Work of Jean Renault shows that, for topologically principal étale groupoids, a diagonal-preserving isomorphism of reduced C ∗-algebras yields an isomorphism of groupoids. Several authors have proved analogues of this result for ample groupoid algebras over integral domains under suitable hypotheses. In this paper, we extend the known results by allowing more general coefficient rings and by we...

Let K be any field, let L n denote the Leavitt algebra of type (1, n − 1) having coefficients in K, and let M d (L n) denote the ring of d × d matrices over L n. In our main result, we show that M d (L n) ∼ = L n if and only if d and n − 1 are coprime. We use this isomorphism to answer a question posed in [14] regarding isomorphisms between various C*-algebras. Furthermore, our result demonstra...

We introduce an order structure on K 0 ( ) 9 K0(; Z/p ) . This group may also be thought of as Ko(; 7z @ Z/p ) . We exhibit new examples of real-rank zero C*-algebras that are inductive limits of finite dimensional and dimension-drop algebras, have the same ordered, graded K-theory with order unit and yet are not isomorphic. In fact they are not even stably shape equivalent. The order structure...

Let K be any field, let L n denote the Leavitt algebra of type (1, n − 1) having coefficients in K, and let M d (L n) denote the ring of d × d matrices over L n. In our main result, we show that M d (L n) ∼ = L n if and only if d and n − 1 are coprime. We use this isomorphism to answer a question posed in [14] regarding isomorphisms between various C*-algebras. Furthermore, our result demonstra...

We develop the theory of Hopf bimodules for a finite rigid tensor category C. Then we use this theory to define a distinguished invertible object D of C and an isomorphism of tensor functors δ : V ∗∗ → D⊗∗∗V ⊗D. This provides a categorical generalization of Radford’s S formula for finite dimensional Hopf algebras [R1], which was proved in [N] for weak Hopf algebras, in [HN] for quasi-Hopf algeb...

Given a directed graph, there exists a universal operator algebra and universal C-algebra associated to the directed graph. In this paper we give intrinsic constructions of these objects. We provide an explicit construction for the maximal C-algebra of an operator algebra. We also discuss uniqueness of the universal algebras for finite graphs, showing that for finite graphs the graph is an isom...

In 1996, Harris and Kadison posed the following problem: show that a linear bijection between C∗-algebras that preserves the identity and the set of invertible elements is a Jordan isomorphism. In this paper, we show that if A and B are semisimple Banach algebras andΦ : A→ B is a linear map onto B that preserves the spectrum of elements, thenΦ is a Jordan isomorphism if either A or B is a C∗-al...

The K-groups, the range of trace on K0, and isomorphism classifications are obtained for simple infinite dimensional quotient C*-algebras of the group C*-algebras of six lattice subgroups, corresponding to each of the six non-isomorphic 5-dimensional connected, simply connected, nilpotent Lie groups. Connes’ non-commutative geometry involving cyclic cocycles and the Chern character play a key r...

In this paper, we study a family of C*-subalgebras defined by fixed points generalized gauge actions Cuntz–Krieger algebra introducing étale groupoids whose associated C*-algebras are these C*-subalgebras. We show that topological conjugacy classes one-sided Markov shifts characterized in terms the isomorphism groupoids.

We provide a complete classification of the class unital graph $C^*$-algebras - prominently containing full family Cuntz-Krieger algebras showing that Morita equivalence in this case is determined by ordered, filtered $K$-theory. The result geometric sense it establishes any between $C^*(E)$ and $C^*(F)$ can be realized sequence moves leading from $E$ to $F$, way resembling role Reidemeister on...