S ep 1 99 7 Quantum Harmonic Analysis and Geometric Invariants ∗

We develop two topics in parallel and show their inter-relation. The first centers on the notion of a fractional-differentiable structure on a commutative or a non-commutative space. We call this study quantum harmonic analysis. The second concerns homotopy invariants for these spaces and is an aspect of non-commutative geometry. We study an algebra A, which will be a Banach algebra with unit, represented as an algebra of operators on a Hilbert space H. In order to obtain a geometric interpretation of A, we define a derivative on elements of A. We do this in a Hilbert space context, taking da as a commutator da = [Q, a]. Here Q is a basic self-adjoint operator with discrete spectrum, increasing sufficiently rapidly that exp(−βQ2) has a trace whenever β > 0. We can define fractional differentiability of order μ, with 0 < μ ≤ 1, by the boundedness of (Q2 + I)μ/2a(Q2 + I)−μ/2. Alternatively we can require the boundedness of an appropriate smoothing (Bessel potential) of da. We find that it is convenient to assume the boundedness of (Q2 + I)−β/2da(Q2 + I)−α/2, where we choose α, β ≥ 0 such that α + β < 1. We show that this also ensures a fractional derivative of order μ = 1− β in the first sense. We define a family of interpolation spaces Jβ,α. Each such space is a Banach algebra of operator, whose elements have a fractional derivative of order μ = 1− β > 0. We concentrate on subalgebras A of Jβ,α which have certain additional covariance properties under a group Z2×G acting onH by a unitary representation γ×U(g). In addition, the derivative Q is assumed to be G-invariant. The geometric interpretation flows from the assumption that elements of A possess an arbitrarily small fractional derivative. We study homotopy invariants of A in terms of equivariant, entire cyclic cohomology. In fact, the existence of a fractional derivative on A allows the construction of the cochain τJLO, which plays the role of the integral of differential forms. We give a simple expression for a homotopy invariant ZQ(a; g), determined by pairing τJLO, with a G-invariant element a ∈ A, such that a is a square root of the identity. This invariant is ZQ(a; g) = 1 √ π ∞ −∞ e −t2Tr ( γU(g)ae−Q 2+itda ) dt. ∗Supported in part by the Department of Energy under Grant DE-FG02-94ER-25228 and by the National Science Foundation under Grant DMS-94-24344.