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Possible restrictions on generators of $M_n(\mathbb{C})$

2013-02-06T15:55:10Z

Suppose matrices $a$ and $b$ generate $M_n(\mathbb{C})$. I would like to know what restrictions this imposes on $a$ and $b$. More concretely, do there exist $a,b\in M_n(\mathbb{C})$, which generate $M_n(\mathbb{C})$, such that the minimal polynomial of $\alpha a+\beta b$ has degree $\leq n-1$ for all $\alpha,\beta\in \mathbb{C}$?

Centralizer in a matrix algebra over commutative polynomials

2012-12-19T21:46:04Z

Let $A=M_n(F[x_{ij}\mid 1\leq i,j\leq n])$ be the matrix algebra over commutative polynomials in $n^2$ variables $x_{11},\dots,x_{nn}$ over a (nice enough) field $F$. I would like to know what is the centralizer of the matrix $x=(x_{ij})$ in $A$.

If we consider $x$ as an element of $M_n(F(x_{ij}\mid 1\leq i,j\leq n))$, it is not difficult to show that its centralizer consists of polynomials in $x$ with coefficients in $F(x_{ij}\mid 1\leq i,j\leq n)$. Does the centralizer of $x$ in $A$ consists of polynomials in $x$ with coefficients in $F[x_{ij}\mid 1\leq i,j\leq n]$?

Centralizers in C*-algebra

2012-01-04T06:20:39Z

Let $a,b\in A$ be self-adjoint elements in $C^*$-algebra $A$ with equal centralizers, $\{x\in A; [a,x]=0\}=\{x\in A; [b,x]=0\}$ . Can we say anything about the correspondence between $a$ and $b$?

For example, if $A=B(H)$ for a separable Hilbert space $H$, then according to the double commutant theorem $b=f(a)$ for some Borel function $f$ on the spectrum of $a$.

spectra of sums in (Banach) algebras

2011-05-06T10:30:36Z

A similar question was already asked in question titled "Spectra of sums and products in (Banach) algebras [was: Spectrum in Banach Algebra]".

Answer there led me to the following question.

If for elements $a,b$ in Banach algebra $A$ hold that $\operatorname{spec}(\lambda a+\mu b)\subseteq \lambda \operatorname{spec}(a)+\mu \operatorname{spec}(b)$ for every $\lambda, \mu\in \mathbb{C}$, can we say something about the subalgebra $B$ of $A$ generated by $a$ and $b$. Might we conclude something about $B/\operatorname{rad}(B)$?

Where to look for (possible) counterexample of $B/\operatorname{rad}(B)$ being commutative?

Thank you for finding a counterexample. Now I would like to add some additional assumptions. Suppose that for a selfadjoint element $a$ and a unitary element $u$ of $C^∗$-algebra holds that $\operatorname{spec}(a+\lambda a^2)=\operatorname{spec}(a+\lambda ua^2u^∗)$ for every $λ∈\mathbb{C}$. Moreover, if $\lambda \in \mathbb{R}$ elements $a+\lambda a^2$ and $a+\lambda ua^2u^∗$ are unitarily equivalent.

positive hermitian elements in $M_n(\mathbb{C})$

2011-10-29T12:01:22Z

Elements of the set $P$ of positive hermitian $n×n$ matrices over complex numbers have some special properties:

(i) they are closed under sum,

(ii) they are closed under multiplication by positive scalars,

(iii) spectrum of every matrix is positive, (all eigenvalues are nonnegative, and not all are equal to 0),

(iv) $P+-P+iP+-iP=M_n(\mathbb{C})$.

Does any other subset of matrix algebra $M_n(\mathbb{C})$ satisfy these properties except for $tPt^{-1}$, where $t$ is an invertible element in $M_n(\mathbb{C})$?

algebraic closure of commuting pairs of matrices

2011-03-07T19:40:33Z

Let $F$ be an arbitrary field of characteristic $0$, $K$ its algebraic closure. Define $M=\{ (x,y)\in M_n(F)×M_n(F) \mid [x,y]=0\}$ and let $N$ be the Zariski closure of $M$ in $K^{2n^2}$.

How can one show that $N$ contains the set $\{(axa^{-1},aya^{-1}) \mid (x,y)\in N, a\in \mathrm{GL}(n,K)\}$?

Thank you.

Comment by

2013-02-06T18:32:23Z

Thanks. I have edited the question.

Comment by

2013-02-06T17:23:06Z

Thank you for your example. But I would like to find $a$ and $b$ (or prove that they do not exist), such that $\alpha a+\beta b$ has the minimal polynomial of degree $<n$ for all possible values $\alpha,\beta \in \mathbb{C}$.

Comment by

2013-02-06T17:18:43Z

I know that there exist generators $a$ and $b$ where one of them has the minimal polynomial of degree $n$ (as also your example shows), but I am asking if it is always the case that at least for some $\alpha,\beta\in\mathbb{C}$ the minimal polynomial of $\alpha a+\beta b$ has degree $n$.

Comment by

2013-02-06T15:46:28Z

Thank you. With this approach we are able to answer the question in affirmative.

Comment by

2012-01-10T14:33:19Z

But it holds a+b=1, so b=f(a).

Comment by

2012-01-06T20:25:53Z

I am interested in the situation of general $C^*$-algebras, and some "nontrivial" correspondence between $a$ and $b$. Since the correspondence in $B(H)$ is quite "strong", I expected that something nonobvious can be said also in general $C^*$-algebras.

Comment by

2012-01-05T09:26:50Z

Since the condition trivially holds if A is commutative, we exclude that case. In the noncommutative case, does it hold the same correspondence as in $B(H)$?

Comment by

2012-01-04T12:35:09Z

Yes, I am assuming that a and b are self-adjoint. Sorry for the mistake and thank you for pointing it out.

Comment by

2011-10-30T16:02:50Z

I cannot find the question in Halmos books. There were some other useful information. Thank you. For case $n=2$, might Mathematica be able to compute this?

Comment by

2011-10-29T13:54:55Z

But $x^∗ax$ is also hermitian matrix if $a$ is. So $x^∗Px⊂P$, and $x^∗M_n(\mathbb{C})x=M_n(\mathbb{C})$ iff $x$ is invertible. So $x^∗Px$ either does not satisfy (iv) or equals $P$.

Comment by

2011-03-07T22:48:46Z

This helps a lot, thank you.

Comment by

2011-03-07T20:18:53Z

@Victor: Sorry, I was reading this in an article where it was left as "easy to check".