Allan Donsig - Algebraic isomorphisms of limit algebras

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Allan Donsig - Algebraic isomorphisms of limit algebras

	ALLAN DONSIG, Department of Mathematics and Statistics, University of Nebraska-Lincoln, Lincoln, Nebraska 68505-0323, USA
	Algebraic isomorphisms of limit algebras

(Joint work with T. D. Hudson and E. G. Katsoulis)

A limit algebra is a closed union of a nested sequence of subalgebras of matrix algebras. The subalgebras are required to contain the diagonal matrices and the inclusion maps from one subalgebra to the next are required to extend to the generated $C^\ast$ -algebra, so that the limit algebra is a subalgebra of an inductive limit of direct sums of matrix algebras, i.e., an $\textrm{AF}$ $C^\ast$ -algebra. Finally, the inclusions are required to respect the normalizers of the diagonal matricies, that is, to send matrix units to sums of matrix units. Examples of such algebras include limits of upper-triangular $2^n \times 2^n$ matrices, T_2ⁿ, with embeddings such as $\sigma_n\colon T_{2^n} \to T_{2^{n+1}}$ given by $\sigma_n(A)=A \oplus A$ and $\rho_n \colon T_{2^n} \to T_{2^{n+1}}$ given by $\rho_n\bigl( (a_{ij}) \bigr) = (a_{ij} I_2 )$ .

We establish automatic continuity for algebraic isomorphisms of limit algebras. It follows that, for inductive limits of finite-dimensional nest algebras, the C^*-envelope is an invariant for algebraic isomorphisms. Also, for strongly maximal TAF algebras generated by their order preserving normalizers, algebraic isomorphism is shown to be equivalent to isometric isomorphism.

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