C ∞(M) as a smooth envelope of its subalgebras

Abstract

A smooth envelope of a topological algebra is introduced, and the following result is announced: the smooth envelope of a given subalgebra A in C ∞ (M ) coincides with C ∞ (M ) if and only if A has the same tangent bundle as M .In [15], J. L. Taylor introduced an operation now called the Arens-Michael envelope. It turns any topological algebra A into the projective limit A ♥ of its Banach quotient algebras (see [8], [14]). The numerous connections of this construction with complex analysis allow one to treat A ♥ as the holomorphic algebra 1 closest to A from the outside. This view is based, in particular, on the formula due to A. Yu. Pirkovskii [14] that connects the algebra P(M ) of polynomials on an affine algebraic variety M with the algebra O(M ) of holomorphic functions on M :(1)In the theory of involutive topological algebras, there is a similar construction which is called the pro-C * -envelope, which turns a topological algebra A into the projective limit Env C A of its C * -quotient algebras (see [7], [9], [13], [10]). The envelope Env C A can be treated as the continuous algebra 2 nearest to A on the outside. The analogue of (1) connects the algebra C(M ) of continuous functions on a paracompact locally compact space M with its arbitrary subalgebra A that has M as the involutive spectrum (see [1]):Both envelopes are used in generalizations of Pontryagin duality from commutative locally compact groups to the classes of noncommutative groups, in particular quantum groups ([3], [10]). Philosophically these theories are in some sense projections of functional analysis into the "noncommutative complex analysis" (considered as a geometric interpretation of the theory of Banach algebras, see [14]) and into the "noncommutative topology" (considered as a geometric interpretation of the theory of C * -algebras, see [6]).The operations ♥ and Env C and the corresponding duality theories give rise to the hypothesis that there are other similar operations, in particular, envelopes that generate by analogy a projection of functional analysis into the "noncommutative differential geometry". In this paper, we describe such an envelope. We define the C ∞ -envelope Env C ∞ A of an involutive topological algebra A and announce a formula analogous to (1) and (2):In this approach, holomorphic algebras are defined by the equality A ♥ = A. In the ideology of this branch of noncommutative geometry, it is assumed that this describes the properties of ordinary algebras O(M ) of holomorphic functions essential for generalizations. 2 Continuous algebras are defined by the equality EnvCA = A, and as in the case of holomorphic algebras, it is assumed that this distinguishes the properties of ordinary algebras C(M ) of continuous functions essential for generalizations. 489

Free and forced bending vibrations of a thin plate in a perfect compressible fluid with energy dissipation taken into account

“…Akbarov, M.I. Ismailov where the motion of the plate is described by three‐dimensional linearized equations of wave propagation in elastic bodies with initial stresses, and the liquid flow is described with linearized Navier‐Stokes equations. The problems of non‐linear oscillations of plates moving in an ideal incompressible liquid were studied in works [].…”