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: [tag:banach-algebras], [tag:von-neumann-algebras], [tag:operator-algebras], [tag:spectral-theory].

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3
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214 views

When is a groupoid the path groupoid of a graph?

I am actually interested in the $C^*$-algebras, so perhaps my question should be: How can you recognize whether a $C^*$-algebra $A$ is isomorphic to $C^*(\Lambda)$ for some (higher-rank) graph ...
7
votes
0answers
325 views

The approximation property of group C*-algebras

Let $G$ be a discrete group. Then the group C*-algebra $C^*(G)$ is nuclear if and only if $G$ is amenable. I am wondering whether nuclearity of $C^*(G)$ can fail for a Banach-space reason, namely due ...
1
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0answers
214 views

Is an exact operator, unitary equivalent to a banded operator?

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Definition : Let $(e_{n})_{n \in \mathbb{N}}$ be an orthonormal basis. $T \in B(H)$ is ...
11
votes
1answer
513 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 ...
3
votes
0answers
363 views

About generator and isomorphism problems for free groups operator algebras

Let $H$ be an infinite dimensional separable Hilbert space. The $C^{*}$-algebras and von Neumann here unital and subalgebras of $B(H)$. Definition : Let $\mathcal{A}$ be $C^{*}$-algebra (resp. a ...
15
votes
2answers
431 views

automorphisms of C*-algebras and partial isometries

Let $A$ be a $C^*$-algebra, let $p$ and $q$ be Murray-von Neumann equivalent projections in $A$, i.e. there is a partial isometry $v$ such that $v^*v = p$ and $vv^* = q$. Suppose $\alpha \in Aut(A)$ ...
2
votes
1answer
337 views

Is there an operator algebraic reformulation of the invariant subspace problem?

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 there a non-trivial closed $T$-invariant ...
4
votes
0answers
457 views

Do the banded operators check the invariant subspace problem?

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 there a non-trivial closed $T$-invariant ...
6
votes
2answers
600 views

Structure theorem for finite dimensional $C^*$-algebras and their representations

I would like a source for some Artin-Wedderburn type facts about these algebras which seem to have easy proofs, and are probably written somewhere. Let $\mathcal{A} \subset M_n(\mathbb{C})$ be an ...
5
votes
0answers
169 views

Does the suspension isomorphism $K_1(A) \to K_0(SA)$ descend from a more refined invariant?

If $A$ is a C*-algebra, denote its minimal unitization by $\tilde A$ and its suspension by $SA$, thought of as all continuous $a:[0,1] \to A$ with $a(0)=a(1)=0$. The unitized suspension ...
5
votes
1answer
350 views

Is there an irreducible, noncompact commuting, nonnormal operator, with spectrum strictly continuous?

Let $H$ be an infinite dimensional separable Hilbert space. Definition: The commutant $\mathcal{S}'$ of a subset $\mathcal{S} \subset B(H)$ is $ \{A \in B(H) : AB=BA \ , \ \forall B \in \mathcal{S} ...
5
votes
0answers
99 views

Equivariant zero dimensional extension recovering a given measure

Let $X$ be a compact metrizable space and $\alpha: \mathbb{Z}^d\curvearrowright X$ a continuous group action. Then it is well known that there exists a zero dimensional compact space $Y$, an action ...
3
votes
2answers
229 views

Tensoring with a CAR-algebra

Let $A$ and $B$ be two unital infinite-dimensional simple separable nuclear $C^{\ast}$-algebras and let $C$ be a CAR-algebra. When does $A\otimes C \simeq B\otimes C$, imply $A\simeq B$? The answer ...
4
votes
2answers
352 views

Projections in a W*-algebra as a continuous lattice?

A continuous lattice is a complete lattice $L$ in which every element $y$ is equal to $\bigvee${$x \in L \mid x \ll y$} where $x \ll y$ ("x approximates y" or "x is way below y") if for any directed ...
0
votes
0answers
103 views

All AI-algebras are AT-algebras

It is known that every AI-algebra (i.e. inductive limit of interval algebras) is an AT-algebra (i.e. inductive limit of circle algebras)? This seems a little bit odd because a building block of an ...
2
votes
1answer
103 views

References on countable W*-algebras

In "Operator algebras with a faithful weakly-closed representation" (1955), Kadison describes a countable W*-algebra as a C*-algebra which has a faithful representation as a countably decomposable ...
3
votes
1answer
444 views

C* Algebras, Foliations and Dynamical Systems

I am a Ph.D student involved in topics like integrability of foliations arising from center stable bundles of partially hyperbolic dynamical systems. These are generally only continuous bundles so one ...
2
votes
1answer
225 views

Inner automorphisms and $K$-theory

It is known that any inner automorphism of a unital $C^{\ast}$-algebra $A$ induces the identity map on $K_{0}(A)$ because unitary equivalence implies Murray-von Neumann equivalence. What is known ...
0
votes
1answer
173 views

Unitary with full spectrum

I have a unitary element $u\in C(\mathbb{T},M_{n}(\mathbb{C}))$ such that $Spec(u)=\mathbb{T}$. Does there exist a unitary $v\in C(\mathbb{T},\mathbb{C})$ such that $Spec(uv)\subsetneqq\mathbb{T}$?
2
votes
1answer
251 views

Tensor product of C*-algebras of bounded, uniformly continuous functions on metric spaces

This is a follow up question to this one. If $X$ is a metric space, denote by $C_u(X)$ the $C^\ast$-algebra of all bounded, uniformly continuous functions on $X$ (with the sup-norm). Do we have ...
22
votes
1answer
456 views

Can nuclearity be determined by tensoring with a single C*-algebra?

A C*-algebra is nuclear if the algebraic tensor product $A\odot B$ ($B$ is any other C*-algebra) admits a unique C*-norm. This definition requires testing the condition for nuclearity with `all' ...
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0answers
309 views

Hans Saar's thesis

I would love to have a look on some results which are claimed by some people to be in Saar's thesis: H. Saar, Kompakte, vollständig beschränkte Abbildungen mit Werten in einer nuklearen ...
7
votes
2answers
267 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 ...
3
votes
1answer
292 views

Amenability at infinity

I have a few questions about amenability at infinity for locally compact, second countable, Hausdorff topological groups. Recall that a locally compact group $G$ is said to be amenable at infinity if ...
7
votes
1answer
358 views

Trace Class Functions on locally compact groups

Let $G$ be a locally compact subgroup, $\mu$ a Haar-measure. For $f \in L^1(G)$, and for $\pi$ a unitary, topology irreducible, representation of $G$ on an Hilbert space $H_\pi$, it is customary to ...
6
votes
1answer
251 views

What are the sub $C^*$-algebras of $C(X,M_n)$?

Let $X$ be a locally compact Hausdorff topological space, denote by $M_n$ the $C^*$-algebra of complex $n\times n$ matrices, by $C_0(X,M_n)$ the $C^*$-algebra of continuous functions on $X$ with ...
7
votes
2answers
450 views

General recipe for building C*-algebras out of combinatorial object

I want to ask what should be a nice way to build C*-algebras out of objects like groups, inverse-semigroups, semigroups, ringgs or graphs. I know there are well known construction of C*-algebras out ...
14
votes
3answers
959 views

Realizing universal C*-algebras as concrete C*-algebras

How do I in general realize a universal C*-algebra generated by some generators and relation as concrete C*-algebras? For example, I know that universal C*-algebra generated by a single unitary is ...
4
votes
1answer
322 views

Kadison-Singer problem in exotic Hilbert spaces

The Kadison-Singer problem is considered in relation to the separable Hilbert space: KS: Does every pure state on the diagonal (atomic) masa of $B(\ell_2)$ has a unique extension to $B(\ell_2)$? ...
9
votes
2answers
397 views

Seeing topological (geom.) properties of the space via corresponding C^*-algebra

Compact Hausdorff spaces bijectively correspond to C^*-algebras with identity. One needs to consider the algebra of continuous functions C(X) to go in one direction and spectrum to go in the other. ...
8
votes
2answers
318 views

Injectivity of the Baum-Connes assembly map for locally compact groups

Skandalis, Tu and Yu in "The coarse Baum-Connes conjecture and groupoids" proved that: Let $\Gamma$ be a countable group with a proper left-invariant metric $d$. If $\Gamma$ admits a uniform ...
2
votes
0answers
104 views

Is the Poincare action on the Klein-Gordon quantum field strongly continuous?

I am interested in checking continuity property of the Poincare group action on the Klein-Gordon quantum field theory defined over the Minkowski spacetime. Maybe the simplest example of QFT out there. ...
2
votes
1answer
181 views

Ideal spanned by matrix units isomorphic to compact operators

Hello, Assume we have $(n+1)$ isometries $S_1,...,S_{n+1}$ in the separable Hilbert space $H$ with the properties that $\sum_{i=1}^{n+1}S_iS_i^*=I, S_i^*S_j=0$ (i.e. $S_i$ are the generators of the ...
0
votes
1answer
280 views

Commutant of a von Neumann algebra as the linear span of unitaries.

I'm reading chapter 4 of Gerard Murphy's C*-algebras book and am confused by a statement in his proof of theorem 4.1.10. In his proof, he says, "$A'$ is the linear span of its unitaries" (where $A'$ ...
10
votes
2answers
380 views

C*-algebras with no nontrivial endomorphisms

Pick a C*-algebra $A$ and call a (*-)endomorphism $\alpha:A\to A$ nontrivial if it is injective and $\alpha(A)\neq A$. Question: Do there exist infinite dimensional C*-algebras with no nontrivial ...
9
votes
0answers
344 views

Tensorial decomposition of $B(H)$

Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I ...
11
votes
3answers
509 views

Von Neumann algebra associated to the infinite Cuntz algebra

The Cuntz algebra $\mathcal{O}_{\infty}$ is the universal $C^*$-algebra generated by countably infinitely many isometries $s_i$ satisfying the relations $s_i^*s_j = \delta_{ij}$ (there is no condition ...
10
votes
1answer
243 views

Which W*-algebras are the duals of C*-coalgebras?

A Banach algebra (assumed associative and unital) is precisely a monoid object in the monoidal category of Banach spaces, short linear maps, and the projective tensor product. A Banach coalgebra is ...
3
votes
2answers
478 views

Do semi-continuous functions generate bounded Borel measurable functions as a $C^*$-algebra?

This question is related to Question 2 of my previous posting. Question. Let $\mu$ be a Radon measure on a compact Hausdorff space $\Omega$ and $L^{\infty}(\Omega,\mu)$ the set of essentially bounded ...
4
votes
0answers
216 views

Extensions of completely positive mappings

I would like to ask the following two questions. Let $1_{\mathcal{H}}\in \mathcal{A}\subset\mathcal{B}\subset\overline{\mathcal{A}}^{SOT}\subset\mathbb{B}(\mathcal{H})$ be a sequence of ...
6
votes
2answers
517 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 $S^m$ ...
2
votes
1answer
209 views

Function spaces over pseudocompact spaces

Let $K$ be a pseudocompact Tychonoff space that is, a Hausdorff $T_{3.5}$ space for which every continuous function $\varphi \colon K\to \mathbb{C}$ is bounded. Let $\beta$ be the Stone-Cech extension ...
5
votes
2answers
414 views

Bounded linear functionals and representations

Suppose that $A$ is a unital C$^*$-algebra and that $\varphi: A \to \mathbb{C}$ is a bounded linear functional. Then there exists a Hilbert space $H$, a representation $\pi: A \to B(H)$ and vectors ...
5
votes
1answer
203 views

States/functionals on crossed product C*-algebras

Let $A$ be a C*-algebra, $\alpha$ a strongly continuous automorphic action by a locally compact group $G$ on $A$, and consider the crossed product $A\rtimes_\alpha G$. I am looking for references ...
11
votes
1answer
423 views

Kuiper's theorem via approximation

Kuiper's theorem says that the unitary group $U(H)$ of a separable infinite dimensional Hilbert space $H$ is contractible, if it is equipped with the norm topology. Let's suppose, I do not know this ...
3
votes
1answer
311 views

When an AW*-algebra is a W*-algebra

In a very old book of Kaplansky "Rings of operators", on p. 123 one can find the following sentence: It is a standing conjecture that an AW${}^\ast$-algebra is W${}^\ast$ if its center is W${}^\ast$. ...
0
votes
1answer
201 views

simultaneously Approximated by self-adjoint elements.

We know that a $C^*$-algebra $A$ has real rank zero iff every self-adjoint element of $A$ can be approximated in norm by self-adjoint element with finite spectra. My question is: If we have two ...
3
votes
2answers
389 views

Polar decomposition in C*-algebras

A very nice feature of W*-algebras is the following: once you have an element $a$ of a W*-algebra $M$, and $a=u|a|$ (the polar decomposition), then $u\in M$. It seems that it carries over to ...
4
votes
1answer
235 views

Abelian sub-W*-algebras

Let $M$ be a von Neumann algebra which acts faithfully on a Hilbert space of density character $\kappa$ but does not on a Hilbert space of density character $\lambda<\kappa$ (that is, the density ...
10
votes
1answer
650 views

Does the hyperfinite II_1 factor admit two irreducible representations that are not unitarily equivalent?

Regarding the hyperfinite $II_{1}$ factor $R$ as $C^{*}$-algebra, is it known whether any two irreducible representations of $R$ are unitarily equivalent? If it is known that there exists a pair of ...