nLab
uniformly hyperfinite algebra

Context

Algebra

Functional analysis

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 iA i\cup_{i \in \mathbb{N}} A_i of a sequence A 0A 1A_0 \hookrightarrow A_1 \hookrightarrow \cdots of an increasing sequence of I n iI_{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)

See also

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

Revised on May 17, 2012 08:37:51 by Urs Schreiber (82.169.65.155)