What might be called the Gelfand Philosophy notation has become popular in the field of 'Woronowicz' quantum groups in the past decade.
The idea starts with the Gelfand theorem that a commutative $\mathrm{C}^*$-algebra $A$ is isometrically isomorphic to $C_0(X)$, for $X$ a particular topological space, certainly compact and Hausdorff if $A$ is unital, in which case $A\cong C(X)$. Restricting now to the unital case, this Gelfand Philosophy says that a noncommutative $\mathrm{C}^*$-algebra $A$ should be thought of as the algebra of continuous functions on a compact quantum space, $\mathbb{X}$, and so we write $A=:C(\mathbb{X})$. Of course $\mathbb{X}$ is not a set, let alone a topological space but a so-called virtual object. A more radical (if only stylistically) approach is not to use blackboard bold to signify that $\mathbb{X}$ is a virtual object but just to use $X$.
For examples of where this goes I want to talk about compact quantum groups. Compact quantum groups are spoken about through what are called algebras of functions on the compact quantum group. For example, a compact quantum group $G$ might be spoken about via it algebra of continuous functions $C(G)$, a (Woronowicz) $\mathrm{C}^*$-algebra. These algebra of functions have Haar states $h$ that are precisely integration against the Haar measure whenever $C(G)$ is commutative/$G$ is classical. Playing a little fast and loose with issues of null sets, in the classical case we can define
$$L^2(G)=\left\{f\in C(G)\mid \int_G |f(t)|^2\,d\mu(t)<\infty\right\},$$
and via $|f|^2:=f^*f$ for $f\in C(G)$ non-commutative, we can also define $L^2(G)$ spaces for compact quantum groups $G$:
$$L^2(G)=\left\{f\in C(G)\mid h(|f|^2)<\infty\right\}.$$
This kind of thing can go in all kinds of directions, the basic principle is if you have a notation for something in commutative algebras of functions/classical groups that makes sense for noncommutative algebra of functions/quantum groups, use that same notation for quantum groups.
This also allows you to, in a strictly nonsensical but useful way, to talk about the quantum group as if it really exists. For example for finite groups at least, with full algebra of functions $F(G)$, there is a bijective correspondence with representation $G\rightarrow L(V)$ and corepresentations $V\rightarrow V\otimes F(G)$. Through this lens one can talk about a representation of a quantum group or in a similar way the action of a quantum group.
To actually answer the question asked I will quote from a recent preprint:
When, for example with the representation theory of compact quantum groups, the noncommutative theory generalises so nicely from the
commutative theory, it can be useful to refer to a virtual object as
if it exists: this approach helps point towards appropriate
noncommutative definitions, and sometimes even towards results, such
as the Peter-Weyl Theorem, that are true in this larger class of
objects. Even when commutative results do not generalise to this
larger class, the Gelfand Philosophy gives a pleasing notation,
helping readers from the commutative world understand better what is
going on in the noncommutative world.
Some examples of this from my own work on finite quantum groups, say given by a $\mathrm{C}^*$-algebra $A$ include:
- referring to $A$ as the algebra of functions on the finite quantum group $G$, and denote it by $F(G)$
- referring to the unit $1_A$ in the algebra of functions as $\mathbf{1}_G$
- referring to the set of states $\mathcal{S}(A)$ as $M_p(G)$, the set of probability measures on the group
- I have started using $f\in F(G)$ for a general "function" rather than the usual $a\in F(G)$ or before $a\in A$
- I have used the notation $2^G$ for the set of projections in $F(G)$
Some of these are pushing the envelope a little on this notation... we will see what the Reviewers say!
I note in this preprint that:
This philosophical approach ramped up in the 2000s, and into the 2010s, and up to 2020.
HOWEVER, when I got my hand on the 1967 paper of Kac and Paljutkin (highly recommended if you can get a copy), the famous eight-dimensional algebra of functions on a finite quantum group, the smallest of which is neither commutative nor cocommutative, the authors refer to it by $\mathfrak{G}_0$ --- not the algebra but the virtual object!
I assume similar notations are at play in other fields.
