## 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 – An affiliated operator is not the identity operator

Suppose that $$T$$ is affiliated with a von Neumann algebra $$M$$. If $$Tneq I$$, then there exits $$S>0$$ and $$Sin M$$ such that $$TS$$ is bounded and $$T^2S^2>T^2$$ or $$T^2S^2.

How to check the above conclusion? Does it mean that $$T$$ and $$I$$ can be comparable?

## 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 $$m, there is a (finite depth) Jones-Wassermann subfactor of index $$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. $$sup_{n,minmathbb{Z}}(1+n^{2}+m^{2})^{k}||a_{n,m}|| for any $$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 – Centralizer of a faithful normal state

The theorem 4 in the screenshot is from Herman’s paper “Centralizers and an ordering for faithful normal states”.

In the proof of (i), I met with troubles, how to conclude that $$varphi=psi$$?. The author listed a reference, but I cannot find the reference in the final page.

## 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 $$A hookrightarrow M(B)$$.