Rieffel projections in certain transformation group C*-algebras

LetA be the transformation groupC ^*-algebra associated with an arbitrary orientation-preserving homeomorphism of . ThisC ^*-algebra contains an infinite family of projections, called Rieffel projections, each of which generates theK ₀-groupK ₀(A). Although these projections must beK-theoretically equivalent, it is easy to see that most are not Murray-von Neumann equivalent. The mystery of how large the matrix algebra must be to implement theK-theory equivalence, is solved by explicitly constructing the equivalence in the smallest possible algebra:A with unit adjoined.Partially supported by NSF Grant DMS 8901923.     

On maximal injective subalgebras in a <Emphasis Type="Italic">w</Emphasis>Γ factor

Let (F _ℚ) × _α ℤ be the crossed product von Neumann algebra of the free group factor (F _ℚ), associated with the left regular representation λ of the free group F _ℚ with the set {u _r : r ∈ ℚ} of generators, by an automorphism α defined by α(λ(u _r )) = exp(2πri)λ(u _r ), where ℚ is the rational number set. We show that (F _ℚ) × _α ℤ is a wΓ factor, and for each r ∈ ℚ, the von Neumann subalgebra generated in (F _ℚ) × _α ℚ by λ(u _r ) and υ is maximal injective, where υ is the unitary implementing the automorphism α. In particular, (F _ℚ) × _α ℣ is a wΓ factor with a maximal abelian selfadjoint subalgebra which cannot be contained in any hyperfinite type II₁ subfactor of (F _ℚ) × _α ℚ. This gives a counterexample of Kadison’s problem in the case of wΓ factor. This work was supported by the National Natural Science Foundation of China (Grant Nos. 10201007, A0324614) and the Natural Science Foundation of Shandong Province (Grant No. Y2006A03)     

The pressure in operator algebras

We introduce two notions of the pressure in operator algebras, one is the pressure Pα（π, T） for an automorphism α of a unital exact C^＊-algebra A at a self-adjoint element T in A with respect to a faithful unital ＊-representation π the other is the pressure Pτ,α（T） for an automorphism α of a hyperfinite von Neumann algebra M at a self-adjoint element T in M with respect to a faithful normal α-invariant state τ. We give some properties of the pressure, show that it is a conjugate invaxiant, and also prove that the pressure of the implementing inner automorphism of a crossed product A×α Z at a self-adjoint operator T in A equals that of α at T.     

Configurations in Coproducts of Priestley Spaces

Let P be a configuration, i.e., a finite poset with top element. Let be the class of bounded distributive lattices L whose Priestley space (L) contains no copy of P. We show that the following are equivalent: is first-order definable, i.e., there is a set of first-order sentences in the language of bounded lattice theory whose satisfaction characterizes membership in ; P is coproductive, i.e., P embeds in a coproduct of Priestley spaces iff it embeds in one of the summands; P is a tree. In the restricted context of Heyting algebras, these conditions are also equivalent to being a variety, or even a quasivariety. Mathematics Subject Classifications (2000) Primary: 06D50, 06D20; Secondary: 06D22.     

Reduced Crossed Products of Pro-Banach Algebras*

This paper gives the concept of the reduced pro-Banach algebra crossed product associated with inversely pro-Banach algebra dynamical system, and shows that the reduced crossed product is an inverse limit of an inverse system of Banach algebra crossed products. Also, the authors show that if the locally compact group is amenable, then the crossed product and the reduced crossed product are isometrically isomorphic.     

The Atiyah-Segal completion theorem for C*-algebras

We generalize the Atiyah-Segal completion theorem to C ^*-algebras as follows. Let A be a C ^*-algebra with a continuous action of the compact Lie group G. If K _* ^G (A) is finitely generated as an R(G)-module, or under other suitable restrictions, then the I(G)-adic completion K _* ^G (A) is isomorphic to RK _*([A C(EG)]^G), where RK _* is representable K-theory for - C ^*-algebras and EG is a classifying space for G. As a corollary, we show that if and are homotopic actions of G, and if K _*(C ^* (G,A,)) and K _*(C ^* (G,A,)) are finitely generated, then K _*(C ^*(G,A,))K_*(C ^* (G,A,)). We give examples to show that this isomorphism fails without the completions. However, we prove that this isomorphism does hold without the completions if the homotopy is required to be norm continuous.This work was partially supported by an NSF Graduate Fellowship and by an NSF Postdoctoral Fellowship.     

Synthesis of α,α′-bis(R-benzylidene)cycloalkanones catalyzed by potassium hydrogen sulfate under solvent-free conditions

Cyclopentanone or cyclohexanone were condensed with aromatic aldehydes to give corresponding α,α′-bis(R-benzylidene)cyclopentanones or α,α′-bis(R-benzylidene)cyclohexanones. The reaction was catalyzed by potassium hydrogen sulfate with a yield of 84–95% under solvent-free conditions.     

冯·诺依曼代数交叉积的一点注记

设α是可数离散群G和H的半直积G■_σH在冯·诺依曼代数M上的作用,则β_h=α_((e,h))AdU_h定义了群H在冯·诺依曼代数交叉积M■_αG上的作用β.本文证明了交叉积冯·诺依曼代数M■_α(G■_σH)与(M■_αG)■_βH是*-同构的,因此在一定条件下,冯·诺依曼代数的交叉积满足结合律.     

Hecke Algebras of Group Extensions

Abstract

We describe the Hecke algebra ?(Γ,Γ₀) of a Hecke pair (Γ,Γ₀) in terms of the Hecke pair (N,Γ₀) where N is a normal subgroup of Γ containing Γ₀. To do this, we introduce twisted crossed products of unital *-algebras by semigroups. Then, provided a certain semigroup S ? Γ/N satisfies S ^?1 S = Γ/N, we show that ? (Γ,Γ₀) is the twisted crossed product of ? (N,Γ₀) by S. This generalizes a recent theorem of Laca and Larsen about Hecke algebras of semidirect products.     

(ω,σ)-Smash积积和和(ν,α)-Smash余余积积

设B,H是两个Hopf代数,构造了(ω,σ)-Smash积Bω#σH和(ν,α)-Smash余积Bν■αH,并给出了Bω#σH是Hopf代数和Bν■αH是双代数的充要条件,证明了许多已知的积和余积是它们的特殊情况.