# Multiplier algebra > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Multiplier_algebra > Markdown URL: https://mediated.wiki/source/Multiplier_algebra.md > Source: https://en.wikipedia.org/wiki/Multiplier_algebra > Source revision: 1312826243 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) In [mathematics](/source/Mathematics), the **multiplier algebra**, denoted by *M*(*A*), of a [C*-algebra](/source/C*-algebra) *A* is a unital C*-algebra that is the largest unital C*-algebra that contains *A* as an [ideal](/source/Ideal_(ring_theory)) in a "non-degenerate" way. It is the [noncommutative](/source/Noncommutative_topology) generalization of [Stone–Čech compactification](/source/Stone%E2%80%93%C4%8Cech_compactification). Multiplier algebras were introduced by Busby (1968). For example, if *A* is the C*-algebra of [compact operators on a separable Hilbert space](/source/Compact_operator_on_Hilbert_space), *M*(*A*) is *B*(*H*), the C*-algebra of all [bounded operators](/source/Bounded_operator) on *H*. ## Definition An ideal *I* in a C*-algebra *B* is said to be **essential** if *I* ∩ *J* is non-trivial for every ideal *J*. An ideal *I* is essential if and only if *I*⊥, the "[orthogonal complement](/source/Orthogonal_complement)" of *I* in the [Hilbert C*-module](/source/Hilbert_C*-module) *B* is {0}. Let *A* be a C*-algebra. Its multiplier algebra *M*(*A*) is any C*-algebra satisfying the following [universal property](/source/Universal_property): for any 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*⊥) = {0}. Uniqueness up to [isomorphism](/source/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. The existence of *M*(*A*) can be shown in several ways. 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. Alternatively, *M*(*A*) can be obtained via representations. The following fact will be needed: **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*. 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](/source/Idealizer) of *π*(*A*) in *B*(*H*). It is immediate that *M*(*K*(*H*)) = *B*(*H*). 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: **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*. Consequently, if *π* is a faithful nondegenerate *-homomorphism of *A* 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*. The C*-algebra *A* is isomorphic to the compact operators on the Hilbert module *A*. Therefore, *M*(*A*) is the adjointable operators on *A*. ## Strict topology Consider the topology on *M*(*A*) specified by the [seminorms](/source/Seminorm) {*la*, *ra*}*a* ∈ *A*, where - l_a (x) = \|ax\|, \; r_a(x) = \| xa \|. The resulting topology is called the **strict topology** on *M*(*A*). *A* is strictly dense in *M*(*A*) . When *A* is unital, *M*(*A*) = *A*, and the strict topology coincides with the [norm topology](/source/Operator_norm). For *B*(*H*) = *M*(*K*(*H*)), the strict topology is the [σ-strong* topology](/source/Topologies_on_the_set_of_operators_on_a_Hilbert_space). It follows from above that *B*(*H*) is complete in the σ-strong* topology. ## Commutative case Let *X* be a [locally compact](/source/Locally_compact) [Hausdorff space](/source/Hausdorff_space), *A* = *C*0(*X*), the commutative C*-algebra of continuous functions that [vanish at infinity](/source/Vanish_at_infinity). Then *M*(*A*) is *C**b*(*X*), the continuous bounded functions on *X*. By the [Gelfand–Naimark theorem](/source/Gelfand%E2%80%93Naimark_theorem), one has the isomorphism of C*-algebras - C_b(X) \simeq C(Y) where *Y* is the [spectrum](/source/Spectrum_of_a_C*-algebra) of *C**b*(*X*). *Y* is in fact homeomorphic to the [Stone–Čech compactification](/source/Stone%E2%80%93%C4%8Cech_compactification) *βX* of *X*. ## Corona algebra The **corona** or **corona algebra** of *A* is the quotient *M*(*A*)/*A*. For example, the corona algebra of the algebra of compact operators on a Hilbert space is the [Calkin algebra](/source/Calkin_algebra). The corona algebra is a noncommutative analogue of the [corona set](/source/Corona_set) of a topological space. ## References - B. Blackadar, *K-Theory for Operator Algebras*, MSRI Publications, 1986. - Busby, Robert C. (1968), ["Double centralizers and extensions of C*-algebras"](https://web.archive.org/web/20200220172805/http://pdfs.semanticscholar.org/6fc8/e18b8f80e808b12cf2a446024d36dcb30555.pdf), *[Transactions of the American Mathematical Society](/source/Transactions_of_the_American_Mathematical_Society)*. **132** (1): 79–99, [doi:10.2307/1994883](https://doi.org/10.2307/1994883). [ISSN 0002-9947](https://www.worldcat.org/issn/0002-9947). [JSTOR 1994883](https://www.jstor.org/stable/1994883). MR 0225175. [S2CID 54047557](https://api.semanticscholar.org/CorpusID:54047557), archived from [the original](http://pdfs.semanticscholar.org/6fc8/e18b8f80e808b12cf2a446024d36dcb30555.pdf) on 2020-02-20 --- Adapted from the Wikipedia article [Multiplier algebra](https://en.wikipedia.org/wiki/Multiplier_algebra) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Multiplier_algebra?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.