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TRIANGULAR UHF ALGEBRAS OVER ARBITRARY FIELDS
Journal article   Open access   Peer reviewed

TRIANGULAR UHF ALGEBRAS OVER ARBITRARY FIELDS

R L Baker
Proceedings of the American Mathematical Society, Vol.123(1), pp.67-79
01/01/1995
DOI: 10.1090/S0002-9939-1995-1215025-4
url
https://doi.org/10.1090/S0002-9939-1995-1215025-4View
Published (Version of record) Open Access

Abstract

Let K be an arbitrary field. Let (q(n)) be a sequence of positive integers, and let there be given a family {Psi(nm)\n greater than or equal to m} of unital K-monomorphisms Psi(nm): T-qn(K) --> T-qn(K) such that Psi(np)Psi(pm) = Psi(nm) whenever m less than or equal to n, where T-qn(K) is the K-algebra of all q(n) x q(n) upper triangular matrices over K. A triangular UHF (TUHF) K-algebra is any K-algebra that is K-isomorphic to an algebraic inductive limit of the form Tau = lim(T-qn(K); Psi nm). The first result of the paper is that if the embeddings Psi(nm) satisfy certain natural dimensionality conditions and if l = lim(T(pn)Phi>(nm)) is an arbitrary TUHF K-algebra, then f is K-isomorphic to Tau, only if the supernatural number, N[(p(n))], of (q(n)) is less than or equal to the super-natural number, N[(pn)],of (p(n)). Thus, if the embeddings Phi(nm) also satisfy the above dimensionality conditions, then f is K-isomorphic to Tau, only if N[(p(n))] = N[(q(n))]. The second result of the paper is a nontrivial ''triangular'' version of the fact that if p, q are positive integers, then there exists a unital K-monomorphism Phi: M(q)(K) --> M(p)(K), only if q/p. The first result of the paper depends directly on the second result. Tau
 Mathematics Physical Sciences Mathematics, Applied Science & Technology

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