An algebra is a vector space with more to do with. Formally, if \(V\) is a vector space over the field \(\mathbb{F}\), then an algebra \(\mathcal{A}\) over \(\mathbb{F}\) is the vector space together with a bilinear multiplication operation defined on it, \(\mathcal{A} = (F,\cdot)\) with \(\cdot: V\times V \to V\), \((a,b) \mapsto a\cdot b \equiv ab\) for \(a,b\in V\). The multiplication is usually associative, \(a(bc) = (ab)c\), hence usally the algebras are associative algebras.
| A norm $$ | \cdot | \(on\)\mathcal{A}\(is said to be _submultiplicative_ if\) | ab | \leq | a | \cdot | b | \(. If so, the pair\)(\mathcal{A}, | \cdot | )$$ is a normed algebra. |
| If further \(\mathcal{A}\) admits a unit element, satisfying \(a1 = 1a = a\) and $$ | 1 | = 1\(we say that\)\mathcal{A}$$ is a unital normed algebra. |
As a reminder, a normed vector space in which every Cauchy sequence converges (w.r.t. that norm) is called complete. A complete normed vector space is a Banach space (if the norm is induced by an inner product, we have a Hilbert space.) Naturally, a complete normed algebra is called a Banach algebra.
For a unital algebra, say that an element \(a\in\mathcal{A}\) is invertible if there exists a \(b\in\mathcal{A}\) such that \(ab = ba = 1\). The set of invertible elements \(\text{Inv}(\mathcal{A})\) of a unital algebra is a group under multiplication. Now, we can define the spectrum of an element \(a\in\mathcal{A}\) in a unital algebra as follows (note that we don’t need the element to “act” on anything)
\begin{equation} \sigma(a) := {\lambda \in \mathbb{C} \mid \lambda 1 - a \not\in\text{Inv}(\mathcal{A})} \end{equation}
Now, we can impose some more structure, because why not. An involution \(*\) on an algebra is a conjugate-linear map \(a\mapsto a^*\) such that \(a^{**} = a\) and \((ab)^* = b^* a^*\). The pair \((\mathcal{A},*)\) is an involutive-algebra or a \(*\)-algebra.
A Banach algebra is a \(*\)-algebra with a complete, submultiplicative norm which obeys \(||a|| = ||a^*||\). A \(C^*\)-algebra is a Banach \(*\)-algebra obeying \begin{equation} ||a^*a|| = ||a||^2. \end{equation}