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With $M = \mathbb{R}$, not even the uniformly continuous functions are dense in $C_b(\mathbb{R})$. As an example, take any bounded continuous functions with faster and faster oscillation, such as $x\m …

In the unital case, it is a result of Choi that a unital 2-positive map $f:A\to B$ is a $*$-homomorphism if and only if $f(a^2) = f(a)^2$ for all self-adjoint $a\in A$. In the present case, this impli …

Let $A$ be a separable C*-algebra and $J\subseteq A$ a closed two-sided ideal. Does this make $J$ into a complemented subspace of $A$? In other words, does the quotient map $A\to A/J$ have a continuou …

Edit: as pointed out in the comments, the following answers the question for unital C-algebras presented in terms of generators and relations. When I say C-algebra, I really mean unital C*-algebra. It …

Proposition: A real Banach space $X$ is isometrically isomorphic to a real subspace of some $M_n(\mathbb{C})$ if and only if the unit ball $X_1$ is a spectrahedron. This is not particularly deep and a …

Here is a partial answer. As has been hinted at in the comments, one should expect that the space of functions under question strongly depends on the continuity assumption made. For example, the Rényi …

For a counterexample, take $\pi$ to be the regular representation and $\rho$ to be a universal representation. Then both $\pi$ and $\rho$ represent $\ell^1(G)$ faithfully; for the regular representati …

I would like to argue that the space of neural networks is a category with finite products, or more concretely a Lawvere theory. This expresses an important piece of structure, namely how neural netwo …

$\DeclareMathOperator\BV{BV}$A Banach algebra $A$ is called Arens regular if the two canonical multiplications on the double dual $A^{**}$ coincide. Let $\BV(\mathbb{R})$ denote the Banach algebra of …