Some operator inequalities for Hermitian Banach $*$-algebras
DOI:
https://doi.org/10.7146/math.scand.a-115624Abstract
In this paper, we extend the Kubo-Ando theory from operator means on C$^{*}$-algebras to a Hermitian Banach $*$-algebra $\mathcal {A}$ with a continuous involution. For this purpose, we show that if $a$ and $b$ are self-adjoint elements in $\mathcal {A}$ with spectra in an interval $J$ such that $a \leq b$, then $f(a) \leq f(b)$ for every operator monotone function $f$ on $J$, where $f(a)$ and $f(b)$ are defined by the Riesz-Dunford integral. Moreover, we show that some convexity properties of the usual operator convex functions are preserved in the setting of Hermitian Banach $*$-algebras. In particular, Jensen's operator inequality is presented in these cases.
References
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Ando, T. and Hiai, F., Operator log-convex functions and operator means, Math. Ann. 350 (2011), no. 3, 611–630. https://doi.org/10.1007/s00208-010-0577-4
Bailey, D. W., On symmetry in certain group algebras, Pacific J. Math. 24 (1968), 413–419.
Bendat, J. and Sherman, S., Monotone and convex operator functions, Trans. Amer. Math. Soc. 79 (1955), 58–71. https://doi.org/10.2307/1992836
Bonsall, F. F. and Duncan, J., Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80, Springer-Verlag, New York-Heidelberg, 1973.
Civin, P. and Yood, B., Involutions on Banach algebras, Pacific J. Math. 9 (1959), 415–436.
El Kinani, A., Holomorphic functions operating in Hermitian Banach algebras, Proc. Amer. Math. Soc. 111 (1991), no. 4, 931–939. https://doi.org/10.2307/2048559
Feng, B. Q., The geometric means in Banach $\ast $-algebras, J. Operator Theory 57 (2007), no. 2, 243–250.
Fujii, M. and Kamei, E., Ando-Hiai inequality and Furuta inequality, Linear Algebra Appl. 416 (2006), no. 2-3, 541–545. https://doi.org/10.1016/j.laa.2005.12.001
Furuta, T., $A\geq B\geq 0$ assures $(B^rA^pB^r)^1/q\geq B^(p+2r)/q$ for $r\geq 0$, $p\geq 0$, $q\geq 1$ with $(1+2r)q\geq p+2r$, Proc. Amer. Math. Soc. 101 (1987), no. 1, 85–88. https://doi.org/10.2307/2046555
Hansen, F., The fast track to Löwner's theorem, Linear Algebra Appl. 438 (2013), no. 11, 4557–4571. https://doi.org/10.1016/j.laa.2013.01.022
Hansen, F. and Pedersen, G. K., Jensen's inequality for operators and Löwner's theorem, Math. Ann. 258 (1981/82), no. 3, 229–241. https://doi.org/10.1007/BF01450679
Kadison, R. V. and Ringrose, J. R., Fundamentals of the theory of operator algebras. Vol. I, Pure and Applied Mathematics, vol. 100, Academic Press, Inc., New York, 1983.
Kaplansky, I., Symmetry of Banach algebras, Proc. Amer. Math. Soc. 3 (1952), 396–399. https://doi.org/10.2307/2031891
Kubo, F. and Ando, T., Means of positive linear operators, Math. Ann. 246 (1979/80), no. 3, 205–224. https://doi.org/10.1007/BF01371042
Moslehian, M. S., Najafi, H., and Uchiyama, M., A normal family of operator monotone functions, Hokkaido Math. J. 42 (2013), no. 3, 417–423. https://doi.org/10.14492/hokmj/1384273390
Najafi, H., Operator means and positivity of block operators, Math. Z. 289 (2018), no. 1-2, 445–454. https://doi.org/10.1007/s00209-017-1958-0
Okayasu, T., The Löwner-Heinz inequality in Banach $\ast $-algebras, Glasg. Math. J. 42 (2000), no. 2, 243–246. https://doi.org/10.1017/S0017089500020097
Shirali, S. and Ford, J. W. M., Symmetry in complex involutory Banach algebras. II, Duke Math. J. 37 (1970), 275–280.
Tanahashi, K. and Uchiyama, A., The Furuta inequality in Banach $\ast $-algebras, Proc. Amer. Math. Soc. 128 (2000), no. 6, 1691–1695. https://doi.org/10.1090/S0002-9939-99-05262-4
Uchiyama, M., Operator monotone functions, positive definite kernel and majorization, Proc. Amer. Math. Soc. 138 (2010), no. 11, 3985–3996. https://doi.org/10.1090/S0002-9939-10-10386-4
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Published
2020-03-29
How to Cite
Najafi, H. (2020). Some operator inequalities for Hermitian Banach $*$-algebras. MATHEMATICA SCANDINAVICA, 126(1), 82–98. https://doi.org/10.7146/math.scand.a-115624
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