Intereting Posts

On sort-of-linear functions
field of algebraic numbers
Stopping time question $\sigma$
Find the first few Legendre polynomials without using Rodrigues' formula
Eigenvalues of product of a matrix and a diagonal matrix
Show that, for any $\epsilon>0$, there exist two rationals such that $q < x < q'$ and $|q-q'|<\epsilon$
Meaning of $\int\mathop{}\!\mathrm{d}^4x$
$f(x-y)$ considered as a function of $(x,y)\in \mathbb{R^{2n}}$ is measurable if $f$ is measurable
Is every irrational number normal in at least one base?
counting Number of matrices
Integers and integer functions
Way to show $x^n + y^n = z^n$ factorises as $(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n$
Analytic solution to the one-compartment model
Solving $x$ for $y = x^x$ using a normal scientific calculator (no native Lambert W function)?
What is a way to do this combinatorics problem that could generalize to do any of problems similar to this but with more path?

We know that there are many matrix representations of the field $\mathbb{C}$.

For $2 \times 2$ real entries matrices, e.g., all the subrings of $M(2,\mathbb{R})$ generated by $I$ and a matrix $J$ such that $J^2=-I$ are matrix representations of $\mathbb{C}$.

This means that any matrix of the form

$$

J=\left[

\begin {array}{cccc}

a&b\\

c&-a

\end{array}

\right]

\qquad bc \le-1 \qquad a=\sqrt{-1-bc}

$$

is a possible representation of the imaginary unity $i$ and a complex number $z=x+iy$ is represented by the matrix $M(z)=Z=xI+yJ$ I think that this fact can be see as a consequence of the Artinâ€“Wedderburn theorem about the matrix representation of rings ( see my answer to: What does it mean to represent a number in term of a $2\times2$ matrix?),

that generalize such kind of representation to all semisimple rings.

Now, consider $\mathbb{C}$ as a $^*$Algebra (with complex conjugation as the involution). If we search a $^*$Algebra isomorphism such that $M(\bar z)=M^T(z)$ (where the involution for the matrix ring is the transpose), we see that there are only two possible representations, with :

$$

J=\left[

\begin {array}{cccc}

0&1\\

-1&0

\end{array}

\right]

\qquad \mbox{or}\qquad

J=\left[

\begin {array}{cccc}

0&-1\\

1&0

\end{array}

\right]

$$

So we have a great restriction of the possible representations.

- $C^*$-algebra which is also a Hilbert space?
- strictly positive elements in $C^*$-algebra
- Unitisation of $C^{*}$-algebras via double centralizers
- Why is $\overline{B(l^2)\odot B(l^2)}^{\| \enspace \|_{op}}\neq B(l^2\otimes l^2)?$
- There are 3 points in the spectrum of some self-adjoint element of a non-unital C*-algebra.
- equivalent? algebraic definition of a partial isometry in a C*-algebra

My question is if this restriction is a consequence of some general representation theorem for *Algebras in the same sense as the matrix representation of the field $\mathbb{C}$ is a consequence of the A-W theorem.

- For which complex $a,\,b,\,c$ does $(a^b)^c=a^{bc}$ hold?
- Does logging infinitely converge?
- A new imaginary number? $x^c = -x$
- 4 dimensional numbers
- A question about pure state
- Sum of every $k$th binomial coefficient.
- Cross Ratio is positive real if four points on a circle
- Universal $C^*$-algebra of countable family of self-adjoint operators have boundedly complete standard Schauder basis
- In a complex vector space, $\langle Tx,x \rangle=0 \implies T = 0$
- Find exact value of $\cos (\frac{2\pi}{5})$ using complex numbers.

- How to calculate $\int_0^\pi \ln(1+\sin x)\mathrm dx$
- $k$-jet transitivity of diffeomorphism group
- Lie algebra-like structure corresponding to noncrystallographic root systems
- Product of simplicial complexes?
- Properties of Weak Convergence of Probability Measures on Product Spaces
- Inverse Nasty Integral
- Unions of matchings in a bipartite graph
- When is $CaCl(X) \to Pic(X)$ surjective?
- Show that rodriques formula is a linear transformation?
- For an outer measure $m^*$, does $m^*(E\cup A)+m^*(E\cap A) = m^*(E)+m^*(A)$ always hold?
- Looking for a logically coherent book for the self-study of differential equations
- Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Prove that the countable complement topology is not meta compact?
- Equality of measures on a generated $\sigma$-algebra
- A subspace $X$ is closed iff $X =( X^\perp)^\perp$