Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [c-star-algebras]

A C*-algebra is a complex Banach algebra together with an isometric antilinear involution satisfying $(ab)^*=b^*a^*$ and the C*-identity $\Vert a^*a\Vert=\Vert a\Vert^2$. Related tags: (banach-algebras), (von-neumann-algebras), (operator-algebras), (spectral-theory).

2
votes
1answer
26 views

Why is the Rational Rotation Algebra not a Matrix Algebra?

Let $A_{\theta}$ be the rotation C$^{*}$-algebra with rotation $\theta$. I.e., $A_{\theta}=C^{*}(u,v)$, where $vu=e^{2\pi i \theta}uv$. Suppose that $\theta=p/q$, where $p$ and $q$ are non-zero ...
1
vote
1answer
32 views

Is it possible $\lVert a\rVert =\lVert 1+a\rVert $ in a C$^*$-algebra?

If $A$ is a unital C$^*$-algebra and $a\in A$, Is it possible $ \lVert a \rVert =\lVert 1+a \rVert $ for an $a\geq 0$ ? I think it's trivial that it's not possible but I can't prove it for even $ ...
1
vote
1answer
21 views

partial isometry

If $A\in M_{r\times m}(\mathbb{C}),B\in M_{m\times r}(\mathbb{C})$,$AB=Id_r$,then $A,B$ is a partial isometry. My question:how to show $A^*A,B^*B$ is a projection?
0
votes
1answer
19 views

nuclear $C^*$ algebra

If $(A_i)$ is a sequence of nuclear $C^*$ algebras,Is $\oplus_{c_0}A_i$ ($c_0$ direct sum)and $\prod A_i$($\ell ^\infty $ direct sum) also nuclear?
-1
votes
1answer
23 views

functional calculus under the $*$ homomorphism

If $A,B$ are two $C^*$ algebras,$\psi:A \rightarrow B$ is a non $*$ homomorphism.Suppose $b$ is a nonzero normal element in $B$,we have a $*$ isometric isomorphism $\phi:C(\sigma_B(b))\to C^*(b,b^*),\;...
1
vote
1answer
12 views

functional calculus for non-unital $C^*$ algebras

If $A$ is a non-unital $C^*$ algebra,$a$ is a normal element of $A$, there is a $*$ isometric isomorphism from $C_0(\sigma_{A}(a))$ to $C^*(a)$. I have a question:what is the set of $C_0(\sigma_{A}(a)...
2
votes
0answers
43 views

Extension of a von Neumann algebra by a von Neumann algebra

Let $A,B,C$ be $3$ unital $C^*$ algebras. Assume that we have the following short exact sequence of $C^*$-algebras: $$0\to A\to C\to B\to 0$$ Assume that $A,B$ are generated by their ...
3
votes
1answer
32 views

trace and operator norm of $\exp^A$

If $A$ is an n by n complex matrix, 1.How to compute $tr(\exp^A)$ .Can we use the Taylor expansion as following: $\exp^A=\sum_{k=0}^{\infty}\frac{A^k}{k!},$then $tr(\exp^A)=\sum_{k=0}^{\infty}tr(\...
0
votes
1answer
21 views

elements in $C^*$ algebra

If $x$ is an element of a $C^*$ algebra $A$,Is $exp^x$ an element of $A$? My thought: compute the Taylor expansion of $exp^x$,but it is a sum of countable terms.Is sum closed in $A$?
1
vote
0answers
22 views

Commuting Elements in Tensor Products of C*-Algebras

I am working on exercise 7.G in the book “K-Theory and C*-Algebras” by Wegge-Olsen. Let $A$ be some unital C*-algebra, $u$ a unitary in $M_n(A)$, and $u’$ a standard unitary (which is defined to be a ...
1
vote
1answer
37 views

infinite dimensional positive matrix

Suppose $A$ is a infinite dimensional positive complex matrix,what is the operator norm of $A$? In the finite dimensional case,we can use the spectral theorm,$\|A\|=sup|\lambda_i|$,where $\lambda_i$ ...
3
votes
1answer
33 views

What Are the Irreducible Representations of the Rational Rotation C$^{*}$-algebra?

Let $m$ and $n$ be integers, with $n>0$ and $\gcd(m,n)=1$. Let $\theta=m/n$ and let $A_{\theta}$ be the rational rotation C$^{*}$-algebra generated by two unitaries $u$ and $v$, satisfying the ...
0
votes
0answers
10 views

An example of Hilbert C$^*$-module $M_A$ in which $A=Mat_{n\times n}(\mathbb{C} )$ and $M$ is not countably generated?

I want to find an example of a Hilbert C$^*$-module $M_A$ in which $A=Mat_{n\times n}(\mathbb{C} )$ and $M$ is not countably generated? I know $\left( \ell_2(A)\right)_A $ is countable generated and ...
0
votes
0answers
11 views

compute the multiplier algebra

If $A$ is a $C^*$ algebra, $\oplus_{c_0} M_{k(n)} (\mathbb{C} )$ is the essential ideal of $A$ ,then we have $\oplus_{c_0} M_{k(n)} (\mathbb{C} ) \subset A \subset \prod M_{k(n)} (\mathbb{C})$. ...
0
votes
0answers
13 views

construct an element in $\prod M_{k(n)} (\mathbb{C})$

Suppose $A$ is a $C^*$ algebra,$\oplus_{c_0} M_{k(n)} (\mathbb{C} )$ is a essential ideal of $A$ and there is an element $(x_n) \in A$ such that $(x_n) \in \prod M_{k(n)} (\mathbb{C})$ and $tr(x_n) \...
1
vote
1answer
31 views

Minimal or Maximal Von Neumann algebra contained in a given $C^*$ algebra

Let $A,B \subset B(H)$ be two concrete von Neumann algebra. Is $A\cap B$ a von Neumann algebra, too? What about the intrinsic analogy of this question, as follows: Let $C$ be a $C^*$ ...
1
vote
1answer
20 views

Classification of $C^*$ algebras whose subalgebra generated by projections is a von neumann algebra

Inspired by this question we ask the following question: Is there a complete classification of all unital $C^*$ algebra $A$ for which the following subalgebra $B$ is a von Neumann algebra? Is ...
2
votes
1answer
23 views

What can we say about the spectrum of the difference of positive elements?

Let $\mathcal{A}$ be a unital $C^*$-algebra with $a,b\in\mathcal{A}_+$ such that $\|a\|<\|b\|$. Does it follow that $a-b\not\in\mathcal{A}_+$? I can find very little about the spectrum of the sum ...
1
vote
1answer
19 views

comparison of multiplier algebras

Suppose $I$ is an essential ideal of a nonunital $C^*$ algebra $A$, can we compare $M(I)$ and $M(A)$,is $M(A)\subset M(I)$,where $M()$ denotes the multiplier algebra.
-1
votes
1answer
37 views

construct a sequence of operators [on hold]

Let $(H_n)$ be a sequence of different finite dimensional complex Hilbert spaces, $A_n \in B(H_n),tr(A_n) \to 0(n \to \infty)$,but the norm of $A_n$ does not converge to 0,where $tr()$ is the standard ...
0
votes
0answers
14 views

strictly positive element in a separable $C^* algebra

If $A$ is a separable $C^*$ algebra,does there exist a strictly positive element in $A$?
1
vote
1answer
46 views

Does a closed right ideal of a C$^*$-algebra have a C$^*$-algebra?

$A$ is an infinite dimensional C$^*$-algebra and $J\subset A$ is a closed right ideal. $A$ and $J$ are infinite dimensional(as a vector space). I want to find an infinite dimensional C$^*$-algebra ...
1
vote
1answer
21 views

Is the set $\overline{\langle \{ f^{*}_{1}f_2: f_1,f_2\in J\}\rangle } $ equal to $B(\ell^2)$?

If $J=\left\lbrace f\in B(\ell^2): f^*(e_1)=0 \right\} $, is the set $\overline{\langle \{ f^{*}_{1}f_2: f_1,f_2\in J\}\rangle } $ equal to $B(\ell^2)$? ( $\ell ^2 $ is the Hilbert space $(\ell^2, \...
1
vote
1answer
20 views

A Hahn-Banach separation theorem argument, claryfying the details

I have a question regarding the proof of proposition 6.1. in https://arxiv.org/pdf/1509.01870.pdf, how exactly Hahn-Banach separation theorem has been used. Proposition 6.1: A discrete group is $C^*$-...
0
votes
0answers
36 views

a norm on a complex matrix

Suppose $A=\prod_n M_{k(n)}(\mathbb{C})$,where $M_{k_n}(\mathbb{C})$ is the space of all $k(n) \times k_n$ complex matrices,does there exist a norm $\| \|_0$ on $A$ (which is different from the ...
0
votes
1answer
18 views

pairwise orthogonal projections in an inseparable $C^* $ algebra

If $A$ is a separable $C^*$ algebra,then there are at most countable pairwise orthogonal projections.If $A$ is inseparable,how many pairwise orthogonal projections in $A$? If it has, is it ...
2
votes
1answer
91 views

Attaining the norm of a C*-algebra quotient by an ideal

The following statement is Exercise I.26 in K. Davidson's book C-Algebras by Example*: Let $\mathfrak{A}$ be a C*-algebra, $\mathfrak{J}$ be an (two-sided and closed) ideal of $\mathfrak{A}$ and $...
-1
votes
0answers
16 views

$C_0$ direct sum of complex matrices

How many elements in the $C_0$ direct sum of complex matrices,say$\oplus_n M_n(\mathbb{C})$? $\oplus_n M_n(\mathbb{C})=\{(x_n)\in \prod_n M_n(\mathbb{C}):\|x_n\| \to 0\}$ I think there are ...
1
vote
3answers
30 views

orthogonal projections in $C^*$ algebra

Suppose $A$ is an arbitrary $C^*$ algebra,can $A$ be linear spanned by all orthogonal projections of in it ? If not,is there a relationship between a $C^*$ algebra and all orthogonal projections in ...
-2
votes
0answers
29 views

compact self-adjoint operator [closed]

If $T$ is a compact self-adjoint operator on a hilbert space $H$.Can $H$ be composed as follows: $H$ is the direct sum of all eigenspaces.That is to say, $H=\oplus H_i$,where $H_i=ker(T–\lambda_i ...
-1
votes
1answer
56 views

finite rank projection

In Murphy's book,there is a statement:If p is a finite rank projection on $H$,then $pB(H)p$ is finite dimensional. My question:Given any $S\subset B(H)$,$S$ does not contain $Id_H$.Does there exist ...
2
votes
2answers
42 views

‎Theorem ‎2.1.15 of‎ ‎Murphy's ‎book

The relevant theorem: Let $\Omega$ be a compact Hausdorff space, and for each $\omega\in\Omega$ let $\delta_\omega$ be the character on $C(\Omega)$ given by evaluation at $\omega$; that is, $\...
0
votes
1answer
15 views

a question on gns construction

If $\pi$ is a zero representation of $C^*$ algebra $A$,there is no state$\tau$ on $A$ such that $\tau(a)=(\pi(a)\xi,\xi)$. When we talk about GNS constuction,Should the zero representation be ...
-1
votes
0answers
23 views

What does $f(a)$ mean for $a\in\mbox{Re}\mathcal{A}$ and $f$ on $\mathbb{R}$?

I have to solve exercise VIII.3.9 of John B. Conway. A Course in Functional Analysis. Springer, 1997 (second edition): If $a,b\in\mbox{Re}\mathcal{A}$, $a\leq b$, and $ab=ba$, then $f(a)\leq f(b)$ ...
1
vote
0answers
22 views

Can C*-envelope introduce the unity?

Let $A$ be a non-unital Banach $*$-algebra with isometric involution. Is it possible that the enveloping $C^*$-algebra of $A$ is unital? I guess not at least when $A$ admits an approximate identity, ...
1
vote
1answer
14 views

When is the intersection of two element-generated hereditary C$^{*}$-algebras non-zero

Let $A$ be a unital C$^{*}$-algebra. Let $b$ and $c$ be non-zero positive elements in $A$. Let's suppose that I can find a unitary $v$ in $A$ such that $bvc\not= 0$ and, hence, such that $z:=bvcv^{*}\...
0
votes
1answer
16 views

nondegenerate representation of a $C^*$ algebra

Every representation $(\pi,H)$ of a $C^*$ algebra $A$ can be reduced to the case of a non-degenerate representation.Usually,we take $K=[\pi(A) H]$,then we get a non-degenerate representation $(\pi_K,...
2
votes
1answer
39 views

proper ideals of $\prod_{i\in I}A_i$

$\oplus_{i\in I}A_i$ denotes the $c_0$ direct sum of $C^*$ algebras $A_i$, $\prod_{i\in I}A_i$ is the $l^\infty$ direct sum of $A_i$.We know that $\oplus_{i\in I}A_i$ is the essential ideal of $\prod_{...
0
votes
1answer
32 views

center of finite dimensional $C^*$ algebra

If $A$ is a finite dimensional $C^*$ algebra,is the center $\mathcal{Z}(A)$ of form $\lambda 1_{A}$,where $\lambda\in \mathbb{C}$?
3
votes
0answers
28 views

Regarding equality of norms in Hilbert bimodules

I was reading the book Elements of noncommutative geometry and in page 160 lemma 4.21 the authors state that in a Hilbert B-A bimodule $E$ the two norms induced by the two inner products coincide. ...
0
votes
1answer
64 views

essential ideal of a $C^*$ algebra

There is a well known fact:$A=\oplus_{i\in I}A_i$($c_0$ direct sum) is an essential ideal $\prod_{i\in I}A_i$($l^\infty$ direct sum),where each $A_i$ is a $C^*$ algebra. I have two questions: 1.If $\...
2
votes
1answer
40 views

Kadison's Theorem for operator subsystems of commutative $C^*$-algebras

By a result of Kadison, every operator subsystem of a commutative $C^*$-algebra is isomorphic to the space of continuous affine functions on its state space. In other words, if $X$ is a compact ...
1
vote
1answer
22 views

About the C*-algebra of the Schrodinger representation of the Weyl C*-algebra

We start with the Weyl C*-algebra $\mathcal{W}$ for a finite dimensional symplectic space and we consider the irreducible Schrodinger representation $\pi:\mathcal{W}\rightarrow \mathcal{B}(\mathcal{H})...
1
vote
1answer
36 views

tracial state on a unital infinite dimensional simple $C^*$ algebra

If $A$ is a finite dimensional simple $C^*$ algebra,then it has the form of $M_n(\mathbb{C})$,which has unique tracial state. My question is:If $A$ is an unital infinite dimensional simple $C^*$ ...
1
vote
2answers
27 views

annihilator of an ideal in $C^*$ algebra

If $H$ is a hilbert space,K is a closed subspace of $H$,then $H=K\oplus K^\perp$. If $A$ is a $C^*$ algebra,$I$ is a closed ideal of $A$.Does there exist a similar decomposition $A=I\oplus I^\perp$?
2
votes
1answer
28 views

Characterization of weak convergence in Hilbert $C^*$-modules?

Assume $M$ is a Hilbert $C^*$-module and $(x_n)^{\infty }_{n=1}$ a bounded sequence in $M$. Are these equivalence? $\langle x_n,y\rangle \to 0 $, for all $y\in M$. $(x_n)$ is convergent to $0$ in ...
2
votes
1answer
32 views

Is an element in a homogeneous C$^{*}$-algebra whose image in each primitive quotient is invertible necessarily invertible?

Let $N\in\mathbb{N}$ and suppose $A$ is a unital C$^{*}$-algebra with the property that each irreducible representation of $A$ has dimension $N$. I.e., $A$ is $N$-homogeneous. Suppose we have an ...
1
vote
1answer
13 views

Is each finite dimensional $C^*$-algebra an $AW^*$-algebra?

I know that a finite dimensional $C^*$-algebra is a $W^*$- algebra(von neumann algebra). But is it true for $AW^*$-algebras?
-2
votes
2answers
39 views

Is $F(H)$ dense in $B(H)$?

Let $F(H)$ be the set of finite rank operators on Hilbert sapce $H$,$K(H)$ is the set of compact operators on $H$,$B(H)$ is the set of bounded linear operators on $H$.I know the fact :$F(H)$ is dense ...
0
votes
1answer
36 views

tracial state on a non-unital simple $C^*$ algebra

I think that there is no tracial state on non-unital simple $C^*$ algebras.Is my thought correct? I 'll appreciate it if anyone can supply me a counterexample.