ABSTRACT CHARACTERIZATIONS OF PSEUDODIFFERENTIAL OPERATORS - - PDF document

ABSTRACT CHARACTERIZATIONS OF PSEUDODIFFERENTIAL OPERATORS References [1]

• H. O. Cordes, On pseuso-differential operators

and smoothness of special Lie-group represen- tations, Manuscripta Math. 28 (1979) 51-69. [2] Marc A. Rieffel, Deformation Quantization for actions of Rd , A.M.S. Memoirs, 506, 1993, ix+93pp. [3]

• M. I. Merklen, Boundedness of pseudodifferential
• perators of a C*-algebra-valued symbol, Proc.

Royal Soc. Edinburg, 135A, 1279-1286 (2005). [4]

• S. T. Melo & M. I. Merklen, C*-algebra-valued-

symbol pseudodifferential operators: abstract characterizations, in preparation. [5]

• A. P. Calder´
• n & R. Vaillancourt, On the bound-

edness of pseudo-differential operators, J. Math.

• Soc. Japan 23 (1971) 374-378.

Pseudodifferential operators H = L2(Rn), u ∈ S(Rn)

Bu(x) = (2π)−n ei(x−y)ξa(x, ξ)u(y)dydξ (*)

a ∈ CB∞(R2n) B ∈ B(H)

H = {B ∈ B(H), B is Heisenberg smooth} In order to make the notation easier, let us consider n = 1. z, ζ ∈ R, Ez,ζ = MζTz, where, if f is a Schwartz function, Ez,ζf(x) = eiζxf(x − z). B ∈ B(H), Bz,ζ = E−1

z,ζBEz,ζ

If (z, ζ) − → Bz,ζ is C∞, we say that B is Heisenberg smooth.

O : CB∞(R2n) − → H a(x, ξ) − → a(x, D)

a(x, D)u(x) = Bu(x), u ∈ S(Rn), as in(∗).

Cordes proved, [1], that O is a bijec- tion, and thus characterized the pseudos in B(H).

Rieffel, in [2], defined a deformed prod- uct in CB∞(Rn, A) depending on an antisymmetric n × n matrix J:

F ×J G(x) = (2π)−n eiu·vF(x+Ju)G(x+v)dudv.

(We are working here with oscilatory integrals.) For g ∈ SA(Rn), LFg = F ×J g and for f ∈ SA(Rn), RGf = f ×J G. Rieffel defined a norm in SA(Rn): f2 =

• f∗(x)f(x)dx
• 1

2

.

If E is the Hilbert A−module obtained by completing SA(Rn) with that norm, Rieffel conjectured that any T ∈ B∗(E), Heisenberg smooth, that commutes with RG for all G ∈ CB∞(Rn, A) is an LF for F ∈ CB∞(Rn, A) and we can see that LF is a pseudo whose symbol is F(x − Jξ).

If we consider a separable C∗−algebra A, a pseudodifferential operator in B∗(E) is given by

a(x, D)u(x) = (2π)−n ei(x−y)ζa(x, ζ)u(y)dydζ, u ∈ S(Rn)

As in the scalar case, we considered

• n B∗(E) the action of the Heisenberg

group,

HA = {T ∈ B∗(E)/T is Heisenberg smooth}

If (z, 0) − → Tz,0 (z ∈ Rn) is C∞, we say that T is translation smooth.

TA = {T ∈ B∗(E)/T is translation smooth}

O : CB∞(R2n, A) − → HA O(a) = a(x, D) is well defined. We proved in [3], section 2, a gener- alized version of Calderon-Vaillancourt theorem, [5], a(x, D) is bounded on E. We also proved that a(x, D) is ad- jointable and that a(x, D)∗ = p(x, D) for a suitable p ∈ CB∞(R2n, A)

O is injective and we also proved in [4] that it has a left inverse, S, if A is commutative. So, if A is commutative, O is a bijec- tion.

We also saw, in section 3, that, if T is translation smooth and commutes with RG, T is Heisenberg smooth. Tz,ζ = Tz−Jζ,0 and we proved that if we are dealing with a C∗−algebra for which O is a bi- jection, then Rieffel’s conjecture is true.

Moreover, we showed that it is enough to assume that T is translation smooth and that G ∈ SA(Rn). Hence, we have that, if T ∈ B∗(E) is translation smooth and commutes with RG for all G ∈ SA(Rn), then there ex- ists F ∈ CB∞(Rn, A) such that T = LF.