An Operator Theoretical Approach to Enveloping ϕ* – and C* – Algebras of Melrose Algebras of Totally Characteristic Pseudodifferential Operators

Abstract

Let X be a compact manifold with boundary. It will be shown (Theorem 3.4) that the small Melrose algebra A := 9i,c, (X, bnl/e) (cf. 1221, [23]) of dmical, totally characteristic peudodifferential operators carries no topology such that it is a topological algebra with an open group of invertible elements, in particular, the algebra A cannot be spectrally invariant in any C 'algebra. On the other hand, the symbolic structure of A can be extended continuously to the C 'algebra 8 generated by A 88 a subalgebra of L(pbL2 (XI *n1I2)) by a generalisation of a method of GOHEERG and KRUPNIK. hrthermore, A is densely embedded in a M e t algebra A C B which is a -algebra in the sense of GRAMSCH [9, Definition 5.11, nfiscting .tso smooth properties of the original algebra A.GRAMSCH [9, Definition 5.11 introduced in connection with a perturbation theory for algebras of pseudodifferential operators the notion of Q' -algebras, i. e., symmetric, spectrally invariant, continuously embedded RBchet subalgebras of C 'algebras.Some of the remarkable consequences of the Q' -property and examples will be briefly discussed in Remark 2.10.Nevertheless, there also exist very interesting algebras in the pseudodifferential analysis on singular manifolds where the q* -property fails; for instance, the algebra 1991 Mafhernaticr Subjcct Clortificotion. Primary: 46H35, 58G 15; Secondary: 47D25, 47B47, Keywordr and phmuu. 1y'algebras, totally characterintic pesudodifferential operators. 47C15.Math. Nachr. 196 (1998) *f,cl (x, *Q1l2) of classical, totally characteristic pseudodifferential operators on a compact manifold X with boundary introduced by MELROSE [22] in 1981 carries even no topology such that (X, *all2) is a topological algebra with an open group of invertible elements (Theorem 3.4). On the other hand, it is an algebra with symbolic structure, i. e., it is possible to characterize the F'redholm property by means of the invertibility of symbols; here we think of !Pt,cl(X,bR1/2) as being embedded in C(ehL2(X,bf11/2)) for some weight b E JR.In order to make the Q' -techniques still available also to the small Melrose calculus, one can try to approximate the algebra from above by !P' -algebras in such a way that the approximating !P* -algebras share as many properties of the small calculus as possible. Following a suggestion of GRAMSCH one can consider the intersection of all !P* -algebras in L ( e b L 2 ( X , *all2)) containing @i,cl (X,*R1/2).The resulting algebra is uniquely determined and still a symmetric, spectrally invariant, continuously embedded subalgebra of L(ebL2 (X, which enjoys also many properties of !P' -algebras, although it is in general not a FMchet algebra, but only a sequentially complete, locally convex topological algebra with jointly continuous multiplication and continuous inversion. It is worth pointing out that in these topological algebras the relative inversion within the algebra (cf. (91, [20]) is an appropriate substitute for the parametrix construction for pseudodifferential operators (Theorem 2.11). Th...

On the Existence and Structure of Ψ*-Algebras of Totally Characteristic Operators on Compact Manifolds with Boundary

“…1Â2 ) be the closed anti *-derivation corresponding to W _ and the minimal closed extension V of V according to [21,Lemma 2.17]. Finally, let for simplicity /=/ _ be the chart diffeomorphism corresponding to U _ .…”

“…It is the third (cf. [21,22] for the first two one's) in a series of papers devoted to the construction and investigation of a 9*-completion of the Melrose algebra 9 0 b, cl (X, b 0 1Â2 ).…”

“…Note that the operator theoretical methods and the general concept of this construction are explained in [21] in detail. However, for the convenience of the reader, let us briefly recall the important steps of the construction.…”

“…In a first step, we extend the well-known symbolic structure of the algebra 9 0 b, cl (X, b 0 1Â2 ) [27] to its C*-closure B (b) (X, b 0 1Â2 ) by an appropriate form of a method of Gohberg and Krupnik [13], in a second step we obtain a 9*-algebra with a smooth symbolic structure simply by pulling back the 9*-algebra Q (b) 9 of symbols, and finally, we refine this algebra by means of commutator methods which were introduced by Beals [2], Coifman and Meyer [4], Cordes [7], Dunau [10] for the characterization of pseudodifferential operators, and later on extended by Gramsch, Ueberberg, and Wagner [18] to a general principle for the construction of 9*-algebras; a suitably adapted version for our construction can be found in [21].…”

“…1Â2 ) will be refined in Subsection 3.2 using the commutator techniques of [21] to a well-behaved 9*-completion…”