nLab uniformly hyperfinite algebra

Definition

A C-star algebra is called a uniformaly hyperfinite algebra or a Glimm algbebra (after Glimm) if it arises as the sequential colimit of an increasing sequence of type In-algebras (finite matrix algebras), hence if it is the norm-closure of the union $\cup_{i \in \mathbb{N}} A_i$ of a sequence $A_0 \hookrightarrow A_1 \hookrightarrow \cdots$ of an increasing sequence of $I_{n_i}$-factors.

Properties

All type III von Neumann algebra factors can be constructed from uniformly hyperfinite algebras (Powers).

Related concepts

These algebras naturally appear as algebras of observables in quantum lattice systems.

References

The notion was introduced in

• J. Glimm, On a certain class of operator algebras, Transactions of the AMS 95 (1960)

See also

• R. Powers, Representations of uniformly hyperfinite algebras and their associated von Neumann rings, Ann. Math. (2) 86 (1967) (Euclid)