## Banach Journal of Mathematical Analysis

### Local matrix homotopies and soft tori

#### Abstract

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 classification and representation theory of $C^{*}$-algebras.

#### Article information

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

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

https://projecteuclid.org/euclid.bjma/1510909220

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

Mathematical Reviews number (MathSciNet)
MR3745579

Zentralblatt MATH identifier
06841270

#### 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

#### References

• [1] M. Ahues, F. D. d’Almeida, A. Largillier, and P. B. Vasconcelos,Spectral refinement for clustered eigenvalues of quasi-diagonal matrices, Linear Algebra Appl.413(2006), no. 2–3, 394–402.
• [2] R. Bhatia,Matrix Analysis, Grad. Texts in Math.169, Springer, New York, 1997.
• [3] O. Bratteli, G. A. Elliott, D. E. Evans, and A. Kishimoto,Homotopy of a pair of approximately commuting unitaries in a simple $C^{*}$-algebra, J. Funct. Anal.160(1998), no. 2, 466–523.
• [4] R. W. Brockett,Least squares matching problems, Linear Algebra Appl.122/123/124(1989), 761–777.
• [5] M. D. Choi,The full $C^{*}$-algebra of the free group on two generators, Pacific J. Math.87(1980), no. 1, 41–48.
• [6] M. D. Choi and E. G. Effros,The completely positive lifting problem for $C^{*}$-algebras, Ann. of Math. (2)104(1976), no. 3, 585–609.
• [7] M. T. Chu,A simple application of the homotopy method to symmetric eigenvalue problems, Linear Algebra Appl.59(1984), 85–90.
• [8] M. T. Chu,Linear algebra algorithms as dynamical systems, Acta Numer.17(2008), 1–86.
• [9] J. E. Dennis, Jr., J. F. Traub, and R. P. Weber,The algebraic theory of matrix polynomials, SIAM J. Numer. Anal.13(1976), no. 6, 831–845.
• [10] S. Eilers and R. Exel,Finite-dimensional representations of the soft torus, Proc. Amer. Math. Soc.130(2002), no. 3, 727–731.
• [11] S. Eilers, T. A. Loring, and G. K. Pedersen,Stability of anticommutation relations: An application of noncommutative CW complexes, J. Reine Angew. Math.499(1998), 101–143.
• [12] L. Elsner,Perturbation theorems for the joint spectrum of commuting matrices: A conservative approach, Linear Algebra Appl.208/209(1994), 83–95.
• [13] M. H. Freedman and W. H. Press,Truncation of wavelet matrices: Edge effects and the reduction of topological control, Linear Algebra Appl.234(1996), 1–19.
• [14] M. B. Hastings and T. A. Loring,Topological insulators and $C^{*}$-algebras: Theory and numerical practice, Ann. Physics326(2011), no. 7, 1699–1759.
• [15] C. J. Hillar and C. R. Johnson,Symmetric word equations in two positive definite letters, Proc. Amer. Math. Soc.132(2004), no. 4, 945–953.
• [16] C. R. Johnson and C. J. Hillar,Eigenvalues of words in two positive definite letters, SIAM J. Matrix Anal. Appl.23(2002), no. 4, 916–928.
• [17] C. J. Ku and T. L. Fine,Testing for stochastic independence: Application to blind source separation, IEEE Trans. Signal Process.53(2005), no. 5, 1815–1826.
• [18] H. Lin,An Introduction to the Classification of Amenable $C^{*}$-Algebras, World Scientific, River Edge, N.J., 2001.
• [19] H. Lin,Approximate homotopy of homomorphisms from $C(X)$ into a simple $C^{*}$-algebra, Mem. Amer. Math. Soc.205(2010), no. 963.
• [20] H. Lin,Approximately diagonalizing matrices over $C(Y)$, Proc. Natl. Acad. Sci. USA109(2012), no. 8, 2842–2847.
• [21] T. A. Loring,$K$-theory and asymptotically commuting matrices, Canad. J. Math.40(1988), no. 1, 197–216.
• [22] T. A. Loring,Lifting Solutions to Perturbing Problems in $C^{*}$-Algebras, Fields Inst. Monogr.8, Amer. Math. Soc., Providence, 1997.
• [23] T. A. Loring and T. Shulman,Noncommutative semialgebraic sets and associated lifting problems, Trans. Amer. Math. Soc.364, no. 2 (2012), 721–744.
• [24] T. A. Loring and F. Vides,Estimating norms of commutators, Experiment. Math.24(2015), no. 1, 106–122.
• [25] T. Maehara and K. Murota,Algorithm for error-controlled simultaneous block-diagonalization of matrices, SIAM J. Matrix Anal. Appl.32(2011), no. 2, 605–620.
• [26] A. McIntosh, A. Pryde, and W. Ricker,Systems of operator equations and perturbation of spectral subspaces of commuting operators, Michigan Math. J.35(1988), no. 1, 43–65.
• [27] A. Pryde, “Inequalities for the joint spectrum of simultaneously triangularizable matrices” inMiniconference on Probability and Analysis (Sydney, 1991), Proc. Centre Math. Appl. Austral. Nat. Univ.29, Austral. Nat. Univ., Canberra, 196–207, 1992.
• [28] M. Rørdam, F. Larsen, and N. J. Laustsen,An Introduction to $K$-Theory for $C^{*}$-Algebras, London Math. Soc. Stud. Texts49, Cambridge Univ. Press, Cambridge, 2000.
• [29] P. Tichavský and A. Yeredor,Fast approximate joint diagonalization incorporating weight matrices, IEEE Trans. Signal Process.57(2009), no. 3, 878–891.