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Proceedings of the American Mathematical Society

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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: Let K be an arbitrary field. Let $ ({q_n})$ be a sequence of positive integers, and let there be given a family $ \{ {\Psi _{nm}}\vert 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} = \mathop {\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} = \mathop {\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\vert p$. The first result of the paper depends directly on the second result.


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DOI: https://doi.org/10.1090/S0002-9939-1995-1215025-4
Keywords: Triangular UHF K-algebras, K-algebras, inductive limits
Article copyright: © Copyright 1995 American Mathematical Society