User jiang - MathOverflow

clarify a question in group cohomology

2013-04-12T19:49:33Z

In page 43 of Kenneth S.Brown's book "Cohomology of Groups", GTM 87, we have a proposition:

If $G=F(S)/R$ then there is an exact sequence $0\to R_{ab}\overset{\theta}{\to} \mathbb{Z}G^{(S)}\overset{\epsilon}{\to}G\to 0$ of $G$-modules, where $\mathbb{Z}G^{(S)}$ is free with basis $(e_s)_{s\in S}$ and $\partial e_s=\bar{s}-1$.

Here according to page 45 Exercise 3(d) on this book, the above proposition implies that we have a partial free resolution

$$\mathbb{Z}G^{(T)}\overset{\partial_2}{\to} \mathbb{Z}G^{(S)}\overset{\partial_1}{\to}\mathbb{Z}G\overset{\epsilon}{\to}G\to 0$$

such that the matrix of $\partial_2$ is the "Jacobian matrix" $\overset{-}{(\partial t/\partial s)}_{t\in T, s\in S}$, which is related to the Fox calculs.

My question is:

In the presentation of $G$ above, could both the generators set $S$ and the relation set $T$ be infinite?

I also want to know in this book, does it always alow that the generator sets and relation sets be infinite in a presentation of a group $G$?

Find a lower bound for a pre-invariant $Fol(L(F_m), X_m)$

2013-04-05T16:31:30Z

In the paper of Bannon and Ravichandran, A Folner invariant for type $\rm{II}_1$ factors, they defined an invariant $Fol(M)$ for a separable type $\rm{II}_1$ factor $M$, especially for the free group factor $M=L(F_m)$, where $F_m$ is the non-abelian free group on $m$ generators ${a_1, \cdots, a_m}$, by measuring how badly $M$ fails to satisfy Connes' Folner-type condition.

Let us focus on the pre-invariant $Fol(L(F_m), X_m)$, where $X_m={U_{a_1},\cdots, U_{a_m}}$.

Here $U_{a_l}: l^2(F_m)\to l^2(F_m)$ are the unitary operators defined by $U_{a_l}(\sum_{g\in F_m}x_g\delta_g)=\sum_{g\in F_m}(x_g\delta_{a_lg})$, for any $\sum_{g\in F_m}x_g\delta_g\in l^2(F_m)$.

Note that $\delta_g\in l^2(F_n)$ denotes the characteristic function on the word $g$.

In their paper, they defined

$$Fol(L(F_m), X_m)=inf \lbrace \epsilon>0: Q(X_m,\epsilon) \rbrace$$

Here, we write $Q(X_m, \epsilon)$ to denote the following property holds:

There exists a nonzero finite-rank, say rank $k$, projection $e\in B(l^2(F_m))$ such that $\forall l\in{1, 2, \cdots, m}$,

$$\sum_{i,j=1}^k|T^l_{i,j}|^2\geq k(1-\frac{\epsilon^2}{2})$$ and $$\sum_{i=1}^k|T^l_{i,j}|\leq k\epsilon$$

where, $T^l:=eU_{a_l}e=(T^l_{i,j}), l=1,\cdots, m$ are the $k\times k$ matrices.

If we pick ${\xi_1,\cdots, \xi_k}$ to be the orthonormal basis with unit length of the dimension $k$ subspace $el^2(F_m)$, then note that $T^l_{i,j}=\langle eU_{a_l}e\xi_j,\xi_i \rangle=\langle U_{a_l}e\xi_j,e\xi_i \rangle=\langle U_{a_l}\xi_j,\xi_i \rangle$.

So, we can first pick any $k>0$ and any orthonormal basis with unit length $\xi_1,\cdots, \xi_k\in l^2(F_m)$, then calculate the $T^l$'s to find the possible value of $\epsilon$ such that $Q(X_m, \epsilon)$. Then, this would give us a upper bound for $Fol(L(F_m), X_m)$.

Now, my question is:

What is the lower bound for $Fol(L(F_m), X_m)$? Does it depend on $m$?

Known Results:

1, In Jon and Mohan's paper, they showed that $Fol(L(F_2), X_2)>0$.

2, Julien has given an elegant argument in http://math.stackexchange.com/questions/347786 , showing that for any $\delta>0$, $Q(X_2, \frac{\sqrt{3}}{2}+\delta)$ holds for some rank 1 projection, so $Fol(L(F_2), X_2)\leq \frac{\sqrt{3}}{2}$. In fact, if you follow the same argument of Julien for $F_m$, you find $Fol(L(F_m), X_m)\leq \sqrt{2}(\frac{m-1}{m})$. Note that we have obtained the upper bound that $Fol(L(F_m), X_m)\leq \sqrt{2(1-\frac{1}{m})}$ by constructing partitions of $F_m$.

Remarks:

1, Note that Julien's construction of the coefficient $x_g's$ for $\delta_g$ happens to be used in the well-known word-length deformation in Popa's deformation/rigidity theory. So, it would be nice to see whether this can be explored further.

2, It might be interesting to see whether we can use Julien's construction locally, i.e., use it for a subset of $F_2$, in other words, combine his construction with the partition of $F_2$, to get a better upper bound.

Not measure equivalent ICC groups $G$ and $H$, but $L(G)\cong L(H)$

2013-02-19T21:50:22Z

Some years ago, I heard about the following problem:

Find two ICC, i.e., infinite conjugacy class groups $G$ and $H$, such that $L(G)\cong L(H)$, but $G$ and $H$ are not measure equivalent, where $L(G)$ denotes the group von Neumann algebra associated to the group $G$.

Recall that $G$ and $H$ are measure equivalent iff they admit stably orbit equivalent actions.

What is the status on this problem now?

Fuglede-Kadison determinants in $L(\mathbb{F}_2)$

2012-11-30T04:39:13Z

Given $M$ a finite von Neumann algebra with trace $\tau$, $T\in M$ invertible.

The Fuglede-Kadison determinant is defined as

$\Delta(T)=e^{\tau(log|T|)}$,

where $|T|=(T^*T)^{\frac{1}{2}}$, and $\tau(log|T|)=\int_{0}^{||T||}log(t)d\mu_{|T|}(t)$, and the probability measure $\mu_{|T|}$ is defined on spectrum(|T|) by requiring $\int_{spec(|T|)}fd\mu_{|T|}=\tau(f(|T|))$ for all $f\in C(spec(|T|))$.

See the reference Determinant theory in finite factors

Especially, let $M=L(\mathbb{F}_2)$ be the free group factor associated to the free group $\mathbb{F}_2$ on two generators $a, b$, and with the canonical trace $\tau$, my question is :

1, Are there any references for the study of the determinant in the case $M=L(\mathbb{F}_2)$ ?

Especially, I also want to know

2, Are there any nontrivial computable examples in this case, i.e., what does $\Delta(T)$ looks like for $T\in \mathbb{C}\Gamma$, invertiable ?

Note: This question is motivated by the paper Li, 2012.

Does a conditional expectation from a von Neumann algebra to its center exist?

2010-11-24T14:49:35Z

In a finite von Neumann algebra, the unique tracial state serves as one, then for a general von Neumann algebra, does it exist?

Group theory required for further study in von Neumann algebra

2011-03-23T16:05:05Z

After over half a year's study on operator algebra (especially on von Neumann algebra) by doing exercises in Fundamentals of the theory of operator algebras 1, 2 --Kadison, I was told that the current research focus is on the Ⅱ1 factor, and certain background on group theory is necessary, such as studying the free product, specific group construction and the ergodic action. Then, I want to know are there any good books on the group theory that might be necessary for further study on von Neumann theory?

Answer by Jiang for Beautiful theorems with short proof

2012-09-11T03:16:25Z

The proof of Brouwer fixed point theorem by using fundamental group of $S^1$ is equal to $\mathbb{Z}$, while the fundamental group of $D^2$ is trivial.

Alternative for Kadison and Ringrose's book

2010-10-18T16:13:10Z

I have read over the book by R. V. Kadison and J. R. Ringrose: Fundamentals of the theory of operator algebras. Vol 1, and have done most of the exercises in it. Now I want to find an alternative book for Vol 2, because I once heard that the content in this book is somewhat out of date and the theories are developed in a rather slow pace. Which book can I choose then?

which class does this function belong to in the fast growing hierarchy?

2011-09-18T03:36:04Z

the class $\mathfrak{F}_k$ of the fast growing hierarchy is the closure under substitution and limited recursion of the constant, sum,projections and $F_n$ functions for $ n\leq k, $ where $F_n$ is defined recursively by

\begin{eqnarray*} F_0(x) &\triangleq & x+1\ F_{n+1}(x) &\triangleq &F_n^{x+1}(x) \end{eqnarray*}

Here, $ F_n^{x+1}(x)=\underbrace {F_n(F_n(\cdots (F_n }_{x+1}(x))) $

The hierarchy is strict for $k\geq 1$, i.e. $ \mathfrak{F}{k}\subsetneq \mathfrak{F}{k+1} $.

Then, if a function $g$ can be written as

\begin{eqnarray} g=\underbrace{f_1^{f_2^{.^{.^{.{f_L}}}}}}_{L} \end{eqnarray}

Here $f_i,L$ are functions with variable $x$, the function $f_2=f_2(x)$ is power of the function $ f_1=f_1(x) $ and so on, and all the functions $ f_i $ belong to $ \mathfrak{F}_3,$ then which class does $g$ belong to? Is $g\in \mathfrak{F}_4 $?

Note: you can find more information on "the fast growing hierarchy" on page 9 in this following paper: http://arxiv.org/abs/1007.2989

existence of the extension of "derivation"

2011-04-07T15:38:13Z

First, we can define a "derivation" (a linear operator) $\delta$ on $\mathbb{C}[x_{1},\cdots\,\,]$ by defining $$\delta(x_{k})=x_{k+1},\forall k\geq 1,\delta(x_{m}x_{n})=\delta(x_{m})x_{n}+x_{m}\delta(x_{n})=x_{m+1}x_{n}+x_{m}x_{n+1},$$ for example, $\delta(x_{2}^2x_{3}+x_{4}+2)=2x_{2}x_{3}^2+x_{2}^2x_{4}+x_{5}$.

Second, define $P_m$ to be the set of all polynomials in $m$ indeterminates, and $C(A),C(B)$ to be the $C^*$ algebras, where $A, B$ denote bounded closed sets in $\mathbb{C}^N,\mathbb{C}^{N+1}$ respectively. Define $P_N|_A$ and $P_{N+1}|_B$ to be the set of $P_N$, $P_{N+1}$ defined on $A,B$ respectively. Then the previous derivation $\delta$ can be considered as a linear operator from $P_N|_A$ to $P_{N+1}|_B$, and the set $P_N|_A$ to $P_{N+1}|_B$ are respectively dense in $C(A)$,$C(B)$ with respect to its norm. I want to ask the following question:

Can we extend the derivation $\delta$ to a "derivation" from $C(A)$ to $C(B)$, i.e., does there exist a derivation $\tilde{\delta}$ from $C(A)$ to $C(B)$, such that the restriction of $\tilde{\delta}$ to $P_N|_A$ is the previous one $\delta$ ? Here, the latter "derivation"$\tilde{\delta}$ means the common one defined for $C^*$ algebras, i.e, a linear operation which satisfies the Lebniz relation: $\tilde{\delta}(fg)=\tilde{\delta}(f)g+f\tilde{\delta}(g),\forall f,g\in C(A)$, and note that all the derivations here are not assumed to be bounded.

Restriction on the coefficients for an operator in the free group factor $ L(\mathbb{F}_2) $

2011-06-29T14:22:14Z

Let $\mathbb{F}_2$ denotes the free group generated by a,b, denote this group by $G$. Then consider the von Neumann algebra $L(G)$ generated by the family $\{L_{x_g} : g \in G\}$ , here, with $g \in G$, we denote by $x_g$ the function on $G$ that takes the value 1 at g and 0 at other elements of $G$. Then, note that we have the following relations:

$(L_{x_g})^*=L_{(x_g)^{-1}} , L_{x_g}L_{x_h}=L_{x_g*x_h}=L_{x_{gh}}$, then, for any $ A \in L(G),$we can set $ A=\sum_{g \in g}\mu_g L_{x_g},$ with $\mu_g \in \mathbb{C}.$

When we calculate $ ||Ax_{h}||^2 $, we find that $ \sum_{g \in G}|\mu_g|^2 < \infty.$

Then, is this condition sufficient for $ A \in L(G) $? Or some stronger condition is necessary? Like $ \sum_{g \in G}|\mu_g| < \infty,$ or something else?

Toeplitz operator used to study the free group factors?

2011-06-25T16:51:49Z

Note that when we construct the 'group von Neumann algebra'$vN(\gamma)$ using a discrete group $\gamma$(especially,the free group on n generators,$n \geq 2$),the elements of $vN(\gamma)$ have matrices(w.r.t, the basis $\epsilon_\gamma$) which are constant along the 'diagonals':{($ \gamma,\tau $):$\gamma \tao^{-1}$ is constant}, which is a Toeplitz matrix.

Recall that a Toeplitz operator can also be represented as a Toeplitz matrix, so I want to know:

Are there known results linked to the free group factors obtained by applying techniques from studying the Toeplitz operator?

the maximal length of a special dicksonian sequence

2011-05-06T05:12:39Z

First, we define a sequence $t_{1},t_{2},\cdots,t_{k}$ of n-tuples dicksonian, if $\forall 1\leq i < j\leq k,$ there does not exist a non-negative n-tuple t such that $t_{i}+t=t_{j}.$ For example, any lexicographically decreasing sequnence is dicksonian. By Dickson's lemma, every dicksonian sequence is finite. Let $(a_{1}^{1},\cdots,a_{n}^{1}),(a_{1}^{2},\cdots,a_{n}^{2}),\cdots,(a_{1}^{k},\cdots,a_{n}^{k})$ be a dicksonian sequence of n-tuples of non-negative integers such that $\sum_{i=1}^{n}(a_{i}^{j})=f(j)$ for all $j,1\leq j\leq k,$ where $f: \mathbb{Z} _{\geq0} \rightarrow \mathbb{Z} _{\geq0}$ is a fixed function.

Note that in the paper, "G. Moreno Socias, An Ackermannian polynomial ideal" it actually considered the maximal length of a dicksonian sequence such that $f(1)=d,f(i+1)=f(i)+1,\forall i\geq 1$, and this result is represented as a Ackermann function. Considering the characteristic of the dicksonian sequence satisfying the requirement with the maximal length with $n=3,d=3,$ given at the end of this paper, I want to ask the following question:

What is the possible maximal length for a dickson sequence such that $f(1)=d,f(i+1)=f(i)+1,\forall i\geq 1$, and the sum of the first two entries of every n-tuple in this dicksonian sequence is a fixed number, say m.?

Note that the position of the two entries with a fixed sum in a n-tuple may further affect the final result, I may further ask the following question:

What is the possible maximal length for a dickson sequence such that $f(1)=d,f(i+1)=f(i)+1,\forall i\geq 1$, and the sum of the two entries at position $i_0,j_0, 1\leq i_0\lt j_0\leq n$ of every n-tuple in this dicksonian sequence is a fixed number, say m.?

Differential ideal membership problem

2010-11-29T09:55:25Z

We know that in the ordinary algebra ideal case, the ideal membership problem can be solved by the Grobner Base theory, then, is there a counterpart theory in the differential ideal case?

To be precise, suppose I is a differential ideal in the differential ring k{y_1,...,y_n}, which is generated by a finite set of differential polynomials{f_1,...,f_m},can we choose a base A={g_1,...g_d} in I ,such that I is also generated by A,and we can use it to determine whether a differential polynomial belongs to I or not by a method that is similar to the use of the Grobner Base .

how to determine whether an ideal is prime or not by an algorithm

2011-04-07T15:49:26Z

Given polynomials $f_{1},\cdots,f_{n}\in \mathbb{C}[x_{1},\cdots,x_{m}]$, do we have an algorithm to determine whether the ideal $I=(f_{1},\cdots,f_{n})$ is prime ideal or not? Of course, we assume the polynomials are irreducible.

seek a theorem to control zero set

2011-03-21T04:37:52Z

I want to know if we can get this type of theorem: Given polynomials $f_{1},\cdots,f_{n}\subset \mathbb{C}[x_{1},\cdots,x_{m}]$, denote their common zeros as a set $V$, suppose $V$ is bounded. Given any point $p\in \mathbb{C}^m$, can we find a number $r(p)>0$, such that $V \subset B(p,r(p)),$ here $B(p,r(p))={q \in \mathbb{C}^m|dist(p,q)< r(p)} $, and $V = \emptyset$ iff $\exists p,q\in \mathbb{C}^m, s.t.,B(p,r(p))\cap B(q,r(q)=\emptyset$.

derivation between two $C^{*}$ algebras

2011-03-03T02:32:07Z

given two $C^{*}$ algebras $A\subset B $, acting on the same Hilbert space $H$, and $\delta $ is a derivation from $A$ into $B$,(in this case, a derivation is a linear mapping such that $\delta(ab)=a\delta(b)+\delta(a)b,\forall a,b \in A$) and assume it is bounded, then, is there an element $h$ of $A^{-}$(the weak operator closure of $A$), such that $\delta(a)=ha-ah, \forall a\in A$? \

Especially, I want to consider the case when $A,B$ are all commucative $C^{*}$ algebras.In other words, is there no nontrivial bounded derivation?

relation between polynomial $f$ and $\delta (f)$

2011-03-01T12:20:31Z

Given a polynomial $f \in \mathrm{C} [y_{0}, y_{1}, \cdots, y_{n-1} ]$, where $n$ is a natural number with $n >1$. Define a derivation $\delta$, s.t., it is linear and $\delta(y_{a}*y_{b})= y_{a+1}*y_{b}+ y_{a}*y_{b+1}, all a, b=1, \cdots, n-1 $. For example, if $f=3*y_{1}*y_{4}-y_{5},$ then, we have $\delta (f)=3*y_{2}*y_{4}+3*y_{1}*y_{5}-y_{6}$. Then, the question is : given a point $ z=(z_{0}, \cdots, z_{n})$, with $z\neq 0$, is there any relation between $|f(z)|$ and $|\delta (f)|$ ? (here, $|\ |$ denotes the absolute value defined by $|z|= \sum_{k=0}^{n} z_{k}$).\

One of my motivation of this problem is that I expect to use the result combining with W.Dale.Brownawell's result on a analytic description of Nullstellensatz Theorem to test a problem in differential algebra, however, I do not know how to get any useful result on the relation of $|f(z)|$ and $|\delta (f)|$ mentioned above. c.f.link text for the differential algebra problem.

Are there known quantitative descriptions of the fact that the common zero set of some polynomials is empty besides Nullstellensatz?

2011-01-23T11:24:01Z

Working in a polynomial ring, if some polynomials has no common zeros, Nullstellensatz tells us a qualitative description of the propertity of these polynomials, are there known quantitative descriptions? For example, these descriptions may tell us information on the degrees or something else of the polynomials.

results on the bound of some nonincreasing sequence?

2010-12-24T15:47:21Z

Hello, everyone. I want to know whether there are nontrivial results of the following type:\

Given an operation $F$, then for any set $A \neq \emptyset$ that satisfies the property: $\exists K(A)>0$,s.t.,$A\supseteq F(A)\supseteq F\circ F(A)\supseteq \cdots \supseteq \underbrace{F\circ \cdots \circ F}_{K(A)}(A)=\emptyset$,then we can find a precise upper bound of $K(A)$ by using the information of $F,A$.

For example,consider A is the variety corresponding to a finite set of differential polynomials$f_{1},\cdots,f_{n}$ such that the differential ideal it generated is the unit ideal,(here A is obtained by regarding the differential polynomials as ordinary ones in polynomial ring) now take $F$ as the operation that putting $\delta f_{1},\cdots,\delta f_{n}$ to the set $f_{1},\cdots,f_{n}$ (here $\delta$ denotes the derivation operation),we obtain a set $A_{1}$ and then taking its variety,we obtain $F(A)$,continue,putting ${\delta}^{2} f_{1},\cdots,{\delta}^{2} f_{n}$ to set $A_{1}$,we obtain $A_{2}$,then $F\circ F(A)$,$\cdots,$. we know that the set sequence is nonincreasing and will become $\emptyset$ in some step.

Since I want to know how to attack this type of problem, any known results linked to this type may be illuminating.

Progress in J.F.Ritt's one problem?

2010-12-19T15:24:51Z

In Ritt's classical book"Differential Algebra",he asked the following question(it can be found in page 177,problem 8):

The decomposition problem: determine the number of times which the d.p.in a finite system $\phi$ must be differentied before elimitions will produce finite systems whose manifolds are the component of $\phi$.One would hope to secure a bound which depends on the number of d.p.in $\phi$,their orders and degrees.

Can anyone tell me the progress in this problem?

Nontrivial criteria for polynomials to have no common zeros?

2010-12-11T04:17:09Z

When we work in $C[x_1,x_2,...,x_n]$,here $C$ denotes the complex field, we know that when polynomials $f_1,f_2,...,f_k$ have no common zeros, then there exists polynomials $g_1,...,g_k$ , such that the sum of $g_i\cdot f_i$ is $1$, (Nullstellensatz). However, this theorem doesn't seem to have any practical value, consider the following problem:

Is there any nontrivial criterion such that we can determine whether some polynomials (chosen randomly) have common zeros or not by using it? Of course, I don't want to use the Grobner base theory, what I need is just a criterion involving information on the algebraic properties of the given polynomials, something just like Eisensten's criterion on irreducibility.

Are there good inequalities on the norm?

2010-09-17T16:13:31Z

It's well known that in a Hilbert space, good inequalities exist concerning the norm due to the existence of inner product.Now let X be a general Banach algebra, are there good inequalities concerning the norm? To be precise, let's consider an example, let X be a commutative Banach algebra with identity I,is the following claim ture or not(especially when X is infinite dimension)? Either for every element b in X with norm 1, we have the norm of b^2 is also 1, or inf ||b^2||=0, with b running over all elements in X with norm 1.

P.S.This problem is derived from a question concerning the existence of a nilpotent element in X, in other words, the linear span of all the multiplicative linear functionals may not equal to the dual space of X.

Comment by Jiang

2013-04-15T02:34:57Z

@Misha, thanks!

Comment by Jiang

2013-02-21T18:17:38Z

We can further ask a weak version of the big one: Does $L(G)\cong L(H)$ and $G$ satisfies property $(T)$ imply that $G, H$ are measure equivalent? We can also replace the property $(T)$ by any other known invariant properties under Measure equivalent. Still, much work to be done on both sides, ME and von Neumann algebra equivalence.

Comment by Jiang

2013-02-21T18:13:04Z

If this type question is sensitive to the separable argument, then a naive approach would be to find ICC groups $\{G_{\alpha}\}_{\alpha\in A}$, where $A$ is a uncountable set, and require $\{L(G_{\alpha})\}$ has at most countable different equivalent classes (maybe we can achieve this by considering the invariant associated to the $L(G_{\alpha})$, dimension, etc.) and at the same time require $G_{\alpha}$ has uncountable many different classes module the ME relation. Of course, these $\{G_{\alpha}\}$ should not have property $(T)$, and it may be not practical.

Comment by Jiang

2013-02-19T22:36:11Z

sorry, I do not quite understand the meaning you say "so that L(G) has some restrictions on the fundamental group". The problem asked for constructing $G, H$ such that $L(G)\cong L(H)$ and $G, H$ are not measure equivalent or show such $G, H$ do not exist. Is the expression misleading or am I miss some point?

Comment by Jiang

2013-01-10T22:09:53Z

Maybe the following links are helpful on this problem: aimath.org/WWN/kadisonsinger math.missouri.edu/~pete/index.html zentralblatt-math.org/portal/en/zmath/search/…

Comment by Jiang

2012-12-02T04:59:08Z

@Andreas, the result of the spectral measure in the above paper looks complicated, it seems unclear whether we could get a similar result as Li's in this case. I guess one reason is that we could not apply finite approximation argument in $L(\mathbb{F}_2)$ as the case when the gorup is amenable?

Comment by Jiang

2012-11-30T16:52:43Z

@ Andreas, thanks for bringing these papers into attention.

Comment by Jiang

2012-01-16T15:47:53Z

One comment, if we could choose one abelian von Neumann subalgebra $A$ such that there exists one unitary $u$ in the normalizer of $A'\cap N$ but not in the normalizer of $A$, then set $B=uAu^*$, this would give us one counterexample.

Comment by Jiang

2011-06-29T15:44:48Z

do you have any suggestions on how to attack this problem or reference, papers linked to this question?

Comment by Jiang

2011-05-06T09:57:42Z

Thanks Vijay, I have not met this paper before.

Comment by Jiang

2011-05-06T09:48:51Z

@Gerry Myerson, the paper can be found here:springerlink.com/content/y36195n14590l4l3. Note that the author first reduced his question to considering only monomial polynomial ideals, then the total degree of $p_d,p_{d+1},\cdots,$ increased by 1 each step.

Comment by Jiang

2011-05-06T09:42:29Z

@Ricky Demer, I have finally made the Tex displayed completely.

Comment by Jiang

2011-05-06T05:18:56Z

Why is the Tex not completely displayed?

Comment by Jiang

2011-04-08T15:00:13Z

Do you mean that $x_{2}=\delta(x_{1})=1\otimes[x_{1},T]$?note that I want $\delta(x_{1})=x_{2}$, and if any such derivation exists, then it is bouned by what you have pointed out before; however, I am not be able to show that $\delta(f)$ is bounded, for any polynomial $f$ in a $C^*$ algebra with norm 1 by direct caculation

Comment by Jiang

2011-04-08T04:11:22Z

Oh,maybe I have to add the condition that $B=A*L$,here $L$ means a bounded interval in $\mathbb{C}$, and regard $C(A)$ as a natural subspace of $C(B)$.But considering the fact that you have mentioned, the answer is indeed "no". However, I really want to know the question:Could the "derivation" $\delta$ in the first paragraph be really considered as a restriction of a derivation from a $C^*$ algebra to another one? Of course, these $C^*$ algebras should be noncommutative.