## On boundedness of sequence of operators in vN algebra

Let $$x_{n}$$ be a sequence of operators in vN algebra $$M$$, $$\Omega$$ is a cyclic vector for $$M$$, if $$x_{n}\Omega$$ converges in $$\mathcal{H}$$, can we say there exist a subsequence $$\{y_{n}\}$$ of $$\{x_{n}\}$$ are uniformly bounded in operator norm?

## Algebra 2 confusing math question

james and katrina are painting a wall. james paints 1/2 the wall in one hour. together they paint the wall in 72 minutes. how long does it take for katrina to paint the wall by herself

## In which conditions x and y in a C*-algebra/von Neumann algebra would be conjugate?

We say $$x$$ and $$y$$ are conjugate if $$x=a^{*}ya$$ for some unitary $$a$$. Is there any criterion?

## Clases of equivalece – Topics ins Algebra Herstein.

Property 2 of an equivalence relation states that if a ~ b then b ~ a; property 3 states that if a a ~ b and b ~ c then a ~ c. What is wrong with the following proof that properties 2 and 3 imply property 1 ? Let a~ b; then b~ a, whence, by property 3 (using a = c), a ~ a.

## Sub Banach spaces (Banach algebras) of the disc algebra which are invariant under the differentiation operator

Is there a complete classification of all Banach subspace of the disc algebra $$\mathcal{A}(\mathbb{D})$$ which are invariants under the differentiation operator? Is the a complete classification of such Banach subspace for which the differentiation is a bounded operator/

Is there a complete classification of all Banach sub algebra of $$\mathcal{A}(\mathbb{D})$$ which are invariant under the differentiation operator?

Having these questions in my mind, I arrived at this question

Is there a holomorphic function on open unit disc with this property?

## Cartan’s magic formula for diffferential graded algebra

Algebra $$A$$ is called graded algebra if it has a direct sum decomposition $$A=\bigoplus_{k\in\Bbb Z} A^k$$ s.t. product satisfies $$(A^k)(A^l)\subseteq(A^{k+l}) \text{ for each } k, l.$$

A differential graded algebra is graded algebra with chain complex structure $$d \circ d = 0$$.

Derivation of degree $$k$$ on $$A$$ means a linear map $$D:A \to A$$ s.t. $$D(A_j)\subset A_{j+k} \text{ and } D(ab)=(Da)b + (-1)^{ik}a(Db), a\in A_i$$

All smooth forms on $$n$$-manifold $$M$$ is a differential graded algebra $$\Omega^{\bullet}(M)=\bigoplus_{k=0}^{n} \Omega^k(M)$$, with wedge product and exterior derivative.

In proving Cartan’s magic formula $$\mathcal{L}_X=i_X \circ d + d\circ i_X$$ holds for $$\Omega^{\bullet}(M)$$, we can use the following steps:

1. Show that two degree $$0$$ derivations on $$\Omega^{\bullet}(M)$$ commuting with $$d$$ are equal iff they agree on $$\Omega^0(M)$$.

2. Show that $$\mathcal{L}_X$$ and $$i_X \circ d + d \circ i_X$$ are derivations on $$\Omega^{\bullet}(M)$$ commuting with $$d$$.

3. Show that $$\mathcal{L}_X f = Xf = i_Xdf+ d i_Xf$$ for all $$f \in C^{\infty}(M)=\Omega^0(M)$$.

My question:

1. Why step 1 is ture? Why commuting with $$d$$ is so important?

2. Can step 1 be extended to any derivations without restriction on degree.

Thank you.

## Relational Algebra with only one operator?

There’s a parlour game of inventing exotic operators for Relational Algebra, and thereby reducing the number of operators needed to be ‘Relationally Complete’. A popular operator for this is ‘Inner Union’ aka SQL’s UNION CORRESPONDING.

I’ve just bumped into a single-operator basis for FOL, due to Schönfinkel. It’s a combo of Sheffer stroke (written infix |) and existential quant (with the bound var superscripted).

P(x) |x Q(x) ≡ ¬∃x.(P(x)∧Q(x))

Q 1. Could there be a Relational Operator corresponding to that?

Q 2. If so, does that mean there could be a version of Relational Algebra with only one operator?

Q 3. If not, in what sense is Codd’s 1972 set “complete”?

My thoughts so far:

Q 1. No. The FOL ∧ corresponds OK to RA ⋈ (Natural Join). The ∃ corresponds OK to ‘Remove’ aka project-away, sometimes written π-hat. But RA can only express correspondence to negation when ¬ is nested inside ∧. I.e. FOL P(x) ∧ ¬Q(x) corresponds to RA P MINUS Q. Whereas this single FOL operator has ¬ at outer level (i.e. absolute complement, not relative).

The reason Codd doesn’t allow absolute complement is it makes queries ‘unsafe’, that is domain-dependent.

Q 2. Then no. Supplementary q: it’s well known Codd omitted RENAME/ρ from his original set. Rename is needed to translate a FOL expression using = between variables:

∃x. P(x) ∧ (x = y)       -- corresponds to  ρ{y := x }(P)            -- relation P with attrib x 

Presumably Schönfinkel’s operator doesn’t avoid the need for =(?).

Q 3. Then how does Codd’s original RA express an equivalent to a FOL expression with outermost ¬? Or outermost ∀, which is the same thing:

∀y.Q(y)≡¬∃y.¬Q(y) 

## Von Neumann algebra such that every state is normal- what can be said about its dimension? What about the other direction?

This is exercise 5 in page 130 of Masamichi Takesaki ‘s Theory of Operator Algebras 1 (TO SHOW THAT IT IS FINITE DIMENSIONAL, AND THAT THE IFF HOLDS).

I know that now every state is weakly continuous on the unit ball. Maybe that means that it must be separable, and then I can show that it is finite dimensional?

## Abstract Algebra – Showing something to be a group

I’ve been posed the question:

“Let $$P$$ be the pairs (a,b) where $$a ∈ Z$$$$4$$, and $$b ∈ Z$$$$2$$.

An operation, $$*$$, is defined by: $$(a,b)*(c,d)=(a+c$$ (mod 4)$$, b+d$$(mod 2)) for all $$(a,c),(b,d)∈P$$

How do I show that this is a group?

I know how to do this with multiplication tables by working through the axioms but I don’t know how to apply these to this question, nor if that’s the best approach

## On cyclicity of fixed point algebra of flip automorphism

Let $$M$$ be a von Neumann algebra having a cyclic vector in $$\mathcal{H}$$, is the fixed point subalgebra under the flip automorphism on $$M\otimes M$$ has a cyclic vector in $$\mathcal{H}\otimes \mathcal{H}$$.