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Ok it's my turn to post a shameless homework help thread  . In the past I have been really surprised/impressed by the background of folks on this site so I'm hoping you guys can help me out ^^.
I'm trying to write complete solutions to the problems in Ramakrishnan and Valenza's book. The problems are generally approachable but given the sheer volume of questions I know I'm going to get stuck in a few places. The first place I'm getting stuck is on part (b) of problem 6 in Chapter 2.
The book uses the convention that a Banach algebra is a (not necessarily commutative) unital algebra whose elements form a complex Banach space and such that
||a times b|| <= ||a|| times ||b||
for any two elements a and b.
I'm supposed to find an example of a Banach algebra together with two elements a and b such that ab is invertible but ba is not invertible.
It would be awesome if anyone could help!
 
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Wow, that's some pretty advanced shit right there. Our math students usually don't get around to this until 3rd year I think. Maybe late 2nd O.o
Can't help you, and I don't think more than 5 people on this site can. Have you tried using the TL Manpower thread?
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lol i don't even know what objects you are describing
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(I'm not totally comfortable with Banach spaces etc., so I'm a little unsure about definitions. But maybe the following will give you ideas.)
Take X to be the Banach space of (square-summable) complex-valued functions on the set of positive integers, with norm something like
(Sum) |f(k)|^2.
(Square-summable meaning I want to take just those functions with finite norm here.)
Look at the algebra of bounded linear operators on X. Take a and b to be the operators
a(f)(n) = f(2n)
b(f)(2n) = f(n) and b(f)(2n+1) = 0.
I think a and b are bounded. b is not surjective, so ba cannot be invertible. On the other hand, ab is the identity already.
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what class did u learn this? lol I took up to calc 5 and i never heard of banach algebra
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It would be better if your defined your terminology first, since it does not seem to be a sophiscated problem requiring sophiscated techniques...
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On July 26 2009 23:42 AoN.DimSum wrote:
what class did u learn this? lol I took up to calc 5 and i never heard of banach algebra
it's algebra, why on earth would you think to learn that on calculus? these are two separate branches of maths
and I'd say it's like third semester of algebra (at least on math studies in Poland I think)
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It's a graduate level course, ie- you'd be working towards a masters or PhD if you needed to do it.
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On July 26 2009 23:42 AoN.DimSum wrote:
what class did u learn this? lol I took up to calc 5 and i never heard of banach algebra
I'm finding it hard to believe you took math at all
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On July 27 2009 01:04 Divinek wrote:Show nested quote +On July 26 2009 23:42 AoN.DimSum wrote:
what class did u learn this? lol I took up to calc 5 and i never heard of banach algebra I'm finding it hard to believe you took math at all
It's not really surprising that he hasn't heard of it. At the undergraduate level (in USA) no one touches anything more advanced than linear algebra except math majors (and the occasional physics major).
It is somewhat surprising that he's surprised that there are topics he hasn't heard of...but I suppose it's good that he's asking to learn more.
Of course I'm guessing that this was just a personal insult based on external posting interactions but you never know.
More on topic I love these threads - I'm hoping to start a course in abstract algebra soon.
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More on topic I love these threads - I'm hoping to start a course in abstract algebra soon.
word, im a math major and it's always nice to see what i get to look forward to  (also i never get to meet many people outside class who like math). only through calc 4 in my UG degree though, so i can't help you OP lol.
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I have eight semesters and a lifetime after of algebra on me, and i am not so familiar with banach spaces... if you post the definition of a banach space im sure i can help you. But, just for the thing you wrote, real numbers are ok, and integers too... and from there a lot of number sets work with triangle unnequality, if a number set has a norm. it will mostly do that.
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double post... i finished reading your post and i suggest you look into the matrix field.
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Thanks so much incnone! That's a really nice example.
With this problem solved I'm finished with chapters 1-3 out of 7 ^^, but I may have to post again if I get stuck somewhere.
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i took intro to linear alegbra :D Btw im not a math major anyway so stop making fun of me
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On July 26 2009 20:34 incnone wrote:
(I'm not totally comfortable with Banach spaces etc., so I'm a little unsure about definitions. But maybe the following will give you ideas.)
Take X to be the Banach space of (square-summable) complex-valued functions on the set of positive integers, with norm something like
(Sum) |f(k)|^2.
(Square-summable meaning I want to take just those functions with finite norm here.)
Look at the algebra of bounded linear operators on X. Take a and b to be the operators
a(f)(n) = f(2n)
b(f)(2n) = f(n) and b(f)(2n+1) = 0.
I think a and b are bounded. b is not surjective, so ba cannot be invertible. On the other hand, ab is the identity already.
wow, epic first post
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EPT
