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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Triangular UHF algebras over arbitrary fields
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by R. L. Baker PDF
Proc. Amer. Math. Soc. 123 (1995), 67-79 Request permission

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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Additional Information
  • © Copyright 1995 American Mathematical Society
  • 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