oa.operator algebras – Trying to understand Haagerup tensor product $B(H)otimes_{rm h}B(K)$

I’m self reading Haagerup tensor product of operator spaces. Understanding it properly, I’m trying some examples. Let $H$ And $K$ be Hilbert space. Let $B(H)$ and $K(H)$ denotes the spaces of bounded and compact operators on $H$?

Can someone explain me what is $B(H)otimes_{rm h}B(K)$ and $B(H)otimes_{rm h}K(H)$? Are these spaces completely isometric to some well known operator space?

Is there any reference/lecture notes where I can find these kind of stuff?

P.S: The same question was first asked on MSE here.

oa.operator algebras – Finite compact quantum groups

Let $(A, Delta)$ be a $C^*$-algebraic compact quantum group (in the sense of Woronowicz). It is called finite if $A$ is a finite-dimensional $C^*$-algebra. By elementary $C^*$-algebra theory, we known that
$$Acong M_{n_1}(mathbb{C}) oplus dots oplus M_{n_k}(mathbb{C})$$ as $C^*$-algebras.

If $X$ is a finite (=> compact) group, then we can consider $C(X)$ with comultiplication
$$Delta(f)(x,y) = f(xy).$$
Then $(C(X), Delta)$ is a finite compact group.

Further, we can consider $C^*(X)= C_r^*(X)= mathbb{C}(X)$, the group algebra, with comultiplication uniquely determined by
$$Delta(x) = x otimes x, quad x in X.$$

Question: What are other examples of finite compact quantum groups? Are the finite compact quantum groups completely classified?

oa.operator algebras – Approximation of certain states by convex combinations of other states

Let $A$ be a $C^ast$-algebra and $phi, psi$ be states on $A$ for which there exists a sequence of elements $(a_i)_{i in mathbb{N}} subseteq A$ with $phi(a_i^ast a_i) neq 0$ for all $i in mathbb N$ and begin{eqnarray} psi(b) = lim_{irightarrow infty } frac{phi(a_i^ast b a_i)}{phi (a_i^ast a_i)} end{eqnarray} for all $b in A$. Is it then true that $psi$ is contained in the weak-$ast$ closure of the convex hull of the set begin{eqnarray} left{phi(u^ast(cdot)u) mid u text{ unitary in }A right}text{?} end{eqnarray} If no, are there conditions on $A$, $phi$ and the $a_i$ which imply this?

oa.operator algebras – What are all the possible indices for the finite depth subfactors?

Jones index theorem (1983) states that the set of all possible (finite) indices of subfactors is exactly $$mathrm{Ind}={ 4 cos(pi/n)^2 | n ge 3 } cup (4, infty),$$ but if we restrict to the finite depth case, the set (say $mathrm{Ind_{fd}}$) must become countable, because then the index is the squared norm of the principal graph (which then is a finite bipartite graph). According to this paper on page 63 (Afzaly-Morrison-Penneys, to appear in MAMS) there are exactly $8$ possible such indices in the interval $(4,5.25)$.

The set $mathrm{Ind_{fd}}$ is multiplicative (i.e. $alpha, beta in mathrm{Ind_{fd}} Rightarrow alpha beta in mathrm{Ind_{fd}}$) because the tensor product keeps the finite depth. Now, by this paper (Wassermann, 1998), for all $m<n$, there is a (finite depth) Jones-Wassermann subfactor of index $frac{sin^2(npi/m)}{sin^2(pi/m)}$, so that the set $mathrm{Ind_{fd}}$ has an accumultation point at $alpha n^2$ for all $n in mathbb{N}_{ge 2}$ and $alpha in mathrm{Ind_{fd}}$.

By this paper (Etingof-Nikshych-Ostrik, 2005), $mathrm{Ind_{fd}}$ is contained in the set of positive cyclotomic integers (i.e. a positive elements of $mathbb{Z}(c_n)$ with $c_n = 2cos(pi/n)$), which is a breakthrough in the understanding of $mathrm{Ind_{fd}}$.

Now, I feel like that we can get even better, because by Theorem 3.2 in this paper (Bisch, 1994): $$alpha in mathrm{Ind_{fd}} setminus mathbb{N} Rightarrow alpha^{-1} mathbb{N} cap mathrm{Ind_{fd}} = emptyset.$$
Example: $n(3-sqrt{5}) not in mathrm{Ind_{fd}}$ because $ 2frac{3+sqrt{5}}{2} cdot n(3-sqrt{5}) = 4n$, and $frac{3+sqrt{5}}{2} = 4cos^2(pi/5)$.

More strongly (see this comment) by extending Bisch’s result to every ring $R$ of cyclotomic integers (for example $mathbb{Z}(c_n)$)
$$alpha in mathrm{Ind_{fd}} setminus R Rightarrow alpha^{-1} R cap mathrm{Ind_{fd}} = emptyset.$$

The purpose of this post is not really to ask what exactly is $mathrm{Ind_{fd}}$ (which seems unreachable), but new results about it, in particular inspired by above observations.

oa.operator algebras – Dense subalgebra that is closed under unbounded derivation on noncommutative torus

Let $A_{theta}$ be the noncommutative torus, we can define:

$A^{infty}_{theta}:={sum_{n,minmathbb{Z}}a_{n,m}U^{n}V^{m}|a_{n,m}in S(mathbb{Z}^{2})}$

where $S(mathbb{Z}^{2})$ is space of sequence with rapidly decreasing property, i.e. $sup_{n,minmathbb{Z}}(1+n^{2}+m^{2})^{k}||a_{n,m}||<infty$ for any $kinmathbb{N}cup{0}$. This is a dense subalgebra of $A_{theta}$ which is closed under unbounded *-derivation $delta$, i.e. $delta(A^{infty}_{theta})subseteq A^{infty}_{theta}$.

My question is, is $A^{infty}_{theta}$ the only dense subalgebra with the property that is closed under unbounded *-derivation? In here, the author seems to identify $A^{infty}_{theta}$ is the only choice when talking about smooth structure on $A_{theta}$. Does there exist other dense subalgebra which have above property?

oa.operator algebras – What’s the matrix of logarithm of derivative operator ($ln D$)? What is the role of this operator in various math fields?

This paper gives some great results:

$(ln D) 1 = -ln x -gamma$

$(ln D) x^n = x^n (psi (n+1)-ln x)$

$(ln D) ln x = -zeta(2) -(gamma+ln x)ln x$

I wonder, what is its matrix, or otherwise, is there a method of applying it to a function?

What is its intuitive role in various fields of math?

oa.operator algebras – When the adjoint of an unbounded operator on a Hilbert space coincides with the formal adjoint on its maximal domain?

This is a copy of https://math.stackexchange.com/questions/3931318/when-the-adjoint-of-an-unbounded-operator-on-a-hilbert-space-coincides-with-the

Suppose we have a closable and densely defined operator $A$ with a domain $dom(A)$ which is a subspace of a Hilbert space $mathcal{H}$.
Let $mathcal{H}$ have an orthonormal basis ${e_n}_{n=1}^infty$.
So the operator $A$ can be viewed as an infinite matrix $A_{ij}$.

We know that there is a usual procedure to define $A^*$ with its domain $dom(A^*)$.
Now consider the formal adjoint operator $A_* = {overline{A_{ji}}}$ with the domain $dom(A_*)$.
Are there some simple conditions on $A$ when these domains coincide: $dom(A^*) = dom(A_*)$?

What can be said on this matter if $A_{ij}$ is a finite-band matrix?
Or when $A$ is formally self-adjoint ($A_{ij} = overline{A_{ji}})$?

oa.operator algebras – Sufficient conditions for a map $phi: M(A) to M(B)$ to have strictly closed image

Let $A$ and $B$ be $C^*$-algebras with multiplier algebras $M(A)$ and $M(B)$. Are there any nice conditions that ensure that a strict (= norm-continuous + strictly continuous on bounded subsets of $M(A)$) linear map $phi: M(A) to M(B)$ has strictly closed image. In particular, I’m interested in the following situations:

(1) $phi$ is a strict $*$-morphism.

(2) $phi$ arises as the unique extension of a strict linear map $A to M(B)$.

(3) $phi$ arises as a unique extension of a strict injective linear map $A hookrightarrow M(B)$.