Jørgensen–Schmitt–Werner conjecture on Wick algebra envelopes

Let TT be the operator of coefficients of a Wick algebra W(T)W(T) on d4c=Cnd4c=\mathbb C^n, and let W(T)=C(W(T))\mathcal{W}(T)=C^*(W(T)) be its universal CC^*-envelope. The operator TT is self-adjoint when T=TT=T^*, braided when

(1T)(T1)(1T)=(T1)(1T)(1T),(\mathbf 1\otimes T)(T\otimes\mathbf 1)(\mathbf 1\otimes T)=(T\otimes\mathbf 1)(\mathbf 1\otimes T)(\mathbf 1\otimes T),

and T<1\lVert T\rVert<1. Jørgensen–Schmitt–Werner conjecture. If TT is self-adjoint, braided and T<1\lVert T\rVert<1, then

W(T)W(0).\mathcal{W}(T)\simeq\mathcal{W}(0).

This conjecture concerns the stability of isomorphism classes of universal CC^*-envelopes of Wick algebras under these deformations. The supplied text attributes it to an earlier work, but gives no evidence that it has been resolved.

Sources & referencesView supporting material

Primary source

Alexey Kuzmin, “CCR and CAR algebras are connected via a path of Cuntz-Toeplitz algebras”, arXiv:2203.10058 (2022).

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.