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 $||{xy-z}|| < \epsilon ||xy||$. 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.)

  1. 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:

  1. 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. $||xy-z||<\epsilon||xy||$.

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.")

  • $\begingroup$ Not that I would know how to answer, but I think you need to make your question more precise. Concretely, what does "nearby... with distance controlleld by $\epsilon$" mean? The problem I see is that if you just require something like the Hausdorff distance, you are not asking $N$ to mimic the pseudo algebraic structure of $M$. So maybe you want a linear isomorphism that is "almost multiplicative"? This sounds more reasonable to me, but then I wouldn't know exactly how to quantify what "almost multiplicative" means. $\endgroup$ Apr 13 '21 at 2:17
  • $\begingroup$ Thanks Martin -- I'm asking for the existence of a nearby subspace $N$ that's a true algebra, but without further requiring that $N$ mimics the (approximate) algebraic structure of $M$, except insofar as $N$ is nearby $M$. That's already a non-trivial request, I think. Were you suggesting asking for something stronger, or using a stronger assumption? $\endgroup$ Apr 13 '21 at 5:15
  • $\begingroup$ I would like to see a concrete meaning for "nearby". Say, why would $N=M_n(\mathbb C)$ not be acceptable? $\endgroup$ Apr 13 '21 at 8:10
  • $\begingroup$ Should have said, I edited the question in response to you: by nearby subspaces, I mean they have small Hausdorff distance between their unit balls (in operator norm). Then a proper subspace will not be "close" to the ambient space. $\endgroup$ Apr 13 '21 at 8:50
  • $\begingroup$ My bad, I had not seen it. $\endgroup$ Apr 13 '21 at 9:19

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