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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 ...
Luuk Stehouwer's user avatar
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 ...
Ken.Wong's user avatar
  • 523
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 ...
user531706's user avatar
2 votes
1 answer
316 views

Decomposition of Hilbert spaces via groups and algebras representations

Let $\mathcal{H}$ be a complex finite dimensional Hilbert space and let $\mathcal{A}\subseteq \mathcal{B}(\mathcal{H})$. I am looking to understand the different decompositions of $\mathcal{H}$ ...
vand's user avatar
  • 23
3 votes
0 answers
197 views

Cuntz semigroups of basic C*-algebras

I am doing some research related to Cuntz semigroups, and I am trying to find concrete examples in simple cases. In one paper that I found, it says the following (p.103): "[...] $A_i$ is ...
Sambo's user avatar
  • 285
9 votes
1 answer
448 views

Reference request: Brown Ozawa and strong completely positive approximation property?

The notion of a $C^*$-algebra being nuclear has many equivalent characterisations. These are considered in the excellent, modern textbook $C^*$-Algebras and Finite-Dimensional Approximations by Brown ...
Matthew Daws's user avatar
  • 18.7k
3 votes
1 answer
119 views

Universal representations of quotient C*-algebras

Suppose that $\mathfrak{J}$ is a closed ideal of a C*-algebra $\mathfrak{A}$. Let $(\pi_u, H_u)$ be the universal representation of $\mathfrak{A}$. Is there a way to use these data to describe the ...
Kumal R.'s user avatar
0 votes
1 answer
124 views

Definition of center of ternary ring of operators

Let $H$ and $K$ be Hilbert spaces and $B(H,K)$ denotes the space of bounded operators from $H$ to $K$. Recall that a ternary ring of operators (TRO) $V$ is a closed subspace of $B(H,K)$ which is ...
Math Lover's user avatar
  • 1,115
5 votes
1 answer
178 views

Hopf C-star algebra/comodules using a Fubini tensor product rather than the minimal tensor product?

Throughout, $\otimes$ denotes the minimal tensor product of $\newcommand{\Cst}{{\rm C}^{\ast}}\Cst$-algebras. and $\odot$ denotes the tensor product of (underlying) vector spaces. Given two $\Cst$-...
Yemon Choi's user avatar
  • 25.8k
1 vote
1 answer
87 views

Projection (or idempotent) graph of a $C^*$ algebra(or a ring)

In the literature, are there some research around a directed graph associated to a $C^*$ algebra or a ring $A$ whose vertices are projections or idempotents of $A$ and $e$ is connected to $f$ iff $ef=...
Ali Taghavi's user avatar
4 votes
1 answer
341 views

On differential equation $Z'=Z^2-Z$ on a $C^*$ algebra

Let $A$ be a Banach or a $C^*$ algebra. We consider the differential equation $$(*)\;\;\;\;Z'=Z^2-Z$$ on $A$. Obviously the singularities of this systems are just the idempotents of the ...
Ali Taghavi's user avatar
7 votes
1 answer
394 views

Inverse limit in the category of $C^{\ast}$-algebras or operator spaces

Does the inverse limits (projective limits) exist in the category of $C^{\ast}$-algebras or operator spaces? I tried to search but could not find a proper reference. Any reference or comments about ...
Math Lover's user avatar
  • 1,115
2 votes
1 answer
189 views

Need a reference of a fact given in B. Blackadar's Operator Algebras

I am reading Blackadar's book on Operator algebras. In $\Pi 9.6.5$ Blackader says that Maximal Tensor products commute with arbitrary limits. In the same book the proof of this fact is not given....
Math Lover's user avatar
  • 1,115
8 votes
1 answer
281 views

Factor traces of the Temperley-Lieb algebra

Given $\delta\in\mathbb C$, let $A(\delta)$ denote the complex unital $*$-algebra generated by an identity $1$ and selfadjoint elements $e_k$, $k\in\mathbb N$, satisfying $e_k^2=\delta e_k$, $e_ke_l=...
Gandalf Lechner's user avatar
4 votes
1 answer
201 views

closure of a separating set of pure states

Let $A$ be a unital C*-algebra, and let $\mathcal R$ be a separating family of irreducible representations of $A$. Each vector state of a representation in $\mathcal R$ is a pure state, and the span ...
Andre Kornell's user avatar
3 votes
1 answer
159 views

K-group properties of quasi-diagonal $C^*$-algebras

Let $A$ be a separable unital quasidiagonal $C^*$-algebra. What can be said about the $K$-theory of $A$, for example some properties? Especially, are there some criterions to decide whether or not $K_*...
Kashvi Ramaprasad's user avatar
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 ...
user10439561's user avatar
5 votes
1 answer
242 views

Spectral decomposition of a C$^*$algebra with respect to an action of a compact abelian group

Let $G$ be a compact abelian group (finite dimensional, but not finite) and $A$ be a $C^*$-algebra. Consider an action $\alpha: G\to Aut(A)$. In analogy with the case of finite abelian group, I ...
John N.'s user avatar
  • 743
10 votes
2 answers
1k views

Kazhdan's property (T) vs. residual finiteness

I have asked this question already on mathstackexchange but got no answer (see https://math.stackexchange.com/questions/1795795/kazhdans-property-t-vs-residual-finiteness) and it was suggested that I ...
M.U.'s user avatar
  • 721
4 votes
0 answers
135 views

References for a lemma about compact operators on a Hilbert module

I am looking for a reference for the following result: If $A$ and $B$ are C* algebras, $H$ is a right Hilbert $A$-modules, $\phi :A \rightarrow B$ is a morphism, and assume that there is a map $\eta :...
Simon Henry's user avatar
  • 42.4k
3 votes
0 answers
255 views

Graded structures for simple $C^{*}$ algebras without nontrivial idempotent

Edit(A confession): I just realized that the question is trivial: Since one can easily prove that the convex hull of the spectrum of every nontrivial homogeneous element of a $\mathbb{Z}_{n}$-graded $...
Ali Taghavi's user avatar
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 $C^{*}$ ...
Ali Taghavi's user avatar
2 votes
1 answer
320 views

Totally non hereditary $C^{*}$-subalgebras

Assume that $B$ is a $C^{*}$ subalgebra of $A$. We say $B$ is totally non hereditary subalgebra of $A$ if not only $B$ is not a hereditary subalgebra but also it is not isomorphic to any ...
Ali Taghavi's user avatar
2 votes
0 answers
234 views

The kernel of $C^{*}(G)\to C_{r}^{*}(G)$

Let $G$ be a locally compact group. Put $I(G)=\ker: C^{*}(G) \to C_{r}^{*} (G)$, the kernel of the canonical morphism. What type of $C^{*}$ algebras can not be isomorphic to $I(G)$, for some ...
Ali Taghavi's user avatar
6 votes
2 answers
429 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 ...
Ali Taghavi's user avatar
10 votes
1 answer
533 views

Who first identified the universal $C^*$-algebra generated by an idempotent of norm at most $C$?

So much is known about hermitian and non-hermitian idempotents in a $C^*$-algebra, that someone must have written down the following. Theorem The universal $C^*$-algebra generated by one element $x$...
Terry Loring's user avatar
  • 1,749
4 votes
0 answers
374 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 C${}^\ast$-...
Lech Roch's user avatar
  • 505
12 votes
2 answers
479 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 ...
Ollie's user avatar
  • 1,411
13 votes
1 answer
1k 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 ...
Jon Bannon's user avatar
  • 7,057
8 votes
1 answer
904 views

The Haar state on compact quantum groups $A_u(Q)$ and $A_o(Q)$

Let $Q\in GL_n(\mathbb{C})$. The free unitary quantum group is the universal $C^*$-algebra $A_u(Q)$ with generators $u_{ij},1\leq i,j\leq n$ and relations making $u=(u_{ij})$ as well as $Q\bar{u}Q^{-1}...
m07kl's user avatar
  • 1,702
7 votes
0 answers
573 views

References for "folklore" on strong amenability of (group) C*-algebras?

[Apologies in advance for the prolixity - but I was unsure how much of the story is familiar.] $\newcommand{\ptp}{\widehat{\otimes}} \newcommand{\co}{\operatorname{co}} \newcommand{\Cst}{\operatorname{...
Yemon Choi's user avatar
  • 25.8k
19 votes
2 answers
1k views

Commutators in the reduced C*-algebra of the free group

Is it known whether any element of trace 0 in the reduced $C^*$-algebra of a non-abelian free group, is a limit of sums of (additive) commutators?
Alain Valette's user avatar
2 votes
1 answer
653 views

Strict positivity in dense subalgebras of $C^{*}$-algebras

Let $A$ be a $C^{*}$-algebra, represented on a Hilbert space $H$, and $D$ a selfadjoint unbounded operator on $H$ (note that we do not impose that $D$ have compact resolvent). Let $\mathcal{A}:=${$a\...
alterationx10's user avatar