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### 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

• 147
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 ...
• 1,841

### 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....
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, ...
• 1,841

### 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 "...
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