Generalized Canonical Isomorphisms on Determinant

In this paper we define tensor modules (sheaves) of Schur type and of generalized Schur type associated with given modules (sheaves), using the so-called Schur functors. According to the functorial property, we give a series of tensor modules (sheaves) of Schur types in a categorical description. The main conclusion is that, by using basic ideas of algebraic geometry, there exists a canonical isomorphism of different tensor modules (sheaves) of Schur types if the original sheaf is locally free, which is in fact a generalization of results in linear algebra into locally free sheaves.

FACTOR MAPS OF LAMBDA-GRAPH SYSTEMS AND INCLUSIONS OF C*-ALGEBRAS

A λ-graph system is a labeled Bratteli diagram with shift transformation. It is a generalization of finite labeled graphs and presents a subshift. In [Doc. Math. 7 (2002), 1–30], the author introduced a C*-algebra [Formula: see text] associated with a λ-graph system [Formula: see text] as a generalization of the Cuntz–Krieger algebras. In this paper, we study a functorial property between factor maps of λ-graph systems and inclusions of the associated C*-algebras with gauge actions. We prove that if there exists a surjective left-covering λ-graph system homomorphism [Formula: see text], there exists a unital embedding of the C*-algebra [Formula: see text] into the C*-algebra [Formula: see text] compatible to its gauge actions. We also show that a sequence of left-covering graph homomorphisms of finite labeled graphs gives rise to a λ-graph system such that the associated C*-algebra is an inductive limit of the Cuntz–Krieger algebras for the finite labeled graphs.

A functorial property of the Aczel-Buchholz-Feferman function

Let Ω be the least uncountable ordinal. Let be the category where the objects are the countable ordinals and where the morphisms are the strictly monotonic increasing functions. A dilator is a functor on which preserves direct limits and pullbacks. Let τ < ΩE ≔ min{ξ > Ω: ξ = ωξ}. Then τ has a unique “term”-representation in Ω. λξη.ωξ + η and countable ordinals called the constituents of τ. Let δ < Ω and K(τ) be the set of the constituents of τ. Let β = max K(τ). Let [β] be an occurrence of β in τ such that τ[β] = τ. Let be the fixed point-free version of the binary Aczel-Buchholz-Feferman-function (which is defined explicitly in the text below) which generates the Bachman-hierarchy of ordinals. It is shown by elementary calculations that is a dilator for every γ > max{β.δ.ω}.