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?
Tag: algebra
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:

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

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

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

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

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 singleoperator 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 projectaway, 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 domaindependent.
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?
Thanks in advance!
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}$ .