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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 known 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 ...
Qingping Zeng's user avatar
6 votes
0 answers
150 views

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 ...
David Gao's user avatar
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1 vote
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96 views

When is the "Gelfand Remainder" compact?

Suppose we have a noncompact Hausdorff space $S$ and a Banach algebra $A \subset C^*(S,\mathbb R)$ of the space of real-valued bounded functions on $S$. For niceness let's assume $A$ separates the ...
Daron's user avatar
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3 votes
0 answers
217 views

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 ...
Ma Joad's user avatar
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3 votes
1 answer
168 views

Kernel of a map of Tate algebras

Let $A$ and $B$ be a pair of noetherian Tate algebras over $\mathbb{Q}_p$, and assume $\text{dim}_{\text{Krull}}(B) > \text{dim}_{\text{Krull}}(A)$. Is it true that any morphism $B \longrightarrow ...
kindasorta's user avatar
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3 votes
1 answer
167 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)...
Ali Hasemi's user avatar
2 votes
0 answers
155 views

Does integration by parts formula hold in $H^1(0,T,L^2(\Omega))$?

Let $\Omega$ be an open set from $\mathbb{R}^N$. How can we prove that if $u,v\in H^1(0,T,L^2(\Omega))$ (in Bochner sense) then $(u\cdot v)'\in L^2(0,T,L^1(\Omega))$ with $(uv)'=u'v+v'u$ and the ...
Bogdan's user avatar
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0 votes
0 answers
74 views

Sequential approximate diagonal

Let $A$ be a unital, amenable Banach algebra. What is the significance of $A$ to have a weakly Cauchy sequential approximate diagonal? A preliminary observation: Let $\displaystyle A\hat{\otimes}_{\...
Onur Oktay's user avatar
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4 votes
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109 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 $L_\infty(...
Norbert's user avatar
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$C^*$ algebra generated by conjugation of an element

Assume $\mathcal{A}$ is a unital $C^*$ algebra and consider some positive-definite element $\Psi\in M_n(\mathcal{A})$. Can we say something about $C^*(\langle \Psi^{-\frac{1}{2}}E_{i,i}\Psi^{\frac{1}{...
Math101's user avatar
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3 votes
1 answer
117 views

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 $...
Pietro Majer's user avatar
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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 ...
Onur Oktay's user avatar
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1 vote
1 answer
138 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 $B$?
user44155's user avatar
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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 ...
Sanae Kochiya's user avatar
1 vote
0 answers
100 views

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 $\...
Onur Oktay's user avatar
  • 2,363
3 votes
1 answer
159 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,+\...
erz's user avatar
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9 votes
3 answers
370 views

Comparison between the operator norm and the $L^1$ norm on group algebras

Consider a discrete group $G$ and its group algebra over $\mathbb{C}$, $\mathbb{C}[G]$. There are four norms on it I wish to consider for this question: The 2-norm given by $||\sum_{g \in G} c_gg||_2^...
David Gao's user avatar
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5 votes
1 answer
194 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 ...
Tobias Fritz's user avatar
  • 6,177
1 vote
1 answer
140 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 ...
Andromeda's user avatar
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1 vote
0 answers
178 views

Trans-universality for finite-dimensional Banach space

In addition to a specific problem Trans-universality for finitely generated groups, I posted also its general form. It should not hurt to provide another special case: QUESTION: does there exist a ...
Wlod AA's user avatar
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0 votes
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Space of non-archimedean characters is nonempty

Let $k$ be an algebraically closed complete non-archimedean field. Let $\mathcal{O}_k$ be its ring of integers. Suppose that $A$ is a $k$-Banach algebra, and $B$ is its closed unitary ball. Note that $...
Luiz Felipe Garcia's user avatar
1 vote
1 answer
112 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 ...
Qingping Zeng's user avatar
2 votes
0 answers
145 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 ...
MrPajeet's user avatar
  • 433
2 votes
1 answer
227 views

Completely contractive Banach algebra structure on the dual of a Hopf $C^*$-algebra

Let $(A, \Delta)$ be a Hopf $C^*$-algebra, i.e. $A$ is a $C^*$-algebra, and $\Delta: A \to M(A\otimes A)$ is an injective non-degenerate $*$-homomorphism that is coassociative: $$(\iota \otimes \Delta)...
Andromeda's user avatar
  • 169
4 votes
1 answer
199 views

When is $W^{1,p}(\Omega)$ a Banach algebra?

Let $\Omega \subseteq \mathbb{R}^d$ be bounded and open with $\partial\Omega$ Lipschitz, $1<p<\infty$. My question: knowing that $f,g\in W^{1,p}(\Omega)$ for what $p$ can we conclude that $f\...
Bogdan's user avatar
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3 votes
1 answer
138 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 ...
 Analyst 's user avatar
1 vote
0 answers
94 views

An example of a non rigid Banach algebra

A Banach algebra $A$ is called a rigid Banach algebra if for every injective Banach algebra morphism $J:A\to A$ we have either $\overline{J(A)}$ is ismorphic to $A$ or it does not contain ...
Ali Taghavi's user avatar
3 votes
1 answer
369 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 ...
Math Lover's user avatar
  • 1,115
0 votes
1 answer
216 views

Bound for the product of Sobolev functions in $W^{s,1}$

I would like to bound the product of two functions $f$, $g$ in the space $W^{s,1}$. $$ \lVert fg\rVert_{W^{s,1}}\leq c\lVert f \rVert \lVert g \rVert. $$ It seems reasonable to want to use Hölder's ...
johann's user avatar
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2 votes
0 answers
70 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 ...
Onur Oktay's user avatar
  • 2,363
1 vote
0 answers
96 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 &...
Onur Oktay's user avatar
  • 2,363
1 vote
0 answers
115 views

A dual Banach algebra question

Let $\Gamma$ be an infinite discrete abelian group and $A=\ell^1(\Gamma)$ denote its group algebra. Clearly, $A_*=c_0(\Gamma)$ is a predual of $\ell^1(\Gamma)$ for which $(A,A_*)$ is a dual Banach ...
Onur Oktay's user avatar
  • 2,363
1 vote
0 answers
88 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 ...
Meisam Soleimani Malekan's user avatar
1 vote
0 answers
88 views

Algebras sitting inside reproducing kernel Hilbert space other than multiplier algebra

Suppose $\mathcal{H}$ is a reproducing kernel Hilbert space. If the kernel is normalized then the multiplier algebra $\mathcal{M}$ is an algebra that is sitting inside $\mathcal{H}$. Is there any ...
Mohan Rahul N's user avatar
0 votes
0 answers
60 views

Weierstrass subdomain of $\DeclareMathOperator\Spm{Spm}\Spm \mathbb{Q}_p$

I am trying to understand Weierstrass subdomains of $\Spm\DeclareMathOperator\QP{\mathbb{Q}_p}\QP$. Recall that a Weierstrass algebra of an affinoid space $\Spm A$, where $A$ is a Banach algebra with ...
kindasorta's user avatar
  • 2,143
2 votes
0 answers
349 views

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 ...
Onur Oktay's user avatar
  • 2,363
0 votes
0 answers
65 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_\...
Meisam Soleimani Malekan's user avatar
3 votes
0 answers
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{ ...
Onur Oktay's user avatar
  • 2,363
2 votes
0 answers
90 views

The group of quasi unitary elements of a (simple) Banach algebra

For a Banach algebra $A$ with invertible group $G(A)$ we define the following group: $$QG(A)=\{u\in G(A)\mid \;\text{the mapping}\; a\mapsto u^{-1} a u \;\text{is an isometry}\}$$ What is an ...
Ali Taghavi's user avatar
1 vote
0 answers
141 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 ...
Meisam Soleimani Malekan's user avatar
10 votes
1 answer
397 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 |...
Aitor Iribar Lopez's user avatar
2 votes
0 answers
189 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 ...
user491354's user avatar
3 votes
0 answers
52 views

Norm under Gelfand map vs norm under left regular representation on $\ell^p$

Let $G$ be a discrete commutative group. Let $p \in [1,\infty)$ and consider the left regular representation $\lambda : \ell^1(G) \to \mathcal{B}(\ell^p(G))$; that is $\lambda(x)y := x*y$, where $$ (x*...
Leo Sera's user avatar
  • 131
1 vote
0 answers
77 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 ...
Onur Oktay's user avatar
  • 2,363
7 votes
0 answers
190 views

Kazhdan's property (T) for Banach algebras?

A locally compact group $G$ has Kazhdan's property (T) if the trivial representation $1_G:G\to\mathbb{C}$, $1_G(x) = 1$ for all $x\in G$, is isolated in $\hat{G}$ with the Fell topology. Bekka took ...
Onur Oktay's user avatar
  • 2,363
0 votes
0 answers
59 views

Are banach space representations of commutative $C^*$ algebras decomposable?

It is well known that, if $\pi:A\to \mathbb B(\mathcal H)$ is a $^*$-representation of a type I $C^*$-algebra, then $\pi$ is unitarily equivalent to a direct integral of irreducible representations. ...
user avatar
1 vote
0 answers
48 views

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 ...
Onur Oktay's user avatar
  • 2,363
2 votes
1 answer
259 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, ...
Sergei Akbarov's user avatar
7 votes
0 answers
295 views

Associativity of the Campbell-Baker-Hausdorff operation on a Banach-Lie algebra

Let $(\mathfrak{g}, [\cdot,\cdot]_\mathfrak{g}, \Vert \cdot \Vert_\mathfrak{g})$ be an infinite-dimensional Banach-Lie algebra, and let us define for any $a,b \in \mathfrak{g}$ the series $$~ Z^\...
Marcos Gonzalez's user avatar
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 ...
Onur Oktay's user avatar
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