Local matrix homotopies and soft tori. (English) Zbl 1394.46056

Loring, Terry A.; Vides, Fredy

Banach J. Math. Anal. 12, No. 1, 167-190 (2018).

Summary: We present solutions to local connectivity problems in matrix representations of the form \(C([-1,1]^{N})\to C^{*}(u_{\varepsilon},v_{\varepsilon})\), with \(C_{\varepsilon}(\mathbb{T}^{2})\twoheadrightarrow C^{*}(u_{\varepsilon},v_{\varepsilon})\) for any \(\varepsilon\in[0,2]\) and any integer \(n\geq1\), where \(C^{*}(u_{\varepsilon},v_{\varepsilon})\subseteq M_{n}\) is an arbitrary matrix representation of the universal \(C^{*}\)-algebra \(C_{\varepsilon}(\mathbb{T}^{2})\) that denotes the soft torus. We solve the connectivity problems by introducing the so-called toroidal matrix links, which can be interpreted as normal contractive matrix analogies of free homotopies in differential algebraic topology.{ }To deal with the locality constraints, we have combined some techniques introduced in this article with some techniques from matrix geometry, combinatorial optimization, and the classification and representation theory of \(C^{*}\)-algebras.

MSC:

46L85	Noncommutative topology
22D25	\(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
20F65	Geometric group theory
65J22	Numerical solution to inverse problems in abstract spaces

Keywords:

matrix homotopy; relative lifting problems; matrix representation; amenable \(C\)-algebra; joint spectrum