I have just started my phd program this semester, I have not worked before on $C^*-$algebras, my master was on Geometry. In my university only reading courses are offered. I have to learn $C^*-$algebras. I have not taken any courses on functional analysis too. I am studying the book " $C^*-$algebras and their automorphism groups" by Pedersen, I usually google each topic and take a look at Book by Murphy too. But I could not find a good resource that give me a good prospective, for example how should I use Gelfand representation in problems. I mean I am learning topics, but I cannot learn with deep depth that I can use stuff like tools in my work. And Also I undestood that Functional analysis is a priority for $C^*-$algebras, but I cannot find a good book for understanding deeply materials like weak and weak$^*$ topology. When I read theorems I understand them, but I could not understand these topics as well as I give ideas for solving theorems on my own, and I cannot enjoy studying. I would appreciate if you recommend me some good books for self studying both Functional analysis and $C^*-$algebras.
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2$\begingroup$ Definitely sounds like you need to read Conway's book, as suggested below. For weak and weak-$\ast$ topologies, have a look at Kesavan's "Functional Analysis". After that, Arveson's book on Spectral theory is worth a read. $\endgroup$– Prahlad VaidyanathanOct 26, 2016 at 9:28
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My suggestion would be to consider Conway's "A Course in Functional Analysis". It even includes a basic chapter on C$^*$-algebras and von Neumann algebras.
Pedersen's book is wonderful in the sense of how concise he could write, but has a couple problems: one is that it is too deep, in a sense, so it is definitely hard for a beginner; and a second problem is that he (naturally) emphasized topics he did research, but that are not of much interest today.
You could also take a look at the first chapter in Davidson's C$^*$-algebras by Example.
answered Oct 23, 2016 at 17:24
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$\begingroup$ Thanks very much for suggestions, is Davidson's book a good apporach for getting familiar with topics of research interest in operator algebras? $\endgroup$ Oct 23, 2016 at 19:20
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1$\begingroup$ It is at least a good sample. It was written almost 30 years ago, though, so of course there are many things that C$^*$-algebra people pay attention to that are not mentioned. But it is important to first a good foundation. For a more comprehensive (but way less friendly) text take a look at Blackadar's Operator Algebras book. Also Rordam's and Lin's. $\endgroup$ Oct 23, 2016 at 19:24