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Relative Inversion in der Störungstheorie von Operatoren und $$\Psi$$- Algebren. (Relative inversions in perturbation theory of operators and $$\Psi$$-algebras). (German) Zbl 0661.47037
In this substantial paper the author considers a separated topological algebra $${\mathcal B}$$ under the assumption that its group of invertible elements is open. A function defined on subsets of $${\mathcal B}^ n$$ is called $${\mathcal B}$$-rational if it is composed of projections onto coordinate spaces of $${\mathcal B}^ n$$ and of constant functions by a finite number of algebraic operations (addition, multiplication and inversion). A topological space is called a locally $${\mathcal B}$$-rational ($$\ell {\mathcal B}r$$-) manifold with an atlas $$\{(U_{\alpha},\phi_{\alpha},V_{\alpha})$$; $$\alpha\in A\}$$ of charts, iff:
1) $$\{U_{\alpha}$$, $$\alpha\in A\}$$ is an open covering of M, for $$\alpha\in A$$, $$V_{\alpha}$$ is an open subset of the topological subspace $$(T_{\alpha},\tau (T_{\alpha}))$$ and $$\phi_{\alpha}: U_{\alpha}\to V_{\alpha}$$ is a homeomorphism,
2) for $$\alpha\in A$$, $$T_{\alpha}$$ is a linear subspace of $${\mathcal B}^{n_{\alpha}}$$ with a topology $$\tau (T_{\alpha})$$ which is finer than the subspace topology of the product topology of $$\tau$$ ($${\mathcal B})$$ on $${\mathcal B}^{n_{\alpha}}$$ and
3) all mappings $$\phi_{\alpha}\phi_{\beta}^{-1}: \phi_{\beta}(U_{\alpha}\cap U_{\beta})\to \phi_{\alpha}(U_{\alpha}\cap U_{\beta})$$ ($$\alpha$$,$$\beta\in A)$$ are $${\mathcal B}$$-rational functions.
A homogeneous space $$(G<\Pi,M)$$ is called an $$\ell {\mathcal B}r$$ homogeneous space and M an $$\ell {\mathcal B}r$$ homogeneous manifold, iff:
1) G and M are $$\ell {\mathcal B}_ r$$-manifolds,
2) the group operations $$G\times G\to G$$ are $$\ell {\mathcal B}r$$-mappings,
3) $$\Pi$$ : $$G\times M\to M$$ is an $$\ell {\mathcal B}r$$-mapping and
4) for all $$p\in M$$ there is an open neighbourhood of p and an $$\ell {\mathcal B}r$$-submanifold $${\mathcal T}_ p$$ of G such that $$\Pi |_{{\mathcal T}_ p}: {\mathcal T}_ p\to U$$ is an $$\ell {\mathcal B}r$$-homeomorphism.
If J is a two-sided ideal in $${\mathcal B}$$, then $$G_ e(J)$$ is the connected component of e (the unit in $${\mathcal B})$$ in $${\mathcal B}^{-1}\cap (e+J)$$. The author shows that the connected component of a projection $$(p=p^ 2)$$ or of a relatively invertible element $$(aba=a)$$ of $${\mathcal B}$$ in its residue class $$p+J$$ or $$a+J$$ resp. is a $${\mathcal B}$$-rational manifold and, together with the group $$G_ e(J)$$, an $$\ell {\mathcal B}r$$- homogeneous space or an $$\ell {\mathcal B}r$$-homogeneus manifold, respectively. The proofs make use of the fact that $${\mathcal B}$$ can be represented as the direct sum of projected subspaces and that the elements of a neighbourhood of e in $$G_ e(J)$$ allow a representation as a product with factors in special subgroups of $$G_ e(J)$$. The group $$G_ e(J)$$ also operates on equivalence classes of projections. This space of equivalence classes, provided with a suitable topology, turns out to be an $$\ell {\mathcal B}r$$-homogeneous manifold with respect to $$G_ e(J)$$. This space is homotopically equivalent to the space of projections, if it is paracompact. The proofs of these statements are performed by deducing explicit algebraic formulas which enable the author to avoid the implicit function theorem. An important example is the algebra $$C^{\infty}(\Omega,L(E))$$ where $$\Omega$$ is a compact differentiable manifold, E a Hilbert space and L(E) is the space of continuous endomorphisms of E.
The article also contains a thorough study of $$\Psi$$-algebras: A Fréchet subalgebra $$\Psi$$ of a Banach algebra $${\mathcal B}$$ is called a $$\Psi$$-algebra iff $$\Psi^{-1}={\mathcal B}^{-1}\cap \Psi$$ where $$\Psi^{-1}$$ and $${\mathcal B}^{-1}$$ denote the group of invertible elements of $$\Psi$$ or $${\mathcal B}$$, respectively; if $${\mathcal B}$$ is a $$C^*$$-algebra and $$\Psi$$ is symmetric, then $$\Psi$$ is called a $$\Psi^*$$-algebra. For such algebras the author proves a series of nontrivial statements, e.g. on connection properties concerning inverse elements, relative inverse elements and orthogonal projections. The concept of $$\Psi$$-algebras is motivated by recent results on pseudodifferential operators, cf. e.g. H. O. Cordes [Elliptic pseudodifferential operators - an abstract theory, Lect. Notes Math. 756 (1979; Zbl 0417.35004)]. For interesting special classes of $$\Psi$$-algebras the author is able to prove a Cartan lemma on holomorphic factorization which enables him to establish an Oka principle in such $$\Psi$$-algebras. The basis for these studies are publications of H. O. Cordes [Manusc. Math. 28, 51-69 (1979; Zbl 0415.35083)] and A. Connes [C. R. Acad. Sci., Paris, Ser. A 290, Ser. A, 599-604 (1980; Zbl 0433.46057)] on pseudodifferential operators. Finally the author shows that in the Fréchet algebra of operators of order 0 the group of invertible elements need not be open.
Reviewer: R.Mennicken

MSC:
 47L10 Algebras of operators on Banach spaces and other topological linear spaces 47A55 Perturbation theory of linear operators 47Gxx Integral, integro-differential, and pseudodifferential operators 58B25 Group structures and generalizations on infinite-dimensional manifolds 58J40 Pseudodifferential and Fourier integral operators on manifolds 47A53 (Semi-) Fredholm operators; index theories
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