All Questions
Tagged with fa.functional-analysis banach-algebras
258 questions
0
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45
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Amenability of locally convex algebras
Let $A$ be an amenable Banach algebra, and let $A_w$ denote $A$ with the weak topology. Clearly, $A_w$ is a Hausdorff locally convex algebra (l.c.a.).
Q0: Is $A_w$ amenable as a l.c.a. in the sense ...
- 2,605
2
votes
1
answer
78
views
Is there a relative projective tensor (cross-)norm for Banach $A$-algebras?
$\newcommand\norm[1]{\lVert#1\rVert}$I am interested in a relative version of the projective tensor product and projective tensor (cross-)norm for Banach algebras. Let $A$, $B$, $C$ be commutative (...
- 7,127
1
vote
0
answers
104
views
Commutative Banach $\mathbb{R}$-algebras without complex structure, but with path-connected group of units
For a finite-dimensional commutative (associative, unital) $\mathbb{R}$-algebra $A$, the condition $\pi_0(A^\times) = 1$ (i.e. the group of units of $A$ being path-connected) is equivalent to $A$ ...
- 7,127
9
votes
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answers
163
views
Moore-Penrose partial isometries and hermitian elements
Let $A$ be a unital Banach algebra. An element $a \in A$ is hermitian if $\|\mathrm{exp}(ita)\|=1$ for every $t \in \mathbb{R}$. An element $a \in A$ is Moore-Penrose invertible if there exists $b \in ...
- 3,497
2
votes
1
answer
127
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Strong Ditkin sets in the Fourier algebra
What is the definition of a Ditkin set (resp. a strong Ditkin set) for the Fourier algebra $A(G)$ of a locally compact (not necessarily abelian) group $G$?
More specifically, let $E$
be a closed ...
- 21
6
votes
3
answers
282
views
Extreme points of the dual unit ball of a Banach algebra
Let $A$ be a unital Banach algebra. Let $f\in A^*$, $\Vert f\Vert=1$ satisfy that there exists a maximal left ideal $L\subset A$ such that $L\subseteq\ker{f}$.
Question: Is $f$ an extreme point of ...
- 2,605
2
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1
answer
113
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Showing $\inf _{a \neq 0} \frac{\left\|a^2\right\|}{\|a\|^2}\leq \inf _{a \neq 0} \frac{\|\hat{a}\|_{\infty}}{\|a\|}$ in a commutative banach algebra
Suppose $A$ is a commutative Banach algebra, and let $u=\inf _{a \neq 0} \frac{\left\|a^2\right\|}{\|a\|^2}$, $v=\inf _{a \neq 0} \frac{r(a)}{\|a\|}$ ($r(a)$ is the spectral radius of $a$). I need to ...
- 775
7
votes
1
answer
252
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Does a Banach algebra version of "the sum of a closed subspace and a finite dimensional subspace is always closed" exist?
In the setting of Banach spaces, it is well know that if $M$ is a closed subspace of a Banach space $X$ and $F$ is a finite dimensional subspace of $X$, then $M+F$ is closed.
Does a Banach algebra ...
- 515
7
votes
2
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349
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Can the Banach algebra structure on $B(E)$ be (almost) retrieved from its Banach space structure?
This is basically just out of curiosity. Also, since my research area is in von Neumann algebras and my knowledge of general Banach algebras as well as general Banach spaces is somewhat limited, I ...
- 2,830
4
votes
0
answers
262
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Spectrum of ring in algebraic geometry vs spectrum of Banach algebra
For a commutative unital Banach algebra $A,$ and $x\in A,$ we have $\lambda \in \sigma_A(x)$ if and only if $\phi(x) = \lambda$ for some algebra homomorphism $\phi:A \to \mathbb C.$ The set of all ...
- 1,755
3
votes
1
answer
169
views
Can homomorphisms be Borel in some weaker topology?
Let $X$ be a Banach space with separable dual and let $A$ be a Banach algebra. Consider a norm continuous homomorphism $h$ from $L(X)$, the Banach algebra of bounded operators on $X$ onto $A$. In $L(X)...
- 31
4
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111
views
Flatness of $C_0(S)$-module $L_\infty(S,\mu)$
Let $S$ be a locally compact Hausdorff space. By $C_0(S)$ we denote the space of continuous functions vanishing at infinity. Let $\mu$ be a finite Borel regular measure om $S$, then consider $L_\infty(...
- 1,697
3
votes
1
answer
120
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Spectra of products variously permutated
Let $x,y$ be elements of a Banach algebra $A$ and $\lambda\in\mathbb C\setminus\{0\}$. If $\lambda-xy $ is invertible, then $\frac1{\lambda}\big[1+y(\lambda-xy)^{-1}x \big]$ is clearly an inverse of $...
- 60.5k
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97
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Amenability of $\textrm{w}_0(A)$ for a $C^*$-algebra $A$
Let $A$ be a $C^*$-algebra with only finite dimensional irreducible representations. As in a previous question, let $\textrm{w}_0(A)$ denote the subspace of $\ell^{\infty}(A)$ consisting of all weakly ...
- 2,605
1
vote
1
answer
142
views
Complemented C*-algebras
Let $A$ and $B$ be unital separable commutative $C^*$ algebras, with $A\subset B$. Is it true that $A$ is complemented in $B$?
- 149
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0
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146
views
Non-degenerate representation of a Banach algebra
Let $\mathcal{A}$ be a non-reflexive Banach algebra. For the definition of Arens product, please refer to this link. Here we let $\square$ denote the first Arens product and $\diamond$ denote the ...
- 307
1
vote
0
answers
104
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Amenability of $\textrm{w}_0(L^1(G))$
Let $G$ be an infinite compact group and $A=L^1(G)$. It is known that $c_0(A)$ is amenable [Runde2020, p.80] while $\ell^{\infty}(A)$ is not [Daws2009] .
Let $\textrm{w}_0(A)$ denote the subspace of $\...
- 2,605
3
votes
1
answer
161
views
Approximating continuous functions from $K\times L$ into $[0,1]$
Let $K$ and $L$ be compact Hausdorff spaces, let $f:K\times L\to [0,1]$ be continuous and let $\varepsilon>0$. Can we find continuous $g_{1},...,g_{n}:K\to[0,+\infty)$ and $h_{1},...,h_{n}:L\to[0,+\...
- 5,529
5
votes
1
answer
221
views
Arens regularity of $\mathrm{BV}(\mathbb{R})$
$\DeclareMathOperator\BV{BV}$A Banach algebra $A$ is called Arens regular if the two canonical multiplications on the double dual $A^{**}$ coincide. Let $\BV(\mathbb{R})$ denote the Banach algebra of ...
- 6,406
1
vote
1
answer
156
views
Do completely bounded maps on an operator space have a completely contractive Banach algebra structure?
Let $X$ be an operator space and $CB(X)$ be the set of all completely bounded linear maps $f: X \to X$. Note that $CB(X)$ becomes a Banach algebra for the composition of operators.
Is the ...
- 175
1
vote
1
answer
115
views
What is the socle of the $2\times 2$ matrix algebra over a Banach algebra?
$\DeclareMathOperator\soc{soc}$Let $\mathcal{A}$ be a unital semisimple Banach algebra. The socle of $\mathcal{A}$, $\soc(\mathcal{A})$ is defined as the sum of the minimal right ideals (which equals ...
- 515
2
votes
0
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157
views
Why do von Neumann algebras possess identity?
My starting point is that a von Neumann algebra is a $C^*$-algebra with a predual. The usual approaches to showing the existence of identity involve spectral theory (for approximate identity), but ...
- 433
3
votes
1
answer
148
views
positive functional on Banach *-algebra (with appro. identity) is continuous?
Theorem (N. Th.Varopoulos): Let $\mathcal{B}$ be a Banach *-algebra
with a bounded approximate identity. Then every positive functional $T$ on $\mathcal{B}$ is continuous.
I think this theorem is ...
- 325
3
votes
1
answer
387
views
Closed prime ideal in $C[0, 1]$
I know that maximal ideals of $C[0, 1]$ corresponds to singleton. Also, using Zorn's lemma one can construct a prime ideal in $C[0, 1]$ which is not maximal.
Is there any $\textbf{closed}$ prime ...
- 1,115
2
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0
answers
71
views
About isometric Banach algebra isomorphisms and WAP functionals
Let $B$ be a Banach algebra and $A\subseteq B$ a subalgebra. It is known that there is a quotient map $\phi: B^*/\textrm{wap}(B)\to A^*/\textrm{wap}(A)$ that is also an $A$-bimodule map.
Let's say ...
- 2,605
1
vote
0
answers
98
views
Representations of the dual Banach algebra pair $(\ell_1,c_0)$
Let $\displaystyle E_p=(\bigoplus_{n\in\mathbb{N}} \ell^1_n)_{\ell^p}$ for some $1<p<\infty$ and $\ell^1 = \ell^1(\mathbb{N})$ be equipped with the convolution. Then, there exists an isometric &...
- 2,605
1
vote
0
answers
93
views
About vector valued measure algebras
Let $G$ be a locally compact group and $A$ be a Banach algebra. By $L^1(G,A)$ and $M(G,A)$ we denote the $A$-valued group, and measure algebra.
Is $M(G,A)$ a Banach algebra (with convolution as the ...
- 2,118
2
votes
0
answers
354
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Weakly null sequences in projective tensor products
First, I'd like to record a question that may still be open. The snippet below is taken from DiestelPuglisi2009.
Second, let $E$ be a Banach space, $(u_n)$ be a weakly null sequence in the projective ...
- 2,605
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votes
0
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66
views
Two questions about the vector-valued Lipschitz algebra
For a commutative Banach algebra $A$ and for any $0<\alpha<1$, let $\text{Lip}_\alpha(K,A)$ consist of all $A$-valued functions $f$ on a metric space $(K,\text d)$ with the property that $\rho_\...
- 2,118
3
votes
0
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295
views
Dunford-Pettis like properties for Banach spaces of operators
Let $E$ be a Banach space and $A\subseteq B(E)$ be a Banach subspace of operators on $E$.
Suppose $A$ satisfies the property (RCC) given below:
$$
\left.\begin{array}{l}
(x_n)\subseteq A \textrm{ ...
- 2,605
1
vote
0
answers
142
views
Could we characterize elements in the second dual by the character space?
Let $A$ and $B$ be two semisimple commutative Banach algebras. Assume that $A\mathbin{\tilde\otimes} B$ is a Banach algebra obtained by completing $A\otimes B$ with respect to a cross norm not ...
- 2,118
11
votes
1
answer
428
views
Maximal ideals of the ring $\mathbb C \{T\}$
Consider the Banach $\mathbb C$-algebra
$$
\mathbb C \{T\} = \left\lbrace \sum_{i \geq 0} a_i T^i : \sum_{i \geq 0} |a_i| < \infty \right\rbrace
$$
With the norm given by $\| \sum a_i T^i\| = \sum |...
2
votes
0
answers
193
views
Commutative Banach algebras with zero-dimensional maximal ideal space and disjoint supports of Gelfand transforms
Let $A$ be a commutative semi-simple unital Banach algebra and let $\Delta$ be the maximal ideal space of $A$. Denote by $\widehat{\cdot}\colon A\to C(\Delta)$ the Gelfand transform.
If $\Delta$ is ...
- 21
1
vote
0
answers
81
views
Tracial linear functionals on an amenable Banach algebra
This post is related to an earlier question about Kazhdan property (T). The purpose of the snippet below is to briefly summarize the background for the question in this post.
Question: Does there ...
- 2,605
1
vote
0
answers
49
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Banach algebras satisfying $pq=qp=q \Rightarrow \|q\|\leq\|p\|$ for all idempotents $p$ and $q$
This question could be way below the level of MO, so apologies in advance. I posted the same question in MS about 10 days ago without a definitive answer so far.
Let $A$ be a Banach algebra with the ...
- 2,605
2
votes
1
answer
261
views
Qualitative difference between "continuous" and "discontinuous" states on $M(G)$
Let $G$ be a locally compact Abelian group (we can think that $G={\mathbb R}$). Let $C_0(G)$ be the space of continuous functions $u:G\to{\mathbb C}$ vanishing at infinity with the usual $\sup$-norm, ...
- 7,426
1
vote
0
answers
99
views
Module homomorphisms modulo compact operators
Let $A$ be a Banach algebra. Let $L_a,R_a:A\to A$ denote the left/right multiplication operators $$L_ax = ax, \hspace{5mm} R_ax = xa$$ for all $a,x\in A$. Assume that no nonzero $L_a$ and $R_a$ is a ...
- 2,605
2
votes
0
answers
81
views
An square root of the multiplicative operator on $\ell^1(\mathbb{Z}_n)$
Let us consider the finite group algebra $\ell^1(\mathbb{Z}_n)$. Let $x=(x_0,\cdots,x_{n-1})$ in $\ell^1(\mathbb{Z}_n)$ and define
$$M_x: \ell^1(\mathbb{Z}_n)\to \ell^1(\mathbb{Z}_n) : M_x(a)=a*x$$
...
- 4,058
0
votes
1
answer
83
views
Do the weakly null sequences in a Banach module factor?
Let $A$ be a Banach algebra with a bounded approximate identity, and let $E$ be a Banach left $A$-module. Suppose neither $A$ nor $E$ has the Schur property.
Question: Given a weakly null sequence $(...
- 2,605
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votes
0
answers
36
views
Regarding significance of spectral variation under algebraic operations
I have been reading the paper Determining elements in $C^∗$-algebras through spectral properties.
The paper discusses about what would be the relation be between two elements $a$ and $b$ of a Banach ...
- 27
2
votes
0
answers
149
views
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=...
- 4,058
1
vote
1
answer
119
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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 ...
- 27
1
vote
0
answers
130
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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)$.
...
- 4,058
4
votes
0
answers
120
views
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}:...
- 81
3
votes
0
answers
108
views
$ 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\...
- 1,219
3
votes
0
answers
131
views
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 ...
- 4,058
7
votes
3
answers
409
views
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 ...
- 9,853
7
votes
1
answer
284
views
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 ...
- 11.3k
3
votes
0
answers
160
views
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 ...
2
votes
0
answers
198
views
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 ...