International Journal of Mathematics

Topological quivers are generalizations of directed graphs in which the sets of vertices and edges are locally compact Hausdorff spaces. Associated to such a topological quiver is a C*-correspondence, and from this correspondence one may construct a Cuntz–Pimsner algebra . In this paper we develop the general theory of topological quiver C*-algebras and show how certain C*-algebras found in the literature may be viewed from this general perspective. In particular, we show that C*-algebras of topological quivers generalize the well-studied class of graph C*-algebras and in analogy with that theory much of the operator algebra structure of can be determined from . We also show that many fundamental results from the theory of graph C*-algebras have natural analogues in the context of topological quivers (often with more involved proofs). These include the gauge-invariant uniqueness theorem, the Cuntz–Krieger uniqueness theorem, descriptions of the ideal structure, and conditions for simplicity.

We consider the Magnus representation of the image of the pure braid group under the generalizations of the standard Artin representation, discovered by M. Wada. We will give a necessary and sufficient condition for the specialization of the reduced Wada's representation G_n(z) : P_n → GL_n-1(ℂ) to be irreducible. It will be shown that for z = (z₁,…,z_n) ∈ (ℂ*)ⁿ, G_n(z) is irreducible if and only if z₁^k⋯z_n^k ≠ 1. This is a generalization of our previous result concerning the irreducibility of the complex specialization of the reduced Gassner representation of P_n.

Let E be a C*-correspondence over a C*-algebra with non-degenerate faithful left action. We show that E admits sufficiently many essential representations (i.e. representations ψ such that ) to recover the Cuntz–Pimsner algebra .

We investigate monotone operator functions of several variables under a trace or a trace-like functional. In particular, we prove the inequality τ(x₁⋯x_n) ≤ τ(y₁⋯y_n) for a trace τ on a C*-algebra and abelian n-tuples (x₁,…,x_n) ≤ (y₁,…,y_n) of positive elements. We formulate and prove Jensen's inequality for expectation values, and we study matrix functions of several variables which are convex or monotone with respect to the weak majorization for matrices.

In this paper we consider coherent systems (E,V) on an elliptic curve which are α-stable with respect to some value of a parameter α. We show that the corresponding moduli spaces, if non-empty, are smooth and irreducible of the expected dimension. Moreover we give precise conditions for non-emptiness of the moduli spaces. Finally we study the variation of the moduli spaces with α.

Let X be a compact metric space and A be a unital simple C*-algebra with TR(A)=0. Suppose that ϕ : C(X) → A is a unital monomorphism. We study the problem when ϕ can be approximated by homomorphisms with finite-dimensional range. We give a K-theoretical necessary and sufficient condition for ϕ being approximated by homomorphisms with finite-dimensional range.