The Arens-Michael envelopes of Laurent Ore extensions

For an Arens-Michael algebra A we consider a class of A-⊗ˆ -bimodules which are invertible with respect to the projective bimodule tensor product. We call such bimodules topologically invertible over A. Given a Fréchet-ArensMichael algebra A and a topologically invertible Fréchet A-⊗ˆ -bimodule M , we construct an Arens-Michael algebra LbA(M) which serves as a topological version of the Laurent tensor algebra LA(M). Also, for a fixed algebra B we provide a condition on an invertible B -bimodule N which allows us to explicitly describe the Arens-Michael envelope of LB(N) as a topological Laurent tensor algebra. In particular, we provide an explicit description of the Arens-Michael envelope of an invertible Ore extension $A[x, x^{−1}; α]$for a metrizable algebra A.

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[1] Akbarov SS. Holomorphic functions of exponential type and duality for Stein groups with an algebraic connected identity component. Fundamental’naya i Prikladnaya Matematika 2008; 14 (1): 3-178. doi: 10.1007/s10958-009- 9646-1
[2] Aristov O. The Relation “Commutator Equals Function” in Banach Algebras. Mathematical Notes 2021; 109 3-4 : 323-334. doi: 10.1134/s0001434621030019
[3] Dales HG. Banach algebras and automatic continuity. Oxford, England: The Clarendon Press, Oxford University Press, 2000.
[4] Helemskii AY. Banach and locally convex algebras. Oxford, England: The Clarendon Press, Oxford University Press, 1993.
[5] Helemskii AY. Lectures and exercises on functional analysis. Providence, RI, USA: American Mathematical Society, 2006. doi: 10.1090/mmono/233
[6] Jarchow H. Locally convex spaces. Stuttgart, Germany: B. G. Teubner, 1981.
[7] Mallios A. Topological algebras. Selected topics. Amsterdam, Netherlands: North-Holland Publishing Co., 1986.
[8] Pirkovskiĭ AY. Arens-Michael envelopes, homological epimorphisms, and relatively quasifree algebras. Trudy Moskovskogo Matematicheskogo Obshchestva 2008; 69: 34-125. doi:10.1090/S0077-1554-08-00169-6
[9] Pirkovskiĭ AY. Homological dimensions and Van den Bergh isomorphisms for nuclear Fréchet algebras. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya 2012; 76 4: 65-124. doi: 10.1070/IM2012v076n04ABEH002603
[10] Pirkovskiĭ AY. Stably flat completions of universal enveloping algebras. Dissertationes Mathematicae 2006; 441:60. doi: 10.4064/dm441-0-1
[11] Taylor JL. A general framework for a multi-operator functional calculus. Advances in Mathematics 1972; 9 2: 183-252. doi: 10.1016/0001-8708(72)90017-5
[12] Trèves F. Topological vector spaces, distributions and kernels. New York-London: Academic Press, 1967.