In mathematics, particularly in the theory of C*-algebras, a uniformly hyperfinite, or UHF, algebra is a C*-algebra that can be written as the closure, in the norm topology, of an increasing union of finite-dimensional full matrix algebras.
Suppressing the connecting maps, one can write If then rkn = kn + 1 for some integer r and where Ir is the identity in the r × r matrices.
The sequence ...kn|kn + 1|kn + 2... determines a formal product where each p is prime and tp = sup {m | pm divides kn for some n}, possibly zero or infinite.
[1] Glimm showed that the supernatural number is a complete invariant of UHF C*-algebras.
It is defined as follows: let H be a separable complex Hilbert space H with orthonormal basis fn and L(H) the bounded operators on H, consider a linear map with the property that The CAR algebra is the C*-algebra generated by The embedding can be identified with the multiplicity 2 embedding Therefore, the CAR algebra has supernatural number 2∞.