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 R d , A.M.S. Memoirs, 506, 1993, ix+93pp. [3] M. I. Merklen, Boundedness of pseudodifferential operators 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´ on & R. Vaillancourt, On the bound- edness of pseudo-differential operators , J. Math. Soc. Japan 23 (1971) 374-378.

Pseudodifferential operators H = L 2 ( R n ) , u ∈ S ( R n ) � � Bu ( x ) = (2 π ) − n e i ( x − y ) ξ a ( x, ξ ) u ( y ) dydξ (*) a ∈ CB ∞ ( R 2 n ) 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 , E z,ζ = M ζ T z , where, if f is a Schwartz function, E z,ζ f ( x ) = e iζx f ( x − z ) . B ∈ B ( H ) , B z,ζ = E − 1 z,ζ BE z,ζ → B z,ζ is C ∞ , If ( z, ζ ) − we say that B is Heisenberg smooth.

O : CB ∞ ( R 2 n ) − → H a ( x, ξ ) − → a ( x, D ) u ∈ S ( R n ) , a ( x, D ) u ( x ) = Bu ( x ) , 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 ∞ ( R n , A ) depending on an antisymmetric n × n matrix J : � � F × J G ( x ) = (2 π ) − n e iu · v F ( x + Ju ) G ( x + v ) dudv. (We are working here with oscilatory integrals.) For g ∈ S A ( R n ) , L F g = F × J g and for f ∈ S A ( R n ) , R G f = f × J G. Rieffel defined a norm in S A ( R n ): 1 � � � 2 f ∗ ( x ) f ( x ) dx � � � f � 2 = . � � � �

If E is the Hilbert A − module obtained by completing S A ( R n ) with that norm, Rieffel conjectured that any T ∈ B ∗ ( E ), Heisenberg smooth, that commutes with R G for all G ∈ CB ∞ ( R n , A ) is an L F for F ∈ CB ∞ ( R n , A ) and we can see that L F 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 ) = (2 π ) − n e i ( x − y ) ζ a ( x, ζ ) u ( y ) dydζ, u ∈ S ( R n )

As in the scalar case, we considered on B ∗ ( E ) the action of the Heisenberg group, H A = { T ∈ B ∗ ( E ) /T is Heisenberg smooth } If ( z ∈ R n ) ( z, 0) − → T z, 0 is C ∞ , we say that T is translation smooth. T A = { T ∈ B ∗ ( E ) /T is translation smooth }

O : CB ∞ ( R 2 n , A ) − → H A 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 ∞ ( R 2 n , 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 R G , T is Heisenberg smooth. T z,ζ = T z − 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 ∈ S A ( R n ). Hence, we have that, if T ∈ B ∗ ( E ) is translation smooth and commutes with R G for all G ∈ S A ( R n ), then there ex- ists F ∈ CB ∞ ( R n , A ) such that T = L F .