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27 votes
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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 ...
Ali Taghavi's user avatar
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 ...
Sebastien Palcoux's user avatar
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 ...
Ali Taghavi's user avatar
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 ...
Sebastien Palcoux's user avatar
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 ...
Ali Taghavi's user avatar
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 ...
geometricK's user avatar
  • 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} \}...
Sebastien Palcoux's user avatar
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 ...
Sebastien Palcoux's user avatar
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 ...
Ali Taghavi's user avatar
5 votes
0 answers
598 views

Do the banded operators check 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 ...
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 ...
Andromeda's user avatar
  • 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 $p \sim q$?
Peg Leg Jonathan's user avatar
4 votes
1 answer
534 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:...
groupoid's user avatar
  • 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 $E$.) ...
geometricK's user avatar
  • 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} \|\...
Andromeda's user avatar
  • 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 ...
Dave Shulman's user avatar
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 $\mathcal{A}^{\ast \ast}$ ...
Alexander Dobrick's user avatar
4 votes
0 answers
239 views

The ideal structure of the Haagerup tensor product of $C^{\ast}$-algebras

I'm currently reading the paper The ideal structure of the Haagerup tensor product of $C^{\ast}$-algebras and having difficulty in understanding the proof of Proposition $4.5$ from the paper. Let $A$ ...
Math Lover's user avatar
  • 1,115
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 ...
Andromeda's user avatar
  • 175
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 ...
A beginner mathmatician's user avatar
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,...
David Gao's user avatar
  • 2,830
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 $...
A beginner mathmatician's user avatar
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, ...
C-star-W-star's user avatar
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 $\sigma$-...
Andromeda's user avatar
  • 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 ...
Andromeda's user avatar
  • 175
3 votes
0 answers
83 views

Reference request for representation theory of TRO

Let $H$ and $K$ be Hilbert spaces. Recall that a Ternary ring of operator(TRO) $V$ is a closed subspace of $B(H,K)$ such that $xy^{\ast}z \in V$ for all $x,y,z \in V$. I have recently started reading ...
Math Lover's user avatar
  • 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 ...
ervx's user avatar
  • 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 ...
ABB's user avatar
  • 4,058
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?
Peg Leg Jonathan's user avatar
2 votes
1 answer
180 views

Does the square root of a finite propagation operator have finite propagation?

Let $X$ be a non-compact manifold and let $C_0(X)$ act on $L^2(X)$ by pointwise multiplication. We say $T\in\mathcal{B}(L^2(X))$ has finite propagation if there exists an $r>0$ such that: for all ...
geometricK's user avatar
  • 1,903
2 votes
1 answer
131 views

Closeness of points in the irreducible decomposition of a C$^{*}$-algebra representation

Suppose $X$ and $Y$ are compact metric 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 ...
ervx's user avatar
  • 267
2 votes
0 answers
203 views

Quasidiagonal C*-algebras

Let $A$ be a nuclear $C^*$-algebra satisfying UCT condition. Then under what assumptions $A$ is quasidiagonal?
Peg Leg Jonathan's user avatar
2 votes
0 answers
120 views

Need help in proving an inclusion between some subspaces of operators

The following question was first posted on Math.Stackexchange.com but unfortunately I didn’t get any answer. This might be obvious for many researchers but I can’t see how this is so, thus I am asking ...
Math Lover's user avatar
  • 1,115
2 votes
0 answers
210 views

The functional equation $T(x\otimes y)=T(x)\otimes T(y)$ on certain $C^{*}$ algebras

Is there a name for the following property of a $C^{*}$ algebra $A$? $$A \simeq A \otimes A\;\;\; \text{The spatial tensor product}$$ Example of this situation is $A=C(X)$ where $X$ is the ...
Ali Taghavi's user avatar
1 vote
1 answer
220 views

Dimension of commutant

Suppose that $A = M_n(\mathbb{C})$ be the algebra of $n*n$ matrices over $\mathbb{C}$. If com(A) = {$B \in M_n(\mathbb{C}); AB = BA$}, then what is the $dim(com(A))?$
Peg Leg Jonathan's user avatar
1 vote
1 answer
180 views

Conditioning a $\mathrm{C}^*$-algebra state with infinite precision

This question (and a second part) have been asked at MSE and gone through two bounties without an answer. I have been beating my head at it for a while without success. Let $\mathcal{A}$ be a unital $\...
JP McCarthy's user avatar
  • 1,037
1 vote
1 answer
136 views

The closure of selfadjoint elements of an algebra whose spectrum consist of rational numbers

Let $H$ be a seperable complex Hilbert space. What is the closure of the set of all self adjoint operators in $B(H)$ whose spectrum is a subset of the rational number $\mathbb{Q}$. Apart from finite ...
Ali Taghavi's user avatar
1 vote
1 answer
284 views

A certainty principle?

Let $\mathcal{A}$ be a unital $\mathrm{C}^*$-algebra with $\varphi\in\mathcal{S}(\mathcal{A})$ a state. Where $$\sigma_\varphi(a):=\sqrt{\varphi((a-\varphi(a)1_{\mathcal{A}})^2)}\qquad (a\in \mathcal{...
JP McCarthy's user avatar
  • 1,037
1 vote
1 answer
118 views

Let $V$ be a TRO such that $A(V)= \mathbb{C}$, what can we say about $V$?

Let $V$ be a TRO i.e. closed subspace of $B(H,K)$ such that $xy^*z \in V$ for all $x,y,z \in V$. Let $C(V)$ and $D(V)$ denotes the $C^{\ast}$-algebra generated by $VV^{\ast}$ and $V^*V$ respectively. ...
Math Lover's user avatar
  • 1,115
1 vote
1 answer
273 views

examples of completely positive order zero maps to demonstrate a theorem

I'm interested explicit examples which can be used to demonstate the theorem: Theorem: Let $A$ and $B$ $C^*-$algebras and $\phi:A\to B$ be a completely postive map of order zero. Set $C:=C^*(\Phi(A))\...
user avatar
1 vote
0 answers
113 views

Is it true that $\omega = \sum_{(k,l)\in I^2}\omega(p_k - p_l)$

Let $\{p_i\}_{i \in I}$ be a collection of projections in a $C^*$-algebra $A$ such that $\sum_{i \in I} p_i = 1$ in the strict topology (note here that $1$ is the unit of the multiplier algebra $M(A)$)...
Andromeda's user avatar
  • 175
1 vote
0 answers
139 views

Reduced twisted $C^*$-algebra and twisted crossed product

Let $G$ be a discrete group. Is it possible to represent $C^*_r(G, \sigma)$, the reduced twisted group $C^*$-algebra as a twisted crossed product?
Peg Leg Jonathan's user avatar
1 vote
0 answers
137 views

A continuous choice of invertible elements

Let $A$ be a simple unital $C^{*}$ algebra with invertible elements $G(A)$. Assume that $A^{*}$ is its dual space, which is equipped with the weak star topology. Is there a continuous map $\alpha:A^...
Ali Taghavi's user avatar
1 vote
0 answers
258 views

Is an exact operator, unitary equivalent to a banded operator?

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Definition : Let $(e_{n})_{n \in \mathbb{N}}$ be an orthonormal basis. $T \in B(H)$ is ...
Sebastien Palcoux's user avatar
0 votes
1 answer
158 views

Abelian twisted reduced group C*-algebra

Let $G$ be an abelian discrete group. Then is $C_r^*(G, \sigma)$ abelian?
Peg Leg Jonathan's user avatar
0 votes
0 answers
92 views

Proof of the isomorphism of the Toeplitz algebra and the algebra generated by the element and the relation

Please tell me where can I see the proof of this well-known fact? enter image description here
Soar Appell's user avatar
0 votes
0 answers
88 views

Is A an amenable $C^{*}$-algebra?

Let $A$ be $C^{*}$-algebra. Suppose that, for any $\epsilon > 0$ and finite subset $F \subset A$, there are an amenable $C^{*}$-subalgebra $B \subset A$, contractive completely positive linear maps ...
Peg Leg Jonathan's user avatar
-1 votes
1 answer
210 views

A commuting pair of isometries

Let $H$ be a Hilbert space and $B(H)$ be the space of all bounded operators on $H$. The Wold decomposition says that: an operator $x$ in $B(H)$ is an isometry if and only if $x=x_u\oplus x_s$ where $...
ABB's user avatar
  • 4,058