Algebraic orders on K0 and approximately finite operator algebras, J. Operator Th., to appear . 51 S.C. Power, On the outer automorphism groups of triangular alternation limit algebras
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چکیده
Approximately finite (AF) C *-algebras are classified by approximately finite (r-discrete principal) groupoids. Certain natural triangular subalgebras of AF C *-algebras are similarly classified by triangular subsemigroupoids of AF groupoids [10]. Putting this in a more intuitive way, such subalgebras A are classified by the topologised fundamental binary relation R(A) induced on the Gelfand space of the masa A A * by the normaliser of A A * in A. (This relation R(A) is also determined by any matrix unit system for A affiliated with A A * .) The fundamental relation R(A) has been useful both in understanding the isomorphism classes of specific algebras and in the general structure theory of triangular and chordal subalgebras of AF C to have more convenient and computable invariants associated with the K 0 group, and we begin such an inquiry in this paper. We introduce the algebraic order and the strong algebraic order on the scale of the K 0 group of a non-self-adjoint subalgebra of a C *-algebra. Analogues (and generalisations) of Elliott's classification of AF C *-algebras are obtained for limit algebras of direct systems A 1 −→ A 2 −→ ... of finite-dimensional CSL algebras (poset algebras) with respect to certain embeddings with C *-extensions which, in a certain sense, preserve the algebraic order. We also require that the systems have a certain conjugacy property. Despite the restrictions there are many interesting applications. For example conjugacy properties prevail for certain embeddings of finite-dimensional nest algebras (block upper triangular matrix algebras) and for systems associated with ordered Bratteli diagrams.
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تاریخ انتشار 1993