A \(C^\ast\text{-algebra}\) is a Banach star algebra whose norm satisfies the \(C^\ast\text{-identity}\).

Let \(A\) be a (no necessarily unital) \(C^\ast\text{-algebra}\).

1. The spectral radius of a normal element is equal to the norm: \[ r(a)= \left\| a \right\|\] for all normal \(a \in A\).
2. A \(\ast\text{-homomorphism}\) \(\pi:A \to B\) between \(C^\ast\text{-algebras}\) is contractive, hence continuous.
3. An injective \(\ast\text{-homomorphism}\) \(\pi:A \to B\) between \(C^\ast\text{-algebras}\) is isometric.

1. Notes on Homology for higher rank graphs and twisted C star algebras