Type II$_1$ factors with arbitrary countable endomorphism group

TL;DR

This paper constructs explicit examples of type II$_1$ factors with prescribed endomorphism semigroups, including trivial invariants, advancing understanding of their structure and invariants.

Contribution

It provides the first explicit construction of type II$_1$ factors with arbitrary countable endomorphism semigroups, including trivial invariants.

Findings

Constructed a type II$_1$ factor with trivial invariants.

For any countable left-cancellative semigroup, built a corresponding type II$_1$ factor.

Demonstrated the diversity of invariants possible in type II$_1$ factors.

Abstract

In \cite{Ioana:vNsuperrigidity}, Ioana introduced three new invariants of type II$_1$ factors: the one-sided fundamental group, the endomorphism semigroup and the set of right-finite bimodules. In \cite{Ioana:vNsuperrigidity}, he does not provide many computations of these invariants. In particular, the question whether these invariants can be trivial is left open. We give an explicit example of a type II$_1$ factor for which all three invariants are trivial. More generally, for any countable left-cancellative semigroup $G$, we construct a type II$_1$ factor $M$ whose endomorphism semigroup is precisely $G$.