requiring the product to be ...

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**1**answer

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### Semi-simple Banach algebra

Is there an example of an unital commutative semi-simple Banach algebra which it is not amenable?

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### Quotient of a Banach algebra [closed]

Let $A$ be a Banach algebra. Is there a Banach algebra $B$ and a non-trivial closed ideal $I$ of $B$ such that $\frac{B}{I}\cong A$?

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287 views

### Is an ultrapower of a faithful Banach algebra always faithful?

Let $A$ be an infinite dimensional faithful Banach algebra and let $\mathcal U$ be a free ultrafilter. Is the ultrapower $(A)_{\mathcal U}$ faithful?

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274 views

### Is there a Gelfand-Naimark-like characterization of group algebras $L_1(G)$?

The Gelfand-Naimark theorem establishes that a complex commutative Banach algebra $A$ with an identity and an involution $x\to x^*$ satisfying $\|x x^*\|=\|x\|^2$ is (isometrically isomorphic to) a ...

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**1**answer

65 views

### Ideal in projective tensor product of Banach algebras [closed]

Let $A,B$ be Banach algebras and $A\hat{\otimes}B$ be projective tensor product of them.
Let $S$ be an ideal of $A\hat{\otimes}B$. Are there ideals $I$ of $A$ and $J$ of $B$ such that
...

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**1**answer

288 views

### Can a finite von Neumann algebra be strongly morita equivalent to a properly infinite von neumann algebra?

Can a finite (by finite I mean when the projection $1$ is finite) von Neumann algebra be strongly morita equivalent to a properly infinite von Neumann algebra?
(Strong morita equivalence is the same ...

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votes

**1**answer

618 views

### Maximal ideals of the rings of Baire-One Functions

A real function $f:X\rightarrow \mathbb{R}$ is called Baire-one function, if there is a sequence $(f_n)_{n=1} ^\infty$ of continuous functions $f_n:X\rightarrow \mathbb{R}$ on $X$ so that for all ...

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### One question about the tensor product of $L^1(G)$ and a Banach space $A$

We know that the tensor product of $L^1(G)$ and a Banach space $A$ is isometric to $L^1(G, A)$, the space of all Bochner-integrable $A$-valued functions on a locally compact group $G$. I am looking ...

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**1**answer

297 views

### A commutative Banach algebra with an abundance of discountinuous functions

Let $A$ be the algebra of all bounded functions from $[0,\;1]$ to $\mathbb{C}$.
For $f\in A,\;$ $\omega_{f}$ is the standard oscillation function.. Each of the following two (equivalent) norms ...

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### questions about the proof of the theorem of completely positive order zero maps

I hope my question is ok for mathoverflow. I first asked on math.stackexchange but received no answer and then delated it.
I want to understand the proof of the theorem (which you can find in the ...

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**1**answer

164 views

### application of factorization theorem

Young's inequlity tells us that $L^{1}(\mathbb R)\ast L^{p}(\mathbb R) \subset L^{p}(\mathbb R)$ with norm inequality
$$\|f\ast g\|_{L^{p}} \leq \|f\|_{L^1}\|g\|_{L^p};$$
and of course this ...

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**1**answer

247 views

### Theory of C* algebras over other fields than the complex numbers

How much of the theory of C*-algebras holds when the complex numbers are replaced by different (algebraically closed) field (possibly with a distinguished ordered subfield that satisfies the same ...

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**1**answer

270 views

### Can states on commutative Banach algebras be understood as probability measures?

Suppose $\mathcal{A}$ is a commutative Banach algebra (over $\mathbb{R}$) or commutative Banach *-algebra (over $\mathbb{C}$). Is there always a measurable space $(\Omega,\mathcal{F})$ such that there ...

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152 views

### Matrix norm on $L^p$ operator algebras

I have a question about the framework of $L^p$ operator algebras ($1<p<\infty$), i.e. norm-closed subalgebras of $B(L^p(X,\mu))$ for some measure space $(X,\mu)$ (see e.g. ...

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### Characterizations of Wiener algebra

The Wiener algebra $\mathcal W$ is defined as $\text{Fourier}(L^1(\mathbb R))$, i.e. the image by the Fourier transform of $L^1(\mathbb R)$. Riemann-Lebesgue's lemma ensures that
$$
\mathcal W\subset ...

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### $\mathbb{Z}G$ (left) Noetherian$\Rightarrow$ $l^1(G)$ is a flat $\mathbb{Z}G$-(right) module?

Let $G$ be a countable discrete group (not necessarily abelian), and suppose the group ring $\mathbb{Z}G$ is a left-Noetherian ring, for example, when $G$ is a polycyclic-by finite group.
Denote the ...

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162 views

### Holomorphic functional calculus and idempotents

One of the applications of the holomorphic functional calculus is with regard to idempotents. For instance, if an element $a$ in a unital Banach algebra $A$ has spectrum contained in two balls, each ...

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### Did Gelfand's theory of commutative Banach algebras influence algebraic geometers?

Guillemin and Sternberg wrote the following in 1987 in a short article called "Some remarks on I.M. Gelfand's works" accompanying Gelfand's Collected Papers, Volume I:
The theory of commutative ...

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### Root of positive function in Fourier algebra

Let $G$ be a locally compact group, let $A(G)$ be the Fourier algebra of $G$. We think of $A(G)$ as a subalgebra of $C_0(G)$.
Question 1: Let $f\in A(G)$ be a function that is pointwise positive. ...

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### Sum-epimorphisms and prod-monomorphisms

Sum-epimorphisms
A longer time ago I have introduced the bi-onto maps for the topological category. Let me formulate here its general categorical definition:
...

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180 views

### Inductive/Projective Limits of Topological Algebras

It is common to form inductive/projective limits of Banach/Frechet spaces in order to come up with natural topologies for common vector spaces. For instance,
For $k \ge 0$ and $K_n$ compact ...

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### How do functions operates in a Fourier algebra $A^{q}(\mathbb T)$?

We put , $A^{q}(\mathbb T)= \{ f\in L^{q}(\mathbb T): \hat{f}\in \ell^{q}(\mathbb Z) \}.$
By Helson-Kahane-Katznelson-Rudin Theorem, it follows that,
"Let $F$ be a function on $\mathbb C$ and if ...

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### Can we obtain a derived category from an additive category? Like a category of Banach modules?

Let $A$ be a Banach algebra, let $A$-mod be the category of left Banach modules (as defined in Helemskii's "Banach and locally convex algebras"), $A$-mod is an additive category, but not abelian ...

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591 views

### When $C(X)$ is an injective $C(X)$-module? Current answer is erroneous

It is an old question if every injective Banach space is isomorphic as Banach space to $C(X)$-space.
I would like to know if the weakened module version of this question is answered. More precisely: ...

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233 views

### Can we extend a multiplicative linear functional of a closed left ideal on whole of the algebra?

Let B be a closed left ideal of a Banach algebra A. Also, B has a right approximate identity (in B).
If g is a nonzero multiplicative linear functional on B, can we always extend g to a ...

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209 views

### Is $\mathcal{K}(H)$ injective $\mathcal{B}(H)$-module?

Does anyone know if the right Banach $\mathcal{B}(H)$-module $\mathcal{K}(H)$ is injective?
The same question for $\ell_\infty$-module $c_0$. Both these modules are not dual, so standard arguments ...

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201 views

### Are norm-continuous representations smooth?

Let $G$ be a real Lie group and $A$ a unital Banach algebra. Let us call a map $\varphi:G\to A$ a (norm-)continuous representation, if it is continuous
$$
x_i\to x\quad\Longrightarrow\quad ...

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229 views

### C*-Algebras: Dynamics vs. Derivations

Problem
Given a C*-algebra $\mathcal{A}$.
Consider dynamics $\tau:\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$ and $\tau':\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$.
(More precisely, strongly continuous ...

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254 views

### What function space does holomorphic functional calculus give us?

Let $A$ be a unital Banach algebra, $U$ be an open subset of $\mathbb{C}$, and $A_U:=\{x\in A:\sigma(x)\subset U\}$. Holomorphic functional calculus says that any holomorphic function ...

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163 views

### Wendel Theorem for center of group algebra

Let $G$ be a locally compact SIN-group. Then $ZL^{1}(G)$ has a bounded approximate identity. I want to prove that the multiplier algebra of $ZL^{1}(G)$ is equal to $ZM(G)$ (center of measure algebra). ...

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### Non commutative topological manifolds

Assume that $A$ is a Banach algebra with two closed two sided ideals $I$ and $J$ such that $I$ and $J$ are commutative and $A=I+J$. Does this implies that $A$ is commutative? For the ...

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### Reasonable crossnorm on Banach algebra tensor product constructed from isometric non-degenerate representations

The following is an abstraction of a more specific problem I've been grappling with, so I might give some extra unnecessary information. (The `bounded approximate
left identities' assumption is ...

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**1**answer

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### ideals in the disk algebra

Let A be the disk algebra, of continuous functions on the closed disk holomorphic on the interior, with sup-norm denoted || . || . Let x be an interior point of the disk. Does there exist a ...

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### Tangent space of the Fourier algebra $A(G)$

Let $G$ be a real Lie group and $A(G)$ be its Fourier algebra. Let us call a linear continuous functional $f:A(G)\to{\mathbb C}$ a tangent vector of $A(G)$ in the point $a\in G$, if it satisfies the ...

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### When is the Fourier algebra $A(G)$ enough close to the Fourier-Stieltjes algebra $B(G)$?

I am reading P.Eymard's paper on the Fourier algebras of locally compact groups, and I have several questions about his constructions. I asked one of them in math.stackexchange, so far without ...

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### The set of matrices with same spectral radius

I am working on an optimization problem over the set of positive matrices (that is, matrices where all entries are positive numbers) that have the same spectral radius. My main problem is how to ...

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### Smallest left bounded approximate identity in $L_1(G)\ast\mu$

Let $G$ be a locally compact group, and $\mu\in M(G)$ be an idempotent measure. If $(e_\alpha)$ is a standard approximate identity of $L_1(G)$, then $(e_\alpha\ast\mu)$ is a left approximate identity ...