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(&;Q) 2 has a trace whenever ;>0.
We can define fractional differentiability of order +, with 0<+ 1, by the boundedness of (Q 2 +I ) +Γ2 a(Q 2 +I ) &+Γ2 . 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 (Q 2 +I ) &;Γ2 da(Q 2 +I) &:Γ2 , where we choose :, ; 0 such that :+;<1. We show that this also ensures a fractional derivative of order +=1&; in the first sense. We define a family of interpolation spaces J ;, : . Each such space is a Banach algebra of operator, whose elements have a fractional derivative of order +=1&;>0.
We concentrate on subalgebras A of J ;, : which have certain additional covariance properties under a group Z 2 _G acting on H by a unitary representation #_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 { JLO , which plays the role of the integral of differential forms. We give a simple expression for a homotopy invariant Z Q (a; g), determined by pairing { JLO , with a G-invariant element a # A, such that a is a square root of the identity. This invariant is Z Q (a; g)= (1Γ-?) & e &t 2 Tr(#U(g) ae &Q 2 +it da ) dt.
This representation of the pairing is reminiscent of the heat-kernel representation for an index. In fact this quantity is an invariant, in the following sense. We isolate a simple condition on a family Q(*) of differentiations that yields a continuouslydifferentiable family { JLO (*) of cochains. Since Z Q (a; g) need not be an integer, continuity of { JLO (*) in * is insufficient to prove the constancy of the pairing. However the existence of the derivative leads to the existence of the homotopy. As