Geodesic
Vector space isomorphisms of non-unital reduced Banach *-algebras
Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 69 (2015) no. 2.

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Let A and B be two non-unital reduced Banach *-algebras and φ: A → B be a vector space isomorphism. The two following statement holds: If φ is a *-isomorphism, then φ is isometric (with respect to the C*-norms), bipositive and φ maps some approximate identity of A onto an approximate identity of B. Conversely, any two of the later three properties imply that φ is a *-isomorphism. Finally, we show that a unital and self-adjoint spectral isometry between semi-simple Hermitian Banach algebras is an *-isomorphism.
Keywords: Reduced Banach algebras, preserving the spectrum. 
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ElHarti, Rachid; Mabrouk, Mohamed. Vector space isomorphisms of non-unital reduced Banach *-algebras. Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 69 (2015) no. 2. http://geodesic.mathdoc.fr/item/AUM_2015_69_2_a6/

[1] Aupetit, B., Spectrum-preserving linear mappings between Banach algebras or Jordan-Banach algebras, J. Lond. Math. Soc. 62 (2000), 917-924.

[2] Bonsall, F. F., Stirling, D. S. G., Square roots in Banach *-algebras, Glasg. Math. J. 13 (1972), 74-74.

[3] Dixon, P. G., Approximate identities in normed algebras, Proc. Lond. Math. Soc. 26 (3) (1973), 485-496.

[4] Doran R. S., Belfi, V. A., Characterizations of C*-algebras. The Gelfand-Naimark Theorems, Marcel Dekker, New York, 1986.

[5] Ford, J. W. M., A square root lemma for Banach (*)-algebras, J. Lond. Math. Soc. 42 (1) (1967), 521-522.

[6] Kadisson, R. V., Isometries of operator algebras, Ann. of Math. 54 (2) (1951), 325-338.

[7] Martin, M., Towards a non-selfadjoint version of Kadison’s theorem, Ann. Math. Inform. 32 (2005), 87-94.

[8] Palmer, T. W., Banach Algebras and the General Theory of *-Algebras. *-Algebras, Vol. II, Cambridge University Press, Cambridge, 2001.

[9] Sakai, S., C*-algebras and W*-algebras, Springer-Verlag, New York-Berlin, 1971.

[10] Ylinen, K., Vector space isomorphisms of C*-algebras, Studia Math. 46 (1973), 31-34.