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| In [[C*-algebra]]s, the '''multiplier algebra''', denoted by ''M''(''A''), of a C*-algebra ''A'' is a unital C*-algebra which is the largest unital C*-algebra that contains ''A'' as an ideal in a "non-degenerate" way. It is the noncommutative generalization of [[Stone–Čech compactification]]. Multiplier algebras were introduced by {{harvtxt|Busby|1968}}.
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| For example, if ''A'' is the C*-algebra of [[compact operator on Hilbert space|compact operators on a separable Hilbert space]], ''M''(''A'') is ''B''(''H''), the C*-algebra of all [[bounded operator]]s on ''H''.
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| == Definition ==
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| An ideal ''I'' in a C*-algebra ''B'' is said to be '''essential''' if ''I'' ∩ ''J'' is non-trivial for all ideal ''J''. An ideal ''I'' is essential if and only if ''I''<sup>⊥</sup>, the "orthogonal complement" of ''I'' in the [[Hilbert C*-module]] ''B'' is {0}.
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| Let ''A'' be a C*-algebra. Its multiplier algebra ''M''(''A'') is the C*-algebra satisfying the following [[universal property]]: for all C*-algebra ''D'' containing ''A'' as an ideal, there exists a unique *-homomorphism φ ''D'' → ''M''(''A'') such that ''φ'' extends the identity homomorphism on ''A'' and ''φ''(''A''<sup>⊥</sup>) = {0}.
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| Uniqueness up to isomorphism is specified by the universal property. When ''A'' is unital, ''M''(''A'') = ''A''. It also follows from the definition that for any ''D'' containing ''A'' as an essential ideal, the multiplier algebra ''M''(''A'') contains ''D'' as a C*-subalgebra.
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| The existence of ''M''(''A'') can be shown in several ways.
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| A '''double centralizer''' of a C*-algebra ''A'' is a pair (''L'', ''R'') of bounded linear maps on ''A'' such that ''aL''(''b'') = ''R''(''a'')''b'' for all ''a'' and ''b'' in ''A''. This implies that ||''L''|| = ||''R''||. The set of double centralizers of ''A'' can be given a C*-algebra structure. This C*-algebra contains ''A'' as an essential ideal and can be identified as the multiplier algebra ''M''(''A''). For instance, if ''A'' is the compact operators ''K''(''H'') on a separable Hilbert space, then each ''x'' ∈ ''B''(''H'') defines a double centralizer of ''A'' by simply multiplication from the left and right.
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| Alternatively, ''M''(''A'') can be obtained via representations. The following fact will be needed:
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| '''Lemma.''' If ''I'' is an ideal in a C*-algebra ''B'', then any faithful nondegenerate representation ''π'' of ''I'' can be extended ''uniquely'' to ''B''.
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| Now take any faithful nondegenerate representation ''π'' of ''A'' on a Hilbert space ''H''. The above lemma, together with the universal property of the multiplier algebra, yields that ''M''(''A'') is isomorphic to the [[idealizer]] of ''π''(''A'') in ''B''(''H''). It is immediate that ''M''(''K''(''H'')) = ''B''(''H'').
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| Lastly, let ''E'' be a Hilbert C*-module and ''B''(''E'') (resp. ''K''(''E'')) be the adjointable (resp. compact) operators on ''E'' ''M''(''A'') can be identified via a *-homomorphism of ''A'' into ''B''(''E''). Something similar to the above lemma is true:
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| '''Lemma.''' If ''I'' is an ideal in a C*-algebra ''B'', then any faithful nondegenerate *-homomorphism ''π'' of ''I'' into ''B''(''E'')can be extended ''uniquely'' to ''B''.
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| Consequently, if ''π'' is a faithful nondegenerate *-homomorphism of ''π'' into ''B''(''E''), then ''M''(''A'') is isomorphic to the idealizer of ''π''(''A''). For instance, ''M''(''K''(''E'')) = ''B''(''E'') for any Hilbert module ''E''.
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| The C*-algebra ''A'' is isomorphic to the compact operators on the Hilbert module ''A''. Therefore ''M''(''A'') is the adjointable operators on ''A''.
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| == Strict topology ==
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| Consider the topology on ''M''(''A'') specified by the [[seminorm]]s {''l<sub>a</sub>'', ''r<sub>a</sub>''}<sub>''a'' ∈ ''A''</sub>, where
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| :<math>l_a (x) = \|ax\|, \; r_a(x) = \| xa \|.</math>
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| The resulting topology is called the '''strict topology''' on ''M''(''A''). ''A'' is strictly dense in ''M''(''A'') .
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| When ''A'' is unital, ''M''(''A'') = ''A'', and the strict topology coincides with the norm topology. For ''B''(''H'') = ''M''(''K''(''H'')), the strict topology is the [[Topologies on the set of operators on a Hilbert space|σ-strong* topology]]. It follows from above that ''B''(''H'') is complete in the σ-strong* topology.
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| == Commutative case ==
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| Let ''X'' be a [[locally compact]] [[Hausdorff space]], ''A'' = ''C''<sub>0</sub>(''X''), the commutative C*-algebra of continuous functions with compact support on ''X''. Then ''M''(''A'') is ''C''<sub>''b''</sub>(''X''), the continuous bounded functions on ''X''. By the [[Gelfand-Naimark theorem]], one has the isomorphism of C*-algebras
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| :<math>C_b(X) \simeq C(Y)</math> | |
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| where ''Y'' is the [[spectrum of a C*-algebra|spectrum]] of ''C''<sub>''b''</sub>(''X''). ''Y'' is in fact homeomorphic to the [[Stone–Čech compactification]] of ''X''.
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| ==Corona algebra==
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| The '''corona''' or '''corona algebra''' of ''A'' is the quotient ''M''(''A'')/''A''.
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| For example, the corona algebra of the algebra of compact operators on a Hilbert space is the [[Calkin algebra]].
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| The corona algebra is a non-commutative analogue of the [[corona set]] of a topological space.
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| ==References==
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| *B. Blackadar, ''K-Theory for Operator Algebras'', MSRI Publications, 1986.
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| *{{Citation | last1=Busby | first1=Robert C. | title=Double centralizers and extensions of C*-algebras | jstor=1994883 | mr=0225175 | year=1968 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=132 | pages=79–99}}
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| *{{eom|id=m/m130260|title=Multipliers of C*-algebras|first=Gert K.|last= Pedersen}}
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| [[Category:C*-algebras|*]]
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