In this paper we analyze states on C*-algebras and their relationship to filter-like structures of projections and positive elements in the unit ball. After developing the basic theory we use this to investigate the Kadison-Singer conjecture, proving its equivalence to an apparently quite weak paving conjecture and the existence of unique maximal centred extensions of projections coming from ultrafilters on the natural numbers. We then prove that Reid's positive answer to this for q-points in fact also holds for rapid p-points, and that maximal centred filters are obtained in this case. We then show that consistently such maximal centred filters do not exist at all meaning that, for every pure state on the Calkin algebra, there exists a pair of projections on which the state is 1, even though the state is bounded strictly below 1 for projections below this pair. Lastly we investigate towers, using cardinal invariant equalities to construct towers on the natural numbers that do and do not remain towers when canonically embedded into the Calkin algebra. Finally we show that consistently all towers on the natural numbers remain towers under this embedding.

Let $A$ be an AI algebra; that is, $A$ is the $\mbox{C}^{*}$-algebra inductive limit of a sequence $$ A_{1}\stackrel{\phi_{1,2}}{\longrightarrow}A_{2}\stackrel{\phi_{2,3}}{\longrightarrow}A_{3} \longrightarrow\cdots\longrightarrow A_{n}\longrightarrow\cdots, $$ where $A_{n}=\bigoplus_{i=1}^{k_n}M_{[n,i]}(C(X^{i}_n))$, $X^{i}_n$ are $[0,1]$, $k_n$, and $[n,i]$ are positive integers. Suppose that $A$ has the ideal property: each closed two-sided ideal of $A$ is generated by the projections inside the ideal, as a closed two-sided ideal. In this article, we give a complete classification of AI algebras with the ideal property.

Every norm $\nu$ on $\mathbf{C}^n$ induces two norm numerical ranges on the algebra $M_n$ of all $n\times n$ complex matrices, the spatial numerical range $$ W(A)= \{x^*Ay : x, y \in \mathbf{C}^n,\nu^D(x) = \nu(y) = x^*y = 1\}, $$ where $\nu^D$ is the norm dual to $\nu$, and the algebra numerical range $$ V(A) = \{ f(A) : f \in \mathcal{S} \}, $$ where $\mathcal{S}$ is the set of states on the normed algebra $M_n$ under the operator norm induced by $\nu$. For a symmetric norm $\nu$, we identify all linear maps on $M_n$ that preserve either one of the two norm numerical ranges or the set of states or vector states. We also identify the numerical radius isometries, {\it i.e.}, linear maps that preserve the (one) numerical radius induced by either numerical range. In particular, it is shown that if $\nu$ is not the $\ell_1$, $\ell_2$, or $\ell_\infty$ norms, then the linear maps that preserve either numerical range or either set of states are ``inner'', {\it i.e.}, of the form $A\mapsto Q^*AQ$, where $Q$ is a product of a diagonal unitary matrix and a permutation matrix and the numerical radius isometries are unimodular scalar multiples of such inner maps. For the $\ell_1$ and the $\ell_\infty$ norms, the results are quite different.