Isometries of absolute order unit spaces. (English) Zbl 1476.46029
Summary: We prove that a unital, bijective linear map between absolute order unit spaces is an isometry if and only if it is absolute value preserving. We deduce that, on (unital) JB-algebras, such maps are precisely Jordan isomorphisms. Next, we introduce the notions of absolutely matrix ordered spaces and absolute matrix order unit spaces and prove that a unital, bijective \(*\)-linear map between absolute matrix order unit spaces is a complete isometry if, and only if, it is completely absolute value preserving. We obtain that on (unital) C*-algebras such maps are precisely C*-algebra isomorphisms.
MSC:
| 46B40 | Ordered normed spaces |
| 46L05 | General theory of \(C^*\)-algebras |
| 46L30 | States of selfadjoint operator algebras |
Keywords:
absolutely ordered space; absolute oder unit space; isometry; absolute value preserving maps; absolute matrix order unit spaceReferences:
| [1] | Alfsen, EM, Compact Convex Sets and Boundary Integrals (1971), Heidelberg: Springer, Heidelberg · Zbl 0209.42601 |
| [2] | Blecher, D.P., Hay, D.M.: Complete isometries into \(\text{C}^*\)-algebras, ArXiv Priprint (2002), https://arxiv.org/abs/math/0203182v1. (The main result is featured in the book “Isometries on Banach spaces: function spaces” by R. J. Fleming and J. E. Jamison, Chapman and Hall/CRC, Published September 5, 2019, 208 Pages) |
| [3] | Blecher, DP; Labuschagne, LE, Logmodularity and isometries of operator algebras, Trans. Am. Math. Soc., 355, 1621-1646 (2002) · Zbl 1026.46046 · doi:10.1090/S0002-9947-02-03195-1 |
| [4] | Choi, MD; Effros, EG, Injectivity and operator spaces, J. Funct. Anal., 24, 156-209 (1977) · Zbl 0341.46049 · doi:10.1016/0022-1236(77)90052-0 |
| [5] | Chu, C-H; Wong, N-C, Isometries between \(\text{ C }^*\)-algebras, Rev. Mat. Iberoamericana, 20, 156-209 (2004) |
| [6] | Gelfand, IM; Naimark, MA, On the embedding of normed rings into the ring of operators in Hilbert space, Mat. Sb., 12, 87-105 (1943) |
| [7] | Gardener, T., Linear maps of \(\text{ C }^*\)-algebras preserving the absolute value, Proc. Am. Math. Soc., 76, 271-278 (1979) · Zbl 0425.46042 |
| [8] | Jana, NK; Karn, AK; Peralta, AM, Contractive linear preservers of absolutely compatible pairs between \(\text{ C }^*\)-algebras, RACSAM, 113, 3, 2731-2744 (2019) · Zbl 1434.46033 · doi:10.1007/s13398-019-00653-0 |
| [9] | Jana, N.K., Karn, A.K., Peralta, A.M.: Absolutely compatible pairs in a von Neumann algebra, (Communicated for publication). (arxiv:1801.01216) · Zbl 1443.46035 |
| [10] | Kadison, RV, Order properties of self-adjoint operators, Trans. Am. Math. Soc, 2, 505-510 (1951) · Zbl 0043.11501 |
| [11] | Kadison, RV, Isometries of operator algebras, Ann. Math., 54, 325-338 (1951) · Zbl 0045.06201 · doi:10.2307/1969534 |
| [12] | Kakutani, S., Concrete representation of abstract \((M)\)-spaces, Ann. Math., 42, 994-1024 (1941) · Zbl 0060.26604 · doi:10.2307/1968778 |
| [13] | Karn, AK, Orthogonality in \(l_p\)-spaces and its bearing on ordered Banach spaces, Positivity, 18, 2, 223-234 (2014) · Zbl 1314.46026 · doi:10.1007/s11117-013-0242-8 |
| [14] | Karn, AK, A p-theory of ordered normed spaces, Positivity, 14, 441-458 (2010) · Zbl 1225.46014 · doi:10.1007/s11117-009-0029-0 |
| [15] | Karn, AK, Orthogonality in \(\text{ C }^*\)-algebras, Positivity, 20, 3, 607-620 (2016) · Zbl 1361.46017 · doi:10.1007/s11117-015-0375-z |
| [16] | Karn, AK, Algebraic orthogonality and commuting projections in operator algebras, Acta Sci. Math. (Szeged), 84, 323-353 (2018) · Zbl 1413.46023 · doi:10.14232/actasm-017-574-3 |
| [17] | Kaup, W., A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces, Math. Z., 138, 503-529 (1983) · Zbl 0519.32024 · doi:10.1007/BF01173928 |
| [18] | Maitland Wright, JD; Youngson, MA, On isometries of Jordan algebras, J. Lond. Math. Soc. (2), 17, 339-344 (1978) · Zbl 0384.46041 · doi:10.1112/jlms/s2-17.2.339 |
| [19] | Pedersen, GK, \( \text{ C }^*\)-algebras and their Automorphism Groups (1979), London: Academic Press, London · Zbl 0416.46043 |
| [20] | Radjabalipour, M.; Siddighi, K.; Taghavi, Y., Additive mappings on operator algebras preserving absolute value, Linear Algebra Appl., 327, 197-206 (2001) · Zbl 0978.15003 · doi:10.1016/S0024-3795(00)00338-4 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
