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A [[C*-algebra|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m1302602.png" />-algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m1302603.png" /> of operators on some [[Hilbert space|Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m1302604.png" /> may be viewed as a non-commutative generalization of a function algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m1302605.png" /> acting as multiplication operators on some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m1302606.png" />-space associated with a measure on the locally compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m1302607.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m1302608.png" /> being compact corresponds naturally to the case where the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m1302609.png" /> is unital. In the non-unital case any embedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026010.png" /> as an essential ideal in some larger unital <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026011.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026012.png" /> (i.e., the annihilator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026013.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026014.png" /> is zero) can be viewed as an analogue of a compactification of the locally compact Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026015.png" />. Thus, the one-point compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026016.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026017.png" /> corresponds to the unitization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026018.png" /> of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026019.png" />. The analogue of the maximal compactification — the [[Stone–Čech compactification|Stone–Čech compactification]] — is the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026020.png" /> of multipliers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026021.png" />, defined by R.C. Busby in 1967 [[#References|[a4]]] and studied in more detail in [[#References|[a2]]]. It is defined simply as the idealizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026022.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026023.png" /> (assuming that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026024.png" /> or, equivalently, that no non-zero vector in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026025.png" /> is annihilated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026026.png" />).
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Linear operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026028.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026029.png" /> are called left and right centralizers if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026031.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026033.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026034.png" />. They are automatically bounded. A double centralizer is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026035.png" /> of left, right centralizers such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026036.png" /> (whence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026037.png" />), and the closed linear spaces of double centralizers becomes a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026038.png" />-algebra when product and involution are defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026040.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026041.png" />). As shown by B.E. Johnson, [[#References|[a8]]], there is an isomorphism between the abstractly defined <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026042.png" />-algebra of double centralizers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026043.png" /> and the concrete <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026044.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026045.png" />. This, in particular, shows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026046.png" /> is independent of the given representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026047.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026048.png" />.
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The strict topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026049.png" /> is defined by the semi-norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026050.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026051.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026052.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026053.png" />, [[#References|[a4]]]. It is used as an analogue of [[Uniform convergence|uniform convergence]] on compact subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026054.png" /> in function algebras. Thus, it can be shown that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026055.png" /> is the strict completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026056.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026057.png" /> and that the strict dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026058.png" /> equals the norm dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026059.png" />, [[#References|[a16]]].
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A [[C*-algebra|$C ^ { * }$-algebra]] $A$ of operators on some [[Hilbert space|Hilbert space]] $\mathcal{H}$ may be viewed as a non-commutative generalization of a function algebra $C _ { 0 } ( \Omega )$ acting as multiplication operators on some $L^{2}$-space associated with a measure on the locally compact space $\Omega$. The space $\Omega$ being compact corresponds naturally to the case where the algebra $A$ is unital. In the non-unital case any embedding of $A$ as an essential ideal in some larger unital $C ^ { * }$-algebra $B$ (i.e., the annihilator of $A$ in $B$ is zero) can be viewed as an analogue of a compactification of the locally compact Hausdorff space $\Omega$. Thus, the one-point compactification $\Omega \cup \{ \infty \}$ of $\Omega$ corresponds to the unitization $\tilde { A } = A \oplus \mathbf{C}$ of the algebra $A$. The analogue of the maximal compactification — the [[Stone–Čech compactification|Stone–Čech compactification]] — is the algebra $M ( A )$ of multipliers of $A$, defined by R.C. Busby in 1967 [[#References|[a4]]] and studied in more detail in [[#References|[a2]]]. It is defined simply as the idealizer of $A$ in $B ( \mathcal{H} )$ (assuming that $A \mathcal{H} = \mathcal{H}$ or, equivalently, that no non-zero vector in $\mathcal{H}$ is annihilated by $A$).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026060.png" /> is the universal Hilbert space for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026061.png" /> (the orthogonal sum of all Hilbert spaces obtained from states of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026062.png" /> via the Gel'fand–Naimark–Segal construction), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026063.png" /> has a more constructive characterization: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026064.png" /> denote the space of self-adjoint operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026065.png" /> that can be obtained as limits (in the [[Strong topology|strong topology]]) of some increasing net of self-adjoint elements from the unitized algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026066.png" /> (cf. also [[Net (directed set)|Net (directed set)]]; [[Self-adjoint operator|Self-adjoint operator]]). Similarly, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026067.png" /> be the space of limits of decreasing nets. Then
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Linear operators $\lambda$ and $\rho$ on $A$ are called left and right centralizers if $\lambda ( x y ) = \lambda ( x ) y$ and $\rho ( x y ) = x \rho ( y )$ for all $x$, $y$ in $A$. They are automatically bounded. A double centralizer is a pair $( \lambda , \rho )$ of left, right centralizers such that $x \lambda ( y ) = \rho ( x ) y$ (whence $\| \lambda \| = \| \rho \|$), and the closed linear spaces of double centralizers becomes a $C ^ { * }$-algebra when product and involution are defined by $( \lambda _ { 1 } , \rho _ { 1 } ) ( \lambda _ { 2 } , \rho _ { 2 } ) = ( \lambda _ { 1 } \lambda _ { 2 } , \rho _ { 2 } \rho _ { 1 } )$ and $( \lambda , \rho ) ^ { * } = ( \rho ^ { * } , \lambda ^ { * } )$ (where $\lambda ^ { * } ( x ) = ( \lambda ( x ^ { * } ) ) ^ { * }$). As shown by B.E. Johnson, [[#References|[a8]]], there is an isomorphism between the abstractly defined $C ^ { * }$-algebra of double centralizers of $A$ and the concrete $C ^ { * }$-algebra $M ( A )$. This, in particular, shows that $M ( A )$ is independent of the given representation of $A$ on $\mathcal{H}$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026068.png" /></td> </tr></table>
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The strict topology on $M ( A )$ is defined by the semi-norms $x \rightarrow \| a x \| + \| a x \|$ on $B ( \mathcal{H} )$ with $a$ in $A$, [[#References|[a4]]]. It is used as an analogue of [[Uniform convergence|uniform convergence]] on compact subsets of $\Omega$ in function algebras. Thus, it can be shown that $M ( A )$ is the strict completion of $A$ in $B ( \mathcal{H} )$ and that the strict dual of $M ( A )$ equals the norm dual of $A$, [[#References|[a16]]].
  
Thus, for every self-adjoint multiplier <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026069.png" /> there are nets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026071.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026072.png" />, one increasing, the other decreasing, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026073.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026074.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026076.png" />-unital, i.e. contains a countable approximate unit, in particular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026077.png" /> is separable (cf. also [[Separable algebra|Separable algebra]]), these nets can be taken as sequences, [[#References|[a2]]], [[#References|[a12]]], p. 12. In the commutative case, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026078.png" />, whence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026079.png" />, this expresses the well-known fact that a bounded, real function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026080.png" /> is continuous precisely when it is both lower and upper semi-continuous.
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If $\mathcal{H}$ is the universal Hilbert space for $A$ (the orthogonal sum of all Hilbert spaces obtained from states of $A$ via the Gel'fand–Naimark–Segal construction), then $M ( A )$ has a more constructive characterization: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026064.png"/> denote the space of self-adjoint operators in $B ( \mathcal{H} )$ that can be obtained as limits (in the [[Strong topology|strong topology]]) of some increasing net of self-adjoint elements from the unitized algebra $\tilde{A}$ (cf. also [[Net (directed set)|Net (directed set)]]; [[Self-adjoint operator|Self-adjoint operator]]). Similarly, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026067.png"/> be the space of limits of decreasing nets. Then
  
For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026081.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026082.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026083.png" /> as an ideal there is a natural morphism (i.e. a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026085.png" />-homomorphism) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026086.png" />, defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026087.png" />, that extends the identity mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026088.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026089.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026090.png" /> is essential in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026091.png" />, one therefore obtains an embedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026092.png" />. Any morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026093.png" /> between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026094.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026095.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026096.png" /> extends uniquely to a strictly continuous morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026097.png" />, provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026098.png" /> is proper (i.e. maps an approximate unit for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026099.png" /> to one for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260100.png" />). Such morphisms are the analogues of proper continuous mappings between locally compact spaces. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260101.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260102.png" />-unital and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260103.png" /> is a quotient morphism, i.e. surjective, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260104.png" /> is also surjective. This result may be viewed as a non-commutative generalization of the Tietze extension theorem, [[#References|[a2]]], [[#References|[a13]]] (cf. also [[Extension theorems|Extension theorems]]).
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<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m13026068.png"/></td> </tr></table>
  
The corona of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260106.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260107.png" /> is defined as the quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260108.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260109.png" />, [[#References|[a13]]]. The commutative analogue is the compact Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260110.png" /> (the corona of the locally compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260111.png" />, [[#References|[a6]]]), but the pre-eminent example of such algebras is the Calkin algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260112.png" />, obtained by taking <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260113.png" /> as the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260114.png" /> of compact operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260115.png" /> (whence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260116.png" />). Corona <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260118.png" />-algebras are usually non-separable and cannot even be represented on separable Hilbert spaces, [[#References|[a14]]]. Nevertheless, they have important roles in the formulation of G. Kasparov's KK-theory and the later variation known as E-theory. The foremost application, however, is to the theory of extensions: An extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260121.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260122.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260123.png" /> is any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260124.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260125.png" /> that fits into a short exact sequence (cf. also [[Exact sequence|Exact sequence]])
+
Thus, for every self-adjoint multiplier $x$ there are nets $( a _\lambda )$ and $( b _ { \mu } )$ in $\widetilde { A } _ { s a }$, one increasing, the other decreasing, such that $a_{\lambda} \nearrow x \swarrow b _ { \mu }$. If $A$ is $\sigma$-unital, i.e. contains a countable approximate unit, in particular if $A$ is separable (cf. also [[Separable algebra|Separable algebra]]), these nets can be taken as sequences, [[#References|[a2]]], [[#References|[a12]]], p. 12. In the commutative case, where $A = C _ { 0 } ( \Omega )$, whence $M ( A ) = C _ { b } ( \Omega )$, this expresses the well-known fact that a bounded, real function on $\Omega$ is continuous precisely when it is both lower and upper semi-continuous.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260127.png" /></td> </tr></table>
+
For any $C ^ { * }$-algebra $X$ containing $A$ as an ideal there is a natural morphism (i.e. a $\square ^ { * }$-homomorphism) $\sigma : X \rightarrow M ( A )$, defined by $\sigma ( x ) a = x a$, that extends the identity mapping of $A \subset X$ onto $A \subset M ( A )$. If $A$ is essential in $X$, one therefore obtains an embedding $X \subset M ( A )$. Any morphism $\alpha : A \rightarrow B$ between $C ^ { * }$-algebras $A$ and $B$ extends uniquely to a strictly continuous morphism $\overline { \alpha } : M ( A ) \rightarrow M ( B )$, provided that $\alpha$ is proper (i.e. maps an approximate unit for $A$ to one for $B$). Such morphisms are the analogues of proper continuous mappings between locally compact spaces. If $A$ is $\sigma$-unital and $\alpha$ is a quotient morphism, i.e. surjective, then $\overline { \alpha }$ is also surjective. This result may be viewed as a non-commutative generalization of the Tietze extension theorem, [[#References|[a2]]], [[#References|[a13]]] (cf. also [[Extension theorems|Extension theorems]]).
  
Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260128.png" /> contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260129.png" /> as an ideal, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260130.png" /> is simply the quotient morphism. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260131.png" /> may be regarded as an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260132.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260133.png" />, and in fact a maximal such. Namely, any other extension will give rise to a commutative diagram
+
The corona of a $C ^ { * }$-algebra $A$ is defined as the quotient $C ^ { * }$-algebra $Q ( A ) = M ( A ) / A$, [[#References|[a13]]]. The commutative analogue is the compact Hausdorff space $\beta \  \Omega \  \backslash  \ \Omega$ (the corona of the locally compact space $\Omega$, [[#References|[a6]]]), but the pre-eminent example of such algebras is the Calkin algebra $B (\mathcal{H} ) / K ( \mathcal{H} )$, obtained by taking $A$ as the algebra $K ( \mathcal{H} )$ of compact operators on $\mathcal{H}$ (whence $M ( A ) = B ( \mathcal{H} )$). Corona $C ^ { * }$-algebras are usually non-separable and cannot even be represented on separable Hilbert spaces, [[#References|[a14]]]. Nevertheless, they have important roles in the formulation of G. Kasparov's KK-theory and the later variation known as E-theory. The foremost application, however, is to the theory of extensions: An extension of $C ^ { * }$-algebras $A$ and $B$ is any $C ^ { * }$-algebra $X$ that fits into a short exact sequence (cf. also [[Exact sequence|Exact sequence]])
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260134.png" /></td> </tr></table>
+
\begin{equation*} 0 \rightarrow A \rightarrow X \stackrel { \pi } { \rightarrow }  B \rightarrow 0. \end{equation*}
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260135.png" /> is the morphism defined above and the induced morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260136.png" /> is known as the Busby invariant for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260137.png" />. This invariant determines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260138.png" /> up to an obvious equivalence, because the right square in the diagram above describes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260139.png" /> as the pull-back of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260140.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260141.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260142.png" />, i.e.
+
Thus, $X$ contains $A$ as an ideal, and $\pi$ is simply the quotient morphism. In particular, $M ( A )$ may be regarded as an extension of $A$ by $Q ( A )$, and in fact a maximal such. Namely, any other extension will give rise to a commutative diagram
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260143.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260134.png"/></td> </tr></table>
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260144.png" /></td> </tr></table>
+
Here $\sigma : X \rightarrow M ( A )$ is the morphism defined above and the induced morphism $\tau : B \rightarrow Q ( A )$ is known as the Busby invariant for $X$. This invariant determines $X$ up to an obvious equivalence, because the right square in the diagram above describes $X$ as the pull-back of $B$ and $M ( A )$ over $Q ( A )$, i.e.
  
One therefore has the identification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260145.png" />, [[#References|[a4]]], [[#References|[a5]]], [[#References|[a15]]].
+
\begin{equation*} X = M ( A ) \bigoplus _ { Q ( A ) } B = \end{equation*}
  
For any quotient morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260146.png" /> between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260147.png" />-algebras one may ask whether an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260148.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260149.png" /> with specific properties is the image of some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260150.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260151.png" /> with the same properties. This is known as a lifting problem, and is the non-commutative analogue of extension problems for functions. Many lifting problems have positive (and easy) solutions: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260152.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260153.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260154.png" />, one can find counter-images in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260155.png" /> with the same properties. However, the properties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260156.png" /> (being idempotent) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260157.png" /> (being normal) are not liftable in general. It follows that the more general commutator relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260158.png" /> is not liftable either. But the orthogonality relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260159.png" /> is liftable (even in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260160.png" />-fold version <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260161.png" />). Using this one may show that the nilpotency relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260162.png" /> is liftable, [[#References|[a1]]], [[#References|[a11]]], [[#References|[a9]]].
+
\begin{equation*} = \{ ( m , b ) \in M ( A ) \bigoplus B : \pi ( m ) = \tau ( b ) \}. \end{equation*}
  
As advocated by T.A. Loring, lifting problems may with advantage be replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260163.png" />-algebra problems concerning projectivity. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260164.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260165.png" /> is projective if any morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260167.png" /> into a quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260168.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260169.png" /> can be factored as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260170.png" /> for some morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260171.png" />, [[#References|[a3]]]. This means that one is lifting a whole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260172.png" />-subalgebra and not just some elements. Projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260173.png" />-algebras are the non-commutative analogues of topological spaces that are absolute retracts, but since the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260174.png" />-algebras is vastly larger than the category of locally compact Hausdorff spaces, projectivity is a rare phenomenon. However, the cone over the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260175.png" />-matrices, i.e. the algebra
+
One therefore has the identification $\operatorname { Ext } ( A , B ) = \operatorname { Hom } ( B , Q ( A ) )$, [[#References|[a4]]], [[#References|[a5]]], [[#References|[a15]]].
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260176.png" /></td> </tr></table>
+
For any quotient morphism $\pi : X \rightarrow B$ between $C ^ { * }$-algebras one may ask whether an element $b$ in $B$ with specific properties is the image of some $x$ in $X$ with the same properties. This is known as a lifting problem, and is the non-commutative analogue of extension problems for functions. Many lifting problems have positive (and easy) solutions: If $b = b ^ { * }$ or $b \geq 0$ or $\| b \| \leq 1$, one can find counter-images in $X$ with the same properties. However, the properties $b ^ { 2 } = b$ (being idempotent) and $b ^ { * } b = b b ^ { * }$ (being normal) are not liftable in general. It follows that the more general commutator relation $b _ { 1 } b _ { 2 } = b _ { 2 } b _ { 1 }$ is not liftable either. But the orthogonality relation $b _ { 1 } b _ { 2 } = 0$ is liftable (even in the $n$-fold version $b _ { 1 } \ldots b _ { n } = 0$). Using this one may show that the nilpotency relation $b ^ { n } = 0$ is liftable, [[#References|[a1]]], [[#References|[a11]]], [[#References|[a9]]].
  
is always projective. This means that although matrix units cannot, in general, be lifted from quotients, there are lifts in the "smeared" form given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260177.png" />, [[#References|[a10]]], [[#References|[a9]]].
+
As advocated by T.A. Loring, lifting problems may with advantage be replaced by $C ^ { * }$-algebra problems concerning projectivity. A $C ^ { * }$-algebra $P$ is projective if any morphism $\alpha : P \rightarrow B$ into a quotient $C ^ { * }$-algebra $B = \pi ( X )$ can be factored as $\alpha = \pi \circ \overline { \alpha }$ for some morphism $\overline { \alpha } : P \rightarrow X$, [[#References|[a3]]]. This means that one is lifting a whole $C ^ { * }$-subalgebra and not just some elements. Projective $C ^ { * }$-algebras are the non-commutative analogues of topological spaces that are absolute retracts, but since the category of $C ^ { * }$-algebras is vastly larger than the category of locally compact Hausdorff spaces, projectivity is a rare phenomenon. However, the cone over the $n \times n$-matrices, i.e. the algebra
  
Corona <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260178.png" />-algebras form an indispensable tool for more complicated lifting problems, because by Busby's theory, mentioned above, it suffices to solve the lifting for quotient morphisms of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260179.png" />. Thus, one may utilize the special properties that corona algebras have. A brief outline of these follows.
+
\begin{equation*} \mathbf{C M} _ { n } = C _ { 0 } ( ]0,1 ] ) \otimes \mathbf{M} _ { n } \end{equation*}
 +
 
 +
is always projective. This means that although matrix units cannot, in general, be lifted from quotients, there are lifts in the "smeared" form given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260177.png"/>, [[#References|[a10]]], [[#References|[a9]]].
 +
 
 +
Corona $C ^ { * }$-algebras form an indispensable tool for more complicated lifting problems, because by Busby's theory, mentioned above, it suffices to solve the lifting for quotient morphisms of the form $\pi : M ( A ) \rightarrow Q ( A )$. Thus, one may utilize the special properties that corona algebras have. A brief outline of these follows.
  
 
==Corona algebras.==
 
==Corona algebras.==
In topology, a compact [[Hausdorff space|Hausdorff space]] is called sub-Stonean if any two disjoint, open, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260180.png" />-compact sets have disjoint closures. Exotic as this may sound, it is a property that any corona set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260181.png" /> will have, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260182.png" /> is locally compact and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260183.png" />-compact. In such a space, every open, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260184.png" />-compact subset is also regularly embedded, i.e. it equals the interior of its closure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260185.png" />, [[#References|[a6]]]. The non-commutative generalization of this is the fact that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260186.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260187.png" />-unital <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260188.png" />-algebra, then every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260189.png" />-unital hereditary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260190.png" />-subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260191.png" /> of its corona algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260192.png" /> equals its double annihilator, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260193.png" />, [[#References|[a13]]]. The analogue of the sub–Stonean property, sometimes called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260195.png" />-condition, is even more striking: For any two orthogonal elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260196.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260197.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260198.png" /> (say <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260199.png" />) there is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260200.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260201.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260202.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260203.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260204.png" />. Even better, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260205.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260206.png" /> are separable subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260207.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260208.png" /> commutes with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260209.png" /> and annihilates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260210.png" />, then the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260211.png" /> can be chosen with the same properties, [[#References|[a11]]], [[#References|[a14]]]. Note that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260212.png" /> could be taken as a projection, e.g. the range projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260213.png" />, this would be a familiar property in [[Von Neumann algebra|von Neumann algebra]] theory. The fact that corona algebras will never be von Neumann algebras (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260214.png" /> is non-unital and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260215.png" />-unital) indicates that the property (first established by G. Kasparov as a "technical lemma" ) is useful. Actually, a potentially stronger version is true: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260216.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260217.png" /> are monotone sequences of self-adjoint elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260218.png" />, one increasing, the other decreasing, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260219.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260220.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260221.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260222.png" /> are separable subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260223.png" />, such that all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260224.png" /> commute with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260225.png" /> and annihilate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260226.png" />, then there is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260227.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260228.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260229.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260230.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260231.png" /> commutes with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260232.png" /> and annihilates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260233.png" />, [[#References|[a11]]]. This has as a consequence that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260234.png" /> is any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260235.png" />-unital <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260236.png" />-subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260237.png" />, commuting with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260238.png" /> and annihilating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260239.png" />, as above, then for any multiplier <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260240.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260241.png" /> there is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260242.png" /> in the idealizer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260243.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260244.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260245.png" />, still commuting with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260246.png" /> and annihilating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260247.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260248.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260249.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260250.png" />, [[#References|[a5]]], [[#References|[a15]]]. In other words, the natural morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260251.png" /> (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260252.png" />) is surjective. This indicates the size of corona algebras, even compared with large multiplier algebras.
+
In topology, a compact [[Hausdorff space|Hausdorff space]] is called sub-Stonean if any two disjoint, open, $\sigma$-compact sets have disjoint closures. Exotic as this may sound, it is a property that any corona set $\beta \  \Omega \  \backslash  \ \Omega$ will have, if $\Omega$ is locally compact and $\sigma$-compact. In such a space, every open, $\sigma$-compact subset is also regularly embedded, i.e. it equals the interior of its closure in $\beta \  \Omega \  \backslash  \ \Omega$, [[#References|[a6]]]. The non-commutative generalization of this is the fact that if $A$ is a $\sigma$-unital $C ^ { * }$-algebra, then every $\sigma$-unital hereditary $C ^ { * }$-subalgebra $B$ of its corona algebra $Q ( A )$ equals its double annihilator, i.e. $B = ( B ^ { \perp } ) ^ { \perp }$, [[#References|[a13]]]. The analogue of the sub–Stonean property, sometimes called the $S A W ^ { * }$-condition, is even more striking: For any two orthogonal elements $x$ and $y$ in $Q ( A )$ (say $x y = 0$) there is an element $e$ in $Q ( A )$ with $0 \leq e \leq 1$, such that $x e = x$ and $e y = 0$. Even better, if $C$ and $N$ are separable subsets of $Q ( A )$ such that $x$ commutes with $C$ and annihilates $N$, then the element $e$ can be chosen with the same properties, [[#References|[a11]]], [[#References|[a14]]]. Note that if $e$ could be taken as a projection, e.g. the range projection of $e$, this would be a familiar property in [[Von Neumann algebra|von Neumann algebra]] theory. The fact that corona algebras will never be von Neumann algebras (if $A$ is non-unital and $\sigma$-unital) indicates that the property (first established by G. Kasparov as a "technical lemma" ) is useful. Actually, a potentially stronger version is true: If $x _ { n }$ and $y _ { n }$ are monotone sequences of self-adjoint elements in $Q ( A )$, one increasing, the other decreasing, such that $x _ { n } \leq y _ { n }$ for all $n$, and if $C$ and $N$ are separable subsets of $Q ( A )$, such that all $x _ { n }$ commute with $C$ and annihilate $N$, then there is an element $z$ in $Q ( A )$ such that $x _ { n } \leq z \leq y _ { n }$ for all $n$, and $z$ commutes with $C$ and annihilates $N$, [[#References|[a11]]]. This has as a consequence that if $B$ is any $\sigma$-unital $C ^ { * }$-subalgebra of $Q ( A )$, commuting with $C$ and annihilating $N$, as above, then for any multiplier $x$ in $M ( B )$ there is an element $z$ in the idealizer $I ( B )$ of $B$ in $Q ( A )$, still commuting with $C$ and annihilating $N$, such that $z b  = x b $ for every $b$ in $B$, [[#References|[a5]]], [[#References|[a15]]]. In other words, the natural morphism $\sigma : I ( B ) \cap C ^ { \prime } \cap N ^ { \perp } \rightarrow M ( B )$ (with $\ker \sigma = B ^ { \perp } \cap C ^ { \prime } \cap N ^ { \perp }$) is surjective. This indicates the size of corona algebras, even compared with large multiplier algebras.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> Ch.A. Akemann, G.K. Pedersen, "Ideal perturbations of elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260253.png" />-algebras" ''Math. Scand.'' , '''41''' (1977) pp. 117–139 {{MR|473848}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> Ch.A. Akemann, G.K. Pedersen, J. Tomiyama, "Multipliers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260254.png" />-algebras" ''J. Funct. Anal.'' , '''13''' (1973) pp. 277–301 {{MR|470685}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> B. Blackadar, "Shape theory for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260255.png" />-algebras" ''Math. Scand.'' , '''56''' (1985) pp. 249–275 {{MR|813640}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> R.C. Busby, "Double centralizers and extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260256.png" />-algebras" ''Trans. Amer. Math. Soc.'' , '''132''' (1968) pp. 79–99 {{MR|225175}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> S. Eilers, T.A. Loring, G.K. Pedersen, "Morphisms of extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260257.png" />-algebras: Pushing forward the Busby invariant" ''Adv. Math.'' , '''147''' (1999) pp. 74–109 {{MR|1725815}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> K. Grove, G.K. Pedersen, "Sub-Stonean spaces and corona sets" ''J. Funct. Anal.'' , '''56''' (1984) pp. 124–143 {{MR|0735707}} {{ZBL|0539.54029}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> K. Grove, G.K. Pedersen, "Diagonalizing matrices over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260258.png" />" ''J. Funct. Anal.'' , '''59''' (1984) pp. 65–89 {{MR|0763777}} {{ZBL|0554.46026}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> B.E. Johnson, "An introduction to the theory of centralizers" ''Proc. London Math. Soc.'' , '''14''' (1964) pp. 299–320 {{MR|0159233}} {{ZBL|0143.36102}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> T.A. Loring, "Lifting solutions to perturbing problems in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260259.png" />-algebras" , ''Fields Inst. Monographs'' , '''8''' , Amer. Math. Soc. (1997) {{MR|1420863}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> T.A. Loring, G.K. Pedersen, "Projectivity, transitivity and AF telescopes" ''Trans. Amer. Math. Soc.'' , '''350''' (1998) pp. 4313–4339 {{MR|1616003}} {{ZBL|0906.46044}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> C.L. Olsen, G.K. Pedersen, "Corona <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260260.png" />-algebras and their applications to lifting problems" ''Math. Scand.'' , '''64''' (1989) pp. 63–86 {{MR|1036429}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> G.K. Pedersen, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260261.png" />-algebras and their automorphism groups" , Acad. Press (1979) {{MR|0548006}} {{ZBL|0416.46043}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> G.K. Pedersen, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260262.png" />-algebras and corona <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260263.png" />-algebras, contributions to non-commutative topology" ''J. Oper. Th.'' , '''4''' (1986) pp. 15–32</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> G.K. Pedersen, "The corona construction" J.B. Conway (ed.) B.B. Morrel (ed.) , ''Proc. 1988 GPOTS-Wabash Conf.'' , Longman Sci. (1990) pp. 49–92 {{MR|1075635}} {{ZBL|0716.46044}} </TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> G.K. Pedersen, "Extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130260/m130260264.png" />-algebras" S. Doplicher (ed.) et al. (ed.) , ''Operator Algebras and Quantum Field Theory'' , Internat. Press, Cambridge, Mass. (1997) pp. 2–35</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> D.C. Taylor, "The strict topology for double centralizer algebras" ''Trans. Amer. Math. Soc.'' , '''150''' (1970) pp. 633–643 {{MR|0290117}} {{ZBL|0204.14701}} </TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top"> Ch.A. Akemann, G.K. Pedersen, "Ideal perturbations of elements in $C ^ { * }$-algebras" ''Math. Scand.'' , '''41''' (1977) pp. 117–139 {{MR|473848}} {{ZBL|}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> Ch.A. Akemann, G.K. Pedersen, J. Tomiyama, "Multipliers of $C ^ { * }$-algebras" ''J. Funct. Anal.'' , '''13''' (1973) pp. 277–301 {{MR|470685}} {{ZBL|}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> B. Blackadar, "Shape theory for $C ^ { * }$-algebras" ''Math. Scand.'' , '''56''' (1985) pp. 249–275 {{MR|813640}} {{ZBL|}} </td></tr><tr><td valign="top">[a4]</td> <td valign="top"> R.C. Busby, "Double centralizers and extensions of $C ^ { * }$-algebras" ''Trans. Amer. Math. Soc.'' , '''132''' (1968) pp. 79–99 {{MR|225175}} {{ZBL|}} </td></tr><tr><td valign="top">[a5]</td> <td valign="top"> S. Eilers, T.A. Loring, G.K. Pedersen, "Morphisms of extensions of $C ^ { * }$-algebras: Pushing forward the Busby invariant" ''Adv. Math.'' , '''147''' (1999) pp. 74–109 {{MR|1725815}} {{ZBL|}} </td></tr><tr><td valign="top">[a6]</td> <td valign="top"> K. Grove, G.K. Pedersen, "Sub-Stonean spaces and corona sets" ''J. Funct. Anal.'' , '''56''' (1984) pp. 124–143 {{MR|0735707}} {{ZBL|0539.54029}} </td></tr><tr><td valign="top">[a7]</td> <td valign="top"> K. Grove, G.K. Pedersen, "Diagonalizing matrices over $C ( X )$" ''J. Funct. Anal.'' , '''59''' (1984) pp. 65–89 {{MR|0763777}} {{ZBL|0554.46026}} </td></tr><tr><td valign="top">[a8]</td> <td valign="top"> B.E. Johnson, "An introduction to the theory of centralizers" ''Proc. London Math. Soc.'' , '''14''' (1964) pp. 299–320 {{MR|0159233}} {{ZBL|0143.36102}} </td></tr><tr><td valign="top">[a9]</td> <td valign="top"> T.A. Loring, "Lifting solutions to perturbing problems in $C ^ { * }$-algebras" , ''Fields Inst. Monographs'' , '''8''' , Amer. Math. Soc. (1997) {{MR|1420863}} {{ZBL|}} </td></tr><tr><td valign="top">[a10]</td> <td valign="top"> T.A. Loring, G.K. Pedersen, "Projectivity, transitivity and AF telescopes" ''Trans. Amer. Math. Soc.'' , '''350''' (1998) pp. 4313–4339 {{MR|1616003}} {{ZBL|0906.46044}} </td></tr><tr><td valign="top">[a11]</td> <td valign="top"> C.L. Olsen, G.K. Pedersen, "Corona $C ^ { * }$-algebras and their applications to lifting problems" ''Math. Scand.'' , '''64''' (1989) pp. 63–86 {{MR|1036429}} {{ZBL|}} </td></tr><tr><td valign="top">[a12]</td> <td valign="top"> G.K. Pedersen, "$C ^ { * }$-algebras and their automorphism groups" , Acad. Press (1979) {{MR|0548006}} {{ZBL|0416.46043}} </td></tr><tr><td valign="top">[a13]</td> <td valign="top"> G.K. Pedersen, "$S A W ^ { * }$-algebras and corona $C ^ { * }$-algebras, contributions to non-commutative topology" ''J. Oper. Th.'' , '''4''' (1986) pp. 15–32</td></tr><tr><td valign="top">[a14]</td> <td valign="top"> G.K. Pedersen, "The corona construction" J.B. Conway (ed.) B.B. Morrel (ed.) , ''Proc. 1988 GPOTS-Wabash Conf.'' , Longman Sci. (1990) pp. 49–92 {{MR|1075635}} {{ZBL|0716.46044}} </td></tr><tr><td valign="top">[a15]</td> <td valign="top"> G.K. Pedersen, "Extensions of $C ^ { * }$-algebras" S. Doplicher (ed.) et al. (ed.) , ''Operator Algebras and Quantum Field Theory'' , Internat. Press, Cambridge, Mass. (1997) pp. 2–35</td></tr><tr><td valign="top">[a16]</td> <td valign="top"> D.C. Taylor, "The strict topology for double centralizer algebras" ''Trans. Amer. Math. Soc.'' , '''150''' (1970) pp. 633–643 {{MR|0290117}} {{ZBL|0204.14701}} </td></tr></table>

Latest revision as of 17:46, 1 July 2020

A $C ^ { * }$-algebra $A$ of operators on some Hilbert space $\mathcal{H}$ may be viewed as a non-commutative generalization of a function algebra $C _ { 0 } ( \Omega )$ acting as multiplication operators on some $L^{2}$-space associated with a measure on the locally compact space $\Omega$. The space $\Omega$ being compact corresponds naturally to the case where the algebra $A$ is unital. In the non-unital case any embedding of $A$ as an essential ideal in some larger unital $C ^ { * }$-algebra $B$ (i.e., the annihilator of $A$ in $B$ is zero) can be viewed as an analogue of a compactification of the locally compact Hausdorff space $\Omega$. Thus, the one-point compactification $\Omega \cup \{ \infty \}$ of $\Omega$ corresponds to the unitization $\tilde { A } = A \oplus \mathbf{C}$ of the algebra $A$. The analogue of the maximal compactification — the Stone–Čech compactification — is the algebra $M ( A )$ of multipliers of $A$, defined by R.C. Busby in 1967 [a4] and studied in more detail in [a2]. It is defined simply as the idealizer of $A$ in $B ( \mathcal{H} )$ (assuming that $A \mathcal{H} = \mathcal{H}$ or, equivalently, that no non-zero vector in $\mathcal{H}$ is annihilated by $A$).

Linear operators $\lambda$ and $\rho$ on $A$ are called left and right centralizers if $\lambda ( x y ) = \lambda ( x ) y$ and $\rho ( x y ) = x \rho ( y )$ for all $x$, $y$ in $A$. They are automatically bounded. A double centralizer is a pair $( \lambda , \rho )$ of left, right centralizers such that $x \lambda ( y ) = \rho ( x ) y$ (whence $\| \lambda \| = \| \rho \|$), and the closed linear spaces of double centralizers becomes a $C ^ { * }$-algebra when product and involution are defined by $( \lambda _ { 1 } , \rho _ { 1 } ) ( \lambda _ { 2 } , \rho _ { 2 } ) = ( \lambda _ { 1 } \lambda _ { 2 } , \rho _ { 2 } \rho _ { 1 } )$ and $( \lambda , \rho ) ^ { * } = ( \rho ^ { * } , \lambda ^ { * } )$ (where $\lambda ^ { * } ( x ) = ( \lambda ( x ^ { * } ) ) ^ { * }$). As shown by B.E. Johnson, [a8], there is an isomorphism between the abstractly defined $C ^ { * }$-algebra of double centralizers of $A$ and the concrete $C ^ { * }$-algebra $M ( A )$. This, in particular, shows that $M ( A )$ is independent of the given representation of $A$ on $\mathcal{H}$.

The strict topology on $M ( A )$ is defined by the semi-norms $x \rightarrow \| a x \| + \| a x \|$ on $B ( \mathcal{H} )$ with $a$ in $A$, [a4]. It is used as an analogue of uniform convergence on compact subsets of $\Omega$ in function algebras. Thus, it can be shown that $M ( A )$ is the strict completion of $A$ in $B ( \mathcal{H} )$ and that the strict dual of $M ( A )$ equals the norm dual of $A$, [a16].

If $\mathcal{H}$ is the universal Hilbert space for $A$ (the orthogonal sum of all Hilbert spaces obtained from states of $A$ via the Gel'fand–Naimark–Segal construction), then $M ( A )$ has a more constructive characterization: Let denote the space of self-adjoint operators in $B ( \mathcal{H} )$ that can be obtained as limits (in the strong topology) of some increasing net of self-adjoint elements from the unitized algebra $\tilde{A}$ (cf. also Net (directed set); Self-adjoint operator). Similarly, let be the space of limits of decreasing nets. Then

Thus, for every self-adjoint multiplier $x$ there are nets $( a _\lambda )$ and $( b _ { \mu } )$ in $\widetilde { A } _ { s a }$, one increasing, the other decreasing, such that $a_{\lambda} \nearrow x \swarrow b _ { \mu }$. If $A$ is $\sigma$-unital, i.e. contains a countable approximate unit, in particular if $A$ is separable (cf. also Separable algebra), these nets can be taken as sequences, [a2], [a12], p. 12. In the commutative case, where $A = C _ { 0 } ( \Omega )$, whence $M ( A ) = C _ { b } ( \Omega )$, this expresses the well-known fact that a bounded, real function on $\Omega$ is continuous precisely when it is both lower and upper semi-continuous.

For any $C ^ { * }$-algebra $X$ containing $A$ as an ideal there is a natural morphism (i.e. a $\square ^ { * }$-homomorphism) $\sigma : X \rightarrow M ( A )$, defined by $\sigma ( x ) a = x a$, that extends the identity mapping of $A \subset X$ onto $A \subset M ( A )$. If $A$ is essential in $X$, one therefore obtains an embedding $X \subset M ( A )$. Any morphism $\alpha : A \rightarrow B$ between $C ^ { * }$-algebras $A$ and $B$ extends uniquely to a strictly continuous morphism $\overline { \alpha } : M ( A ) \rightarrow M ( B )$, provided that $\alpha$ is proper (i.e. maps an approximate unit for $A$ to one for $B$). Such morphisms are the analogues of proper continuous mappings between locally compact spaces. If $A$ is $\sigma$-unital and $\alpha$ is a quotient morphism, i.e. surjective, then $\overline { \alpha }$ is also surjective. This result may be viewed as a non-commutative generalization of the Tietze extension theorem, [a2], [a13] (cf. also Extension theorems).

The corona of a $C ^ { * }$-algebra $A$ is defined as the quotient $C ^ { * }$-algebra $Q ( A ) = M ( A ) / A$, [a13]. The commutative analogue is the compact Hausdorff space $\beta \ \Omega \ \backslash \ \Omega$ (the corona of the locally compact space $\Omega$, [a6]), but the pre-eminent example of such algebras is the Calkin algebra $B (\mathcal{H} ) / K ( \mathcal{H} )$, obtained by taking $A$ as the algebra $K ( \mathcal{H} )$ of compact operators on $\mathcal{H}$ (whence $M ( A ) = B ( \mathcal{H} )$). Corona $C ^ { * }$-algebras are usually non-separable and cannot even be represented on separable Hilbert spaces, [a14]. Nevertheless, they have important roles in the formulation of G. Kasparov's KK-theory and the later variation known as E-theory. The foremost application, however, is to the theory of extensions: An extension of $C ^ { * }$-algebras $A$ and $B$ is any $C ^ { * }$-algebra $X$ that fits into a short exact sequence (cf. also Exact sequence)

\begin{equation*} 0 \rightarrow A \rightarrow X \stackrel { \pi } { \rightarrow } B \rightarrow 0. \end{equation*}

Thus, $X$ contains $A$ as an ideal, and $\pi$ is simply the quotient morphism. In particular, $M ( A )$ may be regarded as an extension of $A$ by $Q ( A )$, and in fact a maximal such. Namely, any other extension will give rise to a commutative diagram

Here $\sigma : X \rightarrow M ( A )$ is the morphism defined above and the induced morphism $\tau : B \rightarrow Q ( A )$ is known as the Busby invariant for $X$. This invariant determines $X$ up to an obvious equivalence, because the right square in the diagram above describes $X$ as the pull-back of $B$ and $M ( A )$ over $Q ( A )$, i.e.

\begin{equation*} X = M ( A ) \bigoplus _ { Q ( A ) } B = \end{equation*}

\begin{equation*} = \{ ( m , b ) \in M ( A ) \bigoplus B : \pi ( m ) = \tau ( b ) \}. \end{equation*}

One therefore has the identification $\operatorname { Ext } ( A , B ) = \operatorname { Hom } ( B , Q ( A ) )$, [a4], [a5], [a15].

For any quotient morphism $\pi : X \rightarrow B$ between $C ^ { * }$-algebras one may ask whether an element $b$ in $B$ with specific properties is the image of some $x$ in $X$ with the same properties. This is known as a lifting problem, and is the non-commutative analogue of extension problems for functions. Many lifting problems have positive (and easy) solutions: If $b = b ^ { * }$ or $b \geq 0$ or $\| b \| \leq 1$, one can find counter-images in $X$ with the same properties. However, the properties $b ^ { 2 } = b$ (being idempotent) and $b ^ { * } b = b b ^ { * }$ (being normal) are not liftable in general. It follows that the more general commutator relation $b _ { 1 } b _ { 2 } = b _ { 2 } b _ { 1 }$ is not liftable either. But the orthogonality relation $b _ { 1 } b _ { 2 } = 0$ is liftable (even in the $n$-fold version $b _ { 1 } \ldots b _ { n } = 0$). Using this one may show that the nilpotency relation $b ^ { n } = 0$ is liftable, [a1], [a11], [a9].

As advocated by T.A. Loring, lifting problems may with advantage be replaced by $C ^ { * }$-algebra problems concerning projectivity. A $C ^ { * }$-algebra $P$ is projective if any morphism $\alpha : P \rightarrow B$ into a quotient $C ^ { * }$-algebra $B = \pi ( X )$ can be factored as $\alpha = \pi \circ \overline { \alpha }$ for some morphism $\overline { \alpha } : P \rightarrow X$, [a3]. This means that one is lifting a whole $C ^ { * }$-subalgebra and not just some elements. Projective $C ^ { * }$-algebras are the non-commutative analogues of topological spaces that are absolute retracts, but since the category of $C ^ { * }$-algebras is vastly larger than the category of locally compact Hausdorff spaces, projectivity is a rare phenomenon. However, the cone over the $n \times n$-matrices, i.e. the algebra

\begin{equation*} \mathbf{C M} _ { n } = C _ { 0 } ( ]0,1 ] ) \otimes \mathbf{M} _ { n } \end{equation*}

is always projective. This means that although matrix units cannot, in general, be lifted from quotients, there are lifts in the "smeared" form given by , [a10], [a9].

Corona $C ^ { * }$-algebras form an indispensable tool for more complicated lifting problems, because by Busby's theory, mentioned above, it suffices to solve the lifting for quotient morphisms of the form $\pi : M ( A ) \rightarrow Q ( A )$. Thus, one may utilize the special properties that corona algebras have. A brief outline of these follows.

Corona algebras.

In topology, a compact Hausdorff space is called sub-Stonean if any two disjoint, open, $\sigma$-compact sets have disjoint closures. Exotic as this may sound, it is a property that any corona set $\beta \ \Omega \ \backslash \ \Omega$ will have, if $\Omega$ is locally compact and $\sigma$-compact. In such a space, every open, $\sigma$-compact subset is also regularly embedded, i.e. it equals the interior of its closure in $\beta \ \Omega \ \backslash \ \Omega$, [a6]. The non-commutative generalization of this is the fact that if $A$ is a $\sigma$-unital $C ^ { * }$-algebra, then every $\sigma$-unital hereditary $C ^ { * }$-subalgebra $B$ of its corona algebra $Q ( A )$ equals its double annihilator, i.e. $B = ( B ^ { \perp } ) ^ { \perp }$, [a13]. The analogue of the sub–Stonean property, sometimes called the $S A W ^ { * }$-condition, is even more striking: For any two orthogonal elements $x$ and $y$ in $Q ( A )$ (say $x y = 0$) there is an element $e$ in $Q ( A )$ with $0 \leq e \leq 1$, such that $x e = x$ and $e y = 0$. Even better, if $C$ and $N$ are separable subsets of $Q ( A )$ such that $x$ commutes with $C$ and annihilates $N$, then the element $e$ can be chosen with the same properties, [a11], [a14]. Note that if $e$ could be taken as a projection, e.g. the range projection of $e$, this would be a familiar property in von Neumann algebra theory. The fact that corona algebras will never be von Neumann algebras (if $A$ is non-unital and $\sigma$-unital) indicates that the property (first established by G. Kasparov as a "technical lemma" ) is useful. Actually, a potentially stronger version is true: If $x _ { n }$ and $y _ { n }$ are monotone sequences of self-adjoint elements in $Q ( A )$, one increasing, the other decreasing, such that $x _ { n } \leq y _ { n }$ for all $n$, and if $C$ and $N$ are separable subsets of $Q ( A )$, such that all $x _ { n }$ commute with $C$ and annihilate $N$, then there is an element $z$ in $Q ( A )$ such that $x _ { n } \leq z \leq y _ { n }$ for all $n$, and $z$ commutes with $C$ and annihilates $N$, [a11]. This has as a consequence that if $B$ is any $\sigma$-unital $C ^ { * }$-subalgebra of $Q ( A )$, commuting with $C$ and annihilating $N$, as above, then for any multiplier $x$ in $M ( B )$ there is an element $z$ in the idealizer $I ( B )$ of $B$ in $Q ( A )$, still commuting with $C$ and annihilating $N$, such that $z b = x b $ for every $b$ in $B$, [a5], [a15]. In other words, the natural morphism $\sigma : I ( B ) \cap C ^ { \prime } \cap N ^ { \perp } \rightarrow M ( B )$ (with $\ker \sigma = B ^ { \perp } \cap C ^ { \prime } \cap N ^ { \perp }$) is surjective. This indicates the size of corona algebras, even compared with large multiplier algebras.

References

[a1] Ch.A. Akemann, G.K. Pedersen, "Ideal perturbations of elements in $C ^ { * }$-algebras" Math. Scand. , 41 (1977) pp. 117–139 MR473848
[a2] Ch.A. Akemann, G.K. Pedersen, J. Tomiyama, "Multipliers of $C ^ { * }$-algebras" J. Funct. Anal. , 13 (1973) pp. 277–301 MR470685
[a3] B. Blackadar, "Shape theory for $C ^ { * }$-algebras" Math. Scand. , 56 (1985) pp. 249–275 MR813640
[a4] R.C. Busby, "Double centralizers and extensions of $C ^ { * }$-algebras" Trans. Amer. Math. Soc. , 132 (1968) pp. 79–99 MR225175
[a5] S. Eilers, T.A. Loring, G.K. Pedersen, "Morphisms of extensions of $C ^ { * }$-algebras: Pushing forward the Busby invariant" Adv. Math. , 147 (1999) pp. 74–109 MR1725815
[a6] K. Grove, G.K. Pedersen, "Sub-Stonean spaces and corona sets" J. Funct. Anal. , 56 (1984) pp. 124–143 MR0735707 Zbl 0539.54029
[a7] K. Grove, G.K. Pedersen, "Diagonalizing matrices over $C ( X )$" J. Funct. Anal. , 59 (1984) pp. 65–89 MR0763777 Zbl 0554.46026
[a8] B.E. Johnson, "An introduction to the theory of centralizers" Proc. London Math. Soc. , 14 (1964) pp. 299–320 MR0159233 Zbl 0143.36102
[a9] T.A. Loring, "Lifting solutions to perturbing problems in $C ^ { * }$-algebras" , Fields Inst. Monographs , 8 , Amer. Math. Soc. (1997) MR1420863
[a10] T.A. Loring, G.K. Pedersen, "Projectivity, transitivity and AF telescopes" Trans. Amer. Math. Soc. , 350 (1998) pp. 4313–4339 MR1616003 Zbl 0906.46044
[a11] C.L. Olsen, G.K. Pedersen, "Corona $C ^ { * }$-algebras and their applications to lifting problems" Math. Scand. , 64 (1989) pp. 63–86 MR1036429
[a12] G.K. Pedersen, "$C ^ { * }$-algebras and their automorphism groups" , Acad. Press (1979) MR0548006 Zbl 0416.46043
[a13] G.K. Pedersen, "$S A W ^ { * }$-algebras and corona $C ^ { * }$-algebras, contributions to non-commutative topology" J. Oper. Th. , 4 (1986) pp. 15–32
[a14] G.K. Pedersen, "The corona construction" J.B. Conway (ed.) B.B. Morrel (ed.) , Proc. 1988 GPOTS-Wabash Conf. , Longman Sci. (1990) pp. 49–92 MR1075635 Zbl 0716.46044
[a15] G.K. Pedersen, "Extensions of $C ^ { * }$-algebras" S. Doplicher (ed.) et al. (ed.) , Operator Algebras and Quantum Field Theory , Internat. Press, Cambridge, Mass. (1997) pp. 2–35
[a16] D.C. Taylor, "The strict topology for double centralizer algebras" Trans. Amer. Math. Soc. , 150 (1970) pp. 633–643 MR0290117 Zbl 0204.14701
How to Cite This Entry:
Multipliers-of-C*-algebras. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multipliers-of-C*-algebras&oldid=24112
This article was adapted from an original article by Gert K. Pedersen (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article