From Matrix to Operator Inequalities

Loring, Terry A.

Canadian mathematical bulletin, Tome 55 (2012) no. 2, pp. 339-350

Résumé

We generalize Löwner's method for proving that matrix monotone functions are operator monotone. The relation $x\,\le \,y$ on bounded operators is our model for a definition of ${{C}^{*}}$ -relations being residually finite dimensional.Our main result is a meta-theorem about theorems involving relations on bounded operators. If we can show there are residually finite dimensional relations involved and verify a technical condition, then such a theorem will follow from its restriction to matrices.Applications are shown regarding norms of exponentials, the norms of commutators, and “positive” noncommutative $*$ -polynomials.

Détail

DOI : 10.4153/CMB-2011-063-8

Mots-clés : 46L05, 47B99, C*-algebras, matrices, bounded operators, relations, operator norm, order, commutator, exponential, residually finite dimensional

Bibliographie

[1] [1] Archbold, R. J., On residually finite-dimensional C*-algebras. Proc. Amer. Math. Soc. 123(1995), no. 9, 2935–2937. Google Scholar

[2] [2] Bendat, J. and Sherman, S., Monotone and convex operator functions. Trans. Amer. Math. Soc. 79(1955), 58–71. Google Scholar

[3] [3] Bhatia, R., Matrix analysis. Graduate Texts in Mathematics, 169, Springer-Verlag, New York, 1997. Google Scholar

[4] [4] Bhatia, R. and Kittaneh, F., Some inequalities for norms of commutators. SIAM J. Matrix Anal. Appl. 18(1997), no. 1, 258–263. Google Scholar | DOI

[5] [5] Blackadar, B., Operator algebras. Theory of C*-algebras and von Neumann algebras. Encyclopaedia of Mathematical Sciences, 122, Operator Algebras and Non-commutative Geometry, III. Springer-Verlag, Berlin, 2006. Google Scholar

[6] [6] Blackadar, B., Shape theory for C*-algebras. Math. Scand. 56(1985), no. 2, 249–275. Google Scholar

[7] [7] Dixmier, J., C*-algebras. North-Holland Mathematical Library, 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Google Scholar

[8] [8] Eilers, S. and Exel, R., Finite-dimensional representations of the soft torus. Proc. Amer. Math. Soc. 130(2002), no. 3, 727–731. Google Scholar | DOI

[9] [9] Exel, R. and Loring, T. A., Finite-dimensional representations of free product C*-algebras. Internat. J. Math. 3(1992), no. 4, 469–476. Google Scholar | DOI

[10] [10] Goodearl, K. R. and Menal, P., Free and residually finite-dimensional C*-algebras. J. Funct. Anal. 90(1990), no. 2, 391–410. Google Scholar | DOI

[11] [11] Hadwin, D., Kaonga, L., and Mathes, B., Noncommutative continuous functions. J. Korean Math. Soc. 40(2003), no. 5, 789–830. Google Scholar | DOI

[12] [12] Helton, J. W., “Positive” noncommutative polynomials are sums of squares. Ann. of Math. (2) 156(2002), no. 2, 675–694. Google Scholar | DOI

[13] [13] Loring, T. A., C*-algebra relations. Math. Scand. 107(2010), no. 1, 43–72. Google Scholar

[14] [14] McCullough, S. and Putinar, M., Noncommutative sums of squares. Pacific J. Math. 218(2005), no. 1, 167–171. Google Scholar | DOI

[15] [15] Pedersen, G. K., C*-algebras and their automorphism groups. London Mathematical Society Monographs, 14, Academic Press Inc., London-New York, 1979. Google Scholar

[16] [16] Pedersen, G. K., A commutator inequality. In: Operator algebras, mathematical physics, and low-dimensional topology (Istanbul, 1991), Res. Notes Math., 5, A K Peters, Wellesley, MA, 1993, pp. 233–235. Google Scholar

[17] [17] Phillips, N. C., Inverse limits of C*-algebras and applications. In: Operator algebras and applications, Vol. 1, London Math. Soc. Lecture Note Ser., 135, Cambridge University Press, Cambridge, 1988, pp. 127–185. Google Scholar