Given an $*$-algebra, when I can construct a codimension 1 subalgbera?

Question

For the case of finite type algebras, the question is answered easily by Proposition 1.1.1 in the first Kadison-Ringrose book on operator algebras, and the answer is obviously yes. Let's stick with countably type algebras. Is there any chance to construct a subalgebra of codimension $1$? In other words, viewing the algebra as a vector space, this would mean that the notion of hyperplane is immediately extendible from finite dimensional algebras to infinite dimensional algebras and, furthermore such an hyperplane would inherit a *-algebra structure. Can someone give me an example in the case of the countable type algebra $\cal K(H)$ of compact operators over an infinite dimensional Hilbert space $H$? I there any chance to construct such objects in other infinite non-countable type algebras?

Answer 1

The result you quote is a result about vector spaces, not about algebras.

For $*$-algebras, what you want fails even in finite-dimension. The easiest example is the fact that $M_2(\mathbb C)$, of dimension $4$, does not admit a $*$-subalgebra of dimension $3$. For a $3$-dimensional $*$-subalgebra of a C$*$-algebra is necessarily isomorphic to $\mathbb C^3$; in particular it has three pairwise orthogonal projections, which is impossible in $M_2(\mathbb C)$.