Banach Journal of Mathematical Analysis

Local matrix homotopies and soft tori

Terry A. Loring and Fredy Vides

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Abstract

We present solutions to local connectivity problems in matrix representations of the form C([1,1]N)C(uε,vε), with Cε(T2)C(uε,vε) for any ε[0,2] and any integer n1, where C(uε,vε)Mn is an arbitrary matrix representation of the universal C-algebra Cε(T2) 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 classification and representation theory of C-algebras.

Article information

Source
Banach J. Math. Anal., Volume 12, Number 1 (2018), 167-190.

Dates
Received: 29 November 2016
Accepted: 22 March 2017
First available in Project Euclid: 17 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1510909220

Digital Object Identifier
doi:10.1215/17358787-2017-0048

Mathematical Reviews number (MathSciNet)
MR3745579

Zentralblatt MATH identifier
06841270

Subjects
Primary: 46L85: Noncommutative topology [See also 58B32, 58B34, 58J22]
Secondary: 22D25: $C^*$-algebras and $W^*$-algebras in relation to group representations [See also 46Lxx] 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 65J22: Inverse problems

Keywords
matrix homotopy relative lifting problems matrix representation amenable C∗-algebra joint spectrum

Citation

Loring, Terry A.; Vides, Fredy. Local matrix homotopies and soft tori. Banach J. Math. Anal. 12 (2018), no. 1, 167--190. doi:10.1215/17358787-2017-0048. https://projecteuclid.org/euclid.bjma/1510909220


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