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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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    [In the above comment I should have written '*complete* $C^*$-norm'.]2012-02-15

1 Answers 1

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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.