Graphs and Ccr Algebras

I introduce yet another way to associate a C*-algebra to a graph and construct a simple nuclear C*-algebra that has irreducible representations both on a separable and a nonseparable Hilbert space. Kishimoto, Ozawa and Sakai have proved in [8] that the pure state space of every separable simple C*-algebra is homogeneous in the sense that for every two pure states φ and ψ there is an automorphism α such that φ ◦ α = ψ. They have shown that this fails for nonseparable algebras and asked whether the pure state space of every nuclear (not necessarily separable) C*-algebra is homogeneous. Theorem 1. There is a simple nuclear C*-algebra B that has irreducible representations both on a separable Hilbert space and on a nonseparable Hilbert space. Corollary 2. There is a simple nuclear algebra whose pure state space is not homogeneous. This algebra moreover has a faithful representation on a separable Hilbert space. As a curious side result, our construction gives a non-obvious equivalence relation on the class of all graphs. For example, among the graphs with four vertices there are three equivalence classes: (1) • • • • • • ~~ ~~ • • • • ~~ ~~ • • • @ @ @ @ • • • • • • • • • • • (2) • • • • • • ~~ ~~ • • • • ~~ ~~ • • • @ @ @ @ • ~~ ~~ • • and the third one containing the null graph. I don’t know whether there is a simple description of this relation or what is its computational complexity (see Question 3.4). In §1 we prove Theorem 1 and in §2 we study some properties of the canonical commutation relation (CCR) algebras associated with graphs of Date: August 27, 2009. 1991 Mathematics Subject Classification. 46L05, 05C90. Partially supported by NSERC. Filename: 2009f04-nonhomogeneous.tex. 1