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It seems to me that to study mathematics is to convert the abstract language into diagrams, graphs and images. It does depend on the subject how much this technique can ease the struggle yet most of the time it is helpful, at least to some extent.

However, studying operator algebras and related fields, I find hardly any images or diagrams can be drawn which makes this subject particularly difficult for me.

I checked almost every book I can find yet none of them contains any technique for visualization.

I wonder whether someone here have advice or experience on visualizing operator algebras. Even just simple diagrams may be helpful.

Many thanks!

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  • $\begingroup$ It seems to me that one should first answer the following question: how do you visualize a ring? And then, of course: how do you visualize a monoid? $\endgroup$
    – Qiaochu Yuan
    Feb 13, 2012 at 18:07
  • $\begingroup$ @QiaochuYuan Good point! So any suggestion on visualizing a ring? $\endgroup$
    – Hui Yu
    Feb 14, 2012 at 2:02
  • $\begingroup$ @QiaochuYuan. Actually I think there are methods to visualize rings. One often uses a diagram to illustrate how elements in a ring interact with others. The difficulty for operator algebras lies in the fact that they have both the algebraic interaction and norm strictures, which can hardly be described in the same image. $\endgroup$
    – Hui Yu
    Feb 14, 2012 at 2:07
  • $\begingroup$ I don't think that's necessarily true. For example, if your operator algebra is a $C^{\ast}$-algebra then the algebra structure already determines the norm. $\endgroup$
    – Qiaochu Yuan
    Feb 14, 2012 at 2:08
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    $\begingroup$ Qiaochu is right (if he meant to include the star operation in the word algebra structure). A $C^*$-algebra is really just a $^*$-algebra which happens to admit a (then necessarily unique) $C^*$-norm. Also, a $^*$-homomorphism between two $C^*$-algebras is automatically continuous. There are other notions of morphisms between $C^*$-algebras, though: for instance, the completely positive maps defined in purely algebraic terms as well, and Woronowicz's morphisms involving multiplier algebras and corresponding to non-proper continuous maps. $\endgroup$
    – Rasmus
    Feb 14, 2012 at 8:47

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I am not sure if this is what you are looking for, but there is a nice class of C*-algebras called graph algebras (standard reference: Iain Raeburn's book) which at this conference were refered to as Operator Algebras We Can See.

A graph algebra is a C*-algebra associated to a graph. By looking at the graph, you can tell many properties of the corresponding C*-algebra. However, not every C*-algebra is a graph algebra.

The short piece Quantum Spaces and Their Noncommutative Topology by Joachim Cuntz in the Notices of the AMS contains several visualizations of ``Quantum Spaces". It might also be worthwhile to take a look at Alain Connes' Noncommutative Geometry.

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