Consider a linear subspace of matrices, $M \subset \textrm{Mat}_n(\mathbb{C})$, which is $\epsilon$-approximately closed under multiplication, i.e. for all $x,y \in M$, there exists $z \in M$ such that $\lVert{xy-z}\rVert < \epsilon \lVert x \rVert \lVert y \rVert$. Assume $M$ is closed under Hermitian adjoints. Then you might say $M$ forms an "$\epsilon$-approximate" $*$-subalgebra of $\textrm{Mat}_n(\mathbb{C})$. (We might also assume $M \ni 1$ and $M$ is approximately closed under inverses.)
- Have approximate $*$-subalgebras of this sort been studied? I am familiar with some notions of Ulam stability, approximate homomorphisms, and approximate representations, but the setup here appears slightly different. ($M$ is not given as the image of some approximate homomorphism from an actual algebra.)
In particular, I'm interested in the following:
- Is it known whether $M$ may be slightly deformed so that it becomes an exact $*$-subalgebra? E.g., does there exist a subspace $N \subset \textrm{Mat}_n(\mathbb{C})$ which is an exact $*$-algebra, and which is nearby to the subspace $M$, with distance controlled by $\epsilon$?
If anyone's interested, I'll offer an example of a conjecture along these lines. For a von Neumann algebra $\mathcal{A}$, call a linear subspace $M \subset \mathcal{A}$ $\epsilon$-approximately multiplicatively closed if it is closed under $*$, contains the identity, and satisfies the property that for all $x,y \in M$, $\exists z \in M$ s.t. $\lVert{xy-z}\rVert < \epsilon \lVert x \rVert \lVert y \rVert$.
Also define a distance between two subspaces: for any linear subspaces $X,Y$, let $d(X,Y)$ be the Hausdorff distance between their unit balls.
Then one might conjecture:
For every $\epsilon>0$, there exists $\delta>0$ such that for any finite-dimensional von Neumann algebra $\mathcal{A}$, for any $\delta$-approximately multiplicatively closed subspace $M \subset \mathcal{A}$, there exists a von Neumann subalgebra $\mathcal{N} \subset \mathcal{A}$ such that $d(M,\mathcal{N})<\epsilon$.
The related literature I know seems to focus on maps between algebras which are approximately multiplicative, e.g. "Approximately Multiplicative Maps Between Banach Algebras" (Johnson, 1986). Here, I'm wondering about linear subspaces that are approximately multiplicatively closed. (So these are like "approximate sub-algebras," rather than "approximate homomorphisms.")