Question 1. For $d > 1$, is there a nonzero commutative and associative product on the real $d \times d$ symmetric matrces $S^d$ preserving the real symmetric $d \times d$ positive semidefinite matrices $S_+^d$, that is, is there a bi-additive map $$ \odot \colon S^d \times S^d \to S^d $$ such that $A \odot B = B \odot A$ and $A \odot (B \odot C) = (A \odot B) \odot C$ for all $A, B, C \in S_+^d$ with is also compatible with nonnegative scalars, that is, $(r A) \odot B = r (A \odot B)$ and so on for all $r \ge 0$ and such that $\odot(S_+^d, S_+^d) \subset S_+^d$?
Question 2. If the answer to Q1 is yes, how many are there (up to scalar factors)?
(Thanks to @Martin Argerami for improving this question.)
I am asking this to determine whether it is possible to give $S^d$ a meaningful structure of a real algebra, which acts a certain sense nicely on positive semidefinite matrices.
Examples. (see also: Bernhard Burgeth et al. “A generic approach to diffusion filtering of matrix-fields”)
- The usual matrix product is not commutative, but associative. Furthermore, it does not preserve symmetry, that is, for $A, B \in S_+^d$ the product $A B$ is only symmetric if $A B = B A$.
- The symmetric (or anticommutator) product $A \cdot B := \frac{1}{2}(A B + B A)$ is commutative and preserves symmetry, but it is not associative and does not preserve positive semidefiniteness (e.g. $A = \tiny\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ and $B = \tiny\begin{pmatrix} 4 & - 1 \\ -1 & 1 \end{pmatrix}$.)
- The product $(A, B) \mapsto A^{\frac{1}{2}} B A^{\frac{1}{2}}$ preserves symmetric positive semidefiniteness but is neither commutative nor associative.
- There is also the Feynman-Kubi product $(A, B) \mapsto \int_{0}^{1} A^{t} B A^{1 - t} \; \text{d}t$, which probably suffers the same drawbacks as the previously mentioned product.
Update.
- @user1551 suggested the Hadamard product $\circ$, i.e the entry-wise product of matrices. It clearly preserves symmetry and is commutative and associative. Due to $$\det(A \circ B) \ge \prod_{k = 1}^{d} \lambda_k(A) \lambda_k(B) \qquad \forall A, B \in S_+^d$$ (by theorem 3 in On majorization and Schur products by Bapat and Sunder), I guess that the Hadamard product preserves positive semidefiniteness as well.
I hope that the answer to the above question(s) can help me find the "obvious" (as stated in Remark 5.2 of The Schrödinger problem on the non-commutative Fisher-Rao space by Léonard Monsaingeon & Dmitry Vorotnikov) C$^*$-algebra structure on $C_b(\Omega;H)$, the continuous bounded function on a locally compact separable metric space $\Omega$ with values in the Hermitian matrices $H$.
Remark. Other products on matrices like the Kronecker or the Khatri–Rao product are not permissible, since the dimension of the product matrix is strictly larger than the ones of the matrices to be multiplied.