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Automatic continuity of \(M\)-norms on \(C^{*}\)-algebras. (English) Zbl 1225.46043
Summary: Elements \(a\) and \(b\) of a \(C^*\)-algebra are called orthogonal \((a\perp b)\) if \(a^*b=ab^*=0\). We say that vectors \(x\) and \(y\) in a Banach space \(X\) are semi-\(M\)-orthogonal \((x\perp_{SM}y)\) if \(\|x\pm y\|\geq \max\{\|x\|,\|y\|\}\). We prove that every linear bijection \(T:A\to X\), where \(X\) is a Banach space, \(A\) is either a von Neumann algebra or a compact \(C^*\)-algebra, and \(T(a)\perp_{SM}T(b)\) whenever \(a\perp b\), must be continuous. Consequently, every complete (semi-)\(M\)-norm on a von Neumann algebra or on a compact \(C^*\)-algebra is automatically continuous.

MSC:
46L05 General theory of \(C^*\)-algebras
47B48 Linear operators on Banach algebras
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[1] Alexander, J., Compact Banach algebras, Proc. London math. soc., 18, 1-18, (1968) · Zbl 0184.16502
[2] Amir, D., Characterizations of inner product spaces, (1986), Birkhäuser Verlag Basel · Zbl 0617.46030
[3] Arendt, W., Spectral properties of Lamperti operators, Indiana univ. math. J., 32, 2, 199-215, (1983) · Zbl 0488.47016
[4] Aronszajn, N.; Panitchpakdi, P., Extensions of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. math., 6, 405-439, (1956) · Zbl 0074.17802
[5] Bikchentaev, A.M., On a lemma of Berezin, Math. notes, 210, 787-791, (2010) · Zbl 1273.47033
[6] Birkhoff, G., Orthogonality in linear metric spaces, Duke math. J., 1, 169-172, (1935) · Zbl 0012.30604
[7] Blanco, A.; Turnšek, A., On maps that preserve orthogonality in normed spaces, Proc. roy. soc. Edinburgh sect. A, 136, 4, 709-716, (2006) · Zbl 1115.46016
[8] Burgos, M.; Fernández-Polo, F.J.; Garcés, J.J.; Martínez Moreno, J.; Peralta, A.M., Orthogonality preservers in C^⁎-algebras, JB⁎-algebras and JB⁎-triples, J. math. anal. appl., 348, 220-233, (2008) · Zbl 1156.46045
[9] Burgos, M.; Garcés, J.; Peralta, A.M., Automatic continuity of biorthogonality preservers between compact C^⁎-algebras and von Neumann algebras, J. math. anal. appl., 376, 221-230, (2011) · Zbl 1216.47068
[10] Cuntz, J., On the continuity of semi-norms on operator algebras, Math. ann., 220, 171-183, (1976) · Zbl 0306.46071
[11] Dales, G., Banach algebras and automatic continuity, (2000), Oxford University Press · Zbl 0981.46043
[12] Dixmier, J., Von Neumann algebras, (1981), North-Holland Amsterdam
[13] Font, J.J.; Hernandez, S., On separating maps between locally compact spaces, Arch. math. (basel), 63, 158-165, (1994) · Zbl 0805.46049
[14] Goldstein, S.; Paszkiewicz, A., Linear combinations of projections in von Neumann algebras, Proc. amer. math. soc., 116, 175-183, (1992) · Zbl 0768.47017
[15] Hewitt, E.; Ross, K.A., Abstract harmonic analysis I, (1963), Springer-Verlag New York, Heidelberg, Berlin · Zbl 0115.10603
[16] James, R.C., Orthogonality and linear functionals in normed linear spaces, Trans. amer. math. soc., 61, 265-292, (1947) · Zbl 0037.08001
[17] Jarosz, K., Automatic continuity of separating linear isomorphisms, Canad. math. bull., 33, 139-144, (1990) · Zbl 0714.46040
[18] Kaplansky, I., Normed algebras, Duke math. J., 16, 399-418, (1949) · Zbl 0033.18701
[19] Koldobsky, A., Operators preserving orthogonality are isometries, Proc. roy. soc. Edinburgh sect. A, 123, 5, 835-837, (1993) · Zbl 0806.46013
[20] Marcoux, L., Sums of small number of commutators, J. operator theory, 56, 111-142, (2006) · Zbl 1111.46033
[21] Martin, M.; Pasnicu, C., Some comparability results in inductive limit C^⁎-algebras, J. operator theory, 30, 137-147, (1993) · Zbl 0816.46053
[22] Pearcy, C.; Topping, D., Sums of small numbers of idempotents, Michigan math. J., 14, 453-465, (1967) · Zbl 0156.38102
[23] Ringrose, J., Linear functionals on operator algebras and their abelian subalgebras, J. London math. soc., 7, 553-560, (1974) · Zbl 0272.46036
[24] Sakai, S., \(C^\ast\)-algebras and \(W^\ast\)-algebras, Ergeb. math. grenzgeb., vol. 60, (1971), Springer-Verlag New York, Heidelberg, Berlin · Zbl 0219.46042
[25] Takesaki, M., Theory of operator algebras 1, (2003), Springer-Verlag New York, Heidelberg, Berlin · Zbl 0990.46034
[26] Wolff, M., Disjointness preserving operators in C^⁎-algebras, Arch. math., 62, 248-253, (1994) · Zbl 0803.46069
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