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2
votes
1answer
198 views

Does Fourier Algebra of locally compact group separate compact sets of the group?

Let $G$ be a locally compact group. Consider the left regular representation $\lambda$ over $L^2(G)$. Then according to Eymard, Fourier algebra of $G$, $A(G)$ is the set of all coefficients of ...
8
votes
3answers
472 views

is every element in a C* algebra a product of normal elements?

I have the following question and since I am not an expert on C*-algebras, I thought I ask it here: I know that in general the sum and product of normal elements need not be normal. It is even true ...
4
votes
0answers
65 views

$p$-operator space structure on Banach algebras

There is an abstract characterization of operator algebras, which says that if $A$ is an operator space that is also an approximately unital Banach algebra, then the following are equivalent: For ...
20
votes
2answers
1k views

In a Banach algebra, do ab and ba have almost the same exponential spectrum?

Let $A$ be a complex Banach algebra with identity 1. Define the exponential spectrum $e(x)$ of an element $x\in A$ by $$e(x)= \{\lambda\in\mathbb{C}: x-\lambda1 \notin G_1(A)\},$$ where $G_1(A)$ is ...
1
vote
2answers
232 views

$C[0,1]$ is Banach-space isomorphic to $c_0(C[0,1])$

$c_0(C[0,1])$ is the $c_0$-direct sum of countably many $C[0,1]$.How to prove $C[0,1]$ is Banach-space isomorphic to $c_0(C[0,1])$. Here,Banach-space isomorphism means a bounded invertible operator ...
2
votes
1answer
165 views

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 ...
3
votes
1answer
219 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 ...
3
votes
2answers
594 views

Banach algebraic proof of the Borsuk Ulam theorem

I am wondering whether there exists a proof of the classical Borsuk Ulam theorem for the Euclidean n-sphere, $n>2$ that is based only on the theory of Banach algebras. I checked on MR but had no ...
3
votes
0answers
62 views

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 ...
1
vote
1answer
89 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 $. ...
7
votes
1answer
140 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 ...
5
votes
1answer
142 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 ...
3
votes
1answer
180 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?
5
votes
1answer
210 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, ...
5
votes
2answers
167 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
3
votes
1answer
162 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 ...
1
vote
0answers
158 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 ...
13
votes
0answers
229 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 ...
2
votes
1answer
109 views

Semi-simple Banach algebra

Is there an example of an unital commutative semi-simple Banach algebra which it is not amenable?
-3
votes
1answer
100 views

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 $\frac{B}{I}\cong A$?
3
votes
1answer
290 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?
11
votes
1answer
281 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 ...
0
votes
1answer
67 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 ...
7
votes
1answer
295 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 ...
6
votes
1answer
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 ...
1
vote
1answer
95 views

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 ...
2
votes
1answer
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 ...
3
votes
1answer
142 views

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 ...
5
votes
1answer
170 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 ...
10
votes
1answer
266 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 ...
3
votes
1answer
274 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 ...
1
vote
0answers
171 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. ...
3
votes
1answer
159 views

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 ...
12
votes
2answers
396 views

$\mathbb{Z}G$ (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 ...
1
vote
2answers
173 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 ...
52
votes
4answers
4k views

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 ...
4
votes
2answers
174 views

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. ...
1
vote
0answers
137 views

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: ...
2
votes
0answers
228 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 ...
1
vote
0answers
91 views

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 ...
4
votes
0answers
214 views

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 ...
7
votes
1answer
597 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: ...
6
votes
1answer
245 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 ...
3
votes
0answers
215 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 ...
3
votes
1answer
206 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 ...
4
votes
1answer
230 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 ...
6
votes
0answers
293 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 ...
2
votes
1answer
169 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). ...
2
votes
1answer
252 views

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 ...
4
votes
1answer
375 views

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 ...