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Have a look at how the Hirzebruch-Riemann-Roch can be deduced as a special case of the Atiyah-Singer index theorem. The idea is to consider the hodge operator $\overline{\partial} + \overline{\partia …

Set $A = L^1([0, \infty))$, equipped with the structure of a Banach $*$-algebra via convolution. The spectrum of this algebra is the half plane $\text{Re}(z) \geq 0$, and the Gelfand transform is the …

I'm can't claim to have studied the relevant history in a lot of detail, but count me a skeptic of Landsman's claim. Let's take this little paper and the companion that it cites as a test case, which …

I think this all falls out of general nonsense with group algebras. Given a locally compact group $G$ there is a homomorphism of Banach algebras $\mathcal{F} \colon L^1(G) \to C_0(\hat{G})$ where: …

There are a number of ways to do this calculation, but at risk of shamelessly plugging my own work there is a nice way to see it using a Mayer-Vietoris principle. Decompose $\mathbb{R}$ as the union …

Perhaps the simplest answer is to use the spectral theorem: $L^2(S)$ decomposes as the orthogonal direct sum of $D$-eigenspaces, and $f(D)$ acts on each $\lambda$-eigenspace as multiplication by $f(\l …

If you're willing to compromise on the operator algebras part then "Introduction to Hilbert Spaces with Applications" is close to optimal. It includes not only a detailed discussion of the spectral t …

Maybe it will help to see how the algebra $B$ is a special case of the general construction in $2.\alpha$. The compact manifold is the space $Y = \{a, b\}$ consisting of two points. The open cover o …

This question is more or less a cross post of https://math.stackexchange.com/questions/1218660/orthogonal-functions-with-shrinking-support. Let $X$ be a metric space (compact, if it helps) and let $Y …

EDIT (much later): My answer below is not quite correct, as pointed out by Dan Ramras, because I ignored the importance of base points in the definition of $\pi_n$. See Ramras' answer for the needed …

One of the major geometric applications of the sort of analysis that you describe is to index theory for elliptic operators on manifolds. Using geometry one can often construct a differential operato …

Let $H$ be a separable Hilbert space and let $\{e_i\}$ be an orthonormal basis. My first question is: Is there a probability measure on $B(H)$ such that for $T$ chosen uniformly randomly the matr …

As discussed in the comments, the statement probably needs to be modified in order for $\langle D \rangle^{-n}$ to be defined. I'm guessing that the correct statement should fit into the following fr …

This is a cross-post from MSE: Elementary applications of Krein-Milman. I'm starting to suspect that the question just doesn't really have a great answer, it's worth a try. Recall that the Krein-Milm …

I have posed this question to some experts at my university who would probably know the answer if there were a complete one, so my expectations are limited. It's possible that the question deserves t …