Smooth \(K\)-groups for monoid algebras and \(K\)-regularity
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 Quillen algebraic \(K\)-theory \(K_n\) 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 Frechét algebra \(A\) with properly uniformly bounded approximate unit. It follows that the polynomial algebra \((A \otimes \mathbb K)[x_1, \dots, x_m]\) and \(A \otimes \mathbb K\) have the same Quillen algebraic \(K\)-theory \(K_n\) for \(n\geq 1\) and that the Laurent polynomial algebra \((A \otimes \mathbb K)[t, t^{-1}]\) has the \(K\)-theory \(K_n\) isomorphic to the direct sum of \(K_n\) and \(K_{n-1}\) of \(A \otimes \mathbb K\) and has the smooth \(K\)-theory \(sK_n\) isomorphic to the direct sum of \(sK_n\) and \(sK_{n-1}\) of \(A \otimes \mathbb K\) both for \(n\geq 1\).
properly uniformly bounded approximate unit
scientific article; zbMATH DE number 6528852
scientific article