Double centralizers and extensions of 𝐶*-algebras
Robert C. Busby · Transactions of the American Mathematical Society · 1968
Introduction.If A, B, and C are associative algebras over a field such that A is a two-sided ideal in B and B/A is isomorphic with C, then we may speak of B as an extension of A by C. It is of interest to be able to find all such extensions (with certain ones identified by a suitable equivalence relation) given A and C.This problem was investigated and essentially solved by G. Hochschild in one of his early papers on cohomology of algebras (see [5]).In this paper we investigate the analogous theory for C*-algebras.An effective solution to this problem should prove useful in the classification of such algebras.One of the tools used by Hochschild in his investigations was what he called the algebra of multiplications, M(A), of an associative algebra A. If A is C*-algebra, then M(A) is exactly the double centralizer algebra of A, a concept developed for semigroups and algebras by B. E. Johnson (see [6]).This concept seems to be very useful in analysis, and is related to the left centralizer concept which J. G. Wendel used to investigate group algebras.The commutative version of this idea was first introduced by S. Helgason under the name of the algebra of multipliers (see [4]), and was recently developed by J. Wang (see [9]) and others.The connection with the extension problem and the work of Hochschild has apparently not been made by those using the centralizer concept.The definitions of double centralizer given in this paper are Johnson's.The proof that M(A) is an involutive algebra is also due to him, as is most of the proof of Propositions 2.5, 3.1.We show that M(A) is a C*-algebra if A is, and the abstract proof presented is new, however the fact can be deduced by using a result of G. A. Reid (see [7, Proposition 3, p. 1021]).We then use what we call the strict topology to investigate M(A) for certain particular A. This topology is a generalization of a topology with the same name which was defined in the commutative case by R. C. Buck, and the resulting proof of 3.9 is new, although the result was proved by Johnson.The main result of the paper is that the classes of extensions of a C*-algebra A by a C*-algebra C are in one to one correspondence with the *-homomorphisms from C to the quotient algebra M(A)/A.The author would like to thank Professor Peter Freyd for the suggestion of using pullback arguments in this section ( §3).This eliminated several pages of long calculations.The remaining sections of this paper are devoted to the description of extensions and their primitive ideal spaces by the use of the main result.This paper formed the basis for the author's doctoral