All Questions
82 questions
27
votes
0
answers
1k
views
Unital $C^{*}$ algebras whose all elements have path connected spectrum
A unital $C^{*}$ algebra is called a "Path connected algebra" if the spectrum of all its elements is a path connected subset of $\mathbb{C}$.
What is an example of a non commutative ...
- 356
17
votes
5
answers
2k
views
If two projections are close, then they are unitarily equivalent
Given two projections $p,q\in B(H)$, it is well-known that if $\|p-q\|<1$, then there exists a unitary $u\in B(H)$ with $q=upu^*$.
The proof that immediately occurs to me uses comparison of ...
- 2,793
17
votes
2
answers
2k
views
The letters of the word "ART"
Edit: According to the Gelfand duality between topological spaces and commutative $C^{*}$algebras, I add some new tags. So the question is that what is the structure of $ Ext (A,A)$ where $A$ is $...
- 356
12
votes
1
answer
901
views
Is there a proof that the $C^{*}$-algebras don't see the invariant subspace problem?
This post is an appendix of this one.
Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators.
Invariant subspace problem: Let $T \in B(H)$. Is ...
9
votes
2
answers
298
views
Two inequalities in $C^*$ algebras
Under what conditions on a $C^*$ algebra $A$ we have the following inequality:
$$x^*a^*ax+a^*x^*xa\leq x^*x+a^*x^*ax+x^*a^*xa\;\;\; \forall x,a\in A$$
The second identity which I am looking for is ...
- 356
8
votes
0
answers
952
views
About generator and isomorphism problems for free groups operator algebras
Let $H$ be an infinite dimensional separable Hilbert space.
The $C^{*}$-algebras and von Neumann here unital and subalgebras of $B(H)$.
Definition : Let $\mathcal{A}$ be $C^{*}$-algebra (resp. a von ...
7
votes
1
answer
329
views
Is this a characterization of commutative $C^{*}$ algebras?
Assume that $A$ is a $C^{*}$ algebra with self adjoint elements $A_{sa}$. Assume that for all $a,b\in A$ we have $$ab\in A_{sa} \iff ba \in A_{sa}$$
Is $A$ necessarily a commutative ...
- 356
7
votes
0
answers
85
views
Analogue of Friedrichs extension for Hilbert $C^*$-modules
Suppose one has a densely defined symmetric operator $T:\mathcal{M}\rightarrow\mathcal{M}$, where $\mathcal{M}$ is a Hilbert $A$-module for a $C^*$-algebra $A$. Suppose that $T$ is non-negative, so ...
- 1,903
6
votes
1
answer
643
views
Is there an irreducible, noncompact commuting, nonnormal operator, with spectrum strictly continuous?
Let $H$ be an infinite dimensional separable Hilbert space.
Definition: The commutant $\mathcal{S}'$ of a subset $\mathcal{S} \subset B(H)$ is $ \{A \in B(H) : AB=BA \ , \ \forall B \in \mathcal{S} \}...
6
votes
2
answers
366
views
Expression of a non-orthogonal projection in a $C^*$ algebra via an orthogonal one
A paper I'm currently reading uses the following fact. If $A$ is a unital $C^*$-algebra, $P=P^2\in A$, then there are $T, F\in A$ s.t. $F$ is an orthogonal projection ($F=F^*=F^2$) and
$$P=F+FT(1-F).\...
- 1,284
6
votes
1
answer
680
views
Is there an operator algebraic reformulation of the invariant subspace problem?
Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators.
Invariant subspace problem: Let $T \in B(H)$. Is there a non-trivial closed $T$-invariant ...
6
votes
0
answers
243
views
For what kind of $C^*$ algebra $A$ every normal element $y\in A$ has a normal lift for every given surjective $C^*$ morphism $\phi:B\to A$
Is there a terminology for the following property of $C^*$ algebra $A$:
For every $C^*$ algebra $B$ and surjective $C^*$ morphism $\phi: B\to A$, every normal element $y\in A$ admits a normal ...
- 356
5
votes
2
answers
988
views
Projections in a W*-algebra as a continuous lattice?
A continuous lattice is a complete lattice $L$ in which every element $y$ is equal to $\bigvee${$x \in L \mid x \ll y$} where $x \ll y$ ("x approximates y" or "x is way below y") if for any directed ...
- 85
5
votes
1
answer
177
views
Approximating a projection by a sum of elementary tensors with a certain property
Let $A$ and $B$ be two C$^{*}$-algebras and suppose we have a non-zero projection $p\in A\otimes B$. (We can assume $A$ is nuclear, so that there is only one possible tensor product.)
Does there ...
- 267
5
votes
1
answer
385
views
hereditary C*-subalgebra of a non-elementary simple C*-algebra
A is said to be elementary if A is isomorphic to some $K(H)$ or $M_n$.
A C*-subalgebra $B$ is said to be hereditary if for every $0≤a≤b∈B$ we have $a∈B$.
I wanted to know that is this statement true?
...
- 581
5
votes
1
answer
321
views
Takesaki's proof of the Kaplansky density theorem
Consider the following fragment from Takesaki's book "Theory of operator algebra I":
Why is the boxed sentence true? It looks like they replace $A$ by its strong$^*$-closure. Is this ...
- 175
5
votes
1
answer
254
views
A nilpotency question on $C^{*}$ algebras
What is an example of a $C^{*}$ algebra $A$ with the property that: for every nilpotent(Quasi nilpotent) $a$ and for every $n\in \mathbb{N}$, there is a $b$ with $b^{n}=a$.
To what extent ...
- 356
4
votes
2
answers
397
views
The closure of all periodic homeomorphisms of circle
Let $X$ be the space of all continuous maps from $S^{1}$ to $S^{1}$. $X$ is equipped with $C^{0}$ topology. Let $Y\subset X$ be the union of all finite order homeomorphisms i.e: all ...
- 356
4
votes
1
answer
489
views
If a completely positive unital map admits a completely positive unital left inverse, it is a complete isometry
Let $T$ be an injective operator system and $U$ be an arbitrary operator system. Let $\varphi: T \to U$ be a unital completely positive map and $\psi: U \to T$ be a unital completely positive map with ...
- 175
4
votes
1
answer
152
views
$\tau (p) = \tau (q)$ for all normalized traces does not imply $p \sim q$
Could you give an example of a unital simple $C^*$-algebra that $\tau (p) = \tau (q)$ for all normalized traces does not imply $p \sim q$?
- 581
4
votes
1
answer
533
views
The C*-envelope of the algebra of continuous functions on a compact topological space is commutative
In my research in operator theory, specifically in C* algebras and enveloping, I came across this strange footnote in a text (locally published in non English where I study) which states the following:...
- 620
4
votes
1
answer
101
views
Functional calculus for "pre-linear" regular operators on a Hilbert module
Let $E$ be a Hilbert module over a $C^*$-algebra $A$. Let $T\colon E\to E$ be a densely defined, unbounded $A$-linear operator. (In particular, the initial domain of $T$ is an $A$-submodule of $E$.) ...
- 1,903
4
votes
1
answer
300
views
$\|t\| = \sup_{\|z\| \le 1} \|\langle tz,z\rangle\|$ when $t=t^*$
Let $A$ be a $C^*$-algebra, $E$ be a (right) Hilbert $A$-module and $t \in \mathcal{L}_A(E)$ be an adjointable operator satisfying $t=t^*$. Is it true that
$$\|t\| = \sup_{z \in E, \|z\| = 1} \|\...
- 175
4
votes
1
answer
110
views
Graded adjointable operators on a graded Hilbert space
Given a graded Hilbert space $\mathbf{H} = \bigoplus_{k \in \mathbb{N}} \mathbf{H}_k$, one might extend the notion of adjoint to a "graded adjoint" defined as follows: $L \in B(\mathbf{H})$ is said to ...
- 969
4
votes
0
answers
242
views
On the Dunford-Pettis property and multiplier algebras
I am not an expert in operator algebras, so if the answer to this question might be trivial, that might be one reason for that:
Let $\mathcal{A}$ be a $C^\ast$-algebra. Then $\mathcal{A}^{\ast \ast}$ ...
4
votes
0
answers
384
views
Extension of Coburn's theorem on isometry and Toeplitz algebra
$\newcommand{\id}{\mathrm{id}}$Let $H$ be a Hilbert space, and $X \in B(H)$ a proper isometry (i.e. $X^{\star}X = \id$ and $XX^{\star} \neq \id$). Coburn's theorem states that ${\rm C}^{\star}(X)$, ...
4
votes
0
answers
146
views
A question on extension of $Z^{*}$ algebras
A $Z^{*}$ algebra is a $C^{*}$ algebra which all elements are(two sided or equivalently one sided) zero divisor.
Are there two $Z^{*}$ algebras $A,B$ such that for every short exact sequence of ...
- 356
3
votes
1
answer
191
views
A possible spectral characterization of commutative $C^*$ algebras
Let $A$ be a $C^*$ algebra. Assume that
the spectrum $Sp(a_1a_2\ldots a_{n-1}a_n)$ is unchanged as a set after a permutation of $a_i$'s. (unless possible emerge or removing 0 from the spectrum)
Does ...
- 356
3
votes
1
answer
236
views
The inequality $a^*ca \le \|c\| a^*a$ in a pre-$C^*$-algebra
Let $A$ be a pre-$C^*$-algebra, i.e. $A$ satisfies all axioms for a $C^*$-algebra except completeness. In other words, $A$ is an involutive algebra with a $C^*$-norm.
We say that $x \in A$ is positive ...
- 175
3
votes
1
answer
481
views
Question on structure of von Neumann algebras, clarification in Conway's "A course in operator theory"
I was reading the section on the structure of type I von Neumann algebras in John B. Conway's "A course in operator theory" and a few questions about certain definitions and references arose, I was ...
- 417
3
votes
1
answer
424
views
A trace inequality between self-adjoint operators
Let $A$ and $B$ be self-adjoint operators on some Hilbert space and $B$ is postive. Suppose we have $-B\leq A\leq B$.Is it true then that $\|A\|_p\leq\|B\|_p$ where $\|.\|_p$ is the Schatten-$p$ norm ...
- 1,879
3
votes
1
answer
388
views
multiplier algebra of a simple $C^*$ algebra
If $A=K(H)$, where $H$ is an infinite dimensional separable Hilbert space, then $A$ is simple and nuclear, and the multiplier algebra $M(A)$ of $A$ is not nuclear.
My question is: can we find a non-...
- 451
3
votes
1
answer
116
views
Does a bounded positive modular sesquilinear form on a $C^\ast$-algebra induces an element of its multiplier algebra?
This is a question that originates from my attempt at this question. Specifically, for a $C^\ast$-algebra $A$, I am attempting to construct a map $\phi: A \times A \to A$ s.t.,
$\phi$ is sesquilinear,...
- 2,830
3
votes
1
answer
161
views
Simple $Z^{*}$ algebra
What is an example of a simple $C^{*}$ algebra which all elements are (two sided or equivalently one sided) zero divisor?
- 356
3
votes
1
answer
306
views
Opposite $C^*$ algebras
$\DeclareMathOperator\op{op}$Let $A$ be a $C^*$-algebra. We know that $A$ admits a natural operator space structure, namely the operator space structure induced by any faithful $*$-representation of $...
- 1,879
3
votes
1
answer
187
views
Algebraic tensor product of C*-algebras extends via ideals? Application to restriction theorem?
Is the following assertion and the proof below correct,
or am I missing something very important?
Moreover, would the corollaries be correct then?
Besides, I would also appreciate a lot any comment, ...
- 598
3
votes
1
answer
173
views
Obstructions for $C^\star$ algebras to contain a $Z^\star$ algebra
As the comment of Andreas Thom indicated here, a separable $C^\star$ algebra $A$ can not contain a $Z^\star$ algebra.(A $Z^\star$ algebra is a $C^\star$ algebra which all elements are zero ...
- 356
3
votes
1
answer
324
views
Example of a ternary $C^{\ast}$-ring which is not an operator space
A ternary $C^{\ast}$-ring is a complex Banach space $X$, equipped with a ternary product $[\cdot,\cdot,\cdot]:X^3 \to X$ which is linear in outer variables and conjugate linear in middle variable. ...
- 1,115
3
votes
1
answer
255
views
Takesaki: Lemma about enveloping von Neumann algebra
Consider the following lemma with proof from Takesaki's book "Theory of operator algebra I" (p121):
It appears to me that Takesaki claims at the end of the proof that $\pi(A)_1$ is $\sigma$-...
- 175
3
votes
0
answers
179
views
Stinespring's theorem: can we choose the dilation to be an isometry?
Let $A$ be a $C^*$-algebra and $\varphi: A \to B(H)$ be a completely positive contractive map. Stinespring's theorem says that there exists a $*$-representation $\pi: A \to B(H')$ and a bounded ...
- 175
3
votes
0
answers
69
views
Trying to understand morphisms in category of ternary $C^*$-rings
Recall that a ternary $C^*$-ring is a complex Banach space $X$, equipped with a associative ternary product $[.,.,.]:X^3 \to X$ which is linear in outer variables and conjugate linear in middle ...
- 1,115
3
votes
0
answers
97
views
Is the set of points in the irreducible decompositions of this C$^{*}$ -algebra's representations closed?
Suppose $X$ and $Y$ are compact Hausdorff spaces. Let $\varphi\colon C(X)\to M_{n}(C(Y))$ be any $*$-homomorphism. If $\pi$ is an irreducible representation of $M_{n}(C(Y))$, then $\pi$ is unitarily ...
- 267
3
votes
0
answers
178
views
A point concerning absolute value of functionals
Let $M$ be a von Neumann sub-algebra in $B(H)$. Let $\phi$ be a normal functional on $M$. Assume $\psi$ is a normal functional on $B(H)$ with $\psi_{|_M}=\phi$ (note that $\phi$ and $\psi$ may have ...
- 4,058
3
votes
0
answers
255
views
Graded structures for simple $C^{*}$ algebras without nontrivial idempotent
Edit(A confession): I just realized that the question is trivial: Since one can easily prove that the convex hull of the spectrum of every nontrivial homogeneous element of a $\mathbb{Z}_{n}$-graded $...
- 356
2
votes
1
answer
279
views
The algebra of continuous functions on Cantor set
Let $C(K)$ be the algebra of continuous functions on Cantor set. Is it possible to prove that $C(K)$ forms an AF-algebra without Bratteli diagram?
- 581
2
votes
1
answer
352
views
K-Theory of $C^{*}(X)$
I'm new to K-Theory for $C^{*}$-algebra and $C^{*}$-algebra of groups.
If $X$ is the group of finite support bijections of natural numbers then what is the K-Theory of $C^{*}(X)$?
I was planning to ...
- 581
2
votes
1
answer
254
views
Is the reduced group $C^*$-algebra quasidiagonal
Let $G$ be an amenable group. I wonder whether it is true that the reduced group $C^*$-algebra $C_r^*(G)$ is quasidiagonal.
- 427
2
votes
1
answer
274
views
Trying to understand Haagerup tensor product $B(H)\otimes_{\rm h}B(K)$
I’m self reading Haagerup tensor product of operator spaces. Understanding it properly, I’m trying some examples. Let $H$ And $K$ be Hilbert space. Let $B(H)$ and $K(H)$ denotes the spaces of bounded ...
- 1,115
2
votes
2
answers
291
views
If either $A$ is exact or $B$ is nuclear then every closed ideal of $A\otimes_{min}B$ is of the form $A \otimes _{min}J$ for some ideal $J$ of $B$
From one of the talks I attended long back, I vaguely seem to remember the following fact:
Let $A$ and $B$ be $C^{\ast}$-algebras. If either $A$ is exact or $B$ is nuclear then every closed ideal of $...
- 1,115
2
votes
1
answer
182
views
A Possible characterization of F.D or AF commutative $C^{*}$ algebras
By F.D or AF $C^{*}$ algebra,we mean finite dimensional or approximately finite dimensional $C^{*}$ algebra.
Let $A$ be a unital commutative $C^{*}$ algebra with the property that for every ...
- 356