# Questions tagged [jordan-algebras]

A Jordan algebra is an algebra with multiplication satisfying two axioms (J1) xy=yx (J2) (xxy)x=xx(yx). They were defined in 1934 by Jordan, von Neuman, and Wigner seeking a better formalism for quantum mechanics. In 1966 McCrimmon proposed to analyze instead the operator Ux(y)=xyx, which lead to a notion of quadratic Jordan algebras. Three axioms (Q1, Q2, Q3) of these objects can be found below.

27 questions
Filter by
Sorted by
Tagged with
0answers
112 views

1answer
165 views

### Left- (right-) multiplications of an algebra that are derivations

Let us say that $A$ is a (finite-dimensional) algebra over a field of characteristic zero. We can assume commutativity but not associativity, if that makes it easier. Indeed, I am mostly interested in ...
1answer
42 views

### Relating classic spectral decomposition with Euclidean Jordan algebras

I'm currently getting into studying optimization problems over symmetric cones (NSCP) and I'm having some trouble to understand something. Let me first give some context, sorry if it is repetitive to ...
0answers
49 views

### Good source for Jordan Fréchet algebras

Is there any good source for Jordan Fréchet (or more generally, Jordan locally convex) algebras? I'm looking for something on the level similar to the level of the book "Banach and Locally Convex ...
0answers
81 views

### Alfsen Shultz theorem-the space of states of $C^*$-algebra depends only on Jordan structure

According to the article on nLab the Alfsen Shultz theorem states that the space of states of a given $C^*$-algebra depends on somehow weaker structure namely on the so called Jordan algebra structure....
0answers
66 views

### Formally real non-Jordan algebras

Jordan, von Neumann and Wigner  showed that for any finite-dimensional real vector space $A$ with a bilinear commutative power-associative operation $\circ : A \times A \to A$, the formal reality ...
1answer
368 views

### Jordan algebra identities

A Jordan algebra is a vector space with a commutative bilinear operation $\circ$ obeying an identity that's often written as $$(x \circ y) \circ (x \circ x) = x \circ (y \circ (x \circ x)) .$$ ...
2answers
81 views

1answer
92 views

2answers
706 views

### How do Jordan algebras help one understand representations of exceptional Lie algebras?

For this question I'm happy to take the complex numbers as the base field. I've been trying to learn a little bit about the exceptional Lie algebras and for a while they seemed inaccessible. I looked ...