Let and be C*-algebras and be a -homomorphism. We obtain necessary and sufficient conditions for injectivity and surjectivity of in terms of properties of . Also, we verify when the quotient group is torsion free. In particular, we deal with the case when and are finite-dimensional, and we obtain a characterization for torsion freeness of the quotient group. Moreover, we show that is injective if is injective and has stable rank one and real rank zero. The quotient group is torsion free if and are commutative and unital, has real rank zero, and is unital and injective.

We say that a -ring R is a generalized quasi-Baer -ring if for any ideal I of the right annihilator of is generated, as a right ideal, by a projection, for some positive integer n depending on I. A unital -ring R is left primary if and only if R is a generalized quasi-Baer -ring with no nontrivial central projections. We study basic properties of such rings and we prove their permanence properties such as the Morita invariance. We show that this notion is well-behaved with respect to polynomial extensions and certain triangular matrix extensions and group rings. A sheaf representation for such -rings is also proved. We obtain algebraic examples which are generalized quasi-Baer -rings but are not quasi-Baer -rings. We show that for pre-C*-

A Cantor minimal system is of finite topological rank if it has a Bratteli-Vershik representation whose number of vertices per level is uniformly bounded. We prove that if the topological rank of a minimal dynamical system on a Cantor set is finite then all its minimal Cantor factors have finite topological rank as well. This gives an affirmative answer to a question posed by Donoso, Durand, Maass, and Petite.

Let G be a finite group, let A be an infinite-dimensional stably finite simple unital C*-algebra, and let\alpha\colon G\to Aut (A) be an action of G on A which has the weak tracial Rokhlin property. Let A^{\alpha} be the fixed point algebra. Then the radius of comparison satisfies rc (A^{\alpha})\leq rc (A) and rc (C*(G, A,\alpha))\leq (1/card (G)) rc (A). The inclusion of A^{\alpha} in A induces an isomorphism from the purely positive part of the Cuntz semigroup Cu (A^{\alpha}) to the fixed points of the purely positive part of Cu (A), and the purely positive part of Cu (C*(G, A,\alpha)) is isomorphic to this semigroup. We construct an example in which G is the two element group, A is a simple unital AH algebra,\alpha has the Rokhlin prope

We introduce the tracial Rokhlin property for finite group actions on simple not necessarily unital C*-algebras which coincides with Phillips' definition in the unital case. We study its basic properties. Our main result is that if $\alpha: G\to\mathrm {Aut (A)} $ is an action of a finite group $ G $ on a simple (not necessarily unital) C*-algebra $ A $ with tracial topological rank zero and $\alpha $ has the tracial Rokhlin property, then $ A\rtimes _ {\alpha} G $ and $ A^{\alpha} $ have tracial topological rank zero. The main idea to show this is to prove that a simple non-unital C*-algebra has tracial topological rank zero if and only if it is Morita equivalent to a simple unital C*-algebra with tracial topological rank zero. Moreover, w

A category structure for ordered Bratteli diagrams is proposed such that isomorphism in this category coincides with Herman, Putnam, and Skau's notion of equivalence. It is shown that the one-to-one correspondence between the category of essentially minimal totally disconnected dynamical systems and the category of essentially simple ordered Bratteli diagrams at the level of objects is in fact an equivalence of categories. In particular, we show that the category of Cantor minimal systems is equivalent to the category of properly ordered Bratteli diagrams. We obtain a model (diagram) for a homomorphism between essentially minimal totally disconnected dynamical systems, which may be useful in the study of factors and extensions of such syste

There are several compactification procedures in topology, but there is only one standard discretization, namely, replacing the original topology with the discrete topology. We give a notion of discretization which is dual (in the categorical sense) to compactification and give examples of discretizations. Especially, a discretization functor from the category of α-scattered Stonean spaces to the category of discrete spaces is constructed, which is the converse of the Stone–Čech compactification functor. The interpretations of discretization in the level of algebras of functions are given.

A category structure for Bratteli diagrams is proposed and a functor from the category of AF algebras to the category of Bratteli diagrams is constructed. Since isomorphism of Bratteli diagrams in this category coincides with Bratteli’s notion of equivalence, we obtain in particular a functorial formulation of Bratteli’s classification of AF algebras (and at the same time, of Glimm’s classification of UHF algebras). It is shown that the three approaches to classification of AF algebras, namely, through Bratteli diagrams, K-theory, and a certain natural abstract classifying category, are essentially the same from a categorical point of view.

In this paper considering a transformation semigroup with finite height we define the notion of indicator sequence in such a way that any two transformation semigroups with the same indicator sequence have the same height. Also related to any transformation semigroup a topological space, called indicator topological space, is defined in such a way that transformation semigroups with homeomorphic indicator topological spaces have the same height. Moreover any two transformation semigroups with homeomorphic indicator topological spaces and finite height have the same indicator sequences.

In the following text a proper subclass of Alexandroff topological spaces, namely functional Alexandroff topological spaces, is introduced. We discuss relation between Alexandroff spaces and functional Alexandroff spaces, functional Alexandroff spaces as dynamical systems, and other related topics.

In the following text we study the product of functional Alexandroff spaces and obtain a theorem on functional Alexandroff topological groups which recognize all functional Alexandroff topologies on a group which made it a topological group, this theorem is parallel to a well-known theorem on Alexandroff topological groups.