Form a commutative algebra M(A) , called the multiplier algebra of A . M(A) is complete under the strong operator topology. A can be algebraically embedded in M(A) as an ideal.nWhen A is semisimple, M(A) is also semisimple; and the maximal ideal space of A can be embedded in that of M(A) .nIf A is a supremum norm algebra, so is also M(A) . In this case there are three natural topologies for M(A) , namely, the norm topology a , the strong operator topology P and the compact-open topology K each being stronger than the following one. cr and, P are equivalent if and only if A has a compact Silov boundary. Under suitable assumptions, B and K are equivalent if and only if every countable union of compact subsets of the Silov boundary of A has a compact closure.
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