An Introduction to Operator Algebras

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Recall that an algebra is a ring which is also a vector space under addition. An algebra is unital if it has a multiplicative identity, which we denote by 1 (or 1 A if A is the algebra and we wish to clarify that this is the identity for A). Unless
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❇❛✐❝❞❡✜♥✐✐♦♥  ◮ ❘❡❝❛❧❧❤❛❛♥  ❛❧❣❡❜❛  ✐❛✐♥❣✇❤✐❝❤✐❛❧♦❛✈❡❝♦♣❛❝❡✉♥❞❡❛❞❞✐✐♦♥✳❆♥❛❧❣❡❜❛✐  ✉♥✐❛❧  ✐❢✐❤❛❛♠✉❧✐♣❧✐❝❛✐✈❡✐❞❡♥✐②✱✇❤✐❝❤✇❡❞❡♥♦❡❜②  1 ✭♦  1  A  ✐❢   A  ✐❤❡❛❧❣❡❜❛❛♥❞✇❡✇✐❤♦❝❧❛✐❢②❤❛❤✐✐❤❡✐❞❡♥✐②❢♦   A  ✮✳ ❯♥❧❡♦❤❡✇✐❡❛❡❞✱❛❧❧♦❢♦✉❛❧❣❡❜❛❛❡❝♦♠♣❧❡①✦  ◮ ❆  ❇❛♥❛❝❤❛❧❣❡❜❛  ✐❛♥❛❧❣❡❜❛✇❤✐❝❤✐❛❇❛♥❛❝❤♣❛❝❡✳❲❡❛✉♠❡❤❛❤❡ ♥♦♠✐✉❜♠✉❧✐♣❧✐❝❛✐✈❡✳ ◮ ❲❡✇✐❡   A  ♯ ❢♦❤❡  ❛❧❣❡❜❛✐❝❞✉❛❧  ♦❢❛♥❛❧❣❡❜❛   A  ❀❤❛✐✱❤❡❝♦❧❧❡❝✐♦♥♦❢❛❧❧❧✐♥❡❛❢✉♥❝✐♦♥❛❧❀❛♥❞✇✐❡   A  ∗ ❢♦❤❡❝♦♥✐♥✉♦✉❧✐♥❡❛❢✉♥❝✐♦♥❛❧✇❤❡♥   A  ✐ ❛♥♦♠❡❞❛❧❣❡❜❛✳◆♦❡❤❛   A  ∗ ✐❛❇❛♥❛❝❤❛❧❣❡❜❛✱❛♥❞✇❡❝❛♥✐♠✐❧❛❧② ❞❡✜♥❡   A  ∗∗ ✱❡❝✳  ❇❛✐❝❞❡✜♥✐✐♦♥✱❝♦♥✐♥✉❡❞  ◮ ❆♥♦♥③❡♦♠✉❧✐♣❧✐❝❛✐✈❡❧✐♥❡❛❢✉♥❝✐♦♥❛❧  χ ♦♥❛♥❛❧❣❡❜❛   A  ✐❝❛❧❧❡❞❛  ❝❤❛❛❝❡  ✳❲❡✇✐❡   X  (  A  ) ❢♦❤❡❡♦❢❝❤❛❛❝❡♦❢   A  ✳ ⋆ ❚❤✐❡♠❛②❜❡ ❡♠♣②✦  ✭❝♦♥✐❞❡  M  2 (  ) ✮✳ ◮ ❆♥  ✭❛❧❣❡❜❛✮❤♦♠♦♠♦♣❤✐♠   ϕ :  A  → B  ✱  A  , B  ❛❧❣❡❜❛✱✐❛❧✐♥❡❛♠❛♣✉❝❤❤❛  ϕ ( ab  ) = ϕ ( a  ) ϕ ( b  ) ✭❤❛✐✱  ϕ ♣❡❡✈❡❛❧❣❡❜❛✉❝✉❡✮✳■❢   A  , B  ❛❡❜♦❤ ✉♥✐❛❧✱❤❡♥✇❡❛②❤❛  ϕ ✐  ✉♥✐❛❧  ✐❢   ϕ ( 1  A  ) = 1 B  ✳ ◮ ❆  ❡♣❡❡♥❛✐♦♥  ♦❢❛❇❛♥❛❝❤❛❧❣❡❜❛   A  ✐❛♥❛❧❣❡❜❛❤♦♠♦♠♦♣❤✐♠❢♦♠   A  ✐♥♦    (   ) ✱❤❡❜♦✉♥❞❡❞❧✐♥❡❛♦♣❡❛♦♦♥❛❍✐❧❜❡♣❛❝❡     ✳❚❤❡  ❞✐♠❡♥✐♦♥  ♦❢❛❡♣❡❡♥❛✐♦♥✐❤❡❞✐♠❡♥✐♦♥♦❢    ✳❆♥✐♥❥❡❝✐✈❡ ❡♣❡❡♥❛✐♦♥✐❛✐❞♦❜❡  ❢❛✐❤❢✉❧  ✳❈❤❛❛❝❡❛❡❥✉  1 ❞✐♠❡♥✐♦♥❛❧❡♣❡❡♥❛✐♦♥✳  ❇❛✐❝❞❡✜♥✐✐♦♥✱❝♦♥✐♥✉❡❞  ◮ ❆  ❧❡❢✐❞❡❛❧   I  ♦❢❛♥❛❧❣❡❜❛   A  ✐❛✉❜❡♦❢   A  ✇❤✐❝❤✐❛✉❜❣♦✉♣✉♥❞❡ ❛❞❞✐✐♦♥✱❛♥❞❤❛✈✐♥❣❤❡♣♦♣❡②❤❛  ax  ∈ I  ✇❤❡♥❡✈❡   a  ∈  A  ❛♥❞   x  ∈ I  ✳■✐❛  ✐❣❤✐❞❡❛❧  ✐❢✐♥❡❛❞  xa  ∈ I  ✱❛♥❞❛  ✇♦✲✐❞❡❞✐❞❡❛❧  ✭♦✐♠♣❧②  ✐❞❡❛❧  ✮✐❢✐✐❜♦❤ ❛❧❡❢❛♥❞✐❣❤✐❞❡❛❧✳■❞❡❛❧❛❡❛✉♠❡❞♦❜❡♥♦♥✐✈✐❛❧✭✐❡✱♥♦  { 0 } ♦   A  ✮✳ ◮ ❆  ♠❛①✐♠❛❧✐❞❡❛❧  ✐❛♥✐❞❡❛❧✇❤✐❝❤✐♥♦♣♦♣❡❧②❝♦♥❛✐♥❡❞✐♥❛♥♦❤❡✐❞❡❛❧✳❚❤❡❡①✐❡♥❝❡♦❢♠❛①✐♠❛❧❧❡❢♦✐❣❤✐❞❡❛❧✐  ⋆ ❛✐♠♣❧❡❛♣♣❧✐❝❛✐♦♥♦❢❩♦♥✬ ❧❡♠♠❛✳❚❤❡❡♦❢♠❛①✐♠❛❧✭✇♦✲✐❞❡❞✮✐❞❡❛❧♦❢❛♥❛❧❣❡❜❛   A  ✐❞❡♥♦❡❞❜②  M  (  A  ) ✭❤✐♠❛②❜❡❡♠♣②✦✮✳ ◮ ❚❤❡  ❛❞✐❝❛❧  ❛❞   (  A  ) ♦❢❛♥❛❧❣❡❜❛   A  ✐❤❡✐♥❡❡❝✐♦♥♦❢❛❧❧♠❛①✐♠❛❧❧❡❢ ✭❡✉✐✈❛❧❡♥❧②✱❛❧❧♠❛①✐♠❛❧✐❣❤✮✐❞❡❛❧✳■❢❤❡❛❞✐❝❛❧✐  { 0 } ✱❤❡❛❧❣❡❜❛✐❛✐❞ ♦❜❡  ✐♠♣❧❡  ✳ ◮ ❆❜❛✐❝❡✉❧✐♥❤❡❤❡♦②♦❢❝♦♠♠✉❛✐✈❡✉♥✐❛❧❇❛♥❛❝❤❛❧❣❡❜❛✐❤❛❢♦ ✉❝❤❛♥❛❧❣❡❜❛   A  ✱❤❡❡   X  (  A  ) ❛♥❞   M  (  A  ) ❛❡♥♦♥❡♠♣②✱❛♥❞✐♥❢❛❝✱❤❡❡✐ ❛♦♥❡♦♦♥❡❝♦❡♣♦♥❞❡♥❝❡❜❡✇❡❡♥❤❡✇♦✭❤❡♠❛①✐♠❛❧✐❞❡❛❧❜❡✐♥❣ ❦❡♥❡❧♦❢❝❤❛❛❝❡✮✳  ❇❛✐❝❞❡✜♥✐✐♦♥✱❝♦♥✐♥✉❡❞  ◮ ■❢   A  ✐♥♦❛✉♥✐❛❧❛❧❣❡❜❛✱✇❡❝❛♥  ✉♥✐✐③❡  ✐✳❚❤❡✉♥✐✐③❛✐♦♥   A  1  =  A  ×  ❤❛  ( a  , α ) + ( b  , β ) = ( a  + b  , α + β ) , β ( a  , α ) = ( β a  , βα ) , ( a  , α )( b  , β ) = ( ab  + β a  + α b  , αβ ) , ❛♥❞   ( a  , α )  =  a   + | α | , ❛♥❞✉♥✐❡✉❛❧♦  ( 0,1 ) ✳■❝♦♥❛✐♥   A  ✐♦♠❡✐❝❛❧❧②❛❛♥✐❞❡❛❧✳❲❤❡♥   A  ✐ ❝♦♠♠✉❛✐✈❡✱♦✐   A  1 ✳  ❚♦♣♦❧♦❣✐❡  ◮ ❙❡✈❡❛❧❞✐✛❡❡♥♦♣♦❧♦❣✐❡✇✐❧❧❜❡♦❢✐♠♣♦❛♥❝❡♦✉✳ ◮ ❚❤❡❡✐❤❡  ♥♦♠♦♣♦❧♦❣②  ♦♥❛♥♦♠❡❞❛❧❣❡❜❛✳❇②❞❡✜♥✐✐♦♥✱❛❇❛♥❛❝❤ ❛❧❣❡❜❛✐❝♦♠♣❧❡❡✐♥❤✐♥♦♠✳ ◮ ❚❤❡❡✐❤❡  ✇❡❛❦♦♣♦❧♦❣②  ✱✇❤✐❝❤✐❞❡✜♥❡❞❛❤❡✇❡❛❦❡♦♣♦❧♦❣②♦♥   A  ✉❝❤❤❛❤❡❧✐♥❡❛❢✉♥❝✐♦♥❛❧✐♥   A  ∗ ❛❡❝♦♥✐♥✉♦✉✳ ◮ ❘❡❝❛❧❧❤❛❤❡❡✐❛♥✐♦♠❡✐❝❡♠❜❡❞❞✐♥❣♦❢❛❇❛♥❛❝❤❛❧❣❡❜❛   A  ✐♥♦   A  ∗∗ ✳❚❤❡  ✇❡❛❦✲  ∗ ♦♣♦❧♦❣②  ✐❤❡✇❡❛❦❡♦♣♦❧♦❣②♦♥   A  ∗ ✉❝❤❤❛❤❡❡❧❡♠❡♥♦❢   A  ✱✈✐❡✇❡❞❛❧✐♥❡❛❢✉♥❝✐♦♥❛❧♦♥   A  ∗ ✱❛❡❝♦♥✐♥✉♦✉✳ ◮ ❚❤❡✐♠♣♦❛♥❝❡♦❢❤❡✇❡❛❦✲   ∗ ♦♣♦❧♦❣②❧✐❡✐♥❤❡❢❛❝❤❛❤❡❝❤❛❛❝❡♦❢   A  ❢♦♠❛❝❧♦❡❞✉❜❡♦❢❤❡✉♥✐❜❛❧❧♦❢   A  ∗ ✳❇②❆❧❛♦❣❧✉✬❤❡♦❡♠✱✐❢♦❧❧♦✇ ❤❛   X  (  A  ) ✭♦   M  (  A  ) ✐❞❡♥✐✜❡❞✇✐❤   X  (  A  ) ✮✐❡❧❛✐✈❡❧②✇❡❛❦✲   ∗ ❝♦♠♣❛❝✳

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