Triangular Uhf Algebras over Arbitrary Fields

Let K be an arbitrary field. Let (qn) be a sequence of positive integers, and let there be given a family \f¥nm\n > m} of unital Kmonomorphisms *F„m. Tqm(K) —► Tq„(K) such that *¥np*¥pm = %im whenever m < n , where Tq„ (K) is the if-algebra of all q„ x q„ upper triangular matrices over K. A triangular UHF (TUHF) K-algebra is any Kalgebra that is A'-isomorphic to an algebraic inductive limit of the form &~ = lim( Tq„ (K) ; W„m ). The first result of the paper is that if the embeddings ^Vnm satisfy certain natural dimensionality conditions and if £? = lim (TPn(K) ; Onm) is an arbitrary TUHF if-algebra, then 5? is A-isomorphic to ^ , only if the supernatural number, N[{pn)], of (q„) is less than or equal to the supernatural number, N[(p„)], of (pn). Thus, if the embeddings i>„m also satisfy the above dimensionality conditions, then S? is À"-isomorphic to F , only if NliPn)] = N[(qn)] ■ 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 A"-monomorphism MP(K), only if q\p . The first result of the paper depends directly on the second result.