7 votes

Can homomorphisms be Borel in some weaker topology?

It doesn't look so. There is a separable reflexive space $X$, so that $L(X)$ has $2^{2^{\aleph_0}}$ homomorphisms to $A = \mathbb C$ (see this thread). As the space is reflexive, the unit ball of $L(X)...
Tomasz Kania's user avatar
  • 11.3k
6 votes
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Spectra of products variously permutated

This is false for other permutations. For example, call $(e_1,e_2,e_3)$ the canonical basis of $\mathbb{R}^3$ and let A,B,C be the $3 \times 3$ real matrices such that $Ae_1=e_2$, $Be_2=e_3$, $Ce_3=...
Christophe Leuridan's user avatar
6 votes
Accepted

Closed prime ideal in $C[0, 1]$

No. Note that an ideal in a commutative ring with identity is prime if and only if the quotient ring is an integral domain. Now consider $C[0,1]$. It is known that the closed ideals in this Banach ...
Yemon Choi's user avatar
  • 25.5k
6 votes

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

This is only a partial answer, but it shows that if G has an element of infinite order then the weakest form of Q2 has a negative answer. We use the so-called Rudin-Shapiro polynomials (really due to ...
Yemon Choi's user avatar
  • 25.5k
5 votes

Arens regularity of $\mathrm{BV}(\mathbb{R})$

Writing this in a hurry on my way to a ritual English social cohesion exercise, so apologies for the lack of formatting and the sketchy nature of claims and references. I think it might be "...
Yemon is logged out right now's user avatar
5 votes
Accepted

Do completely bounded maps on an operator space have a completely contractive Banach algebra structure?

Yes, this is true. More generally:$\newcommand{\CB}{\mathop{\sf{CB}}}\newcommand{\cbnorm}[1]{{\lVert#1\rVert}_{\sf cb}}$ Proposition 1. Let $X$, $Y$, $Z$ be operator spaces. Then composition of ...
Yemon Choi's user avatar
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4 votes

Complemented C*-algebras

No. The classical example is $A$ being the functions on $[0,1]$ that are right continuous, have left limits everywhere, and are left continuous except at dyadic rationals. $B$ is defined similarly but ...
Bill Johnson's user avatar
4 votes
Accepted

Approximating continuous functions from $K\times L$ into $[0,1]$

The following provides a construction of $g_i, h_i$ satisfying the required conditions (the ranges of these functions can even be in $[0, 1]$): Claim: For each $k \in K$, there exists an open ...
David Gao's user avatar
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4 votes
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Comparison between the operator norm and the $L^1$ norm on group algebras

This is an alternative, more ad hoc proof, very closely mimicking a part of section 3 of the paper https://doi.org/10.4007/annals.2013.178.1.4 The start of the argument is the same. Write $M=L(G)$. ...
Stefaan Vaes's user avatar
  • 4,011
4 votes

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

Also for an arbitrary infinite group $G$, the weakest form of Q2 has a negative answer. Denote by $M=L(G)$ the group von Neumann algebra with its canonical trace $\tau$. For every element $a \in M$, ...
Stefaan Vaes's user avatar
  • 4,011
4 votes
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Completely contractive Banach algebra structure on the dual of a Hopf $C^*$-algebra

There's already the idea of this argument in the comments, but let me flesh it out. For any $C^*$-algebra $A$ and any von Neumann algebra $M$ with $A\subseteq M$ non-degenerately we can identify $M(A)...
Matthew Daws's user avatar
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4 votes
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When is $W^{1,p}(\Omega)$ a Banach algebra?

I'll just complete the answers in the comments showing that for $p\le d$ it is not a Banach algebra. Let me take $\Omega=B_1(0) \subset \mathbb{R}^n$ and $p=d$. Assume that $W^{1,d}(B_1) $ is a Banach ...
Michele Caselli's user avatar
4 votes

positive functional on Banach *-algebra (with appro. identity) is continuous?

F. F. Bonsall & J. Duncan, Complete Normed Algebras, Springer 1973: ยง37, Star Representations and Positive Functionals, Theorem 15 (p. 201). Let $A$ (a Banach star algebra) have a bounded two-...
J.J. Green's user avatar
  • 2,497
2 votes

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

I believe that $W^{s,1}(\mathbb{R}^d)$ is a Banach algebra when $s > d$. This is a particular case of Theorem 7.3 of the paper Multiplication in Sobolev spaces by Ali Behzadan and Michael Holst.
cs89's user avatar
  • 971
1 vote

Kernel of a map of Tate algebras

No. Take $U \subset X$ a connected rational domain in a two dimensional affinoid ball, and take $C \subset U$ a Zariski closed affinoid curve which is Zariski dense in $X$. Then $\mathcal{O}(X) \to \...
Satan's Minion's user avatar
1 vote
Accepted

What is the socle of the $2\times 2$ matrix algebra over a Banach algebra?

We have $soc(\mathcal B) =M_2(soc(\mathcal A))$. For $I$ a minimal right ideal of $\mathcal A$, $\{\begin{pmatrix} a & b \\ 0 & 0 \end{pmatrix} \in M_2(\mathcal A) \mid a,b\in I \}$ is a right ...
Will Sawin's user avatar
  • 135k

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