On a Topology and Limits for Inductive Systems of C∗-Algebras over Partially Ordered Sets

Abstract

Motivated by algebraic quantum field theory and our previous work we study properties of inductive systems of C * -algebras over arbitrary partially ordered sets. A partially ordered set can be represented as the union of the family of its maximal upward directed subsets indexed by elements of a certain set. We consider a topology on the set of indices generated by a base of neighbourhoods. Examples of those topologies with different properties are given. An inductive system of C * -algebras and its inductive limit arise naturally over each maximal upward directed subset. Using those inductive limits, we construct different types of C * -algebras. In particular, for neighbourhoods of the topology on the set of indices we deal with the C * -algebras which are the direct products of those inductive limits. The present paper is concerned with the above-mentioned topology and the algebras arising from an inductive system of C * -algebras over a partially ordered set. We show that there exists a connection between properties of that topology and those C * -algebras.

Embedding Semigroup C*-algebras into Inductive Limits

“…We consider the topology on the index set I which was introduced in [7,8]. For the convenience of the reader we recall the definition of this topology and its properties.…”

“…Examples of different topological spaces (I, τ ) are contained in [7,8]. In particular, (I, τ ) may be: a non-Hausdorff space [ Here we give an example of a compact space.…”

“…Now we shall show that in the algebra B a n+1 the following equality holds: (15) L pn a n+1 = τ a n+1 an (L an ). Indeed, firstly, by (8), we can write…”

“…A part of motivation for studying inductive systems of Toeplitz algebras comes from [6,7,8,9]. The paper [6] contains results on limit automorphisms for inductive sequences of Toeplitz algebras which are closely related to the facts on the mappings of topological groups [10,11,12].…”

“…The paper [6] contains results on limit automorphisms for inductive sequences of Toeplitz algebras which are closely related to the facts on the mappings of topological groups [10,11,12]. In [7,8] the authors introduce a topology associated to a partially ordered set and study its relation to properties of inductive limits arising from a system of C * -algebras over the partially ordered set which yields that topology. The existence of an isomorphism between the inductive limit of an inductive system of Toeplitz algebras over a directed set and the reduced semigroup C * -algebra for a semigroup in the group of rational numbers is shown in [7,9].…”