Hyperfinite Type II Factor - Constructions

Constructions

• The von Neumann group algebra of a discrete group with the infinite conjugacy class property is a factor of type II₁, and if the group is amenable and countable the factor is hyperfinite. There are many groups with these properties, as any locally finite group is amenable. For example, the von Neumann group algebra of the infinite symmetric group of all permutations of a countable infinite set that fix all but a finite number of elements gives the hyperfinite type II₁ factor.
• The hyperfinite type II₁ factor also arises from the group-measure space construction for ergodic free measure-preserving actions of countable amenable groups on probability spaces.
• The infinite tensor product of a countable number of factors of type I_n with respect to their tracial states is the hyperfinite type II₁ factor. When n=2, this is also sometimes called the Clifford algebra of an infinite separable Hilbert space.
• If p is any non-zero finite projection in a hyperfinite von Neumann algebra A of type II, then pAp is the hyperfinite type II₁ factor. Equivalently the fundamental group of A is the group of all positive real numbers. This can often be hard to see directly. It is, however, obvious when A is the infinite tensor product of factors of type I_n, where n runs over all integers greater than 1 infinitely many times: just take p equivalent to an infinite tensor product of projections p_n on which the tracial state is either 1 or .