Ola Bratteli

Unbounded Derivations Tangential to Compact Groups of Automorphisms, II*

Let G be a compact abelian group, and τ an action of G on a C∗-algebra U, such that Uτ(γ)Uτ(γ)∗ = Uτ(0) Uτ for all γ ϵ G, where Uτ(γ) is the spectral subspace of U corresponding to the character γ on G. Derivations δ which are defined on the algebra UF of G-finite elements are considered. In the special case δ¦Uτ = 0 these derivations are characterized by a cocycle on G with values in the relative commutant of Uτ in the multiplier algebra of U, and these derivations are inner if and only if the cocycles are coboundaries and bounded if and only if the cocycles are bounded. Under various restrictions on G and τ properties of the cocycle are deduced which again give characterizations of δ in terms of decompositions into generators of one-parameter subgroups of τ(G) and approximately inner derivations. Finally, a perturbation technique is devised to reduce the case δ(UF) ⊆ UF to the case δ(UF) ⊆ UF and δ¦Uτ = 0. This is used to show that any derivation δ with D(δ) = UF is wellbehaved and, if furthermore G = T1 and δ(UF) ⊆ UF the closure of δ generates a one-parameter group of ∗-automorphisms of U. In the case G = Td, d = 2, 3,… (finite), and δ(UF) ⊆ UF it is shown that δ extends to a generator of a group of ∗-automorphisms of the σ-weak closure of U in any G-covariant representation.

Asymptotics of periodic subelliptic operators

We establish that heat diffusion with periodic conductivity is governed by two scales. The small time diffusion is described by the geodesic distance but the large time behaviour is dictated by the distance associated with an homogenized system obtained by a suitable averaging process. Our methods are quite general and apply to diffusion on a stratified Lie group.

Spectral asymptotics of periodic elliptic operators

Abstract. We demonstrate that the structure of complex second-order strongly elliptic operators H on

Derivations commuting with abelian gauge actions on lattice systems

{\bf R}^d

The heat semigroup and integrability of Lie algebras

with coefficients invariant under translation by

Conservative derivations and dissipative Laplacians

{\bf Z}^d

Continuous Quantum Systems. II

can be analyzed through decomposition in terms of versions

OPERATOR ALGEBRAS AND QUANTUM STATISTICAL MECHANICS. 1. C* AND W* ALGEBRAS, SYMMETRY GROUPS, DECOMPOSITION OF STATES

H_z

Cuntz algebras and multiresolution wavelet analysis of scale N

,

C[*]- and W[*]-algebras symmetry groups decomposition of states

z\in{\bf T}^d