Geodesic
A Note on Isomorphisms of Multiplier Algebras
Canadian mathematical bulletin, Tome 22 (1979) no. 2, pp. 243-245

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Let A 1, A 2 be commutative semi-simple Banach algebras and M(A 1), M(A 2) their multiplier algebras. Birtel in [2] has proved that every isomorphism of A 1 onto A 2 induces an isomorphism of M(A 1) onto M(A 2). In this note, we extend this result to the noncommutative case. We also show that if A is a dual A*-algebra which is a dense two-sided ideal of a B*-algebra B, then M(A) is isomorphic to M(B). Thus the converse of the previous result cannot hold. All algebras under consideration are over the complex field. 
Wong, Pak-Ken. A Note on Isomorphisms of Multiplier Algebras. Canadian mathematical bulletin, Tome 22 (1979) no. 2, pp. 243-245. doi: 10.4153/CMB-1979-032-4
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     title = {A {Note} on {Isomorphisms} of {Multiplier} {Algebras}},
     journal = {Canadian mathematical bulletin},
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     year = {1979},
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     number = {2},
     doi = {10.4153/CMB-1979-032-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1979-032-4/}
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