Graphs and Ccr Algebras
نویسنده
چکیده
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
منابع مشابه
NILPOTENT GRAPHS OF MATRIX ALGEBRAS
Let $R$ be a ring with unity. The undirected nilpotent graph of $R$, denoted by $Gamma_N(R)$, is a graph with vertex set ~$Z_N(R)^* = {0neq x in R | xy in N(R) for some y in R^*}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy in N(R)$, or equivalently, $yx in N(R)$, where $N(R)$ denoted the nilpotent elements of $R$. Recently, it has been proved that if $R$ is a left A...
متن کاملZero Product Preserving Linear Maps of Ccr C*-algebras with Hausdorff Spectrum
In this paper, we try to attack a conjecture of Araujo and Jarosz that every bijective linear map θ between C*-algebras, with both θ and its inverse θ−1 preserving zero products, arises from an algebra isomorphism followed by a central multiplier. We show it is true for CCR C*-algebras with Hausdorff spectrum, and in general, some special C*-algebras associated to continuous fields of C*-algebras.
متن کاملDiscretized Ccr Algebras
We discuss how the canonical commutation relations must be modified in order to make appropriate numerical models of quantum systems. The C∗-algebras associated with the discretized CCRs are the non-commutative spheres of Bratteli, Elliott, Evans and Kishimoto. 1991 Mathematics Subject Classification. Primary 46L40; Secondary 81E05.
متن کاملIsomorphisms of Algebras Associated with Directed Graphs
Given countable directed graphs G and G, we show that the associated tensor algebras T+(G) and T+(G ) are isomorphic as Banach algebras if and only if the graphs G are G are isomorphic. For tensor algebras associated with graphs having no sinks or no sources, the graph forms an invariant for algebraic isomorphisms. We also show that given countable directed graphs G, G, the free semigroupoid al...
متن کامل