Stable and topological ranks of real-symmetric continuous function algebras

About 10 years old · traced to

Let XX be a compact Hausdorff space, and let τ\tau be a topological involution of XX. Denote the set of fixed points of τ\tau by EE. The algebra C(X,τ)C(X,\tau) is the algebra of continuous functions compatible with this involution. Stable-rank conjecture. One has

bsr⁡C(X,τ)=tsr⁡C(X,τ)=max⁡{[dim⁡X2],dim⁡E}+1.\operatorname{bsr} C(X,\tau)=\operatorname{tsr} C(X,\tau)=\max\left\{\left[\frac{\operatorname{dim} X}{2}\right],\operatorname{dim} E\right\}+1.

This conjecture gives the expected Bass stable rank and topological stable rank in terms of the covering dimension of XX and its fixed-point set. The paper states that it was confirmed subsequently, so the conjecture is solved.

References

Primary source

Raymond Mortini, “The covering dimension of a distinguished subset of the spectrum M(H^) of H^and the algebra of real-symmetric and continuous functions on M(H^)”, arXiv:1602.06841 (2016).

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.