Strongly asymptotic morphisms on separable metrisable algebras (Q1589667)

\textit{A. Connes} and \textit{N. Higson} [C. R. Acad Sci., Paris, Ser. I 311, No. 2, 101-106 (1990; Zbl 0717.46062)], dealing with the category of separable \(C^{*}\)-algebras, defined an asymptotic morphism to be a family \((f_{t})\) of mappings from \(A\) to \(B\), say \(t \in [0,\infty)\), such that, as \(t \to \infty\), \(f_{t}\) tends to act like a homomorphism, without the assumption that \(f_{t}\) converges as \(t \to \infty\). The methods are closely related to those of continuous fields of \(C^{*}\)-algebras. They defined a composition \( (g \circ_{\varphi} f)_{t}(a) = g_{\varphi(t)}(f_{t}(a))\) of asymptotic morphisms \(f_{t}:A \to B\) and \(g_{t}:A \to \)C where \(\varphi\) is an increasing positive valued function which acts as a change of parameter for \(g\) from \(t\) to \(\varphi(t)\). Two asymptotic morphisms \(f\) and \(g\) with values in \(B\) are called homotopy equivalent if there exists an asymptotic morphism \(F_{t} : A \to C([0,1]) \otimes B\) such that \(F_{t}(a)(0) = f_{t}(a)\) and \(F_{t}(a)(1) = g_{t}(a)\) for all \(t \geq 0\) and \(a \in A\). The author adapts the above to separable metrisable algebras; the difficulty is to ensure that the composition of asymptotic morphisms remains an asymptotic morphism. He gives sufficiency conditions to take account of the metric and the operations of algebraic and scalar multiplication (and involution, if necessary) and defines strong asymptotic continuity of asymptotic morphisms. To construct a reparametrisation \(\varphi\) he uses the fact that for separable metric spaces every cover has a countable subcover, and he shows that strong asymptotic continuity of \(f_{t}\) and \( g_{t}\) implies that of \((g \circ_{\varphi} f)_{t}\) . This leads to a definition of strongly asymptotic morphisms of algebras for which multiplication is continuous with respect to a translation-invariant metric. For the composition to be well-defined at the level of homotopy class a change of parameter should not affect the homotopy class of the composition. Thus, in the case of algebras having also an involution. the author restricts the class of strongly asymptotic morphisms and defines ``valid'' reparametrisations in order that strong homotopy equivalence classes be independent of the choice of \(\varphi\). For separable metrisable involutive algebras which are also locally convex there is a sequence of seminorms the author introduces the concept of ``pointwise boundedness'' as that for all \(t \geq 0\) the set of points \(f_{t}(a) \in B\) is bounded above, pointwise in \(A\), for each of the seminorms. With some extra compatability conditions he defines pointwise bounded asymptotic morphisms. For these the verification of compatability of compositions is simplified. Another result is that when every bounded set has compact closure the compatability conditions are automatically satisfied. A related example based on \textit{G. Elliott, T. Natsume} and \textit{R. Nest} [Commun. Math. Phys. 182, No. 3, 505-533 (1996; Zbl 0876.58047)] and related to the classical limit of quantum mechanics, is a locally convex involutive algebra \(A\), though not a \(C^{*}\)-algebra, constructed from smooth functions \({\mathbb{R}}^{2} \to {\mathbb{R}}\). The limit as \(t\to \infty\) is ''reparametrised'' to that of the value of Planck's constant \(\hbar\) converging to \(0+\). There is a family of mappings \( (\pi_{\hbar})\), \(\pi_{\hbar}:A \to {\mathcal{B}}(L^{2}({\mathbb{R}}))\) which is asymptotically multiplicative and a pointwise bounded asymptotic morphism as \(\hbar \to 0+\). The final application is the construction of an asymptotic integral for a family \((u_t)\) of positive elements in a separable \(C^{*}\)-algebra \(E\) such that \((u_t)\) is asymptotically Abelian in \(E\) as \(t \to +\infty\). The author defines an asymptotic integral \({\mathbb{I}}\) of \(f_{t} \in C([0,1], E)\), and \({\mathbb{I}}_{t}(f) = {\mathbb{I}}(f_{t})\), and this integration is shown to be a strong asymptotic morphism as \(t \to +\infty\). For a closed two-sided ideal \(J\) the asymptotic integral is applied to the result of \textit{M. Dǎdarlat} [K-Theory 8, No. 5, 465-482 (1994; Zbl 0821.19003)] on an exact sequence \( J \to E \to E/J\), following the results of Connes and Higson (loc.cit.). For continuous sections \({\mathcal{S}}: E \to E/J\), \({\mathbb{I}}({\mathcal{S}} \circ e)\) maps \(C_{0}([0,1],E/J) \to E\). This is used to illustrate the asymptotic equivalence between the suspension of the mapping cone of the exact sequence and the suspension of the ideal \(J\).