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Thin \(\text{II}_1\) factors with no Cartan subalgebras. (English) Zbl 1452.46047
Summary: It is a wide open problem to give an intrinsic criterion for a \(\text{II}_1\) factor \(M\) to admit a Cartan subalgebra \(A\). When \(A\subset M\) is a Cartan subalgebra, the \(A\)-bimodule \(L^2(M)\) is simple in the sense that the left and right actions of \(A\) generate a maximal abelian subalgebra of \(B(L^2(M))\). A \(\text{II}_1\) factor \(M\) that admits such a subalgebra \(A\) is said to be \(s\)-thin. Very recently, Popa discovered an intrinsic local criterion for a \(\text{II}_1\) factor \(M\) to be \(s\)-thin and left open the question whether all \(s\)-thin \(\text{II}_1\) factors admit a Cartan subalgebra. We answer this question negatively by constructing \(s\)-thin \(\text{II}_1\) factors without Cartan subalgebras.
MSC:
46L36 Classification of factors
43A46 Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)
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