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Math Help - Calkin-Algebra Definition

  1. #1
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    Calkin-Algebra Definition

    Hello,

    i want to grasp the notion of a calkin-algebra. I have found two definitions, one of my professor and one in a book about functional-anaylsis.

    This definition (of my prof.) i can't understand:
    C(E):=B(E)/K(E) is the definiton of the Calkin algebra. He says, C(E) becomes a B-Algebra with the norm \|T+ K(E)\| = ind_{K\in K(E)} \|T+K\|_{B(E)}

    I think "ind" means here index. But what does it this definition actually mean?
    It makes no sense to me!

    The other one in my book is more reasonable. It says, that C(E) becomes a B-Algebra with the canonical norm induced from B(E). That is \|T+ K(E)\|:=inf_{K \in K(E)} \|T-K\|.

    Or is the first one just a typo?

    Regards
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  2. #2
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    Re: Calkin-Algebra Definition

    Quote Originally Posted by Sogan View Post
    Hello,

    i want to grasp the notion of a calkin-algebra. I have found two definitions, one of my professor and one in a book about functional-anaylsis.

    This definition (of my prof.) i can't understand:
    C(E):=B(E)/K(E) is the definiton of the Calkin algebra. He says, C(E) becomes a B-Algebra with the norm \|T+ K(E)\| = ind_{K\in K(E)} \|T+K\|_{B(E)}

    I think "ind" means here index. But what does it this definition actually mean?
    It makes no sense to me!

    The other one in my book is more reasonable. It says, that C(E) becomes a B-Algebra with the canonical norm induced from B(E). That is \|T+ K(E)\|:=inf_{K \in K(E)} \|T-K\|.

    Or is the first one just a typo?

    Regards
    ind is surely a typo for inf. The Calkin algebra is the quotient of B(E) by the closed ideal of compact operators. So the norm of the coset T+K(E) is equal to \textstyle\inf_{K\in K(E)}\|T+K\|. (Of course, you could equally well write TK in place of T+K there, since K is compact if and only if K is compact.)
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