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64 votes
4 answers
8k views

What is the current status of the Kaplansky zero-divisor conjecture for group rings?

Let $K$ be a field and $G$ a group. The so called zero-divisor conjecture for group rings asserts that the group ring $K[G]$ is a domain if and only if $G$ is a torsion-free group. A couple of good ...
Johan Öinert's user avatar
15 votes
1 answer
1k views

Gelfand-Naimark from the category-theoretic point of view

I was thinking about the Gelfand-Naimark theorem asserting the isometric * isomorphism between a commutative $C^*$-algebra (with unit) $\mathcal{A}$ and the $C^*$ -algebra of continuous complex-valued ...
Gian Maria Dall'Ara's user avatar
14 votes
2 answers
899 views

Do torsion-free groups give projectionless group ($C^\ast$) algebras?

One of the reasons I study von Neumann algebras is that they always have plenty of projections. There are many projectionless $C^\ast$-algebras ($0$ and possibly $1$ are the only projections), but the ...
Dave Penneys's user avatar
  • 5,425
13 votes
2 answers
4k views

Structure theorem for finite dimensional $C^*$-algebras and their representations

I would like a source for some Artin-Wedderburn type facts about these algebras which seem to have easy proofs, and are probably written somewhere. Let $\mathcal{A} \subset M_n(\mathbb{C})$ be an ...
J. E. Pascoe's user avatar
  • 1,429
1 vote
0 answers
126 views

What is a quantum condensed space?

Due to a categorical equivalence involving compact topological spaces and unital commutative $\mathrm{C}^*$-algebras, there is a practise involved in so-called quantum mathematics where a ...
JP McCarthy's user avatar
  • 1,037