# nLab uniformly hyperfinite algebra

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## 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).

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)