Difference between revisions of "Multipliers-of-C*-algebras"
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<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> | <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> | ||
− | is always projective. This means that although matrix units cannot, in general, be lifted from quotients, there are lifts in the | + | 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 <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. | 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. | ||
==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 | + | 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. |
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <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> |
Revision as of 17:34, 31 March 2012
A -algebra
of operators on some Hilbert space
may be viewed as a non-commutative generalization of a function algebra
acting as multiplication operators on some
-space associated with a measure on the locally compact space
. The space
being compact corresponds naturally to the case where the algebra
is unital. In the non-unital case any embedding of
as an essential ideal in some larger unital
-algebra
(i.e., the annihilator of
in
is zero) can be viewed as an analogue of a compactification of the locally compact Hausdorff space
. Thus, the one-point compactification
of
corresponds to the unitization
of the algebra
. The analogue of the maximal compactification — the Stone–Čech compactification — is the algebra
of multipliers of
, defined by R.C. Busby in 1967 [a4] and studied in more detail in [a2]. It is defined simply as the idealizer of
in
(assuming that
or, equivalently, that no non-zero vector in
is annihilated by
).
Linear operators and
on
are called left and right centralizers if
and
for all
,
in
. They are automatically bounded. A double centralizer is a pair
of left, right centralizers such that
(whence
), and the closed linear spaces of double centralizers becomes a
-algebra when product and involution are defined by
and
(where
). As shown by B.E. Johnson, [a8], there is an isomorphism between the abstractly defined
-algebra of double centralizers of
and the concrete
-algebra
. This, in particular, shows that
is independent of the given representation of
on
.
The strict topology on is defined by the semi-norms
on
with
in
, [a4]. It is used as an analogue of uniform convergence on compact subsets of
in function algebras. Thus, it can be shown that
is the strict completion of
in
and that the strict dual of
equals the norm dual of
, [a16].
If is the universal Hilbert space for
(the orthogonal sum of all Hilbert spaces obtained from states of
via the Gel'fand–Naimark–Segal construction), then
has a more constructive characterization: Let
denote the space of self-adjoint operators in
that can be obtained as limits (in the strong topology) of some increasing net of self-adjoint elements from the unitized algebra
(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 there are nets
and
in
, one increasing, the other decreasing, such that
. If
is
-unital, i.e. contains a countable approximate unit, in particular if
is separable (cf. also Separable algebra), these nets can be taken as sequences, [a2], [a12], p. 12. In the commutative case, where
, whence
, this expresses the well-known fact that a bounded, real function on
is continuous precisely when it is both lower and upper semi-continuous.
For any -algebra
containing
as an ideal there is a natural morphism (i.e. a
-homomorphism)
, defined by
, that extends the identity mapping of
onto
. If
is essential in
, one therefore obtains an embedding
. Any morphism
between
-algebras
and
extends uniquely to a strictly continuous morphism
, provided that
is proper (i.e. maps an approximate unit for
to one for
). Such morphisms are the analogues of proper continuous mappings between locally compact spaces. If
is
-unital and
is a quotient morphism, i.e. surjective, then
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 -algebra
is defined as the quotient
-algebra
, [a13]. The commutative analogue is the compact Hausdorff space
(the corona of the locally compact space
, [a6]), but the pre-eminent example of such algebras is the Calkin algebra
, obtained by taking
as the algebra
of compact operators on
(whence
). Corona
-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
-algebras
and
is any
-algebra
that fits into a short exact sequence (cf. also Exact sequence)
![]() |
Thus, contains
as an ideal, and
is simply the quotient morphism. In particular,
may be regarded as an extension of
by
, and in fact a maximal such. Namely, any other extension will give rise to a commutative diagram
![]() |
Here is the morphism defined above and the induced morphism
is known as the Busby invariant for
. This invariant determines
up to an obvious equivalence, because the right square in the diagram above describes
as the pull-back of
and
over
, i.e.
![]() |
![]() |
One therefore has the identification , [a4], [a5], [a15].
For any quotient morphism between
-algebras one may ask whether an element
in
with specific properties is the image of some
in
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
or
or
, one can find counter-images in
with the same properties. However, the properties
(being idempotent) and
(being normal) are not liftable in general. It follows that the more general commutator relation
is not liftable either. But the orthogonality relation
is liftable (even in the
-fold version
). Using this one may show that the nilpotency relation
is liftable, [a1], [a11], [a9].
As advocated by T.A. Loring, lifting problems may with advantage be replaced by -algebra problems concerning projectivity. A
-algebra
is projective if any morphism
into a quotient
-algebra
can be factored as
for some morphism
, [a3]. This means that one is lifting a whole
-subalgebra and not just some elements. Projective
-algebras are the non-commutative analogues of topological spaces that are absolute retracts, but since the category of
-algebras is vastly larger than the category of locally compact Hausdorff spaces, projectivity is a rare phenomenon. However, the cone over the
-matrices, i.e. the algebra
![]() |
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 -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
. 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, -compact sets have disjoint closures. Exotic as this may sound, it is a property that any corona set
will have, if
is locally compact and
-compact. In such a space, every open,
-compact subset is also regularly embedded, i.e. it equals the interior of its closure in
, [a6]. The non-commutative generalization of this is the fact that if
is a
-unital
-algebra, then every
-unital hereditary
-subalgebra
of its corona algebra
equals its double annihilator, i.e.
, [a13]. The analogue of the sub–Stonean property, sometimes called the
-condition, is even more striking: For any two orthogonal elements
and
in
(say
) there is an element
in
with
, such that
and
. Even better, if
and
are separable subsets of
such that
commutes with
and annihilates
, then the element
can be chosen with the same properties, [a11], [a14]. Note that if
could be taken as a projection, e.g. the range projection of
, this would be a familiar property in von Neumann algebra theory. The fact that corona algebras will never be von Neumann algebras (if
is non-unital and
-unital) indicates that the property (first established by G. Kasparov as a "technical lemma" ) is useful. Actually, a potentially stronger version is true: If
and
are monotone sequences of self-adjoint elements in
, one increasing, the other decreasing, such that
for all
, and if
and
are separable subsets of
, such that all
commute with
and annihilate
, then there is an element
in
such that
for all
, and
commutes with
and annihilates
, [a11]. This has as a consequence that if
is any
-unital
-subalgebra of
, commuting with
and annihilating
, as above, then for any multiplier
in
there is an element
in the idealizer
of
in
, still commuting with
and annihilating
, such that
for every
in
, [a5], [a15]. In other words, the natural morphism
(with
) 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 ![]() |
[a2] | Ch.A. Akemann, G.K. Pedersen, J. Tomiyama, "Multipliers of ![]() |
[a3] | B. Blackadar, "Shape theory for ![]() |
[a4] | R.C. Busby, "Double centralizers and extensions of ![]() |
[a5] | S. Eilers, T.A. Loring, G.K. Pedersen, "Morphisms of extensions of ![]() |
[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 ![]() |
[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 ![]() |
[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 ![]() |
[a12] | G.K. Pedersen, "![]() |
[a13] | G.K. Pedersen, "![]() ![]() |
[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 ![]() |
[a16] | D.C. Taylor, "The strict topology for double centralizer algebras" Trans. Amer. Math. Soc. , 150 (1970) pp. 633–643 MR0290117 Zbl 0204.14701 |
Multipliers-of-C*-algebras. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multipliers-of-C*-algebras&oldid=24112