23
votes
Accepted
Are there nice isomorphisms $\operatorname{S}^2(k^n)\cong\Lambda^2(k^{n+1})$?
Let $E$ be a $2$-dimensional $k$-vector space. The Wronksian isomorphism is an isomorphism of $\mathrm{SL}(E)$-modules $\bigwedge^m \mathrm{S}^{m+r-1}(E)\cong \mathrm{S}^m \mathrm{S}^r(E) $. It is ...
- 11.2k
11
votes
Jordan algebra identities
This identity (*) is, indeed, true, and is, in fact, a step in one of the standard ways to prove that Jordan algebras are power-associative: see McCrimmon's 2004 book A Taste of Jordan Algebras, ...
- 32.4k
9
votes
Accepted
Left- (right-) multiplications of an algebra that are derivations
An algebra whose (left) multiplications are derivations is referred to as a (left) Leibniz algebra (or Loday algebra). There is a large literature about this class of non-associative algebras. See e....
- 5,878
5
votes
Can the Banach algebra structure on $B(E)$ be (almost) retrieved from its Banach space structure?
This is by no means a complete answer, but a series of bits that would be too long for the comment space.
By an old result of Eidelheit, if $\phi:B(E)\to B(F)$ is a Banach algebra isomorphism onto $B(...
- 2,605
5
votes
Diagonalization of octonionic Hermitian matrices of size $2\times 2$
Yes, in fact, any $2$-by-$2$ octonionic Hermitian matrix is equivalent under the natural $\mathrm{Spin}(9)$ action to a diagonal $2$-by-$2$ octonionic Hermitian matrix.
This follows from the well-...
- 108k
4
votes
Semisimple Lie algebra and convexity
I'm not super familiar with the theory of Jordan algebras or self-dual homogeneous cones so I might be barking up the wrong tree with this answer but here goes. I don't think the cones I'm about to ...
- 954
4
votes
Jordan isomorphisms of type I von Neumann algebras
I believe two type I von neumann algebras are Jordan isomorphic if and only if they are $*$-isomorphic. Jordan isomorphism preserve order, so if two von Neumann algebras are Jordan isomorphic they are ...
- 42.8k
4
votes
Jordan isomorphisms of type I von Neumann algebras
I have not read through all the details carefully, but I think your questions can probably be answered using the results in Kadison's 1951 paper "On isometries of operator algebras". That paper ...
- 25.8k
4
votes
Irreducible $G$-representations with unital algebra structure
The unit poses a problem here. If you require the multiplication map $\cdot: V\otimes V\to V$ to be $G$-equivariant, then this means $gv\cdot gw = g(v\cdot w)$. Letting $w=1_V$ and using that $g:V\to ...
- 9,650
3
votes
Determinants in Jordan algebras of Euclidean type
We denote by $J_{3}(\mathbb{O})$ be the space:
$$ J_{3}(\mathbb{O}) = \left\{ \begin{pmatrix} \lambda_1 & a_1 & \overline{a_2} \\ \overline{a_1} & \lambda_2 & a_3 \\ a_2 & \...
- 7,300
3
votes
Can the Banach algebra structure on $B(E)$ be (almost) retrieved from its Banach space structure?
It seems that not much is known. I have found a description of isometries on $B(c_0)$ (and some other relevant results) here. A surjective isometry
$T:B(c_0)\to B(c_0)$ is of the form $T(a)=UaV$ where ...
- 151
3
votes
Automorphism group of formally real Jordan algebras of hermitian matrices
In a simple Euclidean Jordan algebra, the Jordan-automorphism group of
the algebra is the subgroup of the cone of squares' automorphism group
that fixes the Jordan unit element. This is written down, ...
- 111
2
votes
Determinants in Jordan algebras of Euclidean type
Makt wrote:
As far as I heard (I am not sure about the precise statement) there is a classification of simple Jordan algebras of Euclidean type.
Yes, in this paper
Pascual Jordan, John von Neumann ...
- 22.2k
2
votes
Accepted
Is the generated subalgebra of a subset of pairwise operator-commuting element in a JB-algebra associative?
Yes, this is true. I couldn't find any proof of the statement you quote in the article, and even after emailing the authors I didn't get any wiser, so I decided to work out the details myself, see my ...
- 136
2
votes
Is it true that any semisimple Jordan algebra has the unit element?
As I was suspecting, the existence of a maximal idempotent is not assured in an infinite-dimensional simple Jordan algebra.
In fact the theorem is valid only in the finite-dimensional case. For ...
- 295
1
vote
Degree 8 multilinear operations on Jordan algebras
I managed to run Albert on a very powerful computer at work, and the computation of the desired dimension converged: it seems equal to 19089. I would very much like to confirm that it is correct (I am ...
- 16.9k
1
vote
Are the automorphism groups of simple symmetric cones algebraic groups?
I think I figure a proof after discussing with Yu Zhao.
First of all, by Koecher-Vinberg theorem, we can put a structure of Euclidean Jordan algebra on $V$ so that the closure $\bar{C}$ of $C$ is the ...
- 329
1
vote
Automorphism group of formally real Jordan algebras of hermitian matrices
To expand on Michael Orlitzky's answer, unpacking the notation the claim is that, with $h_n(R)$ the $n\times n$ Jordan algebra of Hermitian matrices is
$$ \begin{aligned} \mathrm{Aut}(h_n(\mathbb{R})) ...
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