Costel Peligrad

Derivations of C*-Algebras which are Invariant under an Automorphism Group. II

By a C*-dynamical system we mean a triple (B,G,β) consisting of a C*-algebra B, a locally compact group G and a continuous homomorphism B of G into the group Aut (B) of *-automorphism of B equipped with the topology of pointwise convergence.

Commutants of Nest Algebras Modulo the Compact Operators

Any operatorx which commutes modulo the compact operators with a nest algebra is of the form λI+C, where λ is a scalar andC is a compact operator.Any derivation from a nest algebra on a Hilbert spaceH into the compact operators onH is implemented by a compact operator.Any derivation on a quasitriangular operator algebra is inner.

ON THE INVARIANCE PRINCIPLE UNDER MARTINGALE APPROXIMATION

In this paper, we establish continuous Gaussian limits for stochastic processes associated to linear combinations of partial sums. The underlying sequence of random variables is supposed to admit a martingale approximation in the square mean. The results are useful in studying averages of additive functionals of a Markov chain with normal operator.

COMPACT QUANTUM GROUP ACTIONS ON C*-ALGEBRAS AND INVARIANT DERIVATIONS

We define the notion of invariant derivation of a C*-algebra under a compact quantum group action and prove that in certain conditions, such derivations are generators of one parameter automorphism groups.

A strong Connes spectrum for finite group actions of simple rings

Abstract In this paper, we give necessary and sufficient conditions for a skew group ring to be simple. Our conditions are in terms of the irreducible representations of the group G. We also introduce a version of the strong Connes spectrum for finite group actions of simple rings.

Compact derivations of nest algebras

In this paper we determine all the weakly compact derivations of a nest algebra. We also obtain necessary and sufficient conditions in order that a nest algebra admit compact derivations. Finally we prove that every compact derivation of a nest algebra d is the norm limit of finite-rank derivations. 1. Let _ be a Banach algebra and let X be a Banach s-module. By an X-valued derivation of _ we mean a linear mapping 8: sV-X with the property 8(ab) = a8(b) + 8(a)b for all a E :, b E .@. The derivation 8 is called compact if 8 is a compact operator between the Banach space s and X, and weakly compact if 8 is a weakly compact operator from s to X (i.e. 8(s) is relatively weakly compact in X, where SV is the unit ball of sV [4]). Let H be a complex Hilbert space, B(H) the algebra of all bounded operators on H and Y(H) = X the set of compact operators on H. In [7] Johnson and Parrott investigated derivations of a von Neumann subalgebra of B(H) with range contained in Y. They proved that in most cases such derivations are implemented by a compact operator. The general result was recently obtained by Popa [8] who proved that this is the case for all von Neumann subalgebras of B(H). Such derivations are known to be weakly compact [1]. On the other hand, in a series of papers [1, 9, 10], C. A. Akemann, S. K. Tsui, and S. Wright have determined the structure of all compact and weakly compact s-valued derivations of a C*-algebra s, and of all compact B(H) valued derivations of a C *-subalgebra of B(H). In this note we determine the structure of all s-valued compact and weakly compact derivations of a nest algebra S. In particular we prove that every compact derivation of a nest algebra s is the norm limit of finite rank derivations. We need the following result. LEMMA 1 [6]. Let H be an infinite dimensional Hilbert space. If 8 is a compact derivation of B(H) then 8 0. Received by the editors January 16, 1984 and, in revised form, July 1, 1985. 1980 Mathenmatics Subject Classification. Primary 47D25; Secondary 47B05.

The limiting spectral distribution in terms of spectral density

For a large class of symmetric random matrices with correlated entries, selected from stationary random fields of centered and square integrable variables, we show that the limiting distribution of eigenvalue counting measure always exists and we describe it via an equation satisfied by its Stieltjes transform. No rate of convergence to zero of correlations is imposed, therefore the process is allowed to have long memory. In particular, if the symmetrized matrices are constructed from stationary Gaussian random fields which have spectral density, the result of this paper gives a complete solution to the limiting eigenvalue distribution. More generally, for matrices whose entries are functions of independent identically distributed random variables the result also holds.

Spectra for compact quantum group coactions and crossed products

We present definitions of both Connes spectrum and strong Connes spectrum for actions of compact quantum groups on C*-algebras and obtain necessary and sufficient conditions for a crossed product to be a prime or a simple C*-algebra. Our results extend to the case of compact quantum ac- tions the results in (8) which in turn, generalize results by Connes, Olesen and Pedersen and Kishimoto for abelian group actions. We prove in addition that the Connes spectra are closed under tensor products. These results are new for compact nonabelian groups as well.

Smooth derivations commuting with Lie group actions

N. S. Poulsen, motivated in part by questions from relativistic quantum scattering theory, studied symmetric operators S in Hilbert space commuting with a unitary representation U of a Lie group G . (The group of interest in the physical setting is the Poincare group.) He proved ([ 17 ], corollary 2·2) that if S is defined on the space of C ∞ -vectors for U (i.e. D(S) ⊇ ℋ ∞ ( U )), then S is essentially self-adjoint.