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.

1more comment