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We show how to reconstruct a finite directed graph E from its Toeplitz algebra, its gauge action, and the canonical finite-dimensional abelian subalgebra generated by the vertex projections. We also show that if E has no sinks, then we can recover E from its Toeplitz algebra and the generalized gauge action that has, for each vertex, an independent copy of the circle acting on the generators corresponding to edges emanating from that vertex. We show by example that it is not possible to recover E from its Toeplitz algebra and gauge action alone.
We consider the dynamics on the
-algebras of finite graphs obtained by lifting the gauge action to an action of the real line. Enomoto, Fujii and Watatani [KMS states for gauge action on
. Math. Japon.29 (1984), 607–619] proved that if the vertex matrix of the graph is irreducible, then the dynamics on the graph algebra admits a single Kubo–Martin–Schwinger (KMS) state. We have previously studied the dynamics on the Toeplitz algebra, and explicitly described a finite-dimensional simplex of KMS states for inverse temperatures above a critical value. Here we study the KMS states for graphs with reducible vertex matrix, and for inverse temperatures at and below the critical value. We prove a general result which describes all the KMS states at a fixed inverse temperature, and then apply this theorem to a variety of examples. We find that there can be many patterns of phase transition, depending on the behaviour of paths in the underlying graph.
The purpose of this paper is to give a complete description of the primitive ideal space of the C*-algebra [R] associated to the ring of integers R in a number field K in the recent paper . As explained in , [R] can be realized as the Toeplitz C*-algebra of the affine semigroup R ⋊ R× over R and as a full corner of a crossed product C0() ⋊ K ⋊ K*, where is a certain adelic space. Therefore Prim([R]) is homeomorphic to the primitive ideal space of this crossed product. Using a recent result of Sierakowski together with the fact that every quasi-orbit for the action of K ⋊ K* on contains at least one point with trivial stabilizer we show that Prim([R]) is homeomorphic to the quasi-orbit space for the action of K ⋊ K* on , which in turn may be identified with the power set of the set of prime ideals of R equipped with the power-cofinite topology.
We study the Toeplitz algebra 𝒯(ℕ⋊ℕ×) and three quotients of this algebra: the C*-algebra 𝒬ℕ recently introduced by Cuntz, and two new ones, which we call the additive and multiplicative boundary quotients. These quotients are universal for Nica-covariant representations of ℕ⋊ℕ× satisfying extra relations, and can be realised as partial crossed products. We use the structure theory for partial crossed products to prove a uniqueness theorem for the additive boundary quotient, and use the recent analysis of KMS states on 𝒯(ℕ⋊ℕ×) to describe the KMS states on the two quotients. We then show that 𝒯(ℕ⋊ℕ×), 𝒬ℕ and our new quotients are all interesting new examples for Larsen’s theory of Exel crossed products by semigroups.
The graph product of a family of groups lies somewhere between their direct and free products, with the graph determining which pairs of groups commute. We show that the graph product of quasi-lattice ordered groups is quasi-lattice ordered, and, when the underlying groups are amenable, that it satisfies Nica's amenability condition for quasi-lattice orders. The associated Toeplitz algebras have a universal property, and their representations are faithful if the generating isometries satisfy a joint properness condition. When applied to right-angled Artin groups this yields a uniqueness theorem for the C*-algebra generated by a collection of isometries such that any two of them either *-commute or else have orthogonal ranges. The analogous result fails to hold for the nonabelian Artin groups of finite type considered by Brieskorn and Saito, and Deligne.
When S is a discrete subsemigroup of a discrete group G such that
G = S−1S, it is possible to extend
circle-valued multipliers from S to G, to dilate (projective) isometric
representations of S to (projective) unitary representations of G, and to
dilate/extend actions of S by injective endomorphisms of a C*-algebra
to actions of G by automorphisms of a larger C*-algebra. These dilations are unique provided they satisfy
a minimality condition. The (twisted) semigroup crossed product corresponding to an action of S is
isomorphic to a full corner in the (twisted) crossed product by the dilated action of G. This shows that
crossed products by semigroup actions are Morita equivalent to crossed products by group actions, making
powerful tools available to study their ideal structure and representation theory. The dilation of the system
giving the Bost–Connes Hecke C*-algebra from number theory is constructed explicitly as an application:
it is the crossed product C0([Aopf ]f)[rtimes ]ℚ*+,
corresponding to the multiplicative action of the positive rationals
on the additive group [Aopf ]f of finite adeles.
Recently Bost and Connes have studied an interesting C*-algebraic Hecke algebra arising in number
theory. Here it is shown that this algebra can be realised as a semigroup crossed product, and be profitably
studied using methods developed by the authors for analysing Toeplitz algebras. One main result is a
characterisation of faithful representations of the Hecke algebra.
The spectral C*-algebra of the discrete product systems of H.T. Dinh is shown to be a twisted semigroup crossed product whenever the product system has a twisted unit. The covariant representations of the corresponding dynamical system are always faithful, implying the simplicity of these crossed products; an application of a recent theorem of G.J. Murphy gives their nuclearity. Furthermore, a semigroup of endomorphisms of B(H) having an intertwining projective semigroup of isometries can be extended to a group of automorphisms of a larger Type I factor.
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