All Questions
Tagged with fa.functional-analysis c-star-algebras
316 questions
4
votes
1
answer
153
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 ...
- 327
8
votes
1
answer
547
views
Maps which are both completely positive and positive
Definition:A linear map $f:\mathbb C^n\to \mathbb C^n$ is called positive if $\langle fa,a\rangle\ge0$ for all $a\in \mathbb C^n$. Equivalently, $f\in M_{n}(\mathbb C)$ is positive if it can be ...
- 43.2k
6
votes
0
answers
237
views
A characterisation of certain $C^*$-algebras
I was wondering if there is a characterisation for $C^*$-algebras (unital) for which the bidual does not have any central atoms. It is not sufficient for example to demand that the $C^*$-algebra does ...
- 633
1
vote
0
answers
95
views
An example of a sequence of finite projections
Let $A$ be a vn-algebra. Suppose that $x$ is an isometry with $\inf_{n\geq1} x^nx^{*n}=0$ (Note that $x^nx^{*n}$ are all projections). Let $e$ be a (non-zero) finite projection and put $q_n$ to be ...
- 4,058
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 ...
- 267
3
votes
1
answer
309
views
How rich the group of unitary elements in a von Neumann algebra to get "Murray-von Neumann" equivalence?
Denote by $\sim$ the "Murray-von Neumann" equivalence in the projection lattice of a von Neumann algebra. It is well known that in a finite von Neumann algebra the equivalence of projections can be ...
- 2,118
4
votes
1
answer
357
views
Extending maps from dense $*$-algebras of $C^*$-algebras
Given $\cal{A},\cal{B}$ two dense $*$-algebras of two $C^*$-algebras $A$ and $B$ respectively, together with a $*$-algebra homomorphism $f:\cal{A} \to \cal{B}$, is it clear that $f$ extends to a ...
- 993
1
vote
1
answer
787
views
finite dimensional C*-algebras
Let $A$ be a C*-algebra. Suppose that every cyclic representation of $A$ is finite dimensional.
Q. Is $A$ finite dimensional?
- 4,058
4
votes
1
answer
279
views
Pure infiniteness of tensor product $C^\ast$-algebras
I found the following theorem (Proposition 4.5) in the paper of E. Kirchberg and M. Rørdam: Non-simple purely infinite $C^\ast$-algebras. Amer. J. Math. 122(3) (2000), 637-666 (MR1759891, jstor, ...
- 181
3
votes
2
answers
397
views
Is the ideal property of $X^{**}$ inheritable to $X$?
Let $X$ be an operator space such that there is a weak$^*$-continuous complete isometry $\phi$ from its second dual $X^{**}$ into a $W^*$-algebra $M$ in which $\phi(X^{**})$ is a (necessarily weak$^*$-...
- 619
4
votes
1
answer
541
views
Relation between maximal and reduced group $C^*$-algebras
Let $G$ be a Lie group and $C_r^*(G)$ and $C^*(G)$ be its reduced and maximal group $C^*$-algebras respectively. The left-regular representation of a group $G$ induces a surjective map
$$\lambda_G:C^...
- 1,903
7
votes
1
answer
429
views
Open projections and Murray-von Neumann equivalence
Let $\mathcal{A}$ be a $C^*$-algebra and $p\in\mathcal{A}^{**}$ be an open projection, that is, $p=p^*=p^2$ and $p\in\overline{(p\mathcal{A}^{**}p\cap\hat{\mathcal{A}})}^{\operatorname{w}^*}$, where $\...
- 619
3
votes
1
answer
143
views
Solvability of a certain functional equation in simple $C^*$ algebras
For which simple unital $C^*$ algebras does the following functional equation have a solution:
$$ d^2=0,\;{(d+d^*)}^2=1$$
The Calkin algebra and $M_{2n}(\mathbb{C})$ are some examples. It is not ...
- 356
3
votes
1
answer
252
views
Regarding spectral radius
Let $A$ be a $C^*$ algebra. Let $a\in A$ be such that $a^*a-aa^*\geq 0$. Doe this imply that the spectral radius of $a$ is equal to $\|a\|$?
- 175
3
votes
1
answer
130
views
C*-algebras: Existence of an element inducing an injective map
I'm wondering if the following statement is true: Let $A$ be a $C^*$-algebra and $\phi: A\rightarrow A$ be a $*$-homomorphism. Is there always an element $a\in A$ such that the map
$\left\{ \phi^{n}:=...
- 741
4
votes
1
answer
313
views
Full spectrum positive elements of a $C^*$-algebra
I would like to know criteria for a C*-algebra $A$ to have a positive contraction $a$ with full spectrum, ie $\sigma(a) = [0,1]$. I am particularly interested in the simple case. I believe that if a C*...
7
votes
1
answer
311
views
Homomorphism to multiplier algebra of groupoid $C^\ast$-algebra
If I have a functor $X\to Y$ between topological groupoids with appropriate Haar measures, such that $X_0 \to Y_0$ is injective and a homeomorphism onto its image, then I should have (or rather, I ...
- 35.5k
19
votes
1
answer
773
views
Are algebraically isomorphic $C^*$-algebras $*$-isomorphic?
If A and B are C^*-algebras that are algebraically isomorphic to each other, does
this imply that they are *-isomorphic to each other?
- 213
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
2
votes
0
answers
164
views
An operator valued Egoroff's theorem
The following statements suggests $B(H)$-valued Egoroff's theorem when $H$ is a separable Hilbert space. Probably it will be hold even if a von Neumann algebra $M$ whose predual is separable is ...
- 4,058
2
votes
1
answer
645
views
Enveloping $C^*$-algebra
Consider a finite dimensional $C^*$-algebra $\cal{A}$. Is there any enveloping $C^*$-algebra $\cal{C^*(G)}$ such that $\cal{A}\cong C^*(G)$ for some locally compact group $\cal{G}$?
(Note that "$\...
- 565
2
votes
1
answer
341
views
Closed two-sided ideals in $C(X,M_n)$
As is known (see Kadison-Ringrose, 3.4.1) each closed ideal $I$ in the $C^*$-algebra $C(X)$ of continuous functions on a compact space $X$ has the form
$$
I=\{f\in C(X): \ \forall x\in S\quad f(x)=0 \}...
- 7,426
0
votes
0
answers
152
views
Continuity under various topologies for positive linear functionals
It is known that if $\mathcal A$ is a unital $\mathbb C$-$*$-algebra and $A$ is a unital subalgebra closed under $*$, and if $f : A \to \mathbb C$ is linear, then $f$ is positive if and only if $f$ is ...
- 5,407
10
votes
0
answers
201
views
Masas in SAW*-algebras
I asked this question three years ago at MSe but it has no response; let me try here.
Pedersen distilled the following class of C*-algebras which he termed SAW*-algebras (Journal of Operator Theory, ...
- 11.3k
1
vote
0
answers
50
views
Characterizing (minimal) tensor product inside Hilbert C*-module
Let $A$, $B$ be C$^*$-algebras, $\mu$ be a state on $B$ and $\mathcal{I}$ be a family of ideals in $A$. Let $I_0:=\cap_{I\in\mathcal{I}} I$ and put $A_0:=A/I_0$. Consider the minimal tensor product on ...
- 111
2
votes
1
answer
358
views
Existence of an integrable representation
An irreducible continuous unitary representation $\pi$ of $G$ is said to be integrable, if the map $\phi(x)=\langle\pi(x)\zeta,\zeta\rangle$ is integrable on $G$, where that $\zeta\in H(\pi)$.
...
- 399
7
votes
0
answers
222
views
Can C*/W*-algebras be realized as (involutive?) monoid/co-monoid objects?
I would like to know how close one can get to realizing the category of C*-algebras as a category of monoid objects. Related (almost, but not quite, duplicate) questions are:
"Recovering a monoidal ...
- 146
6
votes
4
answers
1k
views
Resource recommendation: Spectral theory and $C^*$ algebras
I have formally studied functional analysis, both as university courses, and by myself, but this is one area of mathematics I find so huge and complicated, I have a hard time properly getting into it.
...
- 2,792
4
votes
1
answer
201
views
closure of a separating set of pure states
Let $A$ be a unital C*-algebra, and let $\mathcal R$ be a separating family of irreducible representations of $A$. Each vector state of a representation in $\mathcal R$ is a pure state, and the span ...
- 974
1
vote
4
answers
367
views
Classification of $C^*$ algebras whose all non scalar elements have disconnected spectrum
To what extent have all unital $C^*$ algebras $A$ with the following property been classified? Is there a simple $C^*$ algebra with this property? Does $C(K)$ satisfy this property, where $K$ is an ...
- 356
5
votes
2
answers
216
views
On the coincidence (or non-coincidence) of two norms defined on the quotient of a given Hilbert $ C^{\ast} $-module by a certain linear subspace
Let $ A $ be a $ C^{\ast} $-algebra, $ I $ a closed two-sided ideal of $ A $, and $ \mathcal{E} $ a Hilbert $ A $-module. Let
$$
\mathcal{E}_{I}
\stackrel{\text{df}}{=}
\{ x \in \mathcal{E} \mid \...
- 1,396
1
vote
1
answer
332
views
Every norm-continuous group of $C^*$-algebra automorphisms weakly inner?
Please, help out of the mind trap. In this prominent paper Kadison and Ringrose prove among other things the following
Corollary 8. Each norm-continuous representation of a connected topological
...
- 1,959
3
votes
1
answer
261
views
CBAP for the full group $C^*$-algebra
Let $G$ be a weakly amenable group, in the sense that it has a net of finitely supported functions $\varphi:G\to \mathbb{C}$ which converge point wise to 1 and their cb norm is bounded uniformly by ...
- 99
3
votes
0
answers
148
views
Full free product of $B(\mathcal H_i)$
It struck me that I know nothing about the full (universal) free product of the $B(\mathcal H_i)$ amalgamated over $\mathbb C$ for Hilbert spaces $\mathcal H_i$ with identified unit vector $\xi_i$. So ...
- 3,984
8
votes
1
answer
586
views
Counterexample to Riesz representation for Hilbert modules
For a Hilbert space $H$, the Riesz representation theorem states that $H$ is isomorphic to its dual $H^*$ via $x \mapsto \langle x, -\rangle$.
It is often stated in the literature that this does not ...
- 659
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
2
votes
2
answers
352
views
Relation between norm of any element of $C^*$-algebra in terms of self adjoint elements
Let $\mathcal{A}$ be a $C^*$-algebra, then for $a\in \mathcal{A}$, there are $b,c \in \mathcal{A}_{SA}$ such that $a=b+ic$, where $\mathcal{A}_{SA}$ is the self adjoint part of $C^*$-algebra.
...
- 139
1
vote
0
answers
109
views
Two tensor product norms inducing different topologies on the space of simple tensors
Are there two Normed spaces $V,W$ for which the algebraic tensor product $V\otimes W$ admits two different norms, both satisfying $\parallel x \otimes y \parallel= \parallel x \parallel. \...
- 356
1
vote
1
answer
109
views
Continuous factors for invertible simple tensors
Our following question is motivated by this very interesting answer
Assume that $A$ is a $C^{*}$ algebra. Put $X=\{a\otimes b \mid a,b \in G(A)\}$ where $G(A)$ is the space of all ...
- 356
0
votes
1
answer
103
views
Suppose that $a \mu = \mu a$ for all $a$ in $C^*$-algebra $A$. Then $\mu \in Z(A^{**})$
Let $A$ is a $C^*$-algebra and $\mu \in A^{**}$. Suppose that $a \mu = \mu a$ for all $a \in A$. Then $\mu \in Z(A^{**})$.
- 19
5
votes
1
answer
307
views
Can an AW*-algebra be recovered from its lattice of projections?
Can an AW*-algebra be recovered (up to Jordan isomorphism) from its lattice of projections? This is possible in the commutative/Boolean case.
- 3,816
7
votes
2
answers
689
views
Which C*-algebras are complemented in their bidual?
Every von Neumann algebra is 1-complemented in its bidual, and so is every injective C*-algebra. Also, if $C_0(X)$ is infinite-dimensional and separable then it is not complemented in its bidual, and $...
- 3,816
5
votes
1
answer
242
views
Spectral decomposition of a C$^*$algebra with respect to an action of a compact abelian group
Let $G$ be a compact abelian group (finite dimensional, but not finite) and $A$ be a $C^*$-algebra. Consider an action $\alpha: G\to Aut(A)$. In analogy with the case of finite abelian group, I ...
- 743
4
votes
0
answers
264
views
Does the Cauchy–Schwarz inequality imply 2-positivity?
Recall the following generalisation of Cauchy–Schwarz.
Theorem. Let $f\colon \mathscr{A} \to \mathscr{B}$ be a linear 2-positive map between C$^*$-algebras. Then for all $a,b \in \mathscr{A}$ we ...
- 355
1
vote
1
answer
269
views
the space of self-adjoint trace class operators over a separable Hilbert space is separable with respect to the trace norm?
My interest is to know whether the assertion
...
- 31
18
votes
3
answers
1k
views
In which sense the GNS-construction is a functor?
I asked this at mathstackexchange a week ago, without success.
I think the Gelfand–Naimark–Segal construction must be a functor in some sense, but I can't find an explicit statement anywhere. Can ...
- 7,426
6
votes
1
answer
320
views
C*-envelope of an operator system by an action
Let $V$ be an operator System in $B(H)$. By Hamana and Ruan theorems, there is an injective envelope $I(V)$ which is minimal injective subspace of $B(H)$ contains $V$.
Thus there is a completely ...
- 81
5
votes
3
answers
292
views
$*$-representation $\pi:A\odot B\to B(H_1\otimes H_2)$ such that $\pi \neq \pi_1\otimes \pi_2$
Let $A$ and $B$ two $C^*$-algebras, $H_1$ and $H_2$ complex Hilbert spaces and $\pi_1:A\to B(H_1)$, $\pi_2:B\to B(H_2)$ two $*$-representations. Then there is a $*$-representation $\pi_1\otimes \pi_2:...
user62639
5
votes
0
answers
314
views
C$^*$-algebras in which the spectral radius is comparable to the norm
For every commutative C$^*$-algebra the spectral radius is equal to the norm. My question is:
For which C$^*$-algebras $\mathcal A$ does there exist a constant $C>0$ such that $$C\|a\| \leq ...
- 3,984
3
votes
1
answer
194
views
Linear independency and compactness of the set of pure states of a $C^*$-algebra
Let $\mathcal{A}$ be a noncommutative $C^*$-algebra and $PS(\mathcal{A})$ be the set of its pure states.
Question 1. Is $PS(\mathcal{A})$ linearly independent (as vectors over $\mathbb{R}$)? (If $\...
- 619