I read the following two sentences in a book about C* algebras:
C*-algebra theory can be thought of as "infinite-dimensional real analysis."
For instance, the study of linear functionals on C*-algebras is "non-commutative measure theory."
I know some basic theory about C*-algebras but I can't see why the above sentences make sense. I would appreciate any clarification.
Any commutative C$^*$-algebra is of the form $C_0(X)$, with $X$ a locally compact Hausdorff space (this is due to Gelfand-Naimark, many years ago).
The Riesz-Markov theorem tells you that the dual of $C(X)$ consists of the set of all complex Borel measures on $X$, where to each measure $\mu$ you associate the functional $f\longmapsto \int_X f\,d\mu$.
There is not a lot more than that, really. In the non-commutative case the above results make no sense, but one can still think of the analogy. Two locally compact Hausdorff spaces $X$ and $Y$ are homeomorphic if and only if $C_0(X)$ and $C_0(Y)$ are isomorphic as C$^*$-algebras. So in a sense, you can think of abelian C$^*$-algebras as an algebraic view of topological spaces. One then goes non-commutative, where you still have C$^*$-algebras, and it is sometimes useful to think you are pushing the analogy.
With measure theory it is the same. The Riesz-Markov theorem tells you that you can see the positive measures on a locally compact Hausdorff space $X$ as the positive bounded linear functionals over $C_0(X)$. In the non-commutative case we don't have a topological space but we still have C$^*$-algebras and their duals.