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].

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Is $\mathcal{B}(\mathcal{H})$ a groupoid $C^*$-algebra?

Let $\mathcal{H}$ be a complex Hilbert space, and $\mathcal{B}(\mathcal{H})$ be the $C^{\ast}$-algebra of bounded operators on $\mathcal{H}$. Is there an étale groupoid $\mathcal{G}$ such that its $C^{...
Luiz Felipe Garcia's user avatar
2 votes
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About normal states in abstract von Neumann algebras

In the book "Fundamental of the theory of operator algebras" (KAdisong and Ringrose, Vol 2) we have the Corollary 7.1.16 but this was state only for concrete von Neumann algebras (because ...
Gabriel Palau's user avatar
2 votes
0 answers
71 views

Commutant of rank one operators over Hilbert module

Let $\mathcal{E}$ be a right Hilbert $C^*$-module over the $C^*$-algebra $A$. Consider for $\xi, \eta\in \mathcal{E}$ the rank-one operator $$\theta_{\xi, \eta}: \mathcal{E}\to \mathcal{E}: \zeta \...
Andromeda's user avatar
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3 votes
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Takesaki's duality in representation theory of $C^*$-algebras

In M.Takesaki's 1967 article titled A Duality in the Representation Theory of C-Algebras*, admissible operator fields are defined in order to generalize Gelfand transform to a non-abelian setting. ...
the_dude's user avatar
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5 votes
1 answer
236 views

Pairwise orthogonality for partitions of unity in a *-algebra

Let $\mathcal{A}$ be a $*$-algebra with unit $1_{\mathcal{A}}$. As in the $\mathrm{C}^*$-setting, a projection is an element $p\in\mathcal{A}$ such that $p=p^2=p^*$. A partition of unity is a finite ...
JP McCarthy's user avatar
2 votes
0 answers
155 views

Question about the ergodic mean

This is a repost from this MathStackExchange question, where unfortunately I was not able to resolve this question. I've read a thesis where there is an example on ergodic mean, where however there is ...
MBlrd's user avatar
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0 votes
0 answers
70 views

$C^*$ algebra generated by conjugation of an element

Assume $\mathcal{A}$ is a unital $C^*$ algebra and consider some positive-definite element $\Psi\in M_n(\mathcal{A})$. Can we say something about $C^*(\langle \Psi^{-\frac{1}{2}}E_{i,i}\Psi^{\frac{1}{...
GBA's user avatar
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1 vote
1 answer
262 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
6 votes
1 answer
197 views

Nuclearity of $C(\partial_F \mathbb{G})$

Let $\mathbb{G}$ be a discrete quantum group, and consider the non-commutative Furstenberg boundary $\partial_F\mathbb{G}$ with function algebra $C(\partial_F \mathbb{G}) = I_{\mathbb{G}}(\mathbb{C})$....
J. De Ro's user avatar
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2 votes
1 answer
128 views

Complemented C*-algebras

Let $A$ and $B$ be unital separable commutative $C^*$ algebras, with $A\subset B$. Is it true that $A$ is complemented in $B$?
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Non-degenerate representation of a Banach algebra

Let $\mathcal{A}$ be a non-reflexive Banach algebra. For the definition of Arens product, please refer to this link. Here we let $\square$ denote the first Arens product and $\diamond$ denote the ...
Sanae Kochiya's user avatar
3 votes
0 answers
180 views

Bochner theorem for (non-abelian) discrete groups

I am interested in Pontryagin duality-like theories for discrete groups, more particularly, whether an analogue to Bochner's theorem for abelian groups exists in the discrete non-finite and non-...
Tomás Pacheco's user avatar
5 votes
1 answer
167 views

Order bounded version of monotone complete $C^*$-algebras

Let $A$ be a $C^*$-algebra with self-adjoint part $A_{\operatorname{sa}}$. Then $A$ is called monotone complete if every increasing norm bounded net in $A_{\operatorname{sa}}$ has a supremum (with ...
Jochen Glueck's user avatar
5 votes
0 answers
95 views

Realize a $K_0$-group homomorphism by a unital $\ast$-homomorphism

This question is inspired by Exercise $7.7$ in *An Introduction to $K$-theory for $C^*$-algebras (available here). Given a unital AF-algebra $A$ and another unital $C^*$-algebra $B$ that has ...
Sanae Kochiya's user avatar
5 votes
1 answer
217 views

Function algebra of Furstenberg boundary $\partial_F \Gamma$: when is it a $W^*$-algebra?

Let $\Gamma$ be a non-amenable discrete group and consider its Furstenberg boundary $\partial_F \Gamma$. It is known that this is a compact topological space which is stonean (equivalently: extremely ...
J. De Ro's user avatar
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9 votes
3 answers
333 views

Comparison between the operator norm and the $L^1$ norm on group algebras

Consider a discrete group $G$ and its group algebra over $\mathbb{C}$, $\mathbb{C}[G]$. There are four norms on it I wish to consider for this question: The 2-norm given by $||\sum_{g \in G} c_gg||_2^...
David Gao's user avatar
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1 vote
1 answer
154 views

Tensor product of faithful normal states is faithful

I know that given C*-algebras $A, B$ with faithful states $\omega,\varpi$, the tensor product state $\omega\otimes\varpi$ on the minimal tensor product $A\otimes_{\text{min}}B$ is faithful. I also ...
J_P's user avatar
  • 439
0 votes
1 answer
120 views

Unitary representation of a group of automorphism on an abelian algebra

Given an abelian C*-algebra $\mathcal{A}$, a state $\omega$, a strongly continuous group of *-automorphism $\{\tau_t : t \in \mathcal{R}\}$, and given a representation $ (\pi(\mathcal{A}), \mid \...
MBlrd's user avatar
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1 vote
1 answer
323 views

Lattices and noncommutative algebras in noncommutative geometry

This a question that I've asked in mathematics stack exchange without having received any response : I am interested in the relation between lattices and noncommutative algebras in the context of ...
Esmond's user avatar
  • 114
1 vote
1 answer
193 views

Borel functions in C*-algebras

Is there a way of defining representations of separable $C^*$-algebras, say $\Phi$, so that $\Phi(A)$ is faithful representation of $A$ on a separable Hilbert space. There is a closure operation $A\...
user52345435's user avatar
1 vote
1 answer
242 views

Intersection of two intermediate subalgebras

Suppose $B\subset A$ is an inclusion of simple $C^*$-algebras with a conditional expectation of (Watatani) index-finite type and $B^{\prime}\cap A=\mathbb{C}$. Then we know $B^{\prime}\cap A_1$ is ...
Keshab Bakshi's user avatar
4 votes
0 answers
219 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
3 votes
1 answer
144 views

Is a compact set of extreme points contained in a compact face?

I have run into the following question in convex analysis, which I haven't found answered in the literature: Suppose that $K$ is a "nice-enough" non-compact convex subset of a Hausdorff ...
Sean's user avatar
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0 answers
64 views

A question regarding certain sequences in hyperfinite type $II_1$ factor

Let us consider the hyperfine type $II_1$ factor $\mathcal M$ arising from the inclusion $M_2\subseteq M_{2^2}\subseteq \dots M_{2^k}\subseteq\dots$ of matrix algebras with the normalised trace $\tau$....
A beginner mathmatician's user avatar
2 votes
1 answer
85 views

Definite negative functions and length functions

$\DeclareMathOperator\ND{ND}$I am reading E. Bedos paper on heat properties for groups. Let's denote, for a group G, $$\ND^+_0(G) := \{d : G \to [0,+\infty[\; : \;d \text{ is negative definite and }d(...
NK777's user avatar
  • 21
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0 answers
70 views

Doubts on convergence of series of operators

Given an operator algebra of bounded operators $\mathcal{A}$ acting on a Hilbert space $\mathbb{H}$, I am interested in the algebra of tensor products $\mathcal{A}^{N} = \otimes_{k=1}^{N} \mathcal{A}...
MBlrd's user avatar
  • 33
3 votes
0 answers
230 views

Two more topologies on unitary groups

Let $H$ be a separable Hilbert space and let $\operatorname{U}(H)$ be the group of unitary transformations of $H$. It is well known that the weak, strong and compact-open topologies on $\operatorname{...
Matthias Ludewig's user avatar
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0 answers
127 views

Maps in the Künneth theorem for K-theory of C*-algebras

The following is named the Künneth theorem for tensor products in the book by Blackadar on K-theory for operator algebras: If $A$ and $B$ are C*-algebras and $A$ is in the bootstrap class, then there ...
AlexE's user avatar
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4 votes
0 answers
202 views

Irreducible representations of $\mathrm{UHF}_n$

I have a question about the irreducible representations of the $C^*$-algebra $\mathrm{UHF}_n = \bigotimes_{k=1}^\infty M_n$. For every sequence of unit vectors $(\xi_k)$ in $\mathbb C^n$ there is a ...
Lau's user avatar
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3 votes
0 answers
107 views

Automorphisms of the injective envelope

Let $A$ be a separable $C^∗$-algebra and $(I(A),\kappa)$ be its injective envelope. WLOG assume that $I(A)$ is a monotone complete $C^*$-algebra, and $\kappa:A\to I(A)$ is the identity map. Let $\...
Onur Oktay's user avatar
  • 2,263
1 vote
0 answers
82 views

Cocycle-conjugacy classes of flows on the C*-algebra of compact operators

A flow on a C*-algebra $A$ is a group homomorphism $\sigma $ from ${\mathbb R}$ into the group of *-automorphisms of $A$ such that the map $$ t\in {\mathbb R}\mapsto \sigma _t(a)\in A $$ is norm-...
Ruy's user avatar
  • 2,233
3 votes
0 answers
78 views

Cohomological counterpart of the K-theory of the Roe C*-algebra in non-periodic systems

In crystalline insulating quantum systems the dynamics of electrons is governed by a Schrödinger operator which is periodic with respect to a Bravais lattice $\Gamma\cong \mathbb{Z}^d$ and whose ...
Rosencrantz's user avatar
3 votes
0 answers
104 views

Reference request for embedding of a tensor product $C^*$-algebra

I am studying Ruy Exel's paper "A new look at the crossed product of a $C^*$-algebra by a semigroup of endomorphisms." In the proof of Theorem 11.7 he writes: Let $G$ be ameanable, thus $C^*(...
Tomás Pacheco's user avatar
1 vote
1 answer
164 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
2 votes
2 answers
182 views

Do positive-definite elements in finite-dimensional $*$-algebras over $\mathbb R$ always admit square roots?

Let $A$ be a finite-dimensional $*$-algebra over $\mathbb R$. We say that an element $x \in A$ is positive definite if $x$ admits an inverse and if $x = y y^*$ for some $y \in A$. Does every such $x$ ...
wlad's user avatar
  • 4,792
3 votes
1 answer
128 views

Property that follows from conditions involving slice maps on Hilbert module

Let $A,B$ be $C^*$-algebras and $E$ be a right $A$-Hilbert $C^*$-module. We can form the Hilbert $A\otimes B$ (minimal tensor product) module $E \otimes B$. If $\omega \in B^*$, there is a unique ...
Andromeda's user avatar
  • 209
2 votes
0 answers
96 views

Morphism of discrete quantum groups

In the paper Kazhdan's Property T for Discrete Quantum Groups , we read the following fragment: First, note that I think there is a typo and that codomain and domain of the dual maps have to be ...
Andromeda's user avatar
  • 209
1 vote
1 answer
78 views

Show that if $V\in M(B_0(H)\otimes A)$, then $V(B(H)\otimes 1)V^*\not\subseteq B(H)\otimes A$ where $A$ is a specific unital $C^*$-algebra

Let $\mathbb{G}$ be a compact (quantum group) with function algebra $(C(\mathbb{G}), \Delta)$ and Haar state $\varphi_{\mathbb{G}}$. Consider the associated GNS-representation $\pi_{\mathbb{G}}: C(\...
Andromeda's user avatar
  • 209
-3 votes
1 answer
319 views

Is the category of $Z^* $ algebra equivalent to the category of $C^*$ algebras

A $Z^*$ algebra is a $C^*$ algebra whose all positive elements are zero divisor. They form a category with usual structures. Question. Is this category equivalent to the category of $C^*$ algebras? ...
Ali Taghavi's user avatar
4 votes
0 answers
228 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
  • 1,065
3 votes
0 answers
92 views

How do I show countable additivity of (proposed) spectral measure in the proof of the spectral theorem?

I'm currently writing a Bachelors thesis based on the following paper: Douglas, R., & Pearcy, C. (1970). On the spectral theorem for normal operators. Mathematical Proceedings of the Cambridge ...
Abhijeet Vats's user avatar
1 vote
0 answers
98 views

Formula for the KK-theory groups $KK(A, C(S))$

I am studying $C^*$-algebras and their KK-theory. Let $A$ be a (unital if you wish) $C^*$-algebra and $S$ be a compact Hausdorff space. I am interested in computing the KK-theory groups $KK(A, C(S))$, ...
Luiz Felipe Garcia's user avatar
1 vote
1 answer
275 views

A subalgebra of $B(H)$ which does not contain a commutator element

Is there a commutative subalgebra $A\subset B(H)$ containing the 1 dimensional scalars with the following property: The algebra $A$ has trivial intersection with the set of commutator ...
Ali Taghavi's user avatar
1 vote
0 answers
118 views

Probabilistic interpretation of von Neumann's approach to quantum mechanics

One of the basic postulates in the mathematical formalism of quantum mechanics is that the probability of a measurement of an observable $A$ in the state $\psi \in \mathscr{H}$ to return a value in a ...
MathMath's user avatar
  • 1,255
1 vote
1 answer
139 views

Convergence of the partial sum of a sequence strictly converging to zero

The following question comes from a statement in Lemma 16.4 in K-theory and $C^{\ast}$-Algebras written by N.E. Wegge-Olsen. Let $A$ be a non-unital $C^*$-algebra, $\{p_n\}_{n\in\mathbb{N}}$ be a ...
Sanae Kochiya's user avatar
3 votes
0 answers
123 views

Relationship between the Penrose groupoid algebra and the group algebra of the symmetry group of a Penrose tiling

Given a Penrose tiling, there are two C*-algebras associated with it: the $\mathcal{O}_\text{Penrose}$ algebra (Penrose groupoid algebra) and the group algebra of the symmetry group of the tiling, ...
Mirco A. Mannucci's user avatar
2 votes
1 answer
170 views

Unifying categorical equivalences and dualities for the functors : Gelfand spectrum and Zariski spectrum, Structural sheaf and Continuous sections

I would be very grateful for any references I might be led to, from a categorical point of view for the functors: $\textsf{Spec}_{\mathscr{Z}\textrm{arisky}}(-)$, related to $\mathcal{O}(-)$, which ...
Julien Dalpayrat-Glutron's user avatar
3 votes
1 answer
283 views

Closed prime ideal in $C[0, 1]$

I know that maximal ideals of $C[0, 1]$ corresponds to singleton. Also, using Zorn's lemma one can construct a prime ideal in $C[0, 1]$ which is not maximal. Is there any $\textbf{closed}$ prime ...
Math Lover's user avatar
  • 1,065
4 votes
1 answer
186 views

Property of pushouts in the category of unital $C^{\ast}$-algebras

Let $A$ be a unital $C^{\ast}$-algebra and $\{ f_i: A \rightarrow A_i \}_i$ a finite collection of morphisms of unital $C^{\ast}$-algebras, such that the associated map $A \rightarrow \prod_i A_i$ is ...
Luiz Felipe Garcia's user avatar
3 votes
1 answer
239 views

Takesaki: question about lemma in section "Left Hilbert algebras and weights"

To make this question relatively self-contained, this post is quite long, but the question itself is rather short. Consider the following fragments in Takesaki's second volume "Theory of operator ...
Andromeda's user avatar
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