# An Introduction to Operator Algebras

Description
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
Categories
Published

View again

All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Related Documents
Share
Transcript
❇❛✐❝❞❡✜♥✐✐♦♥  ◮ ❘❡❝❛❧❧❤❛❛♥  ❛❧❣❡❜❛  ✐❛✐♥❣✇❤✐❝❤✐❛❧♦❛✈❡❝♦♣❛❝❡✉♥❞❡❛❞❞✐✐♦♥✳❆♥❛❧❣❡❜❛✐  ✉♥✐❛❧  ✐❢✐❤❛❛♠✉❧✐♣❧✐❝❛✐✈❡✐❞❡♥✐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✇❡❛❦✲   ∗ ❝♦♠♣❛❝✳

Dec 6, 2018

#### 20-7-2018 Mohamed Ellabib CV

Dec 6, 2018
Search
Similar documents

View more...
Tags

Related Search