Examples of factors which have no Cartan subalgebras

TL;DR

This paper establishes conditions under which certain factors lack Cartan subalgebras, demonstrating that specific quantum group factors and continuous cores of type III_1 factors are semisolid or have no Cartan subalgebras.

Contribution

It introduces new conditions similar to Ozawa's (AO) and proves their implications for the structure of non-injective factors with W*CBAP.

Findings

Non-injective factors satisfying the new condition and W*CBAP have no Cartan subalgebras.

II_1 factors of universal orthogonal and unitary discrete quantum groups lack Cartan subalgebras.

Continuous cores of type III_1 factors with the condition are semisolid.

Abstract

We consider some conditions similar to Ozawa's condition (AO), and prove that if a non-injective factor satisfies such a condition and has the W*CBAP, then it has no Cartan subalgebras. As a corollary, we prove that $\rm II_1$ factors of universal orthogonal and unitary discrete quantum groups have no Cartan subalgebras. We also prove that continuous cores of type $\rm III_1$ factors with such a condition are semisolid as a $\rm II_\infty$ factor.