Algebras of operators on Hilbert space, $C^*-$algebras, von Neumann algebras, non-commutative geometry

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Bott-type projections in $C^*$-algebras

Let $A$ be a unital $C^*$-algebra and $a\in A$. If $aa^*+1$ is invertible in $A$ then the element $$\beta(a)=(aa^*+I)^{-1}\left(\begin{array}{cc}aa^* & a \\a^* & I\end{array}\right)$$ is an ...
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30 views

Sub-matrices with a real spectrum

This question arises from the study of PT-symmetric quantum mechanics. Let $A$ be an $n\times n$ complex-valued matrix with a real spectrum. If $A$ is Hermitian, then any sub-matrix corresponding to ...
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78 views

C$^*$-algebras in which the spectral radius is comparable to the norm

For every commutative C$^*$-algebra the spectral radius is equal to the norm. My question is: For which C$^*$-algebras $\mathcal A$ does there exist a constant $C>0$ such that $$C\|a\| \...
4
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67 views

Coming up with a represenation for sum of functions in the Fourier algebra

This is my first overflow question, so let me apologize in advance if this belongs on http://math.stackexchange.com. Let $G$ be a discrete group. Let $\lambda:G\to B(\ell^{2}(G))$ be the left ...
3
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1answer
94 views

Linear independency and compactness of the set of pure states of a $C^*$-algebra

Let $\mathcal{A}$ be a noncommutative $C^*$-algebra and $PS(\mathcal{A})$ be the set of its pure states. Question 1. Is $PS(\mathcal{A})$ linearly independent (as vectors over $\mathbb{R}$)? (If $\...
11
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Is a C*-algebra with an isomorphic predual a von Neumann algebra?

It is well-known that a C*-algebra $A$ is a von Neumann algebra if and only if it has an isometric predual, that is, if and only if there exists a Banach space $X$ such that $A$ is isometrically ...
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Multiplicativity of the analytic index (or of kernel bundle)

What I want to ask is the multiplicativity of the analytic index of a family of Dirac operators. In the single operator case the analytic index of elliptic operator is multiplicative. This is proved ...
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223 views

Baum Connes Conjecture [closed]

I have recently decided on a topic for my master thesis. I want to compare the Baum Connes conjecture as it is formulated in topology to the conjecture as it is formulated in functional analysis. I ...
7
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1answer
250 views

Why is the Berkovich spectrum of a C*-Algebra the same as the Gelfand spectrum?

Let $A = \mathcal{C}(X)$ be a commutative (unital) C*-Algebra. Let $Spec(A)$ denote its Gelfand spectrum $$ Spec(A) = \{A \rightarrow \mathbb{C} : \text{non-zero *-homomorphism} \} \simeq X. $$ Now ...
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38 views

CP maps or states on the matrix quantum group $C_q[SU_2]$

This question is about the states on the matrix quantum group $C_q[SU_2]$ (generators $a,b,c,d$ with relations...), or possibly about the representations of the $C^*$ algebra $C_q[SU_2]$ - not about ...
3
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1answer
100 views

Classification of $2k$-vectors modulo orthogonal transformations

Consider the following chain $\{A_1,A_2,A_3,\cdots,A_{n}\}$ of orbit spaces of even-rank anti-symmetric tensors, where $$A_k:=\frac{\Lambda^{2k}(\mathbb{R}^{2n})}{e_{i_1}\wedge \cdots \wedge e_{i_{2k}}...
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Countably decomposable von Neumann algebras

A von Neumann algebra is countably decomposable if every family of mutually orthogonal nonzero projections is countable. Even a singly-generated von Neumann algebra need not be countably decomposable; ...
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1answer
97 views

Bounded operators $T: B(H)\to H$ whose Kernel is a Lie algebra

Assume that $H$ is an infinite dimensional Hilbert space.The space of all bounded operators on $H$ is denoted by $B(H)$.We consider the Lie algebra structure $[T,S]=TS-ST$ on $B(H)$. Is there a ...
4
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1answer
127 views

Dye's Theorem for real von Neumann algebras

Dye's classical theorem from 1955 states that if $\theta$ is a projection orthoisomorphism (i.e., a bijection between the projective lattices that preserves orthogonality - it then automatically ...
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1answer
81 views

A nice proof that completely bounded (cb) norm of transpose map on $ M_n $ is n

In my research of operator algebras and their connection with machine learning I of course use the well know result: For the map $ tr:M_n \to M_n $ denoting the transpose map of matrices (meaning ...
3
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1answer
136 views

The C*-envelope of the algebra of continuous functions on a compact topological space is commutative

In my research in operator theory, specifically in C* algebras and enveloping, I came across this strange footnote in a text (locally published in non English where I study) which states the following:...
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Equivalence of questions regarding restrictions of pure states

In Davidson and Szarek's article "Local Operator Theory, Random Matrices and Banach Spaces" in the Handbook of the Geometry of Banach Spaces, the authors discuss the (now solved) Kadison-Singer ...
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69 views

Approximate unit in C*-algebra with additional properties

In the book about K-theory (a friendly approach) by Wegge and Olsen I came across the following notion: in Lemma 16.4 authors assume that the $C^*$-algebra $A$ possess "(commuting) approximate unit $(...
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119 views

On spatial tensor products of von Neumann algebras

Let $H$ be a Hilbert space, and let $A_1,A_2,A_3\subset B(H)$ be three commuting von Neumann algebras. We write $\odot$ for the algebraic tensor product, and $\bar\otimes$ for the spatial tensor ...
4
votes
1answer
139 views

isomorphism of noncommutative tori

I have a kind of vague question. Two non-degenerate symplectic vector space of same dimension (say $\mathbb{R}^{2n}$) are isomorphic. Then why all noncommutative tori of same dimension aren't ...
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Compactum of Banach algebra

What is an example of Banach algebra $A$ and a left non-trivial closed ideal $I$ with all of following properties? There exists a bounded approximate identity in $I$ for $I$ i.e., a net $\{e_\alpha\}...
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1answer
157 views

Convergence of functionals on compact projections on a separable Hilbert space

Let $H$ be a separable Hilbert space over $\mathbb{C}$, say $\ell_2$ for simplicity. Let $\mathcal{K}(H)$ denote the space of all compact operators on $H$ and $\mathcal{P}(H)$ the set of all finite ...
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110 views

On self-dual group-subgroup subfactors

Let $(N \subset M)$ be a finite index inclusion of ${\rm II}_1$ factors, and $N \subset M \subset M_1$ the basic construction. The subfactor $(N \subset M)$ is called self-dual if it is isomorphic to ...
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Generalized singular numbers and the Haagerup $L^p$ spaces

Let $M$ be a semi-finite von Neumann algebra with a trace $\tau$.Let $S(M)$ be the algebra of all affiliated operators measurable with respect to $M$. The $L^p$ norm on $M$ is given by \begin{...
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On finite index infinite depth subfactors and reduction to depth 2

Let $(N \subset M)$ be a finite index irreducible subfactor (with $N$ and $M$, ${\rm II}_1$ factors). Let $N \subset M \subset M_1 \subset M_2 \subset \cdots$ be the tower of basic constructions. ...
9
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1answer
404 views

Real-valued measurable cardinals

A cardinal $\kappa$ is real-valued measurable if there is a probability measure on the $\sigma$-algebra of all subsets of $\kappa$ which is zero on singletons and additive on disjoint families of ...
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110 views

Is there a non-integer in the dimension spectrum for the Heisenberg group?

Let $\Gamma = \langle a,b,c \ | \ c=aba^{-1}b^{-1}, \ ac=ca, \ bc = cb \rangle$ be the discrete Heisenberg group. Let $\ell: \Gamma \to \mathbb{N} $ be the word length on $\Gamma$. This group has a ...
2
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1answer
72 views

Choi type matrix condition for completely positivity on a certain operator system spanned by some unitaries

Let $B_1$ and $B_2$ be $C^*$-algebras. Let $U_1, \ldots, U_n$ be some unitaries in $B_1.$ We consider the operator system $S$ spanned by $U_iU_j^*.$ Let $\phi: S \rightarrow B_2.$ Given that the ...
6
votes
2answers
449 views

Kazhdan's property (T) vs. residual finiteness

I have asked this question already on mathstackexchange but got no answer (see http://math.stackexchange.com/questions/1795795/kazhdans-property-t-vs-residual-finiteness) and it was suggested that I ...
2
votes
2answers
121 views

Multiplier algebra of $A \otimes \mathcal{K}$

If $A$ is unital C$^*$-algebra, is it true that the multiplier algebra of $A \otimes \mathcal{K} $ is $ A \otimes \mathcal{B}(\mathcal{H})$? Where $\mathcal{K}$ is C$^*$-algebra of compact operators ...
2
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99 views

Orthogonality relations for unitary representations of infinite (finitely generated) groups

Let $G$ be a group, and consider the matrix elements of finite dimensional irreducible unitary representations of $G$ over $\mathbb{C}$ as functions $f:G\to \mathbb{C}$. If $G$ is finite, any two ...
4
votes
1answer
182 views

Can we solve the FGF problem by finding an appropriate action?

If we can find an action of the free group $\mathbb{F}_2$ on a measure space $X$ such that the crossed product $M=L^∞(X)⋊\mathbb{F}_2$ is a ${\rm III}_1$ factor with core isomorphic to $L(\mathbb{F}_2)...
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$id:A\to A^{op}$ is completely positive iff $A$ is abelian

Let $A$ be a $C^*$-algebra and $A^{op}$ it's opposite $C^*$-algebra. Let $id:A\to A^{op}$ be the identity map. $id$ is positive. The claim is: $id$ is completely positive iff $A$ is abelian. I need ...
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54 views

Tor functor in the case of algebra of smooth functions

Let $A=C^{\infty}(\mathbb{S}^{1})$, let $B$ be the sub-algebra $C^{\infty}(0,1)$. Here we identify $\mathbb{S}^{1}$ by $\mathbb{R}/2\pi \mathbb{Z}$. I want to ask if there is a way I can decompose $A$ ...
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1answer
81 views

Terminology: jointly completely bounded?

This question has a subjective component but I would like answers that try to stick to concrete observable facts, such as which papers use which terminology. However, the informed impressions of those ...
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70 views

Group actions on principal groupoids

Suppose that $\mathcal{G}$ is etale principal groupoid and that $G$ is a discrete (or finite) group acting freely on the locally compact unit space $\mathcal{G}^0$ (or assuming compactness, if ...
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131 views

Sums of hermitian squares in free abelian group algebras and real positive semidefinite matrices

A little context for the following question, first. As Noah Stein notes in a comment below, the present question is closely related to the free semialgebraic geometry studied by Helton and his ...
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Algebras involved to define spectral triple

A spectral triple consist from $(A,H,D)$ where $A$ is some unital $*$-subalgebra of $B(H)$ and $D$ is unbounded operator with compact resolvent such that for all $a\in A$ the commutatot $[D,a]$ is ...
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Isometry from a representation to the representation tensored with itself

Suppose, the group $ G=S(2^{\infty})$ has a unitary representation $ \pi $ on a separable infinite dimensional Hilbert space $ H $. (The group $ S(2^{\infty}) $ is the direct limit of the following ...
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1answer
119 views

is the maximal tensor product of compact operators an essential ideal?

I'm searching for a counterexample for $C^*$-algebras $A$ and $B$ and essential ideals (I assume an ideal to be closed and only two-sided ideals) $I\subseteq A$, $J\subseteq B$ , such that the ideal $...
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1answer
125 views

is a linear map on an operator system into a $C^*$-algebra (+ extra conditions) positive?

First of all, sorry for my bad english. I tried to find out whether the following statement is true or not: Let $X$ be a operator system, $B$ a $C^*$-algebra and $f:X\to B$ linear such that $f(1)\ge ...
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weak convergence in operator space structure

Let $M$ be von Neumann algebra and $B(H)$ be it's universal representation. Let $(e_i)$ be a Hilbert basis of $H$ and $\zeta_n\xrightarrow{w}\zeta $ in $H$. I know that $[w_{\zeta_n ,e_i}]_{1\times I}\...
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133 views

The multiplier algebra of a Reproducing Kernel Hilbert Space and its commutant

In my research in the theory of Reproducing Kernel Hilbert Spaces I was concerned with this topic which came up but I could not find a reference on: If $ \mathbb{H} $ is an RKHS and we denote the ...
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(Topological) K-theory for commutative $C^*$-algebras: operator and standard approaches

Let $A$ be a commutative unital $C^*$ algebra. Then $A=C(X)$ for some compact Hausdorff space $X$. Topological $K$-theory group (namely $K_0$) is defined in terms of vector bundles as a Grothendieck ...
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Algebraic $K_1$ group for a $C^*$-algebra

Let $A$ be a $C^*$-algebra: then one defines topological $K_1$ group as $GL_{\infty}(A^+)/\Big(GL_{\infty}(A^+)\Big)_0$ where $A^+$ denotes $A$ with the unit adjointed (even if $A$ already had a unit: ...
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380 views

What is the group of automorphisms of $l^{\infty}$?

What is the group of automorphisms of $l^{\infty}$? I think it would be the permutations of the integers. Is this right?
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Time-Energy Uncertainty Relation in relativistic Quantum Mechanics

There is an old intriguing result in non-relativistic QM, stating (roughly) that there is an Heisenberg Time-Energy Uncertainty Relation. Unfortunately, in QM time is not an operator like space, ...
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The functional equation $T(x\otimes y)=T(x)\otimes T(y)$ on certain $C^{*}$ algebras

Is there a name for the following property of a $C^{*}$ algebra $A$? $$A \simeq A \otimes A\;\;\; \text{The spatial tensor product}$$ Example of this situation is $A=C(X)$ where $X$ is the ...
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Primitive ideal space and unitary dual of a [SIN] group - when are they Hausdorff?

Recall that a locally compact group $G$ is said to be an $[FC]^-$ group, if each conjugacy class in $G$ has a compact closure; an $[SIN]$ group, if each neighborhood of the identity includes a ...
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1answer
120 views

Two notions of bundle of C* algebras

One can find in the literature two notions of $C^*$-algebra over a topological space $X$. The first is as the data of an open surjection $ \pi: B \rightarrow X$ together with the structure of a $C^*$-...