Intereting Posts

Log Sine: $\int_0^\pi \theta^2 \ln^2\big(2\sin\frac{\theta}{2}\big)d \theta.$
Cardinality of $\mathbb{R}$ and $\mathbb{R}^2$
Determine the matrix relative to a given basis
Examples of transfinite induction
Stirling numbers with $k=n-2$
Fourier transform of a compactly supported function
Why is this polynomial irreducible?
Sum of reciprocals of binomial coefficients: $ \sum\limits_{k=0}^{n-1}\dfrac{1}{\binom{n}{k}(n-k)} $
Model existence theorem in set theory
“$n$ is even iff $n^2$ is even” and other simple statements to teach proof-writing
Can we prove that there are countably many isomorphism classes of compact Lie groups without appealing to the classification of simple Lie algebras?
Do men or women have more brothers?
Is there a Möbius torus?
expected life absorbing Markov Chain
The collection of all compact perfect subsets is $G_\delta$ in the hyperspace of all compact subsets

I’m missing something in the Gelfand representation. Let’s just say $\mathfrak{A}$ is a Banach algebra. Then it’s a Banach space, and so we have $\mathfrak{A}^\ast$. The multiplicative linear functionals on $\mathfrak{A}$ are continuous and have norm $\leq 1$, and so still sit inside the unit ball of $\mathfrak{A}$, which is weak-$\ast$ compact. The set of multiplicative linear functionals are weak-$\ast$ closed, so the set of mulitplicative linear functionals is then weak-$\ast$ compact.

Where does not having a unit make the above not work? I always see that the multiplicative linear functionals are locally compact in the non-unital case, but I cannot spot where the unit is needed in the above.

- Wiener's theorem in $\mathbb{R}^n$
- C* algebra of bounded Borel functions
- Quaternions as a counterexample to the Gelfand–Mazur theorem
- Why is $GL(B)$ a Banach Lie Group?
- Reference request: Fourier and Fourier-Stieltjes algebras
- Fourier transform as a Gelfand transform

- Strictly convex Inequality in $l^p$
- Dirac's delta in 3 dimensions: proof of $\nabla^2(\|\boldsymbol{x}-\boldsymbol{x}_0\|^{-1})=-4\pi\delta(\boldsymbol{x}-\boldsymbol{x}_0)$
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- Dual of a dual cone
- The dual of subspace of a normed space is a quotient of dual: $X' / U^\perp \cong U'$
- A subspace $X$ is closed iff $X =( X^\perp)^\perp$
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- A Challenge on linear functional and bounding property
- If every $x\in X$ is uniquely $x=y+z$ then $\|z\|+\|y\|\leq C\|x\|$
- Does a Fourier transformation on a (pseudo-)Riemannian manifold make sense?

Assume $A$ is non-unital. Denote by $X(A)$ the space of its characters and denote

$$

X(A)_+=X(A)\cup\{0\}

$$

One can show that $X(A)_+$ is weak-$^*$ closed subset of weak-$^*$ compact unit ball of $A^*$. So $X(A)_+$ is weak-$^*$ compact.

One can show that $X(A)_+$ is a one point compactification of $X(A)$, so $X(A)$ is locally compact.

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- In Linear Algebra, what is a vector?
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- Show that $\pi =4-\sum_{n=1}^{\infty }\frac{(n)!(n-1)!}{(2n+1)!}2^{n+1}$
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- Combinatorial Proof
- Elementary proof that $a_n \to a \implies a_n^r \to a^r$ for $r \in \mathbb{Q}$
- Example of topological spaces where sequential continuity does not imply continuity
- How to calculate expected number of trials of this geometric distribution