Triangular UHF algebras over arbitrary fields

Author:
R. L. Baker

Journal:
Proc. Amer. Math. Soc. **123** (1995), 67-79

MSC:
Primary 46K50; Secondary 16S99

DOI:
https://doi.org/10.1090/S0002-9939-1995-1215025-4

MathSciNet review:
1215025

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Abstract | References | Similar Articles | Additional Information

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.

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Keywords:
Triangular UHF <I>K</I>-algebras,
<I>K</I>-algebras,
inductive limits

Article copyright:
© Copyright 1995
American Mathematical Society