nLab uniformly hyperfinite algebra




Functional analysis



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.


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.


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)

Last revised on May 17, 2012 at 08:37:51. See the history of this page for a list of all contributions to it.