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2 votes
Accepted

Linear map between projective finitely generated Hilbert modules is adjointable

Looking at the proof of Lemma 6.21 in the notes of de Commer that you're reading (https://arxiv.org/pdf/1604.00159.pdf, per the comments), it seems like the relevant property of the modules is that of ...
  • 760
12 votes

For what kind of $C^*$ algebras does the inequality $\frac{(ab+ba)}{2}\leq\frac{ a^p}{p} +\frac{b^q }{q}$ hold for $a,b>0$?

Let me expand slightly on the comments I made above, and give the most general solution. Clearly the inequality $\frac{ab + ba}{2} \leq \frac{a^2}{2} + \frac{b^2}{2}$ holds for all positive elements $...
  • 2,151
1 vote
Accepted

A commuting pair of isometries

Such a pair $(X,Y)$ is constructed as follows. Consider a Hilbert space $M$ with an orthonormal basis $\{e_n:n\in\mathbb Z\}$ and the bilateral shift $U$ on $M$ such that $Ue_n=e_{n+1}$. Denote by $S$...
  • 66
2 votes
Accepted

Unitary in adjointable operators associated with equivariant Hilbert module

The answer is rather easy; I think what is confusing is the sheer number of objects flying around. What is $C(\mathbb X)$? This is Definition 1.4 in the paper: this is a $C^*$-algebra with an ergodic ...
  • 17.7k
2 votes

Algebra objects of $\operatorname{Vec}(\mathscr{C})$ are lax functors $\mathscr{C}^\text{op}\to \operatorname{Vec}$

A somewhat more explicit answer. Details are still to be filled in, but at least you get an idea how to go from one to the other: If $\mathbf{A}$ is an algebra object in $\operatorname{Vec}(\mathcal{...
4 votes
Accepted

Adding finite direct sums to a C*-tensor category

If you care about completeness, you just want to observe that the norm defined this way restricts to the original norms on the direct summands $\operatorname{Mor}(U_i, V_j)$. Since there are only ...
9 votes

Algebra objects of $\operatorname{Vec}(\mathscr{C})$ are lax functors $\mathscr{C}^\text{op}\to \operatorname{Vec}$

This is a special case of the well known result in (enriched) category theory which I believe goes back to Day, for example see Proposition 3.4 of https://ncatlab.org/nlab/show/Day+convolution#Monoids....
3 votes
Accepted

Reference request: decomposability of $\mathbb{G}$-Hilbert modules

Let's assume that $G$ is a reduced compact quantum group, that is, the Haar state on $C(G)$ is faithful. (1): A direct reference is [1, Lemma 4.2]. You can also get this from a careful study of [2, ...
2 votes

Are the ideals in two $C^*$-algebras the same?

Counterexample. Let $H = L^2(\mathbb{T}) \oplus l^2(\mathbb{N})$ and define $V_1 = M_{e^{2\pi it}} \oplus S$ and $V_2 = -I_1 \oplus I_2$. Here $M_{e^{2\pi it}}$ is a multiplication operator, $S$ is ...
  • 39.2k
1 vote
Accepted

Hopf algebras vs. Kac algebras

As pointed out in the comments all semisimple Hopf algebras of dimension up to $23$ are Kac algebras. So it seems like dimension $24$ is the smallest one where this is open. Corollary 9.7 of "...
  • 121

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