Timeline for Gelfand duality in NCG
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
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| Feb 14, 2010 at 7:25 | comment | added | Jonas Meyer | Roger Liu: Theo Johnson-Freyd has typed notes for a C*-algebras class from Rieffel, available here: math.berkeley.edu/~theojf/CstarAlgebras.pdf. | |
| Feb 14, 2010 at 6:48 | comment | added | Roger Liu | Thanks, do you have any electronic material on the lectures that Professor Rieffel gave? | |
| Jan 27, 2010 at 0:19 | history | edited | Dave Penneys | CC BY-SA 2.5 |
deleted 48 characters in body
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| Jan 27, 2010 at 0:18 | comment | added | Dave Penneys | thanks! i will edit the answer to refer to your comment as a counterexample. | |
| Jan 27, 2010 at 0:00 | comment | added | Jonas Meyer | By convention, zero representations don't count as irreducible, or equivalently primitive ideals are assumed to be proper. This is analogous to prime ideals being defined to be proper in ring theory, and for similar reasons. Including the algebra as a primitive ideal causes a problem in the definition of the topology; the empty set wouldn't be closed. (And including it would provide no information.) Not counting the zero rep as irreducible is often only implicitly assumed, which can be annoying, but for example one wants the primitive ideal space of C([0,1]) to just be [0,1]. | |
| Jan 26, 2010 at 23:36 | comment | added | Dave Penneys | Doesn't the primitive ideal space of $B_0(H)$ consist of two points $\\{\\{0\\},B_0(H)\\}$, as the trivial representation is irreducible as well? | |
| Jan 26, 2010 at 9:22 | comment | added | Jonas Meyer | I do have one small quibble (which I am posting as a new comment so that it may later be deleted): The primitive ideal space of the compacts is {{0}}, so it is Hausdorff. An example that comes to mind is the C*-algebra generated by the unilateral shift, whose quotient by the compacts is isomorphic to the continuous functions on the circle. The primitive ideal space "is" the circle unioned with {{0}}, and the closure of {{0}} is the whole space. | |
| Jan 26, 2010 at 9:09 | comment | added | Jonas Meyer | Nice answer, and thanks for the compliment. I wasn't careful with set theoretic considerations when I wrote "space of equivalence classes of irreducible representations" (which of course can be made precise). | |
| Jan 26, 2010 at 0:47 | history | answered | Dave Penneys | CC BY-SA 2.5 |