Banach star-algebra without a required unit

A complex Banach $\ast$-algebra without a required unit is a possibly nonunital complex Banach algebra $A$ together with a map $a\mapsto a^{\ast }$ of $A$ into itself, its involution, such that:

1. $A$ is a complex vector space with an associative complex-bilinear multiplication and a submultiplicative norm under which $A$ is complete (C star algebra, Banach space, A norm on a real vector space, the induced metric, and the dictionary with the metric axioms);
2. the involution is conjugate-linear, $(\alpha a+\beta b{)}^{\ast }=\bar{\alpha }{a}^{\ast }+\bar{\beta }{b}^{\ast }$ for all $\alpha ,\beta \in \mathbb{C}$ and $a,b\in A$;
3. the involution is involutive, $(a^{\ast }{)}^{\ast }=a$ for every $a\in A$;
4. the involution reverses products, $(ab{)}^{\ast }=b^{\ast }{a}^{\ast }$ for all $a,b\in A$;
5. the involution is continuous; when in addition $\mathrm{\parallel }{a}^{\ast }\mathrm{\parallel }=\mathrm{\parallel }a\mathrm{\parallel }$ for every $a\in A$ one says the involution is isometric.

No multiplicative identity is assumed, and none is asserted to exist; when $A$ does have a two-sided identity $1$ with $\mathrm{\parallel }1\mathrm{\parallel }=1$, $A$ is a unital Banach algebra in the sense of Unital Banach algebra and the involution axioms above turn it into a unital Banach $\ast$-algebra.

No ${\mathrm{C}}^{\ast }$-identity. The defining identity $\mathrm{\parallel }{a}^{\ast }a\mathrm{\parallel }=\mathrm{\parallel }a{\mathrm{\parallel }}^2$ of a ${\mathrm{C}}^{\ast }$-algebra (C star algebra) is not imposed here and is not available for the algebras covered by this definition: $L^1(G)$ for a non-discrete $G$ fails it, and its proof below uses only the axioms 1–5. Two properties of this definition are deliberately weaker than the ${\mathrm{C}}^{\ast }$ case: the involution is not assumed isometric, and no norm-uniqueness statement is imported.