Quantum spaces and their noncommutative topology (Q2756778)

The two fundamental ``machines'' of noncommutative geometry are cyclic homology and bivariant topological \(K\)-theory. The author gave a uniform approach to cyclic theory and bivariant \(K\)-theory and constructed the bivariant Chern-Connes character taking bivariant \(K\)-theory to bivariant cyclic theory in full generality [\textit{J. Cuntz}, Documenta Math. J. DMV 2, 139-182 (1997; Zbl 0920.19004)]. This paper sketches these results together with illustrations of author's vision on quantum spaces such as the phase space in quantum mechanics and noncommutative 2-torus.NEWLINENEWLINENEWLINEThe paper begins to give examples of noncommutative spaces with illustrations. Then cyclic theory is defined as the noncommutative analogy of de Rham theory as follows: Let \(A\) be an algebra, \(\Omega^1A\) the space of abstract 1-forms over \(A\) defined as the bimodule consisting of linear combinations of \(xd(y)\), \(x\in \widetilde A\), the unitization of \(A\), \(y\in A\), \(d\) a derivation of \(A\). LetNEWLINENEWLINENEWLINE\(HX^{ev}(A)=\) \{traces on \(A\)\}/\{traces of the form \(f\circ d\), \(f\) a trace on \(\Omega^1 A\} \),NEWLINENEWLINENEWLINE\(HX^{od}(A)=\) \{traces on \(\Omega^1A\), such that \(f\circ d=0\}\).NEWLINENEWLINENEWLINELet \(T\) be a quasi-free algebra with an ideal \(I\) such that \(A=T/I\). Author's definition of the periodic cyclic cohomology \(HP^*(A)\), * stands for \(ev\) or \(od\), \(ev+1= od\), \(od+1=ev\), is NEWLINE\[NEWLINEHP^*(A)= \lim HX^*(T/I^n)=\lim HX^{*+1} (I^n).NEWLINE\]NEWLINE The bivariant periodic cyclic homology \(HP_*(A_1,A_2)\) is defined by the same way. The author remarks to show the coincidence of this definition and the definition of Connes and Tsygan is nontrivial. But this definition provides a striking analogy between periodic cyclic homology and Grothendieck's notion of infinitesimal homology. Then ``excision'' of cyclic theory is stated [\textit{J. Cuntz} and \textit{D. Quillen}, Invent. Math. 127, No. 1, 67-98 (1997; Zbl 0889.46054)]. Next emphasizing the connection between \(K\)-theory and extensions, bivariant \(K\)-theory for complete locally convex algebras is defined as follows: Let \(TA\) be the completion of the algebraic tensor algebra over \(A\), \(J\) the kernel of the map \(x_1\otimes \cdots \otimes x_n\mapsto x_1,\dots, x_n\), \(J^nA=J (J^{n-1}A)\), \(J^1A=JA\), and \(M_\infty(B)\) the algebra of infinite matrices over \(B\) with rapidly decreasing matrix elements (noncommutative space interpretation of \(M_n(\mathbb{C})\) and its illustration is given as example 1). Let * be either 0 or 1, and \([J^{2k+*} A\), \(M_\infty(B)]\) the set of differentiable homotopy classes of homomorphisms \(J^{2k+*} A\to M_\infty(B)\). The bifunctor \(kk_*\) is NEWLINE\[NEWLINEkk_*(A,B)= \lim\bigl[J^{2k+*} A,M_\infty(B) \bigr].NEWLINE\]NEWLINE It is noted that bifunctors \(kk_*\) and \(HP^*\) have same abstract properties and \(kk_* (\mathbb{C},B)\) is the usual \(K\)-group if \(B\) is a Fréchet algebra.NEWLINENEWLINENEWLINEThe paper is concluded to state existence of a multiplicative transformation \(ch\): \(kk_*(A,B) \to HP)*(A,B)\). Interpretations of index theorems in this framework are also stated.