Searcharxiv⌕ Search

arXiv subjects

Takeshi Katsura

Publications and source records attributed to Takeshi Katsura.

35 records · Page 2Linked to original sources
arXiv2008

Graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence

We prove that the classes of graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence. This result answers the long-standing open question of whether every Exel-Laca algebra is Morita equivalent to a graph algebra. Given an ultragraph G we construct a directed graph E such that C*(G) is isomorphic to a full corner of C*(E). As applications, we characterize real rank zero for ultragraph algebras and describe quotients of ultragraph algebras by gauge-invariant ideals.

math.OA↗
arXiv2008

Ultragraph C*-algebras via topological quivers

Given an ultragraph in the sense of Tomforde, we construct a topological quiver in the sense of Muhly and Tomforde in such a way that the universal C*-algebras associated to the two objects coincide. We apply results of Muhly and Tomforde for topological quiver algebras and of Katsura for topological graph C*-algebras to study the K-theory and gauge-invariant ideal structure of ultragraph C*-algebras.

math.OA↗
arXiv2007

Classification of uniformly outer actions of Z^2 on UHF algebras

We give a complete classification up to cocycle conjugacy of uniformly outer actions of Z^2 on UHF algebras. In particular, it is shown that any two uniformly outer actions of Z^2 on a UHF algebra of infinite type are cocycle conjugate. We also classify them up to outer conjugacy.

math.OA↗
arXiv2007

A construction of actions on Kirchberg algebras which induce given actions on their K-groups

We prove that every action of a finite group all of whose Sylow subgroups are cyclic on the K-theory of a Kirchberg algebra can be lifted to an action on the Kirchberg algebra. The proof uses a construction of Kirchberg algebras generalizing the one of Cuntz-Krieger algebras, and a result on modules over finite groups. As a corollary, every automorphism of the K-theory of a Kirchberg algebra can be lifted to an automorphism of the Kirchberg algebra with same order.

math.OA↗
arXiv2006

Permutation presentations of modules over finite groups

We introduce a notion of permutation presentations of modules over finite groups, and completely determine finite groups over which every module has a permutation presentation. To get this result, we prove that every coflasque module over a cyclic p-group is permutation projective.

math.GR↗
arXiv2006

A class of C^*-algebras generalizing both graph algebras and homeomorphism C^*-algebras III, ideal structures

We investigate the ideal structures of the C^*-algebras arising from topological graphs. We give the complete description of ideals of such C^*-algebras which are invariant under the so-called gauge action, and give the condition on topological graphs so that all ideals are invariant under the gauge action. We get conditions for our C^*-algebras to be simple, prime or primitive. We completely determine the prime ideals, and show that most of them are primitive. Finally, we construct a discrete graph such that the associated C^*-algebra is prime but not primitive.

math.OA↗
arXiv2005

A class of C^*-algebras generalizing both graph algebras and homeomorphism C^*-algebras II, examples

We show that the method to construct C^*-algebras from topological graphs, introduced in our previous paper, generalizes many known constructions. We give many ways to make new topological graphs from old ones, and study the relation of C^*-algebras constructed from them. We also prove that our C^*-algebras have a certain characterization. This gives us another definition of our C^*-algebras.

math.OA↗
arXiv2005

A class of $C^*$-algebras generalizing both graph algebras and homeomorphism $C^*$-algebras IV, pure infiniteness

This is the final one in the series of papers where we introduce and study the $C^*$-algebras associated with topological graphs. In this paper, we get a sufficient condition on topological graphs so that the associated $C^*$-algebras are simple and purely infinite. Using this result, we give one method to construct all Kirchberg algebras as $C^*$-algebras associated with topological graphs.

math.OA↗
arXiv2005

Non-separable AF-algebras

We give two pathological phenomena for non-separable AF-algebras which do not occur for separable AF-algebras. One is that non-separable AF-algebras are not determined by their Bratteli diagrams, and the other is that there exists a non-separable AF-algebra which is prime but not primitive.

math.OA↗
arXiv2004

C^*-algebras generated by scaling elements

We investigate C^*-algebras generated by scaling elements. We generalize the Wold decomposition and Coburn's theorem on isometries to scaling elements. We also completely determine when the C^*-algebra generated by a scaling element contains an infinite projection.

math.OA↗
arXiv2003

Ideal structure of $C^*$-algebras associated with $C^*$-correspondences

We study the ideal structure of $C^*$-algebras arising from $C^*$-correspondences. We prove that gauge-invariant ideals of our $C^*$-algebras are parameterized by certain pairs of ideals of original $C^*$-algebras. We show that our $C^*$-algebras have a nice property which should be possessed by generalization of crossed products. Applications to crossed products by Hilbert $C^*$-bimodules and relative Cuntz-Pimsner algebras are also discussed.

math.OA↗
arXiv2003

On $C^*$-algebras associated with $C^*$-correspondences

We study $C^*$-algebras arising from $C^*$-correspondences, which was introduced by the author. We prove the gauge-invariant uniqueness theorem, and obtain conditions for our $C^*$-algebras to be nuclear, exact, or satisfy the Universal Coefficient Theorem. We also obtain a 6-term exact sequence of $K$-groups involving the $K$-groups of our $C^*$-algebras.

math.OA↗
arXiv2001

AF-embeddability of crossed products of Cuntz algebras

We investigate crossed products of Cuntz algebras by quasi-free actions of abelian groups. We prove that our algebras are AF-embeddable when actions satisfy a certain condition. We also give a necessary and sufficient condition that our algebras become simple and purely infinite, and consequently our algebras are either purely infinite or AF-embeddable when they are simple.

math.OA↗