A bit unrelated, but nonetheless interesting:

A finite von Neumann algebra is a von Neumann algebra together with a trace, i.e., a linear functional which is

• positive ($tr(a^*a)\geq 0$ for all $a$)
• faithful ($tr(a^*a)=0$ implies $a=0$)
• normalized ($tr(1)=1$)
• tracial ($tr(ab)=tr(ba)$)

This trace does almost everything for a finite von Neumann algebra. It gives us the standard representation of $M$ on $L^2(M)$, the predual $L^1(M)$, and the noncommutative $L^p$-spaces. In a finite factor (trivial center), it gives us a total ordering on projections (for a $II_1$-factor, it gives a "continuous" version of what Ben is saying). If we have an inclusion of finite von Neumann algebras $N\subset M$, we get a canonical conditional expectation $E\colon M\to N$, which is one of the basic tools. If they are subfactors, if we get a trace on $M_1$, we can iterate the basic construction (nicely that is). The list goes on...

So what does this all have to do with groups? Well von Neumann algebras are exactly the commutants of (unitary) group representations. Moreover, from a unitary group representation, we can form its left regular von Neumann algebra $L(G)$, which is the easiest way to construct an example of a $II_1$-factor (the group must have all infinite conjugacy classes other than that of the identity). Moreover, the study of $II_1$-subfactors generalizes the study of representation theory, including induction-restriction, etc. (see Kate's question).

Basically, every time I see an algebra, I wonder if it has a trace. So from my point of view, or with this hindsight I should say, it's no surprise to me that characters do everything!

$\DeclareMathOperator\tr{tr}$A bit unrelated, but nonetheless interesting:

A finite von Neumann algebra is a von Neumann algebra together with a trace, i.e., a linear functional which is

• positive ($\tr(a^*a)\geq 0$ for all $a$)
• faithful ($\tr(a^*a)=0$ implies $a=0$)
• normalized ($\tr(1)=1$)
• tracial ($\tr(ab)=\tr(ba)$).

This trace does almost everything for a finite von Neumann algebra. It gives us the standard representation of $M$ on $L^2(M)$, the predual $L^1(M)$, and the noncommutative $L^p$-spaces. In a finite factor (trivial center), it gives us a total ordering on projections (for a $II_1$-factor, it gives a "continuous" version of what Ben is saying). If we have an inclusion of finite von Neumann algebras $N\subset M$, we get a canonical conditional expectation $E\colon M\to N$, which is one of the basic tools. If they are subfactors, if we get a trace on $M_1$, we can iterate the basic construction (nicely that is). The list goes on….

So what does this all have to do with groups? Well von Neumann algebras are exactly the commutants of (unitary) group representations. Moreover, from a unitary group representation, we can form its left regular von Neumann algebra $L(G)$, which is the easiest way to construct an example of a $II_1$-factor (the group must have all infinite conjugacy classes other than that of the identity). Moreover, the study of $II_1$-subfactors generalizes the study of representation theory, including induction-restriction, etc. (see Kate's question Why are subfactors interesting?).

Basically, every time I see an algebra, I wonder if it has a trace. So from my point of view, or with this hindsight I should say, it's no surprise to me that characters do everything!

A bit unrelated, but nonetheless interesting:

A finite von Neumann algebra is a von Neumann algebra together with a trace, i.e., a linear functional which is

• positive ($tr(a^*a)\geq 0$ for all $a$)
• faithful ($tr(a^*a)=0$ implies $a=0$)
• normalized ($tr(1)=1$)
• tracial ($tr(ab)=tr(ba)$)

This trace does almost everything for a finite von Neumann algebra. It gives us the standard representation of $M$ on $L^2(M)$, the predual $L^1(M)$, and the noncommutative $L^p$-spaces. In a finite factor (trivial center), it gives us a total ordering on projections (for a $II_1$-factor, it gives a "continuous" version of what Ben is saying). If we have an inclusion of finite von Neumann algebras $N\subset M$, we get a canonical conditional expectation $E\colon M\to N$, which is one of the basic tools. If they are subfactors, if we get a trace on $M_1$, we can iterate the basic construction (nicely that is). The list goes on...

So what does this all have to do with groups? Well von Neumann algebras are exactly the commutants of (unitary) group representations. Moreover, from a unitary group representation, we can form its left regular von Neumann algebra $L(G)$, which is the easiest way to construct an example of a $II_1$-factor (the group must have all infinite conjugacy classes other than that of the identity). Moreover, the study of $II_1$-subfactors generalizes the study of representation theory, including induction-restriction, etc. (see Kate's question).

Basically, every time I see an algebra, I wonder if it has a trace. So from my point of view, or with this hindsight I should say, it's no surprise to me that characters do everything!

A bit unrelated, but nonetheless interesting:

A finite von Neumann algebra is a von Neumann algebra together with a trace, i.e., a linear functional which is

• positive ($tr(a^*a)\geq 0$ for all $a$)
• faithful ($tr(a^*a)=0$ implies $a=0$)
• normalized ($tr(1)=1$)
• tracial ($tr(ab)=tr(ba)$)

This trace does almost everything for a finite von Neumann algebra. It gives us the standard representation of $M$ on $L^2(M)$, the predual $L^1(M)$, and the noncommutative $L^p$-spaces. In a finite factor (trivial center), it gives us a total ordering on projections (for a $II_1$-factor, it gives a "continuous" version of what Ben is saying). If we have an inclusion of finite von Neumann algebras $N\subset M$, we get a canonical conditional expectation $E\colon M\to N$, which is one of the basic tools. If they are subfactors, if we get a trace on $M_1$, we can iterate the basic construction (nicely that is). The list goes on...

So what does this all have to do with groups? Well von Neumann algebras are exactly the commutants of (unitary) group representations. Moreover, from a unitary group representation, we can form its left regular von Neumann algebra $L(G)$, which is the easiest way to construct an example of a $II_1$-factor (the group must have all infinite conjugacy classes other than that of the identity). Moreover, the study of $II_1$-subfactors generalizes the study of representation theory, including induction-restriction, etc. (see Kate's question).

Basically, every time I see an algebra, I wonder if it has a trace. So from my point of view, or with this hindsight I should say, it's no surprise to me that characters do everything!