Smooth \(K\)-groups for monoid algebras and \(K\)-regularity
type: Item
instance of: scholarly article
title: Smooth \(K\)-groups for monoid algebras and \(K\)-regularity
zbMATH Open document ID: 1337.19001
author: Hvedri Inassaridze
published in: Mathematics
publication date: 2016-01-12T00:00:00Z
review text: Without detailed proofs, the paper under review deals with the following. For any topological algebra \(A\), its smooth \(K\)-functors \(sK_*\) are defined as the Quillen (algebraic) \(K\)-theory functor \(K_0\) of \(A\) as degree \(0\) and as a quotient of the group \(\mathrm{GL}(A)\) of invertibles of \(A\) by a kernel as degree \(1\) and as the kernel of the map from the homotopy group of the group of invertibles of the topological algebra of smooth maps from the unit interval \(I\) to \(A\) vanishing at zero, to \(\mathrm{GL}(A)\) as degree \(2\), and as higher homotopy functors of the group of invertibles of the topological algebras of smooth maps from the products of \(I\) to \(A\) as higher degrees more than \(2\). An approximate unit of a Frechét algebra \(A\) is defined to be called properly uniformly bounded approximate unit of \(A\) if it is the image of a bounded approximate unit of a Banach algebra \(B\) under a continuous injective algebra homomorphism from \(B\) to \(A\). In this case \(A\) has the excision property in algebraic \(K\)-theory and the \(H\)-unitality property. As well, if \(A\) has such an approximate unit, then so does any monoid algebra \(A[M]\) over \(A\) with a monoid \(M\). It is stated as a theorem that the (algebraic) \(K\)-theory \(K_1\) and the smooth \(K\)-theory \(sK_1\) are the same on a category if and only if the \(K\)-theory \(K_1\) is a smooth homotopy functor, and that the smooth \(sK_1\) and the Karoubi-Villamayour's algebraic KV-theory \(KV_1\) are the same on the category if and only if \(KV_1\) is a smooth homotopy functor. Moreover, it is stated as theorems that the smooth \(K\)-theory \(sK_n\) and the KV-theory \(KV_n\) are isomorphic for \(n\geq 1\) and for the monoid algebra \((A \otimes \mathbb K)[M]\) over a quasi-stable \(A \otimes \mathbb K\) of any locally convex algebra \(A\) and the \(C^*\)-algebra \(\mathbb K\) of compact operators on a Hilbert space, and that the smooth \(K\)-theory \(sK_n\) and the Qui…
reviewed by: Takahiro Sudo
Mathematics Subject Classification ID: 19D25
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 46H05
Mathematics Subject Classification ID: 46H20
Mathematics Subject Classification ID: 46L80
zbMATH DE Number: 6528852
zbMATH Keywords: algeraic \(K\)-theory
zbMATH Keywords: smooth \(K\)-theory
zbMATH Keywords: \(K\)-regularity
zbMATH Keywords: properly uniformly bounded approximate unit
zbMATH Keywords: stable locally convex algebra
zbMATH Keywords: Frechét algebra
zbMATH Keywords: monoid algebra
MaRDI profile type: Publication
full work available at URL: math3030891
OpenAlex ID: W1758045408
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DOI: 10.3390/MATH3030891
description: scientific article; zbMATH DE number 6528852