Quantum Harmonic Analysis and Geometric Invariants

Abstract

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 studyquantum 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, takingdaas a commutatorda=[Q,a]. HereQis a basic self-adjoint operator with discrete spectrum, increasing sufficiently rapidly that exp(−βQ)2has a trace wheneverβ>0. We can define fractional differentiability of orderμ, 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) ofda. We find that it is convenient to assume the boundedness of (Q2+I)−β/2da(Q2+I)−α/2, where we chooseα,β⩾0 such thatα+β<1. We show that this also ensures a fractional derivative of orderμ=1−βin 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 orderμ=1−β>0. We concentrate on subalgebras A of Jβ,αwhich have certain additional covariance properties under a group Z2×G acting on H by a unitary representationγ×U(g). In addition, the derivativeQis 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 cochainτJLO, which plays the role of the integral of differential forms. We give a simple expression for a homotopy invariant ZQ(a;g), determined by pairingτJLO, with a G-invariant elementa∈A, such thatais a square root of the identity. This invariant is ZQ(a;g)=(1/π)∫∞−∞e−t2Tr(γU(g)ae−Q2+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 familyQ(λ) of differentiations that yields a continuously-differentiable familyτJLO(λ) of cochains. Since ZQ(a;g) need not be an integer, continuity ofτJLO(λ) inλis insufficient to prove the constancy of the pairing. However the existence of the derivative leads to the existence of the homotopy. As τ,a vanishes forτa coboundary, and asdτJLO(λ)/dλis a coboundary, our condition onQ(λ) ensures that ZQ(λ)(a;g) is independent ofλ. Hence it is a homotopy invariant. The theory of ZQ(a;g) reduces to the study of the Radon transform of sequences of certain functions. The fractional differentiability properties of elements of A translate into properties of the asymptotics of the sequences of Radon transforms. The condition thatτJLOfit into the framework of entire cyclic cohomology translates to the existence of some fractional derivative for functions in the algebra under study, and in particular the assumptionα+β<1. Thus the study of fractionally- differentiable structures dove-tails naturally with the theory of homotopy invariants. In our study of quantum harmonic analysis, we introduce spaces T(−β,α) of operator-valued distributions. These spaces are bounded, linear operators between Sobolev spaces. The elements of the interpolation spaces, the Banach algebras Jβ,αhave derivativesdawhich belong to the spaces T(−β,α). For a certain range ofβandα, we extend the theory of the Radon transform from products of regularized, bounded operators to products of regularized, operator-valued distributions. We sometimes wish to evaluate such an invariant at the endpoint of an interval such asλ∈(0,1], where ZQ(λ)(a;g) becomes singular asλ→0. We discuss in brief a procedure to regularize the endpoint, and a method to recover ZQ(λ)(a;g) fully from certain partial information at the endpoint. Finally, we generalize this approach to cover the case whenQcan be split into the sum of “independent” partsQ1+Q2, such that (Q1+Q2)2=(Q1)2+(Q2)2. Here we assume thatQ1and (Q2)2are G-invariant, but not necessarilyQ2. With further assumptions ona, the most important being that (Q1)2−(Q2)2commutes witha, we obtain a modified formula for an invariant, namely Z{Qj}(a;g)=(1/π)∫∞−∞e−t2Tr(γU(g)ae−(Q21+Q22)/2+itd1a)dt.