Just some notes on the pre-print. I am looking at this paper to better understand this pre-print. In particular I am hoping to learn more about the support of a probability on a quantum group. Flags and notes are added but mistakes are mine alone.

Abstract

From this paper I will look at:

  • lattice operations on \mathcal{I}(G), for G a LCQG (analogues of intersection and generation)

1. Introduction

Idempotent states on quantum groups correspond with “subgroup-like” objects. In this work, on LCQG, the correspondence is with quasi-subgroups (the work of Franz & Skalski the correspondence was with pre-subgroups and group-like projections).

Let us show the kind of thing I am trying to understand better.

Let F(G) be the algebra of function on a finite quantum group. Let \nu,\,\mu\in M_p(G) be concentrated on a pre-subgroup S. We can associate to S a group like projection p_S.

Let, and this is another thing I am trying to understand better, this support, the support of \nu be ‘the smallest’ (?) projection p\in F(G) such that \nu(p)=1. Denote this projection by p_\nu. Define p_\mu similarly. That \mu,\,\nu are concentrated on S is to say that p_\nu\leq p_S and p_\mu\leq p_S.

Define a map T_\nu:F(G)\rightarrow F(G) by 

a\mapsto p_\nu a (or should this be ap_\nu or p_\nu a p_\nu?)

We can decompose, in the finite case, F(G)\cong \text{Im}(T_\nu)\oplus \ker(T_\nu)

Claim: If \nu is concentrated on S\nu(ap_S)=\nu(a)I don’t have a proof but it should fall out of something like p_\nu\leq p_S\Rightarrow \ker p_\nu\subseteq \ker p_S together with the decomposition of F(G) above. It may also require that \int_G is a trace, I don’t know. Something very similar in the preprint.

From here we can do the following. That p_S is a group-like projection means that:

\Delta (p_s)(\mathbf{1}_G\otimes p_S)=p_S\otimes p_S

\Rightarrow \sum p_{S(1)}\otimes (p_{S(2)}p_S)=p_S\otimes p_S

Hit both sides with \nu\times \mu to get:

\sum \nu(p_{S(1)})\mu(p_{S(2)}p_S)=\nu(p_S)\mu(p_S).

By the fact that \nu,\,\mu are supported on S, the right-hand side equals one, and by the as-yet-unproven claim, we have

\sum \nu(p_{S(1)})\mu(p_{S(2)})=1.

However this is the same as

(\nu\otimes\mu)\Delta(p_S)=1\Rightarrow (\nu\star \mu)(p_S)=1,

in other words p_{\nu\star \mu}\leq p_S, that is \nu\star \mu remains supported on S. As a corollary, a random walk driven by a probability concentrated on a pre-subgroup S\subset G remains concentrated on S.

Back to the paper, the “subgroup-like” objects are called quasi-subgroups. They are related to group-like projections.

I don’t fully have the functional analysis background (to… read this paper?!) to fully understand these objects, but we are denoting by C_0(G) the reduced \mathrm{C}^*-algebra of the locally compact quantum group G. Let C_0^u(G) be the universal \mathrm{C}^*-algebra, and L^\infty(G) be the von Neumann algebra.

Denote the right Haar measure on G by \psi and L^2(G) the GNS Hilbert space of \psi.

A von Neumann subalgebra N\subset L^\infty(G) is a left coideal if \Delta(N)\subset L^\infty(G)\overline{\otimes} N.

2. Idempotent States, Coideals, and Group-Like Projections

Let \omega be a state on C_0^u(G). The dual of this universal C*-algebra is Banach together with the convolution. Denote the set of idempotent states on the universal C*-algebra by \mathcal{I}(G).

Where \mathbb{W} is the half-lifted Kac-Takesaki Operator (??), the formula

E_\omega (x)=\omega\star x=(I\otimes \omega)(\mathbb{W}(x\otimes \mathbf{1})\mathbb{W}^*),

for x\in L^\infty(G), defines a normal conditional expectation on L^\infty(G) and denote by N_\omega its range:

N_\omega=\{E_\omega(x)\,|\,x\in L^\infty(G)\},

which is a left-coideal in L^\infty(G). Let

P_\omega=(I\otimes \omega)\mathbb{W}\in L^\infty(G).

P_\omega us a group-like projection in L^\infty(G) and is the orthogonal projection onto L^2(N_\omega).

There is a bijective correspondence between:

  • idempotent states on the universal C*-algebra
  • left-coideals (+integrable + scaling group invariant)
  • group like projections P\in L^\infty(\widehat{G}) (+ invariant under the scaling group)

\omega \longleftrightarrow N_\omega\longleftrightarrow P_\omega.

Let \mu,\,\nu\in \mathcal{I}(G). We say that \nu dominates \mu if \mu\star \nu=\nu (I would usually be doing something similar with the support, perhaps… perhaps not) and we write \mu\preceq\nu. This is equivalent to E_\mu\circ E_\nu=E_\nu (which could be something to use in the finite/compact case). The convention is the opposite to that of Franz & Skalski so beware!

Lemma 2.1

Let \mu,\,\nu\in \mathcal{I}(G). TFAE

  1. \mu\preceq \nu
  2. E_\mu\circ E_\nu=E_\nu
  3. N_\nu\subset N_\mu
  4. P_\nu\leq P_\mu \bullet

3. The Lattice of Compact Quasi-Subgroups

Let K\subset G be a compact subgroup. This gives a surjective *-homomorphism \pi_K from the universal C*-algebra of G down to the universal C*-algebra of K. We then have

\int_K \circ\, \pi_K\in\mathcal{I}(G).

This is a Haar type idempotent. Classically, all idempotents arise like this but not in the quantum case. In the quantum case let an idempotent correspond to a quasi-subgroup of G. Under this line of thinking, the coideal N_\omega=\{E_\omega(x)\,|\,x\in L^\infty(G)\} corresponds to the von Neumann algebra of essentially bounded functions on the set of cosets of the quasi-subgroup.

3.1 Intersection of quasi-subgroups

Let \omega,\,\mu\in \mathcal{I}(G). The von Neumann algebra generated by N_\omega and N_\mu, N_\omega\vee N_\mu is an (integrable, \tau-invariant) coideal, and thus of the form N_\nu for some \nu\in \mathcal{I}(G). Denote \nu=\omega\wedge \mu, and the corresponding quasi-subgroup as the intersection of the quasi-subgroups given by \omega and \mu. This \wedge operation is commutative and associative.

3.2 Quasi-subgroup generated by two quasi-subgroups

Theorem 3.1

  1. L^2(N_\omega)\cap L^2(N_\mu)=L^2(N_\omega\cap N_\mu)
  2. ((\omega\star \mu)^{\star n})_{n\in\mathbb{N}} is weak*-convergent to \nu\in \mathcal{I}(G)\cup \{0\}=:\mathcal{I}_0(G)
  3. L^2(N_\omega\cap N_\mu)\neq \{0\} iff \nu\neq 0 iff \nu is the idempotent state corresponding to the (\tau-invariant, integrable, left) coideal N_\omega\cap N_\mu

At this point I have no intuition for the following as I don’t understand what \mathbb{W} is… if we denote the \nu above by \omega\vee\mu, this vee is an associative and commutative map from pairs of \mathcal{I}_0(G) idempotents to \mathcal{I}_0(G)… hang on…

We call the quasi-subgroup corresponding to \omega\vee \mu the quasi-subgroup generated by \omega and \nu. If \omega\vee\mu=0, we say the quasi-subgroup is non-compact.

In particular, if G is compact, then \omega\vee\mu is non-zero, because C_0^u(G)=C^u(G) is unital, and hence the set of states is closed in the weak* topology.

Remark 3.2

For \mu,\,\nu \in \mathcal{I}(G), N_{\mu\wedge \nu} is the smallest (left) coideal containing both N_\mu and N_\nu. In particular, \mu\wedge \nu\preceq \mu,\,\nu. Also \mu\wedge \nu is the largest idempotent dominated by both \mu,\,\nu:

\mu\wedge\nu=\sup\{\omega\in \mathcal{I}(G)\,|\,\omega\preceq\mu,\,\nu\},

with a similar result for the quasi-group generated by \mu,\,\nu.

Theorem 3.3

Let \omega,\,\mu,\,\rho\in\mathcal{I}(G) such that

  1. \rho\preceq \omega
  2. \rho\wedge \mu=\rho\star\mu (weird, no?)
  3. N_{\omega\cap \mu}=\sigma\text{-weakly closed linear span}(N_\omega N_\mu)

Then \omega\wedge (\mu\vee \rho)=(\omega\wedge \mu)\vee\rho.

Note TFAE

  1. \rho\vee\mu=\rho\star\mu
  2. \mu\star\rho\star \mu=\rho\star\mu
  3. \mu\star \rho=\rho\star\mu

Open Compact Quasi-Subgroups

Consider the reducing morphism

\Lambda:C_0^u(G)\twoheadrightarrow C_0(G)\subset L^\infty(G),

we embed the predual L^\infty(G)_* into C_0^u(G)^* (???). The predual is a closed ideal of the Banach algebra given by (C_0^u(G)^*,\star). Consider idempotents that lie in this ideal, called normal idempotents:

\mathcal{I}_{\text{nor}}(G)=\mathcal{I}(G)\cap L^\infty(G)_*.

Such states correspond to open compact quasi-subgroups. The following generalises the fact that if a subgroup of a topological group contains an open subgroup, then it is itself open.

Proposition 4.1

For any \omega\in\mathcal{I}_{\text{nor}}(G), \mu\in \mathcal{I}(G), such that \omega\preceq\mu, \mu\in\mathcal{I}_{\text{nor}}(G).

Proof: Recall that the predual of the von Neumann algebra is an ideal in the convolution algebra. Therefore \mu=\omega\star \mu is in the predual also, and is thus an open compact quasi-subgroup \bullet

Corollary 4.2

If G is a discrete quantum group, then all quasi-subgroups are open compact.

Proof: Presumably \varepsilon is in the predual (something to do with (co)amenability?), and of course any \omega\in\mathcal{I}(G) has \varepsilon\preceq \omega, and thus \omega corresponds to an open compact quasi-subgroup \bullet

For each open compact quasi-subgroup, consider its left kernel

J_\omega=\{x\in L^\infty (G)\,|\,\omega(x^*x)=0\}.

This left-kernel is a \sigma-weakly closed left ideal in the algebra of measurable functions, and hence of the form L^\infty(G)Q_\omega (just what I need?!) for a unique projection Q_\omega. A measurable x is in this left kernel if and only if xQ_\omega=x. Note as a projection, Q_\omega itself satisfies this and thus \omega(Q_\omega)=0, and by Schwarz \omega(yQ_\omega)=\omega(Q_\omega y)=0 for any measurable y (yes, this is very useful). Moreover putting P_\omega=\mathbf{1}-Q_\omega, for all measurable y we have:

\omega(y)=\omega(yP_\omega)=\omega(P_\omega y)=\omega(P_\omega yP_\omega).

Remark 4.3

If x is a positive, measurable, and \omega(x)=0, then P_\omega x=xP_\omega=0 (yelp!). Indeed x=y^*y for some y\in J_\omega=L^\infty(G)Q_\omega, so that y=zQ_\omega for some measurable z. Thus x=Q_\omega z^*zQ_\omega.

Now I think I understand what N_\omega is — in fairness the paper spoke about it. Think classically, indeed finitely. Let \omega=U_H be the uniform distribution on a subgroup H\subset G. The big scary definition for E_\omega above… I am fairly sure is nothing as bad in this easier category than

E_{U_H}=(I_{F(G)}\otimes U_H)\Delta.

Let x=\sum_{s\in G}a_s\delta_s. Note

\Delta(x)=\sum_{s\in G}a_s\sum_{t\in G}\delta_{sh}\otimes \delta_{h^{-1}}

\displaystyle\Rightarrow (I_{F(G)}\otimes U_H)\Delta(x)=\frac{1}{|H|}\sum_{s\in G}a_s\mathbf{1}_{sH}.

Let a_{sH}=\sum_{r\in sH}a_r. Then

\displaystyle E_{U_H}(x)=\frac{1}{|H|}\sum_{sH\in G/H}a_{sH}\mathbf{1}_{sH}.

Therefore this conditional expectation takes functions defined on the group to functions defined on the left cosets of H\subset G.

We want to say that N_{U_H} is the set of functions constant on left cosets. This will feed into the following lemma.

Lemma 4.4

Let \omega be an open compact quasi-subgroup. Then

  1. x\in N_\omega iff \Delta(x)(\mathbf{1}\otimes P_\omega)=x\otimes P_\omega.
  2. \Delta(Q_\omega)(P_\omega\otimes P_\omega)=0
  3. R(Q_\omega)=\tau_t(Q_\omega)=Q_{\omega}

Let us illustrate the first (one direction) in the finite, classical case. Let x be constant on left cosets of H\subset G. That is

x=\sum_{sH\in G/H}a_{sH}\mathbf{1}_{sH}=\sum_{sH\in G/H}a_{sH}\sum_{h\in H}\delta_{sh}.

We have that P_{U_H} is the projection \sum_{h\in G}\delta_h while Q_{U_H}=\sum_{g\not\in H}\delta_g. We have

\Delta(x)=\sum_{sH\in G/H}a_{sH}\sum_{h\in H,\,t\in G}\delta_{sht^{-1}}\otimes \delta_t,

and

\Delta(x)(I_{F(G)}\otimes P_{U_H})=\sum_{sH\in G/H}a_{sH}\sum_{h,t\in H}\delta_{sht^{-1}}\otimes \delta_t.

We can show that

x\otimes P_{U_H}=\sum_{sH\in G/H}a_{sH}\sum_{h,\,t\in H}\delta_{sh}\otimes \delta_t.

Hitting both with \delta^c\otimes\delta^d shows that they are equal. Indeed both are equal to zero if d\not\in H. If d\in H then both are equal to |H|a_{dH}.

Proof: Assume that \Delta(x)(\mathbf{1}_G\otimes P_\omega)=x\otimes P_{\omega}. Hit both sides with I\otimes \omega. What we get (!) is the right-hand side gives

(I\otimes\omega)(x\otimes P_\omega)=x\otimes\omega(P_\omega)=x\omega(\mathbf{1}P_\omega)=x\omega(\mathbf{1})=x.

Now the left-hand side gives:

(I\otimes \omega)\left(\sum x_{(1)}\otimes x_{(2)}P_\omega\right)=\sum x_{(1)}\omega(x_{(2)}P_\omega)=\sum x_{(1)}\omega(x_(2))=E_\omega(x).

That is E_{\omega}(x)=x, that is x is constant on left cosets.

Now assume that x is constant on left cosets. Then so will x^*x (I actually can’t see this… or the next part)

T_\omega(x^*x)=x^*x

Theorem 4.5

Let Q\in L^\infty(G) be a projection, P=I-Q satisfying

\Delta(Q)(P\otimes P)=0,

P\in\text{dom} S, and S(Q)=Q. Then P is group-like.

Recall that Q_\omega is a projection such that for functions x such that \omega(|x|^2)=0, xQ_\omega=x.

Corollary 4.6

Let \omega be an open compact quasi-subgroup. Then P_\omega=\mathbf{1}-Q_\omega is a group-like projection.

Call P_\omega the support projection of \omega.

Theorem 4.8

The support projection of any open compact quasi-subgroup is constant on conjugacy-classes (like one and zero). Moreover P_\omega is minimal (?and central among projections constant on conjugacy classes.

Theorem 4.11

For any open compact quasi-subgroup, for x\in L^\infty(G)

\displaystyle \omega(x)=\frac{\displaystyle \psi(P_\omega xP_\omega)}{\displaystyle \psi(P_\omega)}.

One more result of interest (to me).

Corollary 4.15

Let \omega,\,\nu give open compact quasi-subgroups. Then \omega\preceq \mu iff P_\omega\leq P_{\mu}.

Advertisements