Triangular UHF algebras over arbitrary fields

Abstract:

Let K be an arbitrary field. Let $({q_n})$ be a sequence of positive integers, and let there be given a family $\{ {\Psi _{nm}}|n \geq m\}$ of unital K-monomorphisms ${\Psi _{nm}}:{T_{{q_m}}}(K) \to {T_{{q_n}}}(K)$ such that ${\Psi _{np}}{\Psi _{pm}} = {\Psi _{nm}}$ whenever $m \leq n$, where ${T_{{q_n}}}(K)$ is the K-algebra of all ${q_n} \times {q_n}$ 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 $\mathcal {T} = \lim \limits _ \to ({T_{{q_n}}}(K);{\Psi _{nm}})$. The first result of the paper is that if the embeddings ${\Psi _{nm}}$ satisfy certain natural dimensionality conditions and if $\mathcal {S} = \lim \limits _ \to ({T_{{p_n}}}(K);{\Phi _{nm}})$ is an arbitrary TUHF K-algebra, then $\mathcal {S}$ is K-isomorphic to $\mathcal {T}$, only if the supernatural number, $N[({p_n})]$, of $({q_n})$ is less than or equal to the supernatural number, $N[({p_n})]$, of $({p_n})$. Thus, if the embeddings ${\Phi _{nm}}$ also satisfy the above dimensionality conditions, then $\mathcal {S}$ is K-isomorphic to $\mathcal {T}$, 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 $\Phi :{M_q}(K) \to {M_p}(K)$, only if $q|p$. The first result of the paper depends directly on the second result.