Rieffel deformation of homogeneous spaces

Let G₁⊂G be a closed subgroup of a locally compact group G and let X=G/G₁ be the quotient space of left cosets. Let X=(C₀(X),ΔX) be the corresponding G-C^∗-algebra where G=(C₀(G),Δ). Suppose that Γ is a closed abelian subgroup of G₁ and let Ψ be a 2-cocycle on the dual group . Let GΨ be the Rieffel deformation of G. Using the results of the previous paper of the author we may construct GΨ-C^∗-algebra XΨ - the Rieffel deformation of X. On the other hand we may perform the Rieffel deformation of the subgroup G₁ obtaining the closed quantum subgroup , which in turn, by the results of S. Vaes, leads to the GΨ-C^∗-algebra . In this paper we show that . We also consider the case where Γ⊂G is not a subgroup of G₁, for which we cannot construct the subgroup . Then generically XΨ cannot be identified with a quantum quotient. What may be shown is that it is a GΨ-simple object in the category of GΨ-C^∗-algebras.     

Archimedean operator-theoretic Positivstellensätze

We prove a general archimedean positivstellensatz for hermitian operator-valued polynomials and show that it implies the multivariate Fejer-Riesz theorem of Dritschel-Rovnyak and positivstellensätze of Ambrozie-Vasilescu and Scherer-Hol. We also obtain several generalizations of these and related results. The proof of the main result depends on an extension of the abstract archimedean positivstellensatz for ?-algebras that is interesting in its own right.     

A semi-finite algebra associated to a subfactor planar algebra

We canonically associate to any planar algebra two type II_∞ factors M_±. The subfactors constructed previously by the authors in Guionnet et al. (2010) [6] are isomorphic to compressions of M_± to finite projections. We show that each M_± is isomorphic to an amalgamated free product of type I von Neumann algebras with amalgamation over a fixed discrete type I von Neumann subalgebra. In the finite-depth case, existing results in the literature imply that M₊≅M₋ is the amplification a free group factor on a finite number of generators. As an application, we show that the factors Mj constructed in Guionnet et al. (in press) [6] are isomorphic to interpolated free group factors L(F(rj)), rj=1+2δ−2j(δ−1)I, where δ² is the index of the planar algebra and I is its global index. Other applications include computations of laws of Jones-Wenzl projections.     

On sofic actions and equivalence relations

The notion of sofic equivalence relation was introduced by Gabor Elek and Gabor Lippner. Their technics employ some graph theory. Here we define this notion in a more operator algebraic context, starting from Connes? Embedding Problem, and prove the equivalence of these two definitions. We introduce a notion of sofic action for an arbitrary group and prove that an amalgamated product of sofic actions over amenable groups is again sofic. We also prove that an amalgamated product of sofic groups over an amenable subgroup is again sofic.     

An optimal Steffensen-type family for solving nonlinear equations

In this paper, a general family of Steffensen-type methods with optimal order of convergence for solving nonlinear equations is constructed by using Newton’s iteration for the direct Newtonian interpolation. It satisfies the conjecture proposed by Kung and Traub [H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Math. 21 (1974) 634-651] that an iterative method based on m evaluations per iteration without memory would arrive at the optimal convergence of order 2^m−1. Its error equations and asymptotic convergence constants are obtained. Finally, it is compared with the related methods for solving nonlinear equations in the numerical examples.     

Some <Emphasis Type="Italic">I</Emphasis>-convergent sequence spaces defined by Orlicz functions

In this article we introduce the sequence spaces cI(M),c0I(M),mI(M) and m0I(M) using the Orlicz function M.We study some of the properties like solid,symmetric,sequence algebra,etc and prove some inclusion relations.     

Free boundary problem concerning pricing convertible bond

In this paper, we consider some behaviors of the optimal conversion boundaries (i.e. free boundaries) of American‐style convertible bond with finite horizon in some case. The bond's holder may convert it into the stock of its issued firm at any time before maturity, and the firm may call it at any time before maturity. Its pricing model is a parabolic variational inequality, in which the fundamental variables are time and the stock price of the bond's issuer. We achieve some properties of the free boundary, besides the existence and uniqueness of the solution of the variational inequality, such as: the monotonicity, the boundedness, smoothness and its starting point. Moreover, we analyze the relationship between the free boundary and the parameters in the problem, as well as, obtain the critical condition where the free boundary is a constant independent of time. Copyright © 2011 John Wiley & Sons, Ltd.     

Morita context of weak Hopf algebras

Let H be a finite-dimensional weak Hopf algebra and A a left H-module algebra with its invariant subalgebra A~H.We prove that the smash product A#H is an A-ring with a grouplike character, and give a criterion for A#H to be Frobenius over A. Using the theory of A-rings, we mainly construct a Morita context connecting the smash product A#H and the invariant subalgebra A~H , which generalizes the corresponding results obtained by Cohen, Fischman and Montgomery.     

群在von Neumann代数上作用的自由和遍历性注记

本文研究了群在von Neumann代数上作用的自由性和遍历性问题.利用投影和群SL2(R)的Iwasawa分解,得到了可数离散群在交换von Neumann代数上作用的自由性的等价刻画,证明了SL2(R)在上半平面H上有理作用导出的SL2(R)在极大交换von Neumann代数A={Mf:f∈L2(H,dxdy/y2)}上的作用α是遍历的,但不是自由的.