No 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.

No abstract

In this paper we construct a compact quantum semigroup structure on the Toeplitz algebra T . The existence of a subalgebra, isomorphic to the algebra of regular Borel's measures on a circle with convolution product, in the dual algebra T * is shown. The existence of Haar functionals in the dual algebra and in the above-mentioned subalgebra is proved. Also we show the connection between T and the structure of weak Hopf algebra.

In this paper we study the C * -subalgebras of the Toeplitz algebra T , each element of which is fixed relative to finite subgroup of automorphisms of the algebra T . We prove that such subalgebras have a finite family of unitarily equivalent irreducible representations.