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An expository piece, the watered down version of the madness here… should be on the Arxiv Thursday.

Click here for pdf.

What started about ten months ago as a technical question to an expert, led to a talk, and led to me producing this weird production here.

Now that it is complete, although I really like all its contents (well except for the note to reader and introduction I spilled out very hastily), I can see on reflection it represents rather than a cogent piece of mathematics, almost a log of all the things I have learnt in the process of writing it. It also includes far too much speculation and conjecture. So I am going to post it here and get to work on editing it down to something a little more useful and cogent.

EDIT: Edited down version here.

Giving a talk 17:00, September 1 2020:

See here for more.

Slides.

Finally cracked this egg.

Preprint here.

I thought I had a bit of a breakthrough. So, consider the algebra of a functions on the dual (quantum) group \widehat{S_3}. Consider the projection:

\displaystyle p_0=\frac12\delta^e+\frac12\delta^{(12)}\in F(\widehat{S_3}).

Define u\in M_p(\widehat{S_3}) by:

u(\delta^\sigma)=\langle\text{sign}(\sigma)1,1\rangle=\text{sign}(\sigma).

Note

\displaystyle T_u(p_0)=\frac12\delta^e-\frac12 \delta^{(12)}:=p_1.

Note p_1=\mathbf{1}_{\widehat{S_3}}-p_0=\delta^0-p_0 so \{p_0,p_1\} is a partition of unity.

I know that p_0 corresponds to a quasi-subgroup but not a quantum subgroup because \{e,(12)\} is not normal.

This was supposed to say that the result I proved a few days ago that (in context), that p_0 corresponded to a quasi-subgroup, was as far as we could go.

For H\leq G, note

\displaystyle p_H=\frac{1}{|H|}\sum_{h\in H}\delta^h,

is a projection, in fact a group like projection, in F(\widehat{G}).

Alas note:

\displaystyle T_u(p_{\langle(123)\rangle})=p_{\langle (123)\rangle}

That is the group like projection associated to \langle (123)\rangle is subharmonic. This should imply that nearby there exists a projection q such that u^{\star k}(q)=0 for all k\in\mathbb{N}… also q_{\langle (123)\rangle}:=\mathbf{1}_{\widehat{S_3}}-p_{\langle(123)\rangle} is subharmonic.

This really should be enough and I should be looking perhaps at the standard representation, or the permutation representation, or S_3\leq S_4… but I want to find the projection…

Indeed u(q_{(123)})=0…and u^{\star 2k}(q_{\langle (123)\rangle})=0.

The punchline… the result of Fagnola and Pellicer holds when the random walk is is irreducible. This walk is not… back to the drawing board.

I have constructed the following example. The question will be does it have periodicity.

Where \rho:S_n\rightarrow \text{GL}(\mathbb{C}^3) is the permutation representation, \rho(\sigma)e_i=e_{\sigma_i}, and \xi=(1/\sqrt{2},-1/\sqrt{2},0), u\in M_p(G) is given by:

u(\sigma)=\langle\rho(\sigma)\xi,\xi\rangle.

This has u(\delta^e)=1 (duh), u(\delta^{(12)})=-1, and otherwise u(\sigma)=-\frac12 \text{sign}(\sigma).

The p_0,\,p_1 above is still a cyclic partition of unity… but is the walk irreducible?

The easiest way might be to look for a subharmonic p. This is way easier… with \alpha_\sigma=1 it is easy to construct non-trivial subharmonics… not with this u. It is straightforward to show there are no non-trivial subharmonics and so u is irreducible, periodic, but p_0 is not a quantum subgroup.

It also means, in conjunction with work I’ve done already, that I have my result:

Definition Let G be a finite quantum group. A state \nu\in M_p(G) is concentrated on a cyclic coset of a proper quasi-subgroup if there exists a pair of projections, p_0\neq p_1, such that \nu(p_1)=1, p_0 is a group-like projection, T_\nu(p_1)=p_0 and there exists d\in\mathbb{N} (d>1) such that T_\nu^d(p_1)=p_1.

(Finally) The Ergodic Theorem for Random Walks on Finite Quantum Groups

A random walk on a finite quantum group is ergodic if and only if the driving probability is not concentrated on a proper quasi-subgroup, nor on a cyclic coset of a proper quasi-subgroup.

The end of the previous Research Log suggested a way towards showing that p_0 can be associated to an idempotent state \int_S. Over night I thought of another way.

Using the Pierce decomposition with respect to p_0 (where q_0:=\mathbf{1}_G-p_0),

F(G)=p_0F(G)p_0+p_0F(G)q_0+q_0F(G)p_0+q_0F(G)q_0.

The corner p_0F(G)p_0 is a hereditary \mathrm{C}^*-subalgebra of F(G). This implies that if 0\leq b\in p_0F(G)p_0 and for a\in F(G), 0\leq a\leq b\Rightarrow a\in p_0F(G)p_0.

Let \rho:=\nu^{\star d}. We know from Fagnola and Pellicer that T_\rho(p_0)=p_0 and T_\rho(p_0F(G)p_0)=p_0F(G)p_0.

By assumption in the background here we have an irreducible and periodic random walk driven by \nu\in M_p(G). This means that for all projections q\in 2^G, there exists k_q\in\mathbb{N} such that \nu^{\star k_q}(q)>0.

Define:

\displaystyle \rho_n=\frac{1}{n}\sum_{k=1}^n\rho^{\star k}.

Define:

\displaystyle n_0:=\max_{\text{projections, }q\in p_0F(G)p_0}\left\{k_q\,:\,\nu^{\star k_q}(q)> 0\right\}.

The claim is that the support of \rho_{n_0}, p_{\rho_{n_0}} is equal to p_0.

We probably need to write down that:

\varepsilon T_\nu^k=\nu^{\star k}.

Consider \rho^{\star k}(p_0) for any k\in\mathbb{N}. Note

\begin{aligned}\rho^{\star k}(p_0)&=\varepsilon T_{\rho^{\star k}}(p_0)=\varepsilon T^k_\rho(p_0)\\&=\varepsilon T^k_{\nu^{\star d}}(p_0)=\varepsilon T_\nu^{kd}(p_0)\\&=\varepsilon(p_0)=1\end{aligned}

that is each \rho^{\star k} is supported on p_0. This means furthermore that \rho_{n_0}(p_0)=1.

Suppose that the support p_{\rho_{n_0}}<p_0. A question arises… is p_{\rho_{n_0}}\in p_0F(G)p_0? This follows from the fact that p_0\in p_0F(G)p_0 and p_0F(G)p_0 is hereditary.

Consider a projection r:=p_0-p_{\rho_{n_0}}\in p_0F(G)p_0. We know that there exists a k_r\leq n_0 such that

\nu^{\star k_r}(p_0-p_{\rho_{n_0}})>0\Rightarrow \nu^{\star k_r}(p_0)>\nu^{\star k_r}(p_{\rho_{n_0}}).

This implies that \nu^{\star k_r}(p_0)>0\Rightarrow k_r\equiv 0\mod d, say k_r=\ell_r\cdot d (note \ell_r\leq n_0):

\begin{aligned}\nu^{\star \ell_r\cdot d}(p_0)&>\nu^{\star \ell_r\cdot d}(p_{\rho_{n_0}})\\\Rightarrow (\nu^{\star d})^{\star \ell_r}(p_0)&>(\nu^{\star d})^{\star \ell_r}(p_{\rho_{n_0}})\\ \Rightarrow \rho^{\star \ell_r}(p_0)&>\rho^{\star \ell_r}(p_{\rho_{n_0}})\\ \Rightarrow 1&>\rho^{\star \ell_r}(p_{\rho_{n_0}})\end{aligned}

By assumption \rho_{n_0}(p_{\rho_{n_0}})=1. Consider

\displaystyle \rho_{n_0}(p_{\rho_{n_0}})=\frac{1}{n_0} \sum_{k=1}^{n_0}\rho^{\star k}(p_{\rho_{n_0}}).

For this to equal one every \rho^{\star k}(p_{\rho_{n_0}}) must equal one but \rho^{\star \ell_r}(p_{\rho_{n_0}})<1.

Therefore p_0 is the support of \rho_{n_0}.

Let \rho_\infty=\lim \rho_n. We have shown above that \rho^{\star k}(p_0)=1 for all k\in\mathbb{N}. This is an idempotent state such that p_0 is its support (a similar argument to above shows this). Therefore p_0 is a group like projection and so we denote it by \mathbf{1}_S and \int_S=d\mathcal{F}(\mathbf{1}_S)!

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Today, for finite quantum groups, I want to explore some properties of the relationship between a state \nu\in M_p(G), its density a_\nu (\nu(b)=\int_G ba_\nu), and the support of \nu, p_{\nu}.

I also want to learn about the interaction between these object, the stochastic operator

\displaystyle T_\nu=(\nu\otimes I)\circ \Delta,

and the result

T_\nu(a)=S(a_\nu)\overline{\star}a,

where \overline{\star} is defined as (where \mathcal{F}:F(G)\rightarrow \mathbb{C}G by a\mapsto (b\mapsto \int_Gba)).

\displaystyle a\overline{\star}b=\mathcal{F}^{-1}\left(\mathcal{F}(a)\star\mathcal{F}(b)\right).

An obvious thing to note is that

\nu(a_\nu)=\|a_\nu\|_2^2.

Also, because

\begin{aligned}\nu(a_\nu p_\nu)&=\int_Ga_\nu p_\nu a_\nu=\int_G(a_\nu^\ast p_\nu^\ast p_\nu a_\nu)\\&=\int_G(p_\nu a_\nu)^\ast p_\nu a_\nu\\&=\int_G|p_\nu a_\nu|^2\\&=\|p_\nu a_\nu\|_2^2=\|a_\nu\|^2\end{aligned}

That doesn’t say much. We are possibly hoping to say that a_\nu p_\nu=a_\nu.

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In my pursuit of an Ergodic Theorem for Random Walks on (probably finite) Quantum Groups, I have been looking at analogues of Irreducible and Periodic. I have, more or less, got a handle on irreducibility, but I am better at periodicity than aperiodicity.

The question of how to generalise these notions from the (finite) classical to noncommutative world has already been considered in a paper (whose title is the title of this post) of Fagnola and Pellicer. I can use their definition of periodic, and show that the definition of irreducible that I use is equivalent. This post is based largely on that paper.

Introduction

Consider a random walk on a finite group G driven by \nu\in M_p(G). The state of the random walk after k steps is given by \nu^{\star k}, defined inductively (on the algebra of functions level) by the associative

\nu\star \nu=(\nu\otimes\nu)\circ \Delta.

The convolution is also implemented by right multiplication by the stochastic operator:

\nu\star \nu=\nu P,

where P\in L(F(G)) has entries, with respect to a basis (\delta_{g_i})_{i\geq 1} P_{ij}=\nu(g_jg_{i^{-1}}). Furthermore, therefore

\nu^{\star k}=\varepsilon P^k,

and so the stochastic operator P describes the random walk just as well as the driving probabilty \nu.

The random walk driven by \nu is said to be irreducible if for all g_\ell\in G, there exists k\in\mathbb{N} such that (if g_1=e) [P^k]_{1\ell}>0.

The period of the random walk is defined by:

\displaystyle \gcd\left(d\in\mathbb{N}:[P^d]_{11}>0\right).

The random walk is said to be aperiodic if the period of the random walk is one.

These statements have counterparts on the set level.

If P is not irreducible, there exists a proper subset of G, say S\subsetneq G, such that the set of functions supported on S are P-invariant.  It turns out that S is a proper subgroup of G.

Moreover, when P is irreducible, the period is the greatest common divisor of all the natural numbers d such that there exists a partition S_0, S_1, \dots, S_{d-1} of G such that the subalgebras A_k of functions supported in S_k satisfy:

P(A_k)=A_{k-1} and P(A_{0})=A_{d-1} (slight typo in the paper here).

In fact, in this case it is necessarily the case that \nu is concentrated on a coset of a proper normal subgroup N\rhd G, say gN. Then S_k=g^kN.

Suppose that f is supported on g^kNWe want to show that for Pf\in A_{k-1}Recall that 

\nu^{\star k-1}P(f)=\nu^{\star k}(f).

This shows how the stochastic operator reduces the index P(A_k)=A_{k-1}.

A central component of Fagnola and Pellicer’s paper are results about how the decomposition of a stochastic operator:

P(f)=\sum_{\ell}L_\ell^*fL_{\ell},

specifically the maps L_\ell can speak to the irreducibility and periodicity of the random walk given by P. I am not convinced that I need these results (even though I show how they are applicable).

Stochastic Operators and Operator Algebras

Let F(X) be a \mathrm{C}^*-algebra (so that X is in general a  virtual object). A \mathrm{C}^*-subalgebra F(Y) is hereditary if whenever f\in F(X)^+ and h\in F(Y)^+, and f\leq h, then f\in F(Y)^+.

It can be shown that if F(Y) is a hereditary subalgebra of F(X) that there exists a projection \mathbf{1}_Y\in F(X) such that:

F(Y)=\mathbf{1}_YF(X)\mathbf{1}_Y.

All hereditary subalgebras are of this form.

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This sandbox is going to take from a variety of sources, mostly Shuzhou Wang.

C*-Ideals

Let J\subset C(X) be a closed (two-sided) ideal in a non-commutative unital C^*-algebra C(X). Such an ideal is self-adjoint and so a non-commutative C^*-algebra J=C(S). The quotient map is given by \pi:C(X)\rightarrow C(X)/C(S), f\mapsto f+J, where f+J is the equivalence class of f under the equivalence relation:

f\sim_{J} g\Rightarrow g-f\in C(S).

Where we have the product

(f+J)(g+J)=fg+J,

and the norm is given by:

\displaystyle\|f+J\|=\sup_{j\in C(S)}\|f+j\|,

the quotient C(X)/ C(S) is a C^*-algebra.

Consider now elements j_1,\,j_2\in C(S) and f_1,\, f_2\in C(X). Consider

j_1\otimes f_1+f_2\otimes j_2\in C(S)\otimes C(X)+C(X)\otimes C(S).

The tensor product \pi\otimes \pi:C(X)\otimes C(X)\rightarrow (C(X)/C(S))\otimes (C(X)/ C(S)). Now note that

(\pi\otimes\pi)(j_1\otimes f_1+f_2\otimes j_2)=(0+J)\otimes(f_1+J)+

(f_2+J)\otimes(0+J)=0,

by the nature of the Tensor Product (0\otimes a=0). Therefore C(X)\otimes C(S)+C(S)\otimes C(X)\subset \text{ker}(\pi\otimes\pi).

Definition

A WC*-ideal (W for Woronowicz) is a C*-ideal J=C(S) such that \Delta(J)\subset \text{ker}(\pi\otimes\pi), where \pi is the quotient map C(G)\rightarrow C(G)/C(S).

Let F(G) be the algebra of functions on a classical group G. Let H\subset G. Let J be the set of functions which vanish on H: this is a C*-ideal. The kernal of \pi:F(G)\rightarrow F(G)/J is J.

Let \delta_s\in J so that s\not\in H. Note that

\displaystyle\Delta(\delta_s)=\sum_{t\in G}\delta_{st^{-1}}\otimes\delta_t

and so

\displaystyle(\pi\otimes \pi)\Delta(\delta_s)=\sum_{t\in G}\pi(\delta_{st^{-1}})\otimes \pi(\delta_t).

Note that \pi(\delta_t)=0+J if t\not\in H. It is not possible that both st^{-1} and t are in H: if they were st^{-1}\cdot t\in H, but st^{-1}\cdot t=s, which is not in H by assumption. Therefore one of \pi(\delta_{st^{-1}}) or \pi(\delta_t) is equal to zero and so:

(\pi\otimes\pi)\Delta(\delta_s)=0,

and so by linearity, if f vanishes on a subgroup H,

\Delta(f)\subset \text{ker}(\pi\otimes\pi).

In this way, WC*-ideals generalise functions which vanish on distinguished subgroups. In fact, without checking all the details, I imagine that first isomorphism theorem can show that F(G)/ J=F(H). Let \pi_H:F(G)\rightarrow F(H) be the ring homomorphism

\displaystyle\pi_H\left(\sum_{t\in G}a_t\delta_t\right)=\sum_{t\in H}a_t\delta_t.

Then \text{ker}\,\pi_H=J, \text{im}\,\pi_H=F(H), and so we have the isomorphism of rings, which presumably carries forward to the algebras of functions level…

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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.

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