67 research outputs found
Hecke algebras of semidirect products and the finite part of the Connes-Marcolli C*-algebra
We study a C*-dynamical system arising from the ring inclusion of the 2\times
2 integer matrices in the rational ones. The orientation preserving affine
groups of these rings form a Hecke pair that is closely related to a recent
construction of Connes and Marcolli; our dynamical system consists of the
associated reduced Hecke C*-algebra endowed with a canonical dynamics defined
in terms of the determinant function. We show that the Schlichting completion
also consists of affine groups of matrices, over the finite adeles, and we
obtain results about the structure and induced representations of the Hecke
C*-algebra. In a somewhat unexpected parallel with the one dimensional case
studied by Bost and Connes, there is a group of symmetries given by an action
of the finite integral ideles, and the corresponding fixed point algebra
decomposes as a tensor product over the primes. This decomposition allows us to
obtain a complete description of a natural class of equilibrium states which
conjecturally includes all KMS_\beta-states for \beta\ne 0,1.Comment: 30 pages; minor corrections, final versio
Ground states of groupoid C*-algebras, phase transitions and arithmetic subalgebras for Hecke algebras
We consider the Hecke pair consisting of the group of affine
transformations of a number field that preserve the orientation in every
real embedding and the subgroup consisting of transformations with
algebraic integer coefficients. The associated Hecke algebra
has a natural time evolution , and we describe the corresponding phase
transition for KMS-states and for ground states. From work of
Yalkinoglu and Neshveyev it is known that a Bost-Connes type system associated
to has an essentially unique arithmetic subalgebra. When we import this
subalgebra through the isomorphism of to a corner in the
Bost-Connes system established by Laca, Neshveyev and Trifkovic, we obtain an
arithmetic subalgebra of on which ground states exhibit the
`fabulous' property with respect to an action of the Galois group
, where is the narrow Hilbert class field.
In order to characterize the ground states of the -dynamical system
, we obtain first a characterization of the ground
states of a groupoid -algebra, refining earlier work of Renault. This is
independent from number theoretic considerations, and may be of interest by
itself in other situations.Comment: 21 pages; v2: minor changes and correction
Quantized algebras of functions on homogeneous spaces with Poisson stabilizers
Let G be a simply connected semisimple compact Lie group with standard
Poisson structure, K a closed Poisson-Lie subgroup, 0<q<1. We study a
quantization C(G_q/K_q) of the algebra of continuous functions on G/K. Using
results of Soibelman and Dijkhuizen-Stokman we classify the irreducible
representations of C(G_q/K_q) and obtain a composition series for C(G_q/K_q).
We describe closures of the symplectic leaves of G/K refining the well-known
description in the case of flag manifolds in terms of the Bruhat order. We then
show that the same rules describe the topology on the spectrum of C(G_q/K_q).
Next we show that the family of C*-algebras C(G_q/K_q), 0<q\le1, has a
canonical structure of a continuous field of C*-algebras and provides a strict
deformation quantization of the Poisson algebra \C[G/K]. Finally, extending a
result of Nagy, we show that C(G_q/K_q) is canonically KK-equivalent to C(G/K).Comment: 23 pages; minor changes, typos correcte
von Neuman algebras of strongly connected higher-rank graphs
We investigate the factor types of the extremal KMS states for the preferred
dynamics on the Toeplitz algebra and the Cuntz--Krieger algebra of a strongly
connected finite -graph. For inverse temperatures above 1, all of the
extremal KMS states are of type I. At inverse temperature 1, there is
a dichotomy: if the -graph is a simple -dimensional cycle, we obtain a
finite type I factor; otherwise we obtain a type III factor, whose Connes
invariant we compute in terms of the spectral radii of the coordinate matrices
and the degrees of cycles in the graph.Comment: 16 pages; 1 picture prepared using TikZ. Version 2: this version to
appear in Math. An
Complementarity and the algebraic structure of 4-level quantum systems
The history of complementary observables and mutual unbiased bases is
reviewed. A characterization is given in terms of conditional entropy of
subalgebras. The concept of complementarity is extended to non-commutative
subalgebras. Complementary decompositions of a 4-level quantum system are
described and a characterization of the Bell basis is obtained.Comment: 19 page
A Characterization of right coideals of quotient type and its application to classification of Poisson boundaries
Let be a co-amenable compact quantum group. We show that a right coideal
of is of quotient type if and only if it is the range of a conditional
expectation preserving the Haar state and is globally invariant under the left
action of the dual discrete quantum group. We apply this result to theory of
Poisson boundaries introduced by Izumi for discrete quantum groups and
generalize a work of Izumi-Neshveyev-Tuset on for co-amenable compact
quantum groups with the commutative fusion rules. More precisely, we prove that
the Poisson integral is an isomorphism between the Poisson boundary and the
right coideal of quotient type by maximal quantum subgroup of Kac type. In
particular, the Poisson boundary and the quantum flag manifold are isomorphic
for any q-deformed classical compact Lie group.Comment: 28 pages, Remark 4.9 adde
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