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32
votes
1answer
2k views

What is the current status of the Kaplansky zero-divisor conjecture for group rings?

Let $K$ be a field and $G$ a group. The so called zero-divisor conjecture for group rings asserts that the group ring $K[G]$ is a domain if and only if $G$ is a torsion-free group. A couple of good ...
22
votes
1answer
410 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' ...
20
votes
0answers
1k views

Finite-dimensional subalgebras of $C^\star$-algebras

Let $A$ be a unital $C^\star$-algebra and let $a_1,\dots,a_n$ be a finite list of normal elements in $A$ which (together with their adjoints) generate a norm-dense $\star$-subalgebra $B \subset A$. ...
18
votes
2answers
748 views

Does left-invertible imply invertible in full group C*-algebras (discrete case)?

The following question/problem has been bugging me on and off for some time now: so I thought it might be worth broaching here on MO, as a case of "ask the experts". Let $G$ be a discrete group. ...
17
votes
2answers
380 views

Which groups are the unitary group of a $C^*$-algebra

Which groups are the unitary group of a $C^*$-algebra? Does anyone know anything in this direction?
15
votes
2answers
410 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 $\alpha \in Aut(A)$ ...
14
votes
3answers
815 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 ...
14
votes
7answers
1k views

ubiquity, importance of path algebras

I work in planar algebras and subfactors, where the idea of path algebras on a graph (alternately known as graph algebras, graph planar algebras, etc.) is quite useful. The particular result I'm ...
13
votes
3answers
1k views

Is the group von Neumann algebra construction functorial?

Let $G$ be a group and $CG$ the complex group algebra over the field $C$ of complex number. The group von Neumann algebra $NG$ is the completion of $CG$ wrt weak operator norm in $B(l^2(G))$, the set ...
12
votes
4answers
2k views

Reference: Learning noncommutative geometry and C^* algebras

I am starting to study noncommutative geometry and C^* algebras so my question is Does anyone knows a good reference on this subject? I would like a basic book with intuitions for definitions and ...
12
votes
1answer
172 views

Can a non-commutative C*-algebra be a minimal operator space?

By an operator space structure on a Banach space $X$ I mean a sequence of norms on spaces $M_n \otimes X$ that satisfies Ruan's axioms. Among such admissible sequences there is always the smallest ...
12
votes
0answers
664 views

Must we close weakly to apply the spectral theorem?

Let $H$ be an infinite dimensional separable complex Hilbert space. All C*-subalgebras of $B(H)$ are assumed to be non-degenerate. The spectral projections of a self-adjoint element $T$ of $B(H)$ lie ...
11
votes
3answers
480 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 ...
11
votes
2answers
418 views

Can non-central projections still commute with all other projections?

Let $A$ be a C*-algebra and let $\mathcal{P}(A)$ denote the set of projections in $A$. If $p\in\mathcal{P}(A)$ commutes with everything in $\mathcal{P}(A)$ does it necessarily commute with everything ...
11
votes
1answer
638 views

Gelfand-Naimark from the category-theoretic point of view

I was thinking about the Gelfand-Naimark theorem asserting the isometric * isomorphism between a commutative C* algebra (with unit) A and the C* algebra of continuous complex-valued functions on its ...
11
votes
3answers
375 views

Is the space of *-homomorphisms between two $C^*$-algebras locally path connected

Given the set of *-homomorphisms between two $C^*$-algebras $A$ and $B$, we may define a metric on it by setting $d(f,g):= \sup_{0<\|a\|\le 1}\|f(a)-g(a)\|$. Could it be true that, for each ...
11
votes
1answer
802 views

What is the commutative analogue of a C*-subalgebra?

Using the duality between locally compact Hausdorff spaces and commutative $C^*$-algebras one can write down a vocabulary list translating topological notions regarding a locally compact Hausdorff ...
11
votes
3answers
871 views

The difference between $l^1(G)$ and the reduced group $C^*$ algebra $C_r^*(G)$

Let $G$ be a group and $l^2(G)$ the Hilbert space on $G$. The complex group algebra $CG$ can be imbedded in $B(l^2(G))$, the set of all bounded linear operators, by left translation. The reduced group ...
11
votes
1answer
392 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 ...
10
votes
2answers
719 views

Range of completely positive projection

Let $A$ be a C*-algebra. Suppose that $P:A \rightarrow A$ is a contractive completely positive projection. Does the range $P(A)$ is completely order isomorphic to a $C^*$-algebra? In the case where ...
10
votes
1answer
623 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 ...
10
votes
1answer
471 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 ...
10
votes
2answers
372 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 ...
10
votes
1answer
237 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 ...
10
votes
0answers
416 views

Properties of orthogonality-preserving c.p. maps between $C^*$-algebras

Suppose that $A,C$ are $C^*$-algebras and $\phi:A \to C$ is a completely positive, orthogonality-preserving linear map. (Orthogonality preserving means: if $a,b \in A$ satisfy $ab=0$ then ...
10
votes
0answers
1k views

Schwartz kernel theorem for A-linear operators

Let $X,Y \subset \mathbb{R}^n$ be open subsets. Denote by $C^\infty(X)$ the smooth functions on $X$, let $\mathcal{E}'(Y)$ be its dual space considered as a space of distributions. Let $L(C^\infty(X), ...
9
votes
7answers
991 views

Positive but not completely positive?

The only example I know of a positive map which is not completely positive is the transpose map on $M_n(\mathbb{C})$. Of course, one can come up with minor perturbations of this (compose it with, or ...
9
votes
2answers
518 views

C*-algebras with bizzarre structure of projections

This is probably well-known to the experts but I could not find any answer neither in my head nor in the literature: Is there a (unital) C*-algebra such that its projections do not form a lattice ...
9
votes
2answers
386 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. ...
9
votes
1answer
208 views

Compact Quantum Groups and the Existence of the Classical Haar Measure

Before I state my question, let me provide the definition of a compact quantum group. Definition: An ordered pair $ \mathscr{G} = (\mathscr{A},\Phi) $ is called a compact quantum group if ...
9
votes
1answer
1k views

Algebraic properties of the algebra of continuous functions on a manifold.

Does the algebra of continuous functions from a compact manifold to $\mathbb{C}$ satisfy any specific algebraic property? I'm not sure what kind of algebraic property I expect, but I feel ...
9
votes
1answer
1k views

Does equality of the operator norm and the cb norm for every bimodule map over a C*-subalgebra imply that the subalgebra is matricially norming?

In this post, without further mention all C*-algebras are assumed to have an identity element and subalgebras inherit the identity. Question: Let $\mathcal{C}$ be a C*-subalgebra of $\mathcal{B}$. ...
9
votes
0answers
328 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 ...
8
votes
2answers
298 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 ...
8
votes
1answer
698 views

Matrices with entries in a $C^*$-algebra

Let $\mathcal{A}$ be a $C^\ast$-algebra. Consider vector space of matrices of size $n\times n$ whose entries in $\mathcal{A}$. Denote this vector space $M_{n,n}(\mathcal{A})$. We can define involution ...
8
votes
3answers
1k views

Conjugacy classes and reduced group $C^*$-algebra of an amenable group

The reduced $C^*$-algebra of a non-abelian free group $G$ has a unique trace. Hence, there is no chance to separate conjugacy classes of group elements using traces on $C^\star_{red} G$. On the other ...
8
votes
1answer
311 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 ...
8
votes
1answer
485 views

topology on the automorphism group of a C* algebra

Let $A$ be a $C^*$-algebra. The group $Aut(A)$ of $\ast$-automorphisms of $A$ is usually equipped either with the pointwise norm topology, i.e. the topology generated by the semi-norms $\lVert \varphi ...
8
votes
2answers
797 views

Relative K-theory and split exact sequences of C* algebras

Let $A$ be a C* algebra, $J$ an ideal, $\pi: A \to A/J$ the quotient map. Recall that the relative K theory group $K_0(A, A/J)$ consists of equivalence classes of triples $(p,q,x)$ where $p$ and $q$ ...
7
votes
5answers
927 views

Do subalgebras of C(X) admit a description in terms of the compact Hausdorff space X?

In light of the well-known theorem of Gelfand that, bluntly put, ends up saying that unital abelian C*-algebras are the 'same' as compact Hausdorff topological spaces, I tried to compile a dictionary ...
7
votes
3answers
434 views

Universal $C^*$-algebra with generators and relations

We say that the $C^*$-algebra $A$ generated by $a_1,...,a_n$ is universal subject to relations $R_1,...,R_m$ if for every $C^*$-algebra $B$ with elements $b_1,...,b_n$ satisfying relations ...
7
votes
2answers
397 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 ...
7
votes
2answers
619 views

Solving the equation $xax=b$ in a C*-algebra.

Let $a, b\in A_+$ be positive elements of some C*-algebra $A$. Assume furthermore that $a$ is invertible. Is it true that $$ \exists! x\in A_+\quad:\quad xax=b\quad ? $$ Already in the case ...
7
votes
3answers
1k views

Definition of a von Neumann algebra

Is there a way to equip every C*-algebra A with a functorial topology such that the canonical map A→A** is an isomorphism if and only if A is a von Neumann algebra? Here A** denotes the dual of A* in ...
7
votes
1answer
327 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 ...
7
votes
1answer
425 views

Given a C-star dynamical system and a subgroup of the acting group, is the corresponding map on crossed product algebras necessarily an injection

Let $(A,\alpha, G)$ be a $C^*$-dynamical system, where $G$ is a discrete group. Let $\Gamma$ be a subgroup of $G$, then we can form two universal crossed products $A\rtimes_\alpha \Gamma$ and ...
7
votes
1answer
215 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 ...
7
votes
1answer
549 views

When are certain group C*-algebras exact?

This is somewhere between a "reference request" and "ask an expert", but I hope it is not too trivial or off-topic. Anyway. There has been a lot of attention given to showing that for certain ...
7
votes
0answers
105 views

Do quotients of amenable groups C*-algebras satisfy the UCT?

Let G be a discrete amenable group. General Question: Let $J$ be an ideal of $C^*(G)$, the group C*-algebra of $G.$ Does $C^*(G)/J$ satisfy the universal coefficient theorem (UCT)? I am mainly ...
7
votes
0answers
257 views

Unique maximal ideal in group C*-algebras

Let $G$ be a discrete group. Let $C^*(G)$ denote the full group C*-algebra of $G.$ Let $\pi:C^*(G)\rightarrow \mathbb{C}$ be the *-homomorphism associated with the trivial representation of $G.$ ...