User:DiegoJaureRomero/sandbox

We say that the unital $C^{*}$-algebra ${\mathfrak {C}}$ is ${\textsf {G}}$-graded if there is a family $\{{\mathfrak {C}}_{g}\,|\,g\in {\textsf {G}}\}$ of closed subspaces of ${\mathfrak {C}}$ such that the algebraic direct sum $\bigoplus _{g\in {\textsf {G}}}{\mathfrak {C}}_{g}$ is dense in ${\mathfrak {C}}$ and such that:

• (i) ${\mathfrak {C}}_{g}{\mathfrak {C}}_{h}\subset {\mathfrak {C}}_{gh}$ for every $g,h\in {\textsf {G}}$,
• (ii) ${\mathfrak {C}}_{g}^{*}\subset {\mathfrak {C}}_{g^{-1}}$ for every $g\in {\textsf {G}}$.

For every $h\in G$, we denote by $P_{h}:\bigoplus _{g\in {\textsf {G}}}{\mathfrak {C}}_{g}\to {\mathfrak {C}}_{h}$ the canonical projection (in general it might not be continuous). If $P_{e}$ is continuous, ${\mathfrak {C}}$ is a topologically ${\textsf {G}}$-graded $C^{*}$-algebra. It is usually writen:

${\mathfrak {C}}={\widetilde {\bigoplus }}_{g\in {\textsf {G}}}{\mathfrak {C}}_{g}.$