All Questions
Tagged with oa.operator-algebras c-star-algebras
597 questions
5
votes
2
answers
1k
views
Strong Morita Equivalence and Morphisms Between $ C^{*} $-Algebras
If $ A $ and $ B $ are $ C^{*} $-algebras, then they are strongly Morita equivalent if there exist a $ (B,A) $-bimodule $ E $ and an $ (A,B) $-bimodule $ F $ such that
$$
E \otimes_{A} F \cong B \quad ...
- 743
2
votes
0
answers
82
views
Atkinson-Mingo theorem
I've been read the book "$K$-theory and $C^*$-algebras" by Olsen. In the chapter on the generalized Fredholm index the following definitions are given:
Let $A$ be a $C^*$-algebra and $H_A$ the ...
- 41
7
votes
0
answers
85
views
Analogue of Friedrichs extension for Hilbert $C^*$-modules
Suppose one has a densely defined symmetric operator $T:\mathcal{M}\rightarrow\mathcal{M}$, where $\mathcal{M}$ is a Hilbert $A$-module for a $C^*$-algebra $A$. Suppose that $T$ is non-negative, so ...
- 1,903
11
votes
2
answers
636
views
Quasinilpotent elements of group C-star algebras
If $G$ is a discrete torsion-free group, can its (reduced or full) group C-star algebra contain non-zero quasinilpotent elements? I've seen various examples in the group von Neumann algebra setting (...
- 25.8k
5
votes
1
answer
307
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Can an AW*-algebra be recovered from its lattice of projections?
Can an AW*-algebra be recovered (up to Jordan isomorphism) from its lattice of projections? This is possible in the commutative/Boolean case.
- 3,816
6
votes
1
answer
272
views
Non-invertible version of unitary intertwiners between correspondences of $C^\ast$-algebras
There is a version of the Eilenberg-Watts theorem for $C^\ast$-algebras, where functors between appropriate module categories correspond to what are called 'correspondences' of $C^\ast$-algebras. ...
- 35.5k
5
votes
1
answer
370
views
explicit description of the product map in K-theory
Let $A$ and $B$ be unital $C^*$-algebras. Let $u\in M_n(A)$ be a unitary representing an element $[u]\in K_1(A)$ and $p\in M_m(B)$ be a projection representing an element $[p]\in K_0(B)$. Then the ...
user62639
3
votes
1
answer
143
views
Solvability of a certain functional equation in simple $C^*$ algebras
For which simple unital $C^*$ algebras does the following functional equation have a solution:
$$ d^2=0,\;{(d+d^*)}^2=1$$
The Calkin algebra and $M_{2n}(\mathbb{C})$ are some examples. It is not ...
- 356
5
votes
3
answers
292
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$*$-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
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
0
answers
146
views
A relation between Hochschild cohomology of a $C^*$ algebra and its bidual
Let $A$ be a $C^*$ algebra and $A''$ be its bidual with the Arens product.
Is there any relation between the Hochschild cohomolgy of $A$ with complex coefficients and the Hochschild cohomology of $A''$...
- 356
17
votes
1
answer
515
views
K-theory space of a C*-algebra
Let $A$ be a unital C*-algebra.
Let me define its "$K$-theory space" to be the image of its $K$-theory spectrum under the functor $\Omega^\infty:$ Spectra $\to$ Spaces.
I denote the $K$-theory space ...
- 43.2k
7
votes
2
answers
742
views
$H^{*}$ algebras as a generalization of $C^{*}$ algebras
Let $H$ be the quaternions algebra. An $H^{*}$ algebra is a normed ring $A$ which is simultaneously a unital left $H$ module and has an involution $*$ with the following properties:
$\forall \lambda \...
- 356
2
votes
0
answers
116
views
Closable operators on Hilbert modules
For $T:{\frak{Dom}}(T) \to H$ a densely defined operator, with $H$ a (separable) Hilbert space, we know that $T$ is closable if its adjoint $T^*$ has dense domain in $H$.
Does this extend to the (...
- 993
2
votes
1
answer
221
views
Non-perfect type one C^*-algebra, and a lemma in Fourier analysis
I would like to know if the following is true :
Let $\mathcal{H}$ be the complex Hilbert space $L^2([0,1])$ for the Lebesgue measure.
Let $q$ be the orthogonal projection on the subspace of $\mathcal{...
- 42.4k
1
vote
1
answer
171
views
A $C^{*}$ algebra associated to a graded $C^{*}$ algebra
A $C^{*}$ algebra $A$ is graded by $\mathbb{Z}_{n}$ iff it can be acted by $\mathbb{Z}_{n}$. So we associate the $C^{*}$ algebra $A\rtimes \mathbb{Z}_{n}$ to a $\mathbb{Z}_{n}$-graded $C^{*}$ ...
- 356
13
votes
3
answers
742
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Is the space of *-homomorphisms between two $C^*$-algebras locally path connected
Given the set of *-homomorphisms between two $C^*$-algebras $A$ and $B$, we may define a metric on it by setting $d(f,g):= \sup_{0<\|a\|\le 1}\|f(a)-g(a)\|$. Could it be true that, for each *-...
- 613
-1
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1
answer
90
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Strictly increasing approximation of the identiy
Is there always a strictly increasing approximation of the identity in a separable $C^*$-algebra?
- 49
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
3
votes
2
answers
1k
views
$C^*$ algebras and states
I have a question arising from von Neumann's C*-algebra formulation of quantum mechanics. In it, a state is a $\mathbb{C}$-linear functional on a C*-algebra $A$
$\rho:A\rightarrow\mathbb{C}$
...
- 103
10
votes
1
answer
373
views
Real rank zero of group $C^*$-algebras
The concept of real rank zero of a $C^*$-algebra is introduced as non-commutative analogue of dimension ( topological dimension ). For example, it shown (by Brown-Pedersen) such that, if $X$ is a ...
- 399
15
votes
1
answer
2k
views
Matrices with entries in a $C^*$-algebra
Let $\mathcal{A}$ be a $C^\ast$-algebra. Consider vector space of matrices of size $n\times n$ whose entries in $\mathcal{A}$. Denote this vector space $M_{n,n}(\mathcal{A})$. We can define involution ...
- 1,697
2
votes
1
answer
97
views
Approximation of unity by projectors
Let $A$ be a $\sigma$-unital $C^*$-algebra and $A_s:=A\otimes K$ its stabilization (where $K$ is the algebra of compact operators on a separable Hilbert space). Is it true that there exist an ...
user95283
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
1
vote
0
answers
220
views
Idempotents in Group Algebras
What is known about idempotents in Lie group algebras (such as on the classical Lie groups)? Specifically the self-adjoint ones. Is there anything interesting to say? I haven't been able to find much ...
- 1,198
6
votes
2
answers
351
views
$id:A\to A^{op}$ is completely positive iff $A$ is abelian
Let $A$ be a $C^*$-algebra and $A^{op}$ it's opposite $C^*$-algebra. Let $id:A\to A^{op}$ be the identity map. $id$ is positive.
The claim is: $id$ is completely positive iff $A$ is abelian.
I need ...
user62639
24
votes
2
answers
1k
views
Does left-invertible imply invertible in full group C*-algebras (discrete case)?
The following question/problem has been bugging me on and off for some time now: so I thought it might be worth broaching here on MO, as a case of "ask the experts".
Let $G$ be a discrete group. ...
- 25.8k
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
4
votes
1
answer
157
views
Geometric Motivation for Hilbert $C^*$-Bimodules
I'm trying to get an understanding of Hilbert $C^*$-bi-modules from a geometric point of view. As is well-known, we have that
i) Commutative unital $C^*$-algebras correspond to compact Hausdorff ...
- 167
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
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 ...
1
vote
0
answers
86
views
A cross product on $C^*_{red} G$
For every group $G$, the reduced group $C^*$-algebra $C^*_{red}G$ is equipped with the inner product $\langle a,b\rangle=tr(ab^*)$ where "$tr$" is the standard trace on group $C^*$-algebras.
For ...
- 356
2
votes
0
answers
125
views
Computing the $K$-theory of the free inverse semigroup $C^*$-algebra
A typical example where I know the $K$-theory of the quotient, and wonder if this could help to compute the $K$-theory of the extension. (Or other way round: knowing one part out of three ones.)
I ...
- 685
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:...
- 620
8
votes
1
answer
307
views
Induced map of an approximately inner automorphism on the multiplier algebra of $A\otimes\mathcal{K}$
Suppose that $A$ is a separable, simple, non-unital C*-algebra. Let $\varphi$ be an approximately inner automorphism on $A\otimes\cal K$, meaning that there exists a sequence of unitaries $v_n$ in the ...
- 1,023
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
12
votes
1
answer
607
views
Almost idempotent approximate units in C*-algebras
As in Blackadar's "Operator Algebras" Definition II.4.1.1., call an approximate unit $(a_\lambda)$ in the positive unit ball of a C*-algebra almost idempotent if $a_\lambda a_\gamma=a_\lambda$ ...
- 1,307
0
votes
1
answer
142
views
A property of compact topological space via certain $C^*$ embedding in operator algebras
Assume that $A$ is a unital $C^*$ algebra. Is there a $C^*$ embedding of $A$ in some $B(H)$ whose image is a hereditary $C^*$ subalgebra of $B(H)$?
If not, is the answer affirmative when $A$ is ...
- 356
3
votes
1
answer
232
views
unital embedding into the coner $C^*$-algebra
Let $A$ be a $C^*$-algebra, we denote with $V(A)$ the semigroup of Murray-von Neumann equivalence classes of projections in matrices over $A$ (as usual). In https://arxiv.org/pdf/math/0310340.pdf, ...
- 1,188
16
votes
3
answers
2k
views
Is the group von Neumann algebra construction functorial?
Let $G$ be a group and $CG$ the complex group algebra over the field $C$ of complex number. The group von Neumann algebra $NG$ is the completion of $CG$ wrt weak operator norm in $B(l^2(G))$, the set ...
- 1,373
6
votes
0
answers
169
views
Characterizing fullness of a von Neumann algebra by the topology of its bimodules
Let $\mathcal{M}$ be a $\mathrm{II}_1$ factor. Among other characterizations, it is said to be full iff the adjoint map:
$$
\mathrm{Ad}: U(\mathcal{M})/\mathbb{T} \longrightarrow \mathrm{Aut}(\...
- 1,090
8
votes
2
answers
961
views
The monotone closure of a $C^*$-algebra
Related to Jon's question, I have two questions. Let $\mathcal{A}$ be a concrete $C^*$-algebra on a Hilbert space $\mathcal{H}$. For any selfadjoint subset $S$ of $\mathbb{B}(\mathcal{H})$, let $S^m$ ...
- 619
5
votes
1
answer
269
views
Equivalence of two pictures of odd $K$-theory
One can show that two functors $K^0$ and $K_0(C(-))$ from the category of compact topological spaces to the category of abelian groups are naturally equivalent. The first one is topological $K$-theory ...
- 9,330
7
votes
0
answers
555
views
maximal tensor product commutes with inductive limits
Let $(A_n, \phi_n)$ be an inductive system of $C^*$ algebras and let $B$ be an arbitary $C^*$ algebra.
I want to prove $(\varinjlim A_n)\otimes_{max} B \cong \varinjlim (A_n \otimes_{max} B)$. This ...
- 1,188
6
votes
2
answers
430
views
Metrics on the space of $C^{*}$ algebras
I think that there is a metric on the huge space of all $C^{*}$ algebras. What is the explicit
definition of this metric?may you introduce me a reference?
Moreover is the restriction of this ...
- 356
14
votes
1
answer
616
views
How "nondegenerate" are amalgamated free products of C*-algebras?
In the following, I assume all algebras are unital. Let $A$ and $B$ be C*-algebras that each contain (isomorphic copies of) a common C*-subalgebra $C$. Let $A *_C B$ denote the amalgamated free ...
- 5,407
0
votes
0
answers
54
views
On cyclicity of a module
Let $A$ be a $\text{ von Neumann algebra }$, $\mathcal{H}$ is a cyclic $A$ module, $G$ be a finite group acting on $A$, is $\mathcal{H}$ cyclic module over fixed point subalgebra of the action? ...
- 677
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
7
votes
1
answer
434
views
How the modular theory of von Neumann algebras, deal with generating C*-algebras?
Let $H$ be a separable infinite dimensional Hilbert space, $M \subset B(H)$ a von Neumann algebra and $A \subset M$ a separable ${\rm C}^*$-algebra such that $A''=M$. Suppose the existence of a ...
5
votes
1
answer
249
views
$C^{*}$-correspondences viewed as generalized endomorphisms
I've heard that $C^{*}$-correspondences (over a $C^{*}$-algebra) can be viewed as generalized endomorphisms of the algebra. I would like to understand this, and be pointed towards books or papers ...
- 367