A dilation theorem for ⋅-preserving maps of 𝒞*-algebras

Gardner, L. Terrell

doi:10.1090/S0002-9939-1979-0518516-5

A dilation theorem for $\cdot$-preserving maps of $\mathcal {C}^*$-algebras
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by L. Terrell Gardner PDF

Proc. Amer. Math. Soc. 73 (1979), 341-345 Request permission

Abstract:

A linear map of a (unital) ${\mathcal {C}^\ast }$-algebra into a ${\mathcal {C}^\ast }$-algebra, which preserves the absolute value, is analyzed as the composition of a (unital) $^\ast$-homomorphism into a usually larger target algebra, followed by a suitable multiplication $x \mapsto xb = {b^{1/2}}x{b^{1/2}}$.

References

Jacques Dixmier, Les algèbres d’opérateurs dans l’espace hilbertien (Algèbres de von Neumann), Cahiers Scientifiques, Fasc. XXV, Gauthier-Villars, Paris, 1957 (French). MR 0094722
Irving Kaplansky, A theorem on rings of operators, Pacific J. Math. 1 (1951), 227–232. MR 50181, DOI 10.2140/pjm.1951.1.227

The second adjoint of a

algebra

Jan A. van Casteren, A characterization of $C^{\ast }$-subalgebras, Proc. Amer. Math. Soc. 72 (1978), no. 1, 54–56. MR 503530, DOI 10.1090/S0002-9939-1978-0503530-5
Richard V. Kadison, A generalized Schwarz inequality and algebraic invariants for operator algebras, Ann. of Math. (2) 56 (1952), 494–503. MR 51442, DOI 10.2307/1969657