All Questions
Tagged with c-star-algebras oa.operator-algebras
597 questions
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
1
vote
1
answer
165
views
Convergent bounded net of positive operators converges to a positive operator
Let $A$ be a $C^*$-algebra. Endow $A$ with the strict topology for which a net $\{a_i\}_{i \in I}$ converges to $a \in A$ if $$\|a_i b-ab\| + \|ba_i-ba\| \to 0$$
for all $b \in A$. Is it true that if $...
- 175
1
vote
1
answer
178
views
A completely positive equivariant map $\varphi: A \to B$ induces a map on the full crossed products
Let $G$ be a discrete group. Let $(A,\alpha)$ and $(B,\beta)$ be $G$-$C^*$-algebras and $\varphi: A \to B$ be $G$-equivariant and completely positive. All crossed products in this post are full (= ...
- 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
4
votes
0
answers
126
views
Can the injective envelope ever be injective for $*$-homomorphisms?
The answers to the question "Is the injective envelope functorial" resoundingly remind us that the injective envelope of a C$^*$-algebra really belongs in the category of completely positive ...
- 3,984
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
2
votes
1
answer
143
views
$(\iota \otimes f)(X) = 0$ for all $f \in B^*$ implies $X=0$
Let $A$ and $B$ be $C^*$-algebras. Given $f \in B^*$, we can form the right slice map
$$\iota \otimes f: A \otimes B \to A: a \otimes b \mapsto af(b)$$
which extends uniquely to a bounded linear map
$$...
- 175
5
votes
0
answers
136
views
C^*-algebra theory with all the Koszul signs
I was wondering if someone knows of a reference in which $\mathbb{Z}_2$-graded $C^*$-algebra theory is developed using the sign convention $(ab)^* = (-1)^{|a||b|}b^* a^*$. I would be most enthusiastic ...
- 357
15
votes
0
answers
283
views
Stable isomorphism of group C$^*$-algebras
For a discrete group $G$, let $C^*_r(G)$ be its reduced group C$^*$-algebra.
Question: Do there exist discrete, torsion-free non-isomorphic groups $G,H$ such that $C^*_r(G)$ and $C^*_r(H)$ are stably ...
- 2,689
1
vote
1
answer
161
views
$C\lVert\sum_i a_{ii}\rVert \ge \lVert(a_{ij})\rVert$ for matrices with entries in a $C^*$-algebra
Let $A$ be a $C^*$-algebra and $(a_{ij}) \in M_n(A)$ be a positive matrix. Does there exist a constant $C \ge 0$ (not depending on the $a_{ij}$) such that
$$\lVert(a_{ij})\rVert \le C \Bigl\lVert\...
- 175
0
votes
1
answer
77
views
Is $N_\phi = \{x \in E: \phi(\langle x,x\rangle)=0\}$ a Hilbert submodule of $E$?
Let $E$ be a (right) Hilbert module over the $C^*$-algebra $B$. Let $\phi$ be a state on the $C^*$-algebra $B$. Then consider
$$N_\phi:= \{x \in E: \phi(\langle x,x\rangle)=0\}.$$
I want to show that $...
- 175
3
votes
1
answer
241
views
Monotone approximation of elements in AF-algebras
Suppose that we are given an AF-algebra $A$ and a sequence of finite-dimensional subalgebras $\mathbb{C}=A_0\subset A_1\subset A_2\subset\ldots$ such that $A=\overline{\bigcup\limits_{n\geq 0}A_n}$. ...
- 321
5
votes
1
answer
318
views
Two densely defined traces on a $C^*$-algebra coinciding on a dense subalgebra are equal
Let $t_1$ and $t_2$ be lower semicontinuous semifinite densely defined traces on a $C^*$-algebra $A$. Let us denote by $\mathcal{R}_1$ and $\mathcal{R}_2$ their ideals of definition, i.e. $\mathcal{R}...
- 321
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
4
votes
2
answers
448
views
A completely positive equivariant map $\varphi: A \to B$ induces a map $A \rtimes_r \Gamma \to B \rtimes_r \Gamma.$
Recall the construction of the reduced crossed product:
Let $\Gamma$ be a discrete group and $A$ be a $C^*$-algebra with an action $\alpha: \Gamma\to \operatorname{Aut}(A)$. Consider the $*$-algebra $...
user160032
0
votes
1
answer
419
views
What are the maximal ideals of $C_0 (X),$ where $X$ is a locally compact Hausdorff space?
Crossposted from MSE
How do the maximal ideals of $C_0(X)$ look like where $X$ is a locally compact Hausdorff space?
I know that if $X$ is a compact Hausdorff space then the maximal ideals of $C(X)$ ...
- 141
1
vote
1
answer
367
views
finitely generated C*-algebra as $C(X)$
In the question ($C(X)$ as finitely generated $C^*$-algebra), the answer show that spectrum of an abelian unital finitely generated C*-algebra is homeomorphic to compact subset of $\mathbb{C}^{n}$. I ...
- 523
0
votes
1
answer
208
views
Trying to recognise a $C^*$-algebra
Let $H$ and $K$ be infinite dimensional (separable) Hilbert spaces and $X=B(H,K)$ denote the space of bounded linear operators. For $T_1, T_2$ in $X$, we define $D_{T_1,T_2}:X \to X$ as $D_{T_1,T_2}(T)...
- 1,115
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
2
votes
0
answers
141
views
Is it true that $\varinjlim A_n^{**}=(\varinjlim A_n)^{**}$?
Let $(A_n, \phi_n)$ be an inductive system of $C^*$ algebras. It's well known double dual of $C^*$-algebra is again a $C^*$ algebra.
Is it true that $\varinjlim A_n^{**}=(\varinjlim A_n)^{**}$
Can ...
- 1,115
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
$$\|t\| = \sup_{z \in E, \|z\| = 1} \|\...
- 175
0
votes
0
answers
119
views
Subalgebras of $B(H)$ consisting of all operators leaving a given finite dimensional space invariant
Let $H$ be an infinite dimensional separable Hilbert space. Let $V$ be a finite dimensional subspace of $H$.
Put $$A=\{T\in B(H)\mid T(V)\subseteq V\}.$$
So $A$ is a Banach algebra.
Can we equip $A$ ...
- 356
7
votes
2
answers
869
views
Amenable action intuition
Let $\Gamma$ be a discrete group and $A$ be a $C^*$-algebra. Consider an action $\alpha: \Gamma \to \operatorname{Aut}(A)$. There is a notion of amenability for such an action (see e.g. Brown and ...
- 175
0
votes
1
answer
158
views
Abelian twisted reduced group C*-algebra
Let $G$ be an abelian discrete group. Then is $C_r^*(G, \sigma)$ abelian?
- 581
2
votes
0
answers
203
views
Quasidiagonal C*-algebras
Let $A$ be a nuclear $C^*$-algebra satisfying UCT condition. Then under what assumptions $A$ is quasidiagonal?
- 581
4
votes
1
answer
224
views
Direct sum of multiplier algebras
Consider a collection of $C^*$-algebras $\{A_i\}_{i \in I}$. We can form the direct sum $$\bigoplus_{i \in I}^{c_0} A_i:= \left\{(a_i)_{i \in I} \in \prod_{i\in I} A_i: \lim_{i \in I} \|a_i\| = 0\...
- 175
0
votes
1
answer
154
views
Why is $q(f,g) = (f-g,0)$ not adjointable?
Let $A= C([0,1])$ and $J= \{f \in A: f(0) = 0\}$. Consider the Hilbert $C^*$-module
$E:= A \oplus J$ (with the obvious right $A$-action and inner product). I want to prove that
$$q: E \to E: (f,g) \...
- 175
2
votes
0
answers
243
views
Reference request: definiitions of exact C* algebra and group C* algebra
I am writing my Ph.D. thesis and I would like to cite the specific papers where the concept of exact $C^*$ algebra and group $C^*$ algebra was defined.
In the book of Brown and Ozawa "$C^*$-...
- 115
0
votes
1
answer
163
views
Regarding socle of a C* algebra
I wanted to know if the socle of a complex C*-algebra is essential?
Can anyone suggest a text where the socle is studied in detail. I tried reading it from the book by Bernard Aupetit, A Primer in ...
- 149
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
2
votes
0
answers
119
views
Does this groupoid have a quasi-diagonal reduced $C^*$-algebra?
Let $H$ be a discrete group, and let $X$ be the one-point compactification of $\mathbb{N}$. Consider the étale groupoid $G = H \times \{\infty\} \sqcup \mathbb{N}$, whose unit space is $X$, and with ...
- 733
7
votes
0
answers
113
views
Does the following tracial inequality (involving certain function applications) hold for positive semi-definite matrices?
Given $n \in \mathbb{N}$ we define the function $f_{i,n}: [0,1] \rightarrow \mathbb{R}$ for $i \in \{1,..., n\}$ by $f_{i,n} = 0$ on the interval $[0,(i-1)/n]$, $f_{i,n} = 1$ on $[i/n,1]$, and $f_{i,n}...
- 71
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 $dim(com(A))?$
- 581
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?
- 581
2
votes
2
answers
189
views
Unconditional Convergence of Positive Terms in a $C*$-algebra
I am reading the paper Frames and Outer Frames for Hilbert $C^*$-modules by L.J. Arambasic and D. Bakic. They have mentioned in passing, the following:
"...Since in each $C^*$-algebra, a ...
- 155
1
vote
0
answers
283
views
Continuous fields of Hilbert spaces arising from representations of abelian C*-algebras
This is a followup to a previous question [1] on MO.
Let $X$ be a second-countable, locally compact, Hausdorff space, and let $\mathscr H =\{H_x\}_{x\in X}$, be a
measurable field of Hilbert spaces ...
- 483
1
vote
2
answers
313
views
On solvability of equation $D(x)=1$ where $D:A\to A$ is a bounded outer derivation on a $C^*$ algebra
Let $A$ be a unital $C^*$ algebra. Assume that $D:A\to A$ is a bounded derivation.
Can one say that $1$ can not be in the image of $D$?
If the answer is no:
What is a counter example? What kind of $...
- 356
-1
votes
1
answer
246
views
Density of normal elements in a C*- algebra [closed]
Let $A$ be a unital C*-algebra.
I wanted to know if there is a necessary and sufficient condition for normal elements to be dense in $A$?
- 149
7
votes
1
answer
264
views
Can the intersection of a unitary and an irreducibly represented injective $C^*$-algebra be $\{0\}$?
Let $\mathcal{A}$ be an injective $C^*$-algebra irreducibly acting on a Hilbert space $\mathcal{H}$, and let $\phi$ be a completely positive idempotent from $\mathbb{B}(\mathcal{H})$ onto $\mathcal{A}$...
- 619
1
vote
2
answers
148
views
Show convergence of net associated to GNS-triplet associated to state on a $C^*$-algebra
Let $A\subseteq B \subseteq B(H)$ be an inclusion of $C^*$-algebras where $H$ is some Hilbert space. We have the following conditions:
B is a von Neumann algebra with $A'' = B$.
The inclusion $A \...
- 175
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 $...
- 1,879
13
votes
1
answer
451
views
Factor states on C*-algebras
Which C$^*$-algebras admit factor states for which the von Neumann algebra it generates in the corresponding GNS representation is a type III$_1$ factor? For example, do all purely infinite algebras ...
- 771
3
votes
2
answers
376
views
Elements of the minimal tensor product of a finite dimensional operator system and a $C^*$-algebra
I am trying to prove something that seemed simple to me at first sight but apparently it is giving me a hard time. Here is the same question on MSE.
Let $E\subset A$ be a finite dimensional operator ...
- 267
11
votes
1
answer
2k
views
Motivation for $C^*$-algebras
I just gave a presentation on exotic group $C^*$-algebras and someone asked why these are studied. I could answer that they can be used to construct $C^*$-algebras with certain properties. However, I ...
- 111
5
votes
1
answer
394
views
Polar decomposition in abstract von Neumann algebra
Probably an easy question, but here goes:
In a concrete von Neumann algebra $M \subseteq B(H)$, every element $m \in M$ has a polar decomposition $m= p|m|$ where $p$ is a partial isometry and $|m|= \...
user167952
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
5
votes
1
answer
303
views
Non-unital Russo-Dye Theorem
Let $A$ be a C$^*$-algebra and let $\phi$ be a positive linear map from $A$ to $B(H)$ (bounded linear operators on Hilbert's
space). If $A$ is unital, then the Russo-Dye Theorem implies that $\|\phi\...
- 483
9
votes
1
answer
372
views
Simplicity of group $C^\ast$-algebra implies fullness of group-von Neumann algebra?
Let $\Gamma$ be a discrete group whose reduced group $C^\ast$-algebra is simple. Can we conclude that the corresponding group-von Neumann algebra $\mathcal{L}(G)$ is a full $\text{II}_1$-factor, ...
- 741
9
votes
1
answer
585
views
Finite compact quantum groups
Let $(A, \Delta)$ be a $C^*$-algebraic compact quantum group (in the sense of Woronowicz). It is called finite if $A$ is a finite-dimensional $C^*$-algebra. By elementary $C^*$-algebra theory, we ...
user167952
3
votes
1
answer
112
views
Is restriction to the center an open map?
Given a type one $C^*$-algebra $A$, its center $Z$ acts by scalars on each irreducible representation space. Mapping a representation to its central character yields a continuous map from the ...
user130903