Questions tagged [banach-algebras]

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3 votes
1 answer
150 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 ...
3 votes
1 answer
166 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)...
1 vote
1 answer
132 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 $B$?
4 votes
0 answers
107 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(...
2 votes
0 answers
152 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 ...
0 votes
0 answers
73 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}_{\...
0 votes
0 answers
71 views

$C^*$ 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}{...
79 votes
4 answers
7k 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 ...
0 votes
0 answers
90 views

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 ...
3 votes
1 answer
114 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 $...
4 votes
2 answers
589 views

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 success,...
5 votes
3 answers
395 views

Examples of amenable Banach algebras which have non-amenable subalgebra

I am looking for examples of amenable Banach algebras which have non-amenable subalgebra I know 1: Each amenable Banach algebra has a bounded approximate identity 2: If $I$ be a closed ideal in an ...
0 votes
0 answers
135 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 ...
1 vote
0 answers
99 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 $\...
3 votes
1 answer
145 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,+\...
9 votes
3 answers
334 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^...
5 votes
1 answer
181 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 ...
4 votes
2 answers
946 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 ...
1 vote
1 answer
125 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 ...
2 votes
1 answer
220 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)...
1 vote
0 answers
176 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 ...
0 votes
0 answers
67 views

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 $...
1 vote
1 answer
105 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 ...
2 votes
0 answers
140 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 ...
4 votes
1 answer
170 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\...
0 votes
0 answers
59 views

About injective and projective tensor products of commutative Banach algebras

Let $A_i$, $B_i$, $i=1,2$, be semisimple commutative Banach algebras such that $A_i$ is isomorphic to $B_i$, $i=1,2$. Is the injective tensor product of $A_1$ and $A_2$ is isomorphic to the injective ...
3 votes
1 answer
128 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 ...
1 vote
0 answers
93 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 ...
3 votes
1 answer
283 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 ...
0 votes
1 answer
155 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 ...
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 ...
1 vote
0 answers
95 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 votes
1 answer
249 views

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 ...
1 vote
0 answers
111 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 ...
1 vote
0 answers
86 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 votes
0 answers
342 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 ...
1 vote
0 answers
82 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 ...
0 votes
0 answers
58 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 ...
0 votes
0 answers
63 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_\...
3 votes
0 answers
291 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 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 ...
1 vote
0 answers
140 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 votes
0 answers
185 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 ...
10 votes
1 answer
374 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 |...
78 votes
3 answers
8k views

Norms of commutators

If an $n$ by $n$ complex matrix $A$ has trace zero, then it is a commutator, which means that there are $n$ by $n$ matrices $B$ and $C$ so that $A= BC-CB$. What is the order of the best constant $\...
3 votes
0 answers
279 views

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 $...
3 votes
0 answers
48 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*...
1 vote
0 answers
74 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 ...
7 votes
0 answers
182 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 ...
6 votes
1 answer
317 views

Does there exist any massive proper $C^*$-subalgebra?

Definition 1: Suppose $B$ is a $C^* $-algebra. $A$ is massive $C^* $-subalgebra of $B$ iff 1. $A$ is a subalgebra of $B$; 2. for each irreducible representation $\pi$ of $B$ representation $\pi|_A$ is ...

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