Results 1  10
of
96
Probability laws related to the Jacobi theta and Riemann zeta functions, and the Brownian excursions
 Bulletin (New series) of the American Mathematical Society
"... Abstract. This paper reviews known results which connect Riemann’s integral representations of his zeta function, involving Jacobi’s theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to onedimensional ..."
Abstract

Cited by 57 (11 self)
 Add to MetaCart
Abstract. This paper reviews known results which connect Riemann’s integral representations of his zeta function, involving Jacobi’s theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to onedimensional Brownian motion and to higher dimensional Bessel processes. We present some characterizations of these probability laws, and some approximations of Riemann’s zeta function which are related to these laws. Contents
Noncommutative geometry, quantum fields and motives
 Colloquium Publications, Vol.55, American Mathematical Society
, 2008
"... ..."
From endomorphisms to automorphisms and back: dilations and full corners
 J. London Math. Soc
"... Abstract. When S is a discrete subsemigroup of a discrete group G such that G = S −1 S, it is possible to extend circlevalued 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 e ..."
Abstract

Cited by 25 (5 self)
 Add to MetaCart
Abstract. When S is a discrete subsemigroup of a discrete group G such that G = S −1 S, it is possible to extend circlevalued 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(Af) ⋊ Q ∗ +, corresponding to the multiplicative action of the positive rationals on the additive group Af of finite adeles.
KMS states and complex multiplication
 the proceedings of the Abel Symposium
, 2005
"... The following problem in operator algebra has been open for several years. Problem 1.1. For some number field K (other than Q) exhibit an explicit quantum statistical mechanical system (A, σt) with the following properties: (1) The partition function Z(β) is the Dedekind zeta function of K. ..."
Abstract

Cited by 23 (6 self)
 Add to MetaCart
The following problem in operator algebra has been open for several years. Problem 1.1. For some number field K (other than Q) exhibit an explicit quantum statistical mechanical system (A, σt) with the following properties: (1) The partition function Z(β) is the Dedekind zeta function of K.
Milman phenomenon, Urysohn metric spaces, and extremely amenable groups
 Israel J. Math
"... Abstract. In this paper we further study links between concentration of measure in topological transformation groups, existence of fixed points, and Ramseytype theorems for metric spaces. We prove that whenever the group Iso(U) of isometries of Urysohn’s universal complete separable metric space U, ..."
Abstract

Cited by 20 (9 self)
 Add to MetaCart
Abstract. In this paper we further study links between concentration of measure in topological transformation groups, existence of fixed points, and Ramseytype theorems for metric spaces. We prove that whenever the group Iso(U) of isometries of Urysohn’s universal complete separable metric space U, equipped with the compactopen topology, acts upon an arbitrary compact space, it has a fixed point. The same is true if U is replaced with any generalized Urysohn metric space U that is sufficiently homogeneous. Modulo a recent theorem by Uspenskij that every topological group embeds into a topological group of the form Iso(U), our result implies that every topological group embeds into an extremely amenable group (one admitting an invariant multiplicative mean on bounded right uniformly continuous functions). By way of the proof, we show that every topological group is approximated by finite groups in a certain weak sense. Our technique also results in a new proof of the extreme amenability (fixed point on compacta property) for infinite orthogonal groups. Going in the opposite direction, we deduce some Ramseytype theorems for metric subspaces of Hilbert spaces and for spherical metric spaces from existing results on extreme amenability of infinite unitary groups and groups of isometries of Hilbert spaces. 1.
FUN WITH F1
"... Abstract. We show that the algebra and the endomotive of the quantum statistical mechanical system of Bost–Connes naturally arises by extension of scalars from the “field with one element ” to rational numbers. The inductive structure of the abelian part of the endomotive corresponds to the tower of ..."
Abstract

Cited by 19 (10 self)
 Add to MetaCart
Abstract. We show that the algebra and the endomotive of the quantum statistical mechanical system of Bost–Connes naturally arises by extension of scalars from the “field with one element ” to rational numbers. The inductive structure of the abelian part of the endomotive corresponds to the tower of finite extensions of that “field”, while the endomorphisms reflect the Frobenius correspondences. This gives in particular an explicit model over the integers for this endomotive, which is related to the original Hecke algebra description. We study the reduction at a prime of the endomotive and of the corresponding noncommutative crossed product algebra. Contents
Quantum statistical mechanics over function fields
"... It has become increasingly evident, starting from the seminal paper of Bost and Connes [3] and continuing with several more recent developments ([8], [10], [12], [13], [22], [24]), that there is a rich interplay between quantum statistical mechanics and arithmetic. In the case of number fields, the ..."
Abstract

Cited by 15 (9 self)
 Add to MetaCart
It has become increasingly evident, starting from the seminal paper of Bost and Connes [3] and continuing with several more recent developments ([8], [10], [12], [13], [22], [24]), that there is a rich interplay between quantum statistical mechanics and arithmetic. In the case of number fields, the symmetries and equilibrium states of the Bost–Connes system are closely