Strongly Solid II1 Factors with an Exotic MASA

Using an extension of techniques of Ozawa and Popa, we give an example of a non-amenable strongly solid II1 factor M containing an “exotic” maximal abelian subalgebra A: as an A,A-bimodule, L²(M) is neither coarse nor discrete. Thus, we show that there exist II1 factors with such property but without Cartan subalgebras. It also follows from Voiculescu's free entropy results that M is not an interpolated free group factor, yet it is strongly solid and has both the Haagerup property and the complete metric approximation property.