Questions tagged [c-star-algebras]
A C*-algebra is a complex Banach algebra together with an isometric antilinear involution satisfying (a b)* = b* a* and the C*-identity ‖a* a‖ = ‖a‖². Related tags: [banach-algebras], [von-neumann-algebras], [operator-algebras], [spectral-theory].
853
questions
5
votes
1
answer
309
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}...
3
votes
1
answer
297
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. ...
4
votes
2
answers
424
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 $...
2
votes
1
answer
159
views
Gelfand-Naimark and Peter-Weyl for the unitary group
Consider the compact Lie groups $U(l)$ (the unitary group) and $U(1) \times SU(l)$ for some natural number $l$. Both the groups have the same Lie algebra $\frak{gl}_l$. Which means that they both have ...
0
votes
1
answer
336
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 $C(X)$ ...
1
vote
1
answer
272
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 ...
0
votes
1
answer
207
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)...
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 ...
9
votes
1
answer
297
views
Do extensions of pure states separate points?
Let $B$ be a unital C*-algebra and let $A⊆B$ be a closed *-subalgebra containing the unit of $B$. I am mostly
interested in the case that $A$ is abelian but, for the strict purpose of stating my ...
5
votes
2
answers
224
views
Hermitian vector bundles and Hilbert $C^*$-modules
Let $X$ be a compact Hausdorff space and $C(X)$ its algebra of continuous complex valued functions. The Gelfand-Naimark theorem tells us that we have a duality between commutative $C^*$-algebras and ...
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 ...
4
votes
0
answers
146
views
Drinfeld center of a tensor category
Firstly, apologies for the imprecise language, I'm a physicist trying to understand anyonic excitations from the lens of category theory.
If I have a category (say $\operatorname{Rep}(\mathbb{Z}_2)$) ...
4
votes
1
answer
296
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
$$\|t\| = \sup_{z \in E, \|z\| = 1} \|\...
0
votes
0
answers
119
views
Subalgebras of $B(H)$ consisting of all operators leaving a given finite dimensional space invariant
Let $H$ be an infinite dimensional separable Hilbert space. Let $V$ be a finite dimensional subspace of $H$.
Put $$A=\{T\in B(H)\mid T(V)\subseteq V\}.$$
So $A$ is a Banach algebra.
Can we equip $A$ ...
7
votes
2
answers
755
views
Amenable action intuition
Let $\Gamma$ be a discrete group and $A$ be a $C^*$-algebra. Consider an action $\alpha: \Gamma \to \operatorname{Aut}(A)$. There is a notion of amenability for such an action (see e.g. Brown and ...
0
votes
1
answer
155
views
Abelian twisted reduced group C*-algebra
Let $G$ be an abelian discrete group. Then is $C_r^*(G, \sigma)$ abelian?
2
votes
0
answers
191
views
Quasidiagonal C*-algebras
Let $A$ be a nuclear $C^*$-algebra satisfying UCT condition. Then under what assumptions $A$ is quasidiagonal?
4
votes
1
answer
196
views
Direct sum of multiplier algebras
Consider a collection of $C^*$-algebras $\{A_i\}_{i \in I}$. We can form the direct sum $$\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\...
1
vote
1
answer
105
views
Complemented submodules of a Hilbert C*-module
Let $A$ be a $C^*$-algebra and $E$ be a (right) Hilbert $C^*$-module over $A$. Assume $F$ is a closed submodule of $E$ such that $F^\perp := \{x \in E: \langle x, F\rangle=0\}$ is orthogonally ...
0
votes
1
answer
141
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
$$q: E \to E: (f,g) \...
2
votes
1
answer
224
views
Primitive ideals of minimal tensor product
Let $A$ and $B$ be $C^{\ast}-$ algebras and $A \otimes B$ denotes minimal(spatial) tensor product.
Is there any classification of primitive ideals of $A \otimes B$? (I'm mainly interested in the ...
2
votes
0
answers
183
views
Reference request: definiitions of exact C* algebra and group C* algebra
I am writing my Ph.D. thesis and I would like to cite the specific papers where the concept of exact $C^*$ algebra and group $C^*$ algebra was defined.
In the book of Brown and Ozawa "$C^*$-...
3
votes
1
answer
327
views
A $*$-homomorphism $C(X) \to C(Y)$ gives a continuous map $Y \to X$
Given a $C^*$-algebra $A$, we write $\Omega(A)$ for its space of characters, i.e. its non-zero algebra homomorphisms $A \to \mathbb{C}$. If $X$ is a compact Hausdorff space, it is well-known that
$$X \...
3
votes
0
answers
178
views
$C^*$-algebras over an extension of $\mathbb{Q}_p$?
I'm wondering to what extent it might be possible for the theory of $C^*$-algebras to be translated into the $p$-adic context i.e. to define 'p-adic $C^*$-algebras' over some extension of $\mathbb{Q}...
3
votes
1
answer
377
views
Strict topology on the multiplier algebra
Let $A$ be a $C^*$-algebra. Let $M(A)$ be its multiplier $C^*$-algebras. The strict topology on $M(A)$ is given by
$$x_\lambda \to x \iff \forall a\in A: (\|x_\lambda a-xa\| + \|ax_\lambda - ax\| \to ...
0
votes
1
answer
162
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 ...
3
votes
1
answer
225
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 ...
2
votes
0
answers
114
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 ...
3
votes
1
answer
137
views
Nonstandard Podles spheres as $U_c(\frak{h})$ invariants
In this paper Podles introduced a $2$-parameter family of $q$-deformed spheres $S_{q,c}$ that are now called the "Podles spheres". The case of $c=0$ is very special and is known as the "...
7
votes
0
answers
111
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}...
1
vote
1
answer
188
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))?$
1
vote
0
answers
120
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?
2
votes
2
answers
178
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 ...
1
vote
0
answers
230
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 ...
1
vote
2
answers
312
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 $...
-1
votes
1
answer
237
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$?
7
votes
1
answer
260
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}$...
1
vote
2
answers
138
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 \...
3
votes
1
answer
260
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 $...
13
votes
1
answer
417
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 ...
3
votes
2
answers
307
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 ...
8
votes
1
answer
1k
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 ...
2
votes
1
answer
121
views
Is it possible to characterize the elements of the C$^*$-algebra of an open subgroupoid?
$\newcommand{\Cstar}{C^*_{\text{red}}}\newcommand{\G}{\mathscr G}\newcommand{\H}{\mathscr H}$Let
$\G$ be an etale groupoid, let $U$ be an open subset of $\G^{(0)}$, and let
$$
\H = \{\gamma \in \G:...
5
votes
1
answer
333
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|= \...
2
votes
1
answer
235
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 ...
5
votes
1
answer
267
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\...
8
votes
1
answer
319
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, ...
4
votes
0
answers
120
views
Restricting a function defined on an étale groupoid to an isotropy group
Let $\mathcal G$ be an étale groupoid, let $x$ be a point in the unit space of $\mathcal G$, and let $\mathcal G(x)$
be the isotropy group of $x$.
If $f$ is a continuous, complex valued, compactly ...
2
votes
0
answers
72
views
Example of a ternary Lie ideal which is not a Lie ideal
Let $H$ and $K$ be Hilbert spaces and $V\subset B(H,K)$ be a ternary ring of operators i.e. $xy^*z \in V$ for all $x,y,z \in V$. Let $I$ be a closed subspace of $V$. $I$ is called a ternary Lie ideal ...
6
votes
1
answer
163
views
Morphisms between compact quantum groups
Let $(A, \Delta_A)$ and $(B, \Delta_B)$ be two compact quantum groups (in the sense of Woronowicz). I would be tempted to define a morphism $(A, \Delta_A) \to (B, \Delta_B)$ to be a unital $*$-...